NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS …people.math.harvard.edu/~gaitsgde/RepRealRed_2017/Notes.pdf · NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS MATH 224, SPRING

NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS

MATH 224, SPRING 2017

DENNIS GAITSGORY

1. Week 1, Day 1 (Tue, Jan 24)

1.1. Action of distributions.

1.1.1. LetK be a compact topological group. Let C(K) denote the space of continuous complex-valued functions on K. Unless specified otherwise, integration over K will be with respect tothe Haar measure µHaar on K, normalized so that the volume of K equals 1.

Let Meas(K) be the topological dual of C(K). Unless specified otherwise, we will regard itas equipped with the weak topology.

For k ∈ K we let δk ∈ Meas(K) denote the corresponding δ-function, i.e., 〈δk, f〉 = f(k).

Note that we have a canonical embedding

(1.1) L1(K) → Meas(K), f 7→ f · µHaar

with dense image (with respect to the weak topology on Meas(K)).

Remark 1.1.2. Note that, being the topological dual of a Banach space, Meas(K) has a naturalnorm, isn which it is a Banach space. The corresponding topology (called the norm topology)is stronger than the weak topology. The map (1.1) is an isometric embedding with respect tothe corresponding norms (in particular, the image of L1(K) in Meas(K) is not dense in thenorm topology).

The above discussion is applicable to (K,µHaar) replaced by an arbitrary compact topologicalset equipped with a Borel measure.

1.1.3. Pushforward of measures along the multiplication map K ×K → K (i.e., the operatordual to that of pullback of functions) makes Meas(K) into an associative algebra. We denotethe corresponding operation by

f1, f2 7→ f1 ? f2

and refer to it as the convolution product. The unit for this algebra is δ1. We have

δk1 · δk2 = δk1·k2 .

This algebra structure is compatible with the embedding (1.1), i.e., it induces a structure ofassociative algebra on L1(K). We have

µHaar ? µHaar = µHaar.

Date: March 9, 2017.

1

2 DENNIS GAITSGORY

1.1.4. Let V be a finite-dimensional continuous representation of K. For k ∈ K we let Tk thecorresponding element of End(V ).

Since for every v ∈ V, v∗ ∈ V ∗, the matrix coefficient function

k 7→ 〈v∗, Tk(v)〉

is continuous, we obtain that the assignment k 7→ Tk uniquely extends to a continuous linearmap

(1.2) Meas(K)→ End(V ), f 7→ Tf ,

determined by the condition that Tδk = Tk.

The map (1.2) is a homomorphism of associative algebras.

1.2. Matrix coefficients.

1.2.1. We view C(K) as a representation of K ×K by

((k1, k2) · f)(k) = f(k−11 · k · k2).

By adjunction, we obtain action of K ×K on Meas(K). We have

(k1, k2) · δk = δk1·k·k−12.

1.2.2. For a finite-dimensional continuous representation V of K, we view End(V ) as a repre-sentation of K ×K via

(k1, k2) · S = Tk1 S Tk−12.

The action map

(1.3) ActV : Meas(K)→ End(V ), f 7→ Tf

and the matrix coefficient map

(1.4) MCV : End(V )→ C(K), MCS,V (k) = Tr(S Tk−1 , V )

are maps of K ×K-representations.

1.2.3. Note that we can canonically identify End(V ) with the dual of End(V ∗) as K × K-representations by setting

〈S1, S2〉 = Tr(S1 S∗2 , V ), S1 ∈ End(V ), S2 ∈ End(V ∗),

where S 7→ S∗ denotes the operation of taking the dual linear operator.

In terms of this identification, the maps (1.3) and (1.4) are each other’s duals.

Remark 1.2.4. Note that the map S 7→ S∗ defines an isomorphism of vector spaces

End(V ) ' End(V ∗).

This isomorphism is compatible with the K ×K-actions up to the swap of factors in K ×K.

1.3. Orthogonality formulas.

NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS 3

1.3.1. Let U be a finite-dimensional continuousrepresentation of K. Note that the operatorP invU = TµHaar is an idempotent projection onto UK , the subspace of K-invariant vectors.

For S ∈ End(U) we have

(1.5)

∫k∈K

MCS,U (k) = Tr(S P invU , U) = Tr(P inv

U S P invU , UK).

Proposition 1.3.2. Let V1 and V2 be finite-dimensional continuous irreducible representations.Then we have: ∫

k∈K

MCS1,V1(k) ·MCS2,V2

(k) = 0

unless V2 ' V ∗1 (both reps are irreducible), and for V1 = V and V2 = V ∗, we have∫k∈K

MCS1,V (k) ·MCS2,V ∗(k) =1

dim(V )· Tr(S1 S∗2 , V ).

Proof. Consider U = V1 ⊗ V2, and apply (1.5) to S = S1 ⊗ S2. If V2 is non-isomorphic to V ∗1 ,then UK = 0, and the assertion follows.

If V1 = V and V2 = V ∗, we identify

V ⊗ V ∗ ' End(V ),

and under this identification, for S1 ∈ End(V ) and S2 ∈ End(V ∗), the corresponding endomor-phism S1 ⊗ S2 of V ⊗ V ∗ corresponds to the endomorphism of End(V ) given by

(1.6) S′ 7→ S1 S′ S∗2 .

By Schur’s lemma, the map

C→ End(V )K , 1 7→ IdV

is an isomorphism. The corresponding projector

P invEnd(V ) : End(V )→ End(V )K

identifies with

S′ 7→ Tr(S′, V ).

The operator

S′ 7→ P invEnd(V ) S

′ P invEnd(V ), End(End(V ))→ C

sends the endomorphism of End(V ) given by (1.6) to

1

dim(V )· Tr(S1 S∗2 , V ).

This establishes the desired identity.

4 DENNIS GAITSGORY

1.3.3. Note thatMCS∗,V ∗(k) = MCS,V (k−1).

Hence, the orthogonality relation can also be interpreted as a formula for

(1.7)

∫k∈K

MCS1,V1(k) ·MCS2,V2

(k−1).

Namely, (1.7) vanishes unless V1 ' V2, and for V1 = V = V2, it equals

1

dim(V )· Tr(S1 S2, V ).

1.3.4. As a corollary of Proposition 1.3.2, we obtain:

Corollary 1.3.5. For V1 and V2 irreducible, the composite

End(V1)ActV1−→ C(K) → Meas(K)

MCV2−→ End(V2)

is zero unless V1 ' V2 and equals 1dim(V ) · IdEnd(V ) for V1 = V = V2.

1.3.6. Using an invariant Hermitian form, we also have

MCS,V (k−1) = MCS†,V (k).

Thus, we obtain that the images of MCV for distinct V ’s are pair-wise orthogonal in L2(K),and that for an individual V , the map MCV is an isometric embedding of End(V ) into L2(K),where we endow End(V ) with a Hermitian structure by the formula

(1.8) (S1, S2) =1

dim(V )· Tr(S1 S†2).

1.4. Characters.

1.4.1. Let V be a finite-dimensional continuous representation of K. Let χV ∈ C(K) be equalMCV (IdV ). From Proposition 1.3.2 we obtain:

Corollary 1.4.2. Let V and W be irreducible. Then∫k∈K

χV (k) · χW (k−1) =

∫k∈K

χV (k) · χW∗(k) =

∫k∈K

χV (k) · χW (k)

equals 1 if V 'W , and vanishes otherwise.

1.4.3. For V irreducible, set ξV := dim(V ) · χV . From Corollary 1.3.5, we obtain:

Corollary 1.4.4. For W irreducible, the element TξV ∈ End(W ) equals IdW for W ' V andzero otherwise.

From Corollary 1.7, we obtain:

Corollary 1.4.5. ξV ? ξW = 0 unless W ' V and ξV ? ξV = ξV .


2.1. Continuous representations.

2.1.1. Let G be a topological group and V a topological vector space. We shall say that arepresentation of G on V is continuous if the action map

G× V → V

is continuous.


2.1.2. In this course we will restrict our attention to the following class of topological vectorspaces: we will assume that they are Hausdorff, 2nd countable, locally convex and complete.

Locally convex is equivalent to saying that there is a fundamental system of neighborhoodsof 0 ∈ V of the form

v ∈ V , ρ(v) < 1,

where ρ : V → R is a continuous map satisfying

ρ(c · v) = |c| · ρ(v), ρ(v1 + v2) ≤ ρ(v1) + ρ(v2).

Such maps are called semi-norms.

2.1.3. Suppose that G is locally compact. Here is a (tautological) way to reformulate thecondition of continuity of action. It amounts to the combination of the following two:

(i) For every v ∈ V , a semi-norm ρ and ε, there exists a neighborhood 1 ∈ U ⊂ G such that

ρ(Tg(v)− v) < ε for g ∈ U.

(ii) For every compact Ω ⊂ G and every semi-norm ρ, there exists a semi-norm ρ′ so that

ρ(Tg(v)) ≤ ρ′(v), for all v ∈ V and g ∈ Ω.

2.1.4. Here are some examples of continuous representations. Let G act continuously on atopological space X.

(a) Let C(X) be the space of continuous functions on G. We topologize by convergence oncompact subsets. I.e., the fundamental system of neighborhoods of 0 is given by semi-norms

maxx∈Ω|f(x)|,

where Ω is a compact subset of X.

We consider the action of G on V = C(X) by

(Tg · f)(x) = f(g−1 · x).

Conditions (i) and (ii) are evident in this case.

(b) Let µ be a (positive) measure on X. I.e., consider Cc(X) equipped with the sup norm, µ isa continuous functional. Assume that G acts on µ continuously: i.e., for every compact Ω ⊂ Gthere exists a scalar c ∈ R+ such that

g(µ) ≤ c · µ.

Take V = Cc(X), but equip it with the topology induced by the Lp norm with respect to µ.Conditions (i) and (ii) are again evident.

(b’) Take V = Lp(X,µ), i.e., the completion of the space of from the previous example in itsnorm. Conditions (i) and (ii) follow formally from the fact that they hold on a dense subset.

6 DENNIS GAITSGORY

2.1.5. Let V be a topological vector space as in Sect. 2.1.2. Let X be a topological space, andlet

f : X → V

be a continuous function.

Proposition 2.1.6. The assignment

δx 7→ f(x)

uniquely extends to a continuous linear map

Measc(X)→ V, µ 7→∫µ

f

where Measc(X) = C(X)∗, equipped with the weak topology.

Proof. Note that the linear span of the elements δx, denoted Meas0c(X), is dense in Measc(X),

in the weak topology. By the completeness of V , we need to show that the map

µ 7→∫µ0

f, Meas0c(X)→ V

is continuous (in the topology on Meas0c(X), induced by the weak topology on Measc(X)).

This is equivalent to showing that for every semi-norm ρ on V , there exists a neighborhood0 ∈ U0 ⊂ Meas0

c(X), such that

ρ(

∫µ0

f) < 1 for µ0 ∈ U0.

However, we have

ρ(

∫µ0

f) ≤ 〈µ0, ρ(f)〉,

by convexity.

Hence, we can take U0 to correspond to the neighborhood U ⊂ Measc(X) given by

µ , |〈µ, ρ(f)〉| < 1.

2.1.7. As a corollary, we obtain that for a continuous representation V of G and v, we have awell-defined linear map

µ 7→ µ ? v : Measc(G)→ V,

which is continuous in the weak topology on Measc(G), and such that δg ? v = Tg(v).

Moreover, it is easy to see that the resulting map

Measc(G)× V → V, (µ, v) 7→ µ ? v

is continuous. For a fixed µ, we will denote the corresponding endomorphism of V by Tµ, andthe map

Measc(G)→ End(V )

by ActV ; it is continuous at least for the weak topology on End(V ).


2.1.8. By a Dirac sequence we will mean a family of continuous compactly supported functionsfn on G such that fn · µHaar → δ1 in the weak topology on Measc(G). The latter conditionfollows from the following two:

∫fn → 1 and supp(fn)→ 1.

We obtain that for a continuous representation V and v ∈ V ,

(2.1) fn ? v → v.

2.2. The notion of K-finite vector.

2.2.1. Let K be compact, and V a continuous representation of K. We let V K -fin denote thesubspace of V equal to the union of finite-dimensional K-subrepresentations of V .

Clearly, an element v ∈ V belongs to V K -fin if and only if the elements Tk(v) span a finite-dimensional subspace of V .

2.2.2. For an irreducible finite-dimensional representation ρ, we let

V ρ ⊂ V K -fin

be the ρ-isotypic component, i.e., the union of finite-dimensional subrepresentations that areisomorphic to direct sums of copies of ρ.

Clearly, we have

V ρ ' ρ⊗HomK(ρ, V ),

where HomK(ρ, V ) ⊂ Hom(ρ, V ) inherits a topology from V .

We have

V K -fin = ⊕ρ∈Irrep(K)

V ρ.

2.2.3. We claim:

Lemma 2.2.4. Let ρ be an irreducible finite-dimensional representation. Then for any S ∈End(ρ∗), the image

MCS,ρ ·µHaar ∈ Meas(K)

acting on V belongs to V ρ.

Proof. Follows from the fact that

δk ? (MCS,ρ ·µ) = MCTg·S,ρ .

Corollary 2.2.5. For an irreducible representation ρ, the element ξρ · µHaar ∈ Meas(K) actsin any continuous representation V as an idempotent with image equal to V ρ.

Proof. Follows from Lemma 2.2.4 and Corollary 1.4.4.

2.3. Peter-Weyl theorem.

8 DENNIS GAITSGORY

2.3.1. Recall that we have an isometric embedding

(2.2) ⊕ρ∈Irrep(K)

End(ρ)→ L2(K),

where each End(ρ) is endowed with a Hermitian structure by formula (1.8).

The Peter-Weyl theorem says:

Theorem 2.3.2.

(a) The map is an isomorphism.

(b) The subspace ⊕ρ∈Irrep(K)

End(ρ) ⊂ L2(K) identifies with the set of K-finite vectors with

respect to the action by left translations.

2.3.3. We will first prove point (b). This consists of the following two statements:

Proposition 2.3.4. The map MCρ : End(ρ)→ C(K) is an isomorphism onto C(K)ρ,l, wherethe superscript l indicates that we are considering C(K) as equipped with the action by lefttranslations.

Proof. This is just Frobenius reciprocity:

HomK(ρ, C(K)) ' ρ∗.

Proposition 2.3.5. The inclusion C(K)K -fin,l → L2(K)K -fin,l is an isomorphism.

Proof. We will need the following lemma:

Lemma 2.3.6. Let W be a finite-dimensional representation of K. Then IdW lies in the imageof C(K) ⊂ Meas(K) under ActW .

Proof. On the one hand, the closure of the image of C(K) contains IdW by (2.1). On the otherhand, the said image is a linear subspace of End(W ), and hence is closed.

Let f be a (left) K-finite element in L2(K). By the above lemma, we can find f1 ∈ C(K)so that f1 ? f = f (note that the operator Tf1 for V = L2(K) amounts to the convolutionf1 ?−). However, as we shall see below (in Sect. 2.4.3), for any f1 ∈ C(K) and f ∈ L1(K), theconvolution f1 ? f belongs to C(K).

2.4. Proof of density.

2.4.1. To prove point (a) of Theorem 2.3.2, we need to show that for any v ∈ L2(K) and ε thereexists v′ ∈ L2(K)K -fin,l such that ||v − v′|| < ε.

We recall that an operator T : V1 → V2 between Banach spaces is called compact if theclosure of the image of the unit ball in V1 is compact in V2.

We will use the following key observation:

Theorem 2.4.2. Let X and Y be compact metric spaces, and let F (−,−) be a continuousfunction on X × Y . Then for µ ∈ Meas(X), the function

Tµ,F (y) =

∫x∈X,µ

F (x, y)


is continuous. Moreover, the resulting operator

TF : Meas(X)→ C(Y ),

is compact, when Meas(X) is considered as equipped with its natural norm.

Proof. We will show that Tµ,F is bounded and uniformly continuous with the estimates de-pending only on ||µ||. This will prove the theorem in view of the Arzela-Ascoli theorem.

Indeed, for boundedness we note that for any y ∈ Y

|Tµ,F (y)| ≤ ||µ|| · sup(F ).

For uniform continuity fix an ε. Since F is uniformly continuous, we can find a δ such that

ρ(y1, y2) < δ ⇒ |F (x, y1)− F (x, y2)| < ε

||µ||∀x ∈ X.

However, this implies that

|Tµ,F (y1)− Tµ,F (y2)| < ε.

2.4.3. We apply the above observation to X = Y = K, and F (k1, k2) = f(k−11 · k2), where

f ∈ C(K) We note that the corresponding operator

TF : Meas(K)→ C(K)

equals

µ 7→ µ ? f.

In particular, we obtain that the latter operator is compact. Hence, the operator

T rf : L2(K)→ L2(K),

being equal to the composition

L2(K)→ Meas(K)µ7→µ?f−→ C(K)→ L2(K),

is also compact.

2.4.4. For a given v ∈ L2(K), let us choose f to be a continuous function on K such that||v − v ? f || < ε

2 ; it exists by (2.1), where we regard L2(K) as equipped with the action on Kby right translations.

We can furthermore assume that f is such that f is real-valued and satisfies f(g) = f(g−1).In this case the operator T rf : L2(K)→ L2(K) is self-adjoint.

We now quote the following theorem (which is proved in the same way as its finite-dimensionalcounterpart):

Theorem 2.4.5. Let T be a compact self-adjoint operator on a Hilbert space H. Then H

admits a direct sum decomposition

H ' H0 ⊕⊕λ

Hλ,

where Hλ is the λ-eigenspace of T . Moreover: (i) for λ 6= 0 the space Hλ is finite-dimensional;(ii) for every ε all but finitely many λ’s satisfy |λ| < ε.

10 DENNIS GAITSGORY

We apply this theorem for H = L2(K) and T = T rf . Note that since left and right multipli-

cations commute, we obtain that each Hλ is left K-invariant. In particular, we obtain that forλ 6= 0, we have Hλ ⊂ L2(K)K -fin,l.

Decompose v as v0 + Σλ6=0

vλ with each vλ ∈ Hλ. Note that

v − v ? f = v0 + Σλ6=0

(1− λ) · vλ.

Hence,

||v − v ? f || < ε

2⇒ ||v − Σ

|λ|> ε2

vλ|| < ε.

Thus, the (finite) sum Σ|λ|> ε

2

vλ gives the desired approximation to v by left K-finite vectors.

This finishes the proof of Theorem 2.3.2(a).

2.5. Density in other representations. We will now prove:

Theorem 2.5.1. For any continuous representation V of K, the subset V K -fin ⊂ V is dense.

Proof. Let v be a vector in V . Choose a Dirac sequence pf continuous functions fn so that

fn ? v → v.

By Peter-Weyl, we can choose K-finite functions f ′n such that ||fn − f ′n||L2 <1n . Then the

sequence f ′n also converges to δ1 in the weak topology on Meas(K). Hence,

f ′n ? v → v.

However, each f ′n ? v is K-finite by Lemma 2.2.4.

Corollary 2.5.2. Matrix coefficients of finite-dimensional representations are dense in the maxnorm in C(K).

2.5.3. As another corollary we obtain:

Corollary 2.5.4. Any irreducible continuous representation of K is finite-dimensional.

Proof. Let V be an irreducible representation. Since V K -fin is dense in V , it is non-zero. HenceV K -fin = V . By irreducibility, V K -fin is isomorphic to some ρ.

2.6. Plancherel’s formula.

2.6.1. Recall that the Peter-Weyl theorem says that the map

⊕V ∈Irrep(K)

End(V )→ L2(K)

(where ⊕ means the Hilbert space direct sum) is an isomorphism. Since we already know thatit is an isometric embedding, the assertion amounts to the fact that the above map has a denseimage.

It follows from Corollary 1.3.5 that the inverse map

L2(K)→ ⊕V ∈Irrep(K)

End(V ),

sends

(2.3) f 7→ ⊕ dim(V ) · Tf .In particular, we have:


Theorem 2.6.2. For f1, f2 ∈ L2(K), we have

(2.4)

∫k∈K

f1(k) · f2(k) =∑

V ∈Irrep(K)

dim(V ) · Tr(Tf1 T†f2, V ),

where the series on the right is absolutely convergent.

The latter identity is one form Plancherel’s formula for compact groups.

2.6.3. Here is another identity (also sometimes referred to as Plancherel’s formula):

Theorem 2.6.4. For a continuous function f on K, we have

(2.5) f(1) = Σρ∈Irrep(K)

dim(ρ) · Tr(Tf , ρ),

where the series on the right is an absolutely convergent.

In order to prove this assertion, one need to recall the notion of trace class endomorphismof a Hilbert space. By definition, a compact endomorphism T : H → H is said to be of traceclass if for some choice of orthonormal bases ei, the series

(T (ei), ei)

is absolutely convergent.

If this happens, one (easily) shows that the above condition and the resulting sum

Tr(T,H) := (T (ei), ei)

does not depend on the choice of a basis. Moreover, if H is represented as⊕i

Hi, then

(2.6) Tr(T,H) = Σi

Tr(Ti,Hi),

where Ti is the orthogonal projection of T onto Hi (part of the statement is that each Ti istrace class and the above series is absolutely convergent).

Let us be in the situation of Theorem 2.4.2 with X = Y . Let X be equipped with a Borelmeasure µ; consider the corresponding embedding

C(K)f 7→f ·µ−→ L2(K,µ)

and its dualL2(K,µ) → Meas(K,µ).

For a continuous function F on X×X consider the corresponding composite (also abusivelydenoted TF ):

L2(K,µ) → Meas(K)TF−→ C(K) → L2(K,µ).

Proposition 2.6.5. Under the above circumstances, TF is trace class and

Tr(TF , L2(X,µ)) =

∫x∈X,µ

F (x, x).

Proof of Theorem 2.6.4. We apply Proposition 2.6.5 to X = K and F (k1, k2) = f(k1 · k−12 ), so

that TF = T lf . By Proposition 2.6.5, the operator T lf is trace class and its trace equals f(1).

However, by (2.3) and (2.6), the RHS in (2.5) also equals Tr(T lf , L2(K)).

12 DENNIS GAITSGORY


3.1. Some differential calculus.

3.1.1. Let X be a differentiable manifold, and let V be a topological vector space as inSect. 2.1.2. We shall say that a function

F : X → V

is differentiable at x, of there exists a linear map dFx : TxX → V such that for every vectorξx ∈ TxX and some/any differentiable path

γ : (−1, 1)→ X, γ(0) = x, dγ0 = ξx

we haveF (γ(t))− F (x)

t→ dFx(ξx).

Let ξ1, ..., ξn be a vector fields on X such that ξ1x, ..., ξ

nx form a basis of TxX for every

x ∈ X (such a frame of vector field exists locally on X). We shall say that F is continuouslydifferentiable if the functions

Lieξi(F ); x 7→ dFx(ξix), i = 1, ..., n

are continuous. This definition is easily seen to be independent of the choice of the frame.

We let C1(X,V ) ⊂ C(X,V ) be the subspace of continuously differentiable functions. Wedefine the spaces Cm(X,V ) inductively, by setting

Cm(X,V ) = F ∈ Cm−1(X,V ) , Lieξi(F ) ∈ Cm−1(X,V )∀i = 1, ..., n.

Set

C∞(X,V ) = ∩mCm(X,V ).

3.1.2. We recall the notion of the ring of differential operators on X. Namely, we define

Diff≤n(X) ⊂ EndC(C∞(X))

inductively as follows:

(i) Diff≤n(X) = 0 for n < 0;

(ii) T ∈ Diff≤n(X) if and only if [T, f ] ∈ Diff≤n−1(X) = 0 for every f ∈ C∞(X), viewed as anendomorphism g 7→ f · g.

Clearly,

Diff(X) := ∪n

Diff≤n(X)

is a subring of EndC(C∞(X)). Each Diffn(X) is a bimodule over C∞(X).

Thus, Diff≤0(X) = C∞(X). One can show that Diff≤1(X) = C∞(X)⊕Vect(X). Moreover,for a choice of a frame ξ1, ..., ξn as above, every differential operator can be uniquely written as

Σi1,...,in≥0

fi1,...,in · ξi11 · ... · ξinn , fi1,...,in ∈ C∞(X).

Canonically

Diff≤n(X)/Diff≤n−1(X) ' SymnC∞(X)(Vect(X)).


3.1.3. In what follows,for x ∈ X, it will be convenient to consider the vector space

Distrx(X) := C ⊗C∞(X)

Diff(X),

which is the union of its subspaces Distr≤nx (X) := C ⊗C∞(X)

Diffn(X).

Note that for f ∈ C∞(X) and d ∈ Diff(X), the value of d(f) at x only depends on the imageof d in C ⊗

C∞(X)Diff(X).

Let Germnx be the quotient ring of C∞(X) by the ideal of functions that vanish to the order

n at x (so that evaluation at x defines an isomorphism Germ0x ' C).

The map

d, f 7→ (d(f))(x)

defines a pairing

(3.1) Distr≤nx (X)⊗Germnx → C.

It is easy to see that (3.1) is a perfect pairing, i.e., identifies Distr≤nx (X) as the dual ofGermn

x .

Note that for a smooth map

F : X → Y, F (x) = y,

we have a canonically defined maps

(3.2) Diffnx(X)→ Diffny (Y ), Distrx(X)→ Diffy(Y ),

where the former map is the dual of the pullback map F ∗ : Germny → Germn

x .

Consider the vector space Cn(X) equipped with its natural topology. Consider its topologicaldual Cn(X)∗. The Taylor expansion at x defines a map

Cn(X)→ Germnx .

Hence, we obtain a naturally defined map

(3.3) Distr≤nx (X) → Cn(X)∗.

By definition, C∞(X) := ∩nCn(X), and it is equipped with the inverse image topology. Set

Distrc(X) := C∞(X)∗.

From (3.3), we obtain an embedding

(3.4) Distrx(X) → Distrc(X).

3.2. Differentiable vectors.

14 DENNIS GAITSGORY

3.2.1. Let G be a Lie group, and let V be a continuous representation. We shall say that avector v ∈ V is differentiable if the function

(3.5) F v : G→ V, , F v(g) := Tg(v)

is differentiable at 1.

Lemma 3.2.2. If v ∈ V is differentiable, then the function (3.5) is continuously differentiable.

Proof. For ξ ∈ g, let ξr denote the corresponding left-invariant vector field. Note that suchfields span the tangent space at every point of G. Hence, it is enough to show that the functionsLieξr (F ) are continuous on G. However, it is easy to see that

Lieξr (Fv) = F v

′, v′ := dF v1 (ξ).

Let V 1 ⊂ V be the subspace consisting of differentiable vectors. By construction, we have awell defined map

g⊗ V 1 → V, ξ, v 7→ dF v1 (ξ).

For ξ ∈ g denote the corresponding map V 1 → V by Tξ.

Define V n inductively by

V n = v ∈ V n−1 , Tξ(v) ∈ V n−1 ∀ξ ∈ g

Set

V∞ := ∩nV n.

We topologize V n inductively as follows as follows. If ρ is a system of semi-norms for V n−1

and ξi is a basis for g, we introduce semi-norms ρi on V n by

ρi(v) = ρ(Tξi(v)).

We give V∞ the inverse limit topology.

In the same way as in Proposition 2.1.6, one shows:

Proposition 3.2.3. For any m,n there exists a canonically defined continuous map

Cn(G)∗ × V m+n → V m, µ 7→ Tµ.

It is uniquely determined by the requirement that for v ∈ V m+n and v∗ ∈ (V m)∗ we have

〈v∗, Tµ(v)〉 = 〈µ, 〈v∗, F v〉〉,

where the function 〈v∗, F v〉 on G is easily seen to be in Cn(G).

As a corollary, we have:

Corollary 3.2.4. There is a canonically defined continuous map

Distrc(G)× V∞ → V∞, µ 7→ Tµ,

compatible with the convolution algebra structure on Distrc(G).


3.2.5. Recall the vector space Distr1(G) = ∪n

Distr≤n1 (G). The group law onG endows Distr1(G)

with a structure of associative algebra via (3.2).

We can alternatively view this structure as follows: identify Distr1(G) with left (or right)invariant differential operators on G. Then the associative algebra structure on Diff(G) inducesone on Distr1(G).

The assignment

(ξ ∈ g) 7→ (f 7→ df1(ξ))

defines a map

g→ Distr≤11 (G) ⊂ Distr1(G),

and it is easy to see that it respects the commutator relation. Hence, we obtain a homomor-phisms of associative algebras

(3.6) U(g)→ Distr1(G).

It is easy to see that (3.6) is an isomorphism. Alternatively, we can view (3.6) as an identi-fication between U(g) with left (or right) invariant differential operators on G.

Combining with (3.2.3) we obtain canonical maps

U(g)≤n ⊗ V m+n → V m, u 7→ Tu

as well as an action of U(g) on V∞.

By construction, for ξ ∈ g ⊂ U(g), this notation is consistent with the notation Tξ introducedearlier.

3.2.6. Here comes the arch-important smoothing construction:

Lemma 3.2.7. Let f be an element of Ckc (G), and consider f µHaar as an element of C(G)∗,where µHaar is a left-invariant Haar measure. Then for any v ∈ V , the vector Tf ·µHaar(v) belongsto V k.

Proof. It is easy to see that for ξ ∈ g,

Tξ(Tf ·µHaar(v)) = TLieξ(f)·µHaar

(v).

Corollary 3.2.8. For f ∈ C∞c (G) and any v ∈ V , the vector Tf ·µHaar(v) ∈ V belongs to V∞.

Corollary 3.2.9. For any V , the subspace V∞ ⊂ V is dense.

Proof. We can choose a Dirac sequence of smooth functions fn. By (2.1), we have

Tfn(v)→ v.

However, by Corollary 3.2.8, each Tfn(v) is smooth.

Corollary 3.2.10. Let V be a continuous finite-dimensional representation. Then V∞ = V .Equivalently, the image of

MCV : End(V )→ C(G)

lies in C∞(G).

Proof. A dense subspace of a finite-dimensional vector space is everything.

3.3. Compact real algebraic groups.

16 DENNIS GAITSGORY

3.3.1. Let G be an algebraic group over R. We will say that G is compact, if G(R), endowedwith its natural structure of Lie group, is compact. Thus, we obtain a functor

(3.7) Compact algebraic groups over R → Compact Lie groups.

The functor (3.7) is not an equivalence of categories. Indeed, non-isomorphic compact alge-braic groups over R can give rise to the same Lie group.

Example: Take the group of 3rd roots of unity x3 = 1. The group of its real points is thesame as that of the trivial algebraic group.

3.3.2. We shall say that a compact algebraic groups over R is relevant if the map

π0(G(R))→ π0(G)

is surjective, where π0 is the algebro-geometric group of connected components.

We will prove:

Theorem 3.3.3. The restriction of the functor (3.7) to the subcategory of relevant compactalgebraic groups is an equivalence from the category of relevant compact algebraic groups overR to that of compact Lie groups.

3.3.4. As a first step, we will prove:

Theorem 3.3.5. Every compact Lie group admits an injective homomorphism into GLn(R)for some n.

Proof. Let K be a compact Lie group. The statement of the theorem is equivalent to fact thatK admits a faithful finite-dimensional representation. Note that if K is a compact Lie groupand

K ⊃ K1 ⊃ K2...

is sequence of closed subgroups with trivial intersection, then then exists an n such that Kn =1.

The set of isomorphism classes of irreducible representations of K is countable. Enumeratethem by ρ1, ρ2, ..., etc. Set

Vi := ρ1 ⊕ ...⊕ ρiand Ki = ker(K → GL(Vi)).

We claim that ∩iKi = 1. Indeed, if k ∈ K belongs to the above intersection, then its acts

trivially in every irreducible representation, and hence, by Corollary 2.5.2, multiplication by kacts as identity on C(K). The latter means that k = 1.

Hence, by the above Kn = 1 for some n.

3.3.6. To prove Theorem 3.3.3 we will need the following construction. Let Z be an affinealgebraic variety over R, and let X be a subset of Z(R). Let IX be the ideal of regularfunctions on Z that vanish on X, Let Z ′ := V (IX) be the corresponding algebraic subvarietyin Z. Clearly, X ⊂ Z ′(R).

Moreover, by construction for every connected component of Z ′0 ⊂ Z ′, we have

(3.8) X ∩ Z ′0(R) 6= ∅.

3.3.7. Assume now that Z is acted on by a compact Lie group K (i.e., the algebra of regularfunctions on Z is an algebraic1 representation of K). Assume that X is a single K-orbit.

1algebraic=K-finite


Proposition 3.3.8. Under the above circumstances, the inclusion X ⊂ Z ′(R) is an isomor-phism.

Proof. Let x′ be a point in Z(R) that is not in X. We claim that we can find a regular functionthat vanishes on X and doesn’t vanish at x′. Let X ′ be the K-orbit of x′. We claim that wecan find a K-invariant regular function that takes value 0 on X and value 1 on K ′. Indeed, byStone-Weierstrass, the map

C[Z]→ C(X tX ′)is K-equivariant and has a dense image (indeed, embed Z in the affine space Rn so that X tX ′becomes a compact subset of Rn). Averaging with respect to K, we obtain that the map

C[Z]K → C(X tX ′)K

also has a dense image. However, C(X tX ′)K ' C⊕C, so the latter map is surjective, and inparticular, its image contains the element (0, 1).

3.3.9. We are now ready to prove Theorem 3.3.3:

Let now G be a real algebraic group and K a compact subgroup of G(R). Consider thesubscheme G′ ⊂ G, obtained by the construction in Sect. 3.3.6. By the functoriality of thisconstruction, we obtain that G′ is an algebraic subgroup of G. By Proposition 3.3.8, the mapK → G′(R) is an isomorphism, so that G′ is a compact real algebraic group, and K is thegroup of its R-points. Moreover, by (3.8), the compact real group G′ is relevant. Combinedwith Theorem 3.3.5, this implies that that functor in Theorem 3.3.3 is essentially surjective.

Let us now show that the functor in Theorem 3.3.3 is fully faithful. Let G1 and G2 berelevant compact real groups, and let φ : G1(R) → G2(R) be a homomorphism. We wish toshow that φ comes from a (unique) homomorphism of algebraic groups. Let

K ⊂ G1(R)×G2(R)

be the graph of φ. Let G′1 be the subgroup of G1×G2 corresponding to K via the constructionof Sect. 3.3.6. It suffices to show that the projection ψ : G′1 → G1 is an isomorphism.

By Proposition 3.3.8, the map G′1(R) → G1(R) is an isomorphism. Hence, ψ induces anisomorphism at the level of Lie algebras, and hence also of the (algebraic) connected componentsof the identity. Since both groups are relevant, this implies that ψ is itself an isomorphism.

Note that the above proof actually showed:

Corollary 3.3.10. Let G1 and G2 be real algebraic groups with G1 compact and relevant. Thenthe map

HomAlgGrp(G1, G2)→ HomLieGrp(G1(R), G2(R))

is an isomorphism.

In particular, taking G2 to be GLn,C|R (here the operation Z 7→ Z|R is the operation ofrestriction of scalars a la Weil, i.e., the standard way of viewing a complex algebraic variety asa real one), we obtain:

Corollary 3.3.11. For a compact relevant real algebraic group G and its complexification GC,the map

HomAlgGrp /C(GC, GLn,C)→ HomLieGrp(G(R), GLn(C))

is a bijection.

18 DENNIS GAITSGORY

The last corollary says that for a compact relevant real algebraic group G, the categoryof algebraic representations of its complexification is canonically equivalent to the category ofcomplex representations of the underlying compact Lie group.

4. Week 2, Day 2 (Thurs, Feb 2)

4.1. Compact Lie groups vs complex reductive algebraic groups.

4.1.1. Recall that an algebraic group is said to be reductive if its unipotent radical is trivial.

Proposition 4.1.2. Let G be a compact real algebraic group. Then its compexification GC isreductive.

Proof. Let U(G) be the unipotent radical of G. Consider the center of U(G). This is a groupover R, whose complexification is isomorphic to Gna for some n ≥ 1; by Hilbert 90, we obtainthat this isomorphism is valid also over R. However, the group of its R-points is then isomorphicto Rn, which cannot be contained as a closed subgroup in a compact group.

4.1.3. From the above proposition we obtain that complexification defines a functor

Compact real algebraic groups → Complex reductive groups.We will study how close this functor is to be an equivalence.

4.2. The polar decomposition.

4.2.1. We will prove the following theorem:

Theorem 4.2.2. Let G a complex algebraic group. Let K be a compact Lie subgroup of G(C).Assume that:

(i) The mapkC := C⊗

Rk→ g

is an isomorphism (i.e., g is a complexification of k).

(ii) K intersects non-trivially every connected component of G(C);

Then the group G carries a unique real structure σ for which K = G(C)σ. Moreover: letp ⊂ g be the subspace

ξ ∈ g, , σ(ξ) = −ξ.Then the map

(4.1) K × p→ G(C), k, p 7→ k · exp(p)

is a diffeomorphism.

In the situation of the theorem we denote by P ⊂ G(C) the image of p ⊂ g under theexponential map. By (4.1), this is a closed submanifold of G(C). By construction

(4.2) P ⊂ P := g ∈ G(C), , σ(g) = g−1.

Corollary 4.2.3.

(a) The product map defines a diffeomorphism

tk∈K,k2=1

(k × pk)→ P .

(b) For every g ∈ P , we have g2 ∈ P .


Corollary 4.2.4. Let G be as in Theorem 4.2.2. Then G is reductive.

Proof. Follows from Proposition 4.1.2.

Corollary 4.2.5. The map K → G(C) is a homotopy equivalence. In particular π0(K) 'π0(G(C)), and π1(K) ' π1(G(C)).

Note that for an algebraic variety Z over C, the topological π0(Z(C)) identifies with thealgebro-geometric π0(Z). Also, for a reductive algebraic group G one can define its algebraicπ1(G), and for G over the field of complex numbers we have π1(G(C)) ' π1(G).

Corollary 4.2.6. In the situation of Theorem 4.2.2, we have

(4.3) ZG(C) = CentrG(C)(K) = ZK × (ZG(C) ∩ P ).

If G is semi-simple, ZG(C) = ZK .

Proof. By the isomorphism (4.1), in order to prove (4.3), we only need to show that the inclusionZG(C) ⊂ CentrG(C)(K) is an equality. If g ∈ Centr(K), then its adjoint action on k is trivial, andhence its adjoint action on g is also trivial (by condition (i)). Therefore g ∈ CentrG(C)(G0(C)).To prove that the adjoint action of g on all of G is trivial, it is therefore enough to show thatevery connected component of G(C) contains a point that commutes with g. However, everyconnected component contains a point of K by (ii).

Let us show that if G is semi-simple, then ZG(C) ∩ P is trivial. Indeed, by the above,ZG(C)∩P equals the set of AdK-invariants in P , and the exponential map identifies the latterwith pK . However, pK ⊂ gK = gG, and the latter is trivial, since G is semi-simple.

Corollary 4.2.7. In the situation of Theorem 4.2.2, we have

NormG(C)(K) = K × (ZG(C) ∩ P ).

Proof. It is clear that NormG(C)(K) = K · (Norm(K)∩ P ). However, from (4.1) it follows thatNorm(K) ∩ P ⊂ CentrG(C)(K), and the latter equals ZG(C).

4.2.8. Example. Let G = Gnm. Consider the subgroup K = Un1 ⊂ (C×)n. It obviously satisfiesthe assumption of Theorem 4.2.2. The corresponding real structure is

σ(g) = g−1.

We claim that this is the unique compact real form of Gnm.

Indeed, the quotient (C×)n/Un1 is isomorphic to Rn, and hence contains no non-trivial com-pact subgroups. Hence, if σ′ is another compact real structure, then the corresponding subgroupK ′ is contained in K. However, since dimR(k′) = n, we obtain that K ′ = K. Hence, σ′ = σ.

4.3. Proof of Theorem 4.2.2.

4.3.1. We first consider the particular case whenG = GL(V ), where V is a complex vector space.Assume that V is equipped with a Hermitian form. Define a real structure on G = GL(V ) by

σ(S) = (S†)−1.

Then K = U(V ) and p = E(V ) is the space of Hermitian operators. The exponential mapdefined an isomorphism

E(V )→ E+(V ),

20 DENNIS GAITSGORY

where the latter is the space of positive Hermitian operators. The statement that

U(V )× E+(V )→ GL(V,C)

is an isomorphism is well-known. The inverse map is constructed as follows:

For S ∈ GL(V,C) consider S S†. This is a positive Hermitian operator; hence it admits a

well-defined square root (S S†) 12 . In sought-for inverse map sends S to the pair

(S S†)− 12 S ∈ U(V ), (S S†) 1

2 ∈ E+(V ).

4.3.2. Let us now return to the general case of Theorem 4.2.2.

First off, it is clear that the real structure σ is unique: indeed, condition (i) fixes what it ison g, and hence on G0, and condition (ii) fixes it on the rest of G.

Let G be as in the theorem, and let G→ GL(V ) be a faithful representation. Choose a K-invariant Hermitian structure on V . We claim that the real structure on GL(V ) from Sect. 4.3.1induces one on G.

We have to show that the operation S 7→ (S†)−1 preserves the image of G(C) in GL(V,C).By construction, it acts as identity on K, and hence preserves k, and hence by (i) all of g, andhence G0(C). By (ii) it preserves all of G(C).

Denote K = G(C)σ. We obviously have K ⊂ K, and this inclusion is an isomorphism at thelevel of Lie algebras, and hence on the neutral connected components (we will soon prove thatit is an isomorphism also at the group level).

4.3.3. We claim that the mutually inverse maps

U(V )× E+(V ) GL(V )

send the subsets

G(C) ⊂ GL(V )

and

K × p ⊂ U(V )× E+(V )

to one another.

To show this, we only need to see that for g ∈ G(C) ∩ E+(V ), the elements gt ∈ GL(V,C),t ∈ R and log(g) ∈ End(V ) belong to G(C) and g, respectively.

If n ∈ Z, then gn clearly belongs to G(C). However, for any S ∈ GL(V,C), the Zariskiclosure of the elements Sn contains the elements St, t ∈ R and log(S). This implies that gt

and log(g) belong to G(C), since G(C) is Zariski-closed in GL(V,C).

4.3.4. Thus, we have established that the map

K × p→ G(C).


In particular π0(K)→ π0(G(C)) is an isomorphism. However, by assumption, the compositemap

π0(K)→ π0(K)→ π0(G(C))

is surjective. Hence, the map

K → K

is a surjective at the level of π0 components, and hence is an isomorphism.


4.3.5. Example. Let V be a real vector space equipped with a positive-definite scalar product.Consider the corresponding algebraic group O(V ). Then this is a relevant compact real formof its complexification O(V )C. Indeed, we only need to see that O(V,R) hits every connectedcomponent of O(V,C), but the latter is obvious.

5. Week 3, Day 1 (Tue, Feb. 7)

5.1. Compact Lie groups vs compact Lie algebras.

5.1.1. Recall that a Lie algebra g is semi-simple if and only if the Killing form

Kil(ξ, η) := Tr(adξ adη, g)

is non-degenerate. (This is insensitive to what field we are working over, as long as it is ofcharacteristic 0.)

5.1.2. Let k be a real Lie algebra. We shall say that it is compact if its Killing form is negativedefinite. By the above, if k is compact, then it is semi-simple.

Let Aut(k) be the real algebraic group of automorphisms of k. We claim:

Lemma 5.1.3.

(a) If k is compact, then the Lie group Aut(k)(R) is compact.

(b) If K is a compact Lie group with trivial center, then its Lie algebra is compact.

Proof. By definition, Aut(k)(R) is a closed subgroup of the group GLn(R) of linear automor-phisms of k as a vector space, where n = dim(k). However, it is actually contained in theorthogonal group On(R) corresponding to Kil. Since the latter is negative-definite, the orthog-onal group in question is compact. Hence, Aut(k)(R) is compact as well, proving (a).

For (b), since K is compact, we can choose a K-invariant positive-definite scalar product onk. We have an embedding

K → On,

Consider the corresponding maps of Lie algebras k → on.

The Killing form on k is obtained by restriction form the standard form on on:

(S1, S2) 7→ Tr(S1 S2),

while the above form is easily see to be negative-definite (skew-symmetric matrices have imag-inary eigenvalues).

Corollary 5.1.4. For a compact Lie algebra k, the neutral (algebraic) connected componentAut(k)0 of Aut(k) is a relevant compact real algebraic group.

5.1.5. Let g be a complex semi-simple Lie algebra, with a chosen Cartan and Boreal subalgebrash ⊂ b ⊂ g. For vertex root α let hα ∈ h be corresponding simple coroot element. Pick a non-zeroelement eα in the corresponding root space. Let fα be the unique element in the correspondingnegative root space such that [eα, fα] = hα.

Then it is easy to see that there exists a unique real structure σ on g such that

σ(hα) = −hα; σ(eα) = −fα; σ(fα) = −eα.

22 DENNIS GAITSGORY

We claim that this real structure is compact. Indeed, it is well-known that the Killing formis positive definite on SpanR(hα) and

Kil(eα, fα) =Kil(hα, hα)

2> 0.

Hence, it is negative definite on i · h as well as on eα− fα and i · (eα + fα), while the rest of thescalar products are zero.

5.2. Existence and uniqueness of the compact real form.

5.2.1. Let σ and τ be two real structures on a complex Lie algebra. We shall that they are com-patible if the (sesqui)-linear automorphisms σ and τ commute. This is equivalent to requiring:

(i) gσ = (gσ ∩ gτ )⊕ (gσ ∩ i · gτ );

(i’) gτ = (gσ ∩ gτ )⊕ (i · gσ ∩ gτ );

(ii) σ(gτ ) = gτ ;

(ii’) τ(gσ) = gσ.

(iii) The linear automorphism θ = τ σ satisfies θ2 = 1.

Note that if both σ and τ are compact, then σ = τ (indeed, if the Killing form is negative-definite on gσ, then it is positive-definite on i · gσ, so the intersection i · gσ ∩ gτ is zero).

5.2.2. We have:

Theorem 5.2.3. Let g be a complex semi-simple Lie algebra, and let σ be a compact realstructure on g. Let τ be some other real structure on g.

(a) There exists an element g ∈ Aut(g)0(C) such that Adg(σ) is compatible with τ .

(b) If σ1 and σ2 be two compact real structures on g compatible with a real form ψ, then thereexists g ∈ Aut(gψ)(R)0 such that Adg(σ1) = σ2.

Corollary 5.2.4. If σ1 and σ2 are two compact real structures on a semi-simple Lie algebra gare conjugate by an element of Aut(g)0(C).

Proof. Set G = Aut(g). Let K ⊂ G(C) be the group of fixed points of the involution inducedby σ. By Lemma 5.1.3, K is compact. We’d like to apply Theorem 4.2.2. For this we need toknow that K meets every connected component of Aut(g). This is obvious for the compact realform from Sect. 5.1.5. So we argue as follows: we first prove the theorem in this case (below);then deduce Corollary 5.2.4, thereby verifying the condition on π0 for any given compact form.Then the proof given below applies to this compact form.

Consider the decompositionG(C) ' K × P.

Recall also the subsetP ⊂ G(C),

see (4.2)

The element θ = τ σ ∈ G(C). We have g ∈ P . Hence, by Corollary 4.2.3, θ2 ∈ P . Set

g = θ14 ∈ P . Then it is easy to see that σ′ = Adg(σ) satisfies

τ σ′ = σ′ τ.This proves point (a).

For point (b), we apply the above construction for σ = σ1 and τ = σ2. Then all the elementsgt (for t ∈ R) commutes with ψ, and hence belong to (the identity component of) Aut(gτ )(R).


5.2.5. We will now prove:

Theorem 5.2.6. Let G be a complex reductive group.

(a) G admits a relevant compact real form, and such forms are in bijection with those of G0/ZG0.

(b) Any two such real forms are conjugate by an element of G0(C).

Proof. Note that if G is adjoint (i.e., maps isomorphically to Aut(g)0), then the assertion ofTheorem 5.2.6 follows from Sect. 5.1.5 and Corollary 5.2.4.

Assume that G is such that G0 is adjoint. Choose a compact real form of G0; let K0 denotethe corresponding compact Lie group. Set K = NormG(C)(K0). Then K · G0(C) = G(C) andK ∩G0(C) = K0, by Corollaries 5.2.4 and 4.2.7. Applying Theorem 4.2.2, we obtain that theresult follows from the adjoint case.

Let G be a torus. Then G admits a unique compact real form by Sect. 4.2.8.

Let G be such that G0 is a torus. In this case it is still true that G has a unique compact realform. Indeed, the quotient of G(C) by the maximal compact subgroup of G0(C) is canonicallyof the form

π0(G) nRn.

Let G be an arbitrary reductive group. Consider the (surjective) map

G→ G := G/ZG0 ×π0(G)

G/(G0)′.

Choose a relevant compact real structure on G/ZG0 and consider the unique compact real

structure on G/(G0)′. Let K be the corresponding compact Lie subgroup of G(C). Let K be

the preimage of K in G(C). Applying Theorem 4.2.2, we obtain the result. (Indeed, any realstructure on G induces one on G/ZG0 and G/(G0)′ by the canonicity of these quotients.)

5.3. Polar decomposition of a real reductive group.

5.3.1. Let G be a complex reductive group, equipped with a real form τ . We will write GR forthe corresponding algebraic group over R and G(R) for its set of real points, i.e.,

G(R) = G(C)τ .

We shall say that a relevant compact real form σ is compatible with τ if θ = τ σ = σ τas automorphisms of G. Note that in this case θ defines an involution of GR as a real algebraicgroup.

From Theorem 5.2.6 and Theorem 5.2.3 we obtain:

Corollary 5.3.2. Let G be as above. Then G admits a relevant compact real form compatiblewith τ ; such forms are in bijection with those on G0/ZG0

. Any two such real forms are conjugateby an element of G(R)0.

24 DENNIS GAITSGORY

5.3.3. Let gR be the Lie algebra of GR, i.e., gR := gτ . For a compatible compact real form σ,consider the subspaces

kτ = gθR = (gR ∩ kC) = gσR = kθ

and

pτ = ξ ∈ gR , θ(ξ) = −ξ = (gR ∩ pC) = ξ ∈ gR , σ(ξ) = −ξ = ξ ∈ pC , θ(ξ) = −ξ,

where g ' kC ⊕ pC is the decomposition corresponding to σ.

We have:

gR ' kτ ⊕ pτ .

If g is semi-simple, then the form

−Kil(ξ, θ(η))

is positive-definite on gR.

5.3.4. Let K = G(C)σ. Consider the subgroup

Kτ = Kθ = G(R) ∩K = G(R)θ = G(R)σ.

From Theorem 4.2.2 we obtain:

Theorem 5.3.5. The map

(k, p) 7→ k · exp(p)

defines a diffeomorphism

Kτ × pτ → G(R).

5.3.6. Example. Let V be a real vector space, and GR = GL(V ). Choose a positive-definitescalar product on V , and endow VC with the corresponding Hermitian structure.

The real form τ on G := GL(VC) is given by S 7→ S. The compatible compact form σ isgiven by S 7→ (S†)−1. We have:

K = U(V ) and Kτ = O(V ).

5.3.7. As in the complex case, we obtain:

Corollary 5.3.8. The inclusion Kτ → G(R) is a homotopy equivalence. In particularπ0(Kτ ) ' π0(G(R)), and π1(Kτ ) ' π1(G(R)).

Corollary 5.3.9. NormG(R)(Kτ ) = Kτ × (ZG(R) ∩ PR).

Proof. We only need to show that (pτ )Kτ

is contained in the center of gR. Thus, we can assumethat gR is semi-simple. Let ξ be an element of (pτ )K

τ

. We need to show that [ξ, ξ′] = 0 for anyξ′ ∈ pτ . However, [ξ, ξ′] ∈ kτ , and it is sufficient to show that Kilg([ξ, ξ′], η) = 0 for any η ∈ kτ .However,

Kilg([ξ, ξ′], η) = Kilg(ξ′, [ξ, η]),

while [ξ, η] = 0 by assumption.

6. Week 3, Day 2 (Thurs, Feb. 9)

6.1. The maximal compact subgroup.


6.1.1. Let GR be a real reductive group, and let σ be a compatible relevant compact real formof GC. Consider the corresponding subgroup

Kτ = K ∩G(R) = G(R)σ

and the subsetP τ = P ∩G(R).

We will prove:

Theorem 6.1.2. For any compact subgroup K ′ ⊂ G(R), there exists an element g ∈ P τ suchthat Adg(K

′) ⊂ Kτ .

We will deduce this theorem from the next one.

6.1.3. Note that G(R) acts on P τ by the formula

g(p) = g · p · σ(g−1).

The stabilizer of 1 ∈ P τ is by definition G(R)σ = Kτ .

We will prove:

Theorem 6.1.4. Any compact subgroup K ′ ⊂ G(R) has a fixed point on P τ .

Let us see how Theorem 6.1.4 implies Theorem 6.1.2:

Proof of Theorem 6.1.2. Let p ∈ P τ be the fixed point of K ′. Conjugating K ′ by p we canassume that K ′ stabilizes 1 ∈ P τ . But this means that K ′ ⊂ Kτ .

6.2. Proof of Theorem 6.1.4. To prove the theorem, we can quotient G out by ZG0, so we

can assume that G is semi-simple.

We will define a certain continuous function

r : P τ × P τ → R>0

with the following properties:

(i) r is G(R)-invariant with respect to the diagonal action of G(R) on P τ × P τ ;

(ii) For every fixed p′ ∈ P τ and sufficiently large R ∈ R>0 (depending on p′), the function

p 7→ r(p, p′) : P τ → R>0

takes values > R away from a compact subset.

(iii) For every fixed p′, p ∈ P τ , the function

r(pt, p′) : R→ R>0

is strictly convex.

6.2.1. Let us prove Theorem 6.1.4 assuming the existence of such a function.

Proof of Theorem 6.1.4. For every compact subset Ω ⊂ P τ , define the function

ρΩ : P τ → R>0, ρΩ(p) = maxp′∈Ω

r(p, p′).

Lemma 6.2.2. The function ρΩ has a unique point of minimum.

Let us prove the theorem assuming this lemma. Indeed, take Ω to be the orbit of 1 ∈ P τunder K ′. Let p ∈ P τ its (unique) point of minimum. Since Ω is K ′-invariant, by condition (i),the function ρΩ is also K ′-invariant. Hence, p is K ′-invariant.

26 DENNIS GAITSGORY

6.2.3. Proof of Lemma 6.2.2. The function ρΩ is easily seen to be continuous. By condition(ii), for all sufficiently large R ∈ R>0 such that the function ρΩ takes values > R outside acompact set. Hence, it attains a minimum.

Assume that it has two minima, p1 and p2. Translating by means of p1, we can assume thatp1 = 1 and p2 = p. Consider the function

ρΩ(pt), R→ R.

By assumption, it has a minimum at t = 0 and t = 1. However, it follows from condition (iii)that the above function is strictly convex, which is a contradiction.

6.2.4. We will now construct the desired function r; we will obtain it by restriction on G(C).Choose a G→ GL(n), and choose a K-invariant Hermitian structure on Cn, so that we are inthe situation of Sect. 4.3.2. Thus, we can assume that G = GL(n) and P ' E+(n).

Since G was semi-simple, its image in GL(n) in fact belongs to SL(n).

We set

r(S1, S2) = Tr(S1 S−12 ).

Property (i) is evident. For property (ii), we claim that for a fixed S2, we have

r(S1, S2) > b · ||S1||

for some constant b > 0. Indeed, choose an orthonormal basis in which S1 is diagonal. Then

r(S1, S2) = Σiai,i · bi,i,

where ai,i are the diagonal entries of S1 and bi,i are the diagonal entries of S−12 . Note that all

bi,i are strictly positive, and

||S1|| = max ai,i.

This implies the claim: take b = min bi,i and take the minimum over all possible orthonormalbasis (the set of such is compact).

Now, we note that on the set SL(n,C), the sets

S , ||S|| ≤ c

are compact (indeed, SL(n,C) is a closed subset of Mat(n× n,C).

Property (iii) follows similarly:

r(St1, S2) = Σiati,i · bi,i.


7.1. Elements in a reductive group/algebra.


7.1.1. Let G be a connected reductive group. We will prove:

Theorem 7.1.2. Every element in a reductive group G (over an algebraically closed field) canbe conjugated into a given Borel.

Proof. Let X be the flag variety (i.e., X = G/B), thought of as the variety of Borel subgroups.

Over it we consider the bundle, denoted G (called the Grothendieck alteration) that attachesto a given x ∈ X the corresponding subgroup Bx. We have the natural forgetful map

π : G→ G.

Since G acts on X transitively, our assertion is equivalent to the fact that π is surjective. Thelatter will follow from the combination of the following three observations:

(i) G is reduced and irreducible (obvious);

(ii) The image of π is closed. This follows from the fact that π is proper; in fact, it’s the

composition of the closed embedding G → X ×G and the projection X ×G→ G.

(iii) π is generically smooth. To see the latter, take the point of X of the form (x, ξ), where ξis regular semi-simple. Consider the map

bx ⊕ g→ T(x,ξ)(X),

where the first component is the tangent space to the fiber of G→ X and the second component

is given by the action of G on G. We claim that the composition

bx ⊕ g→ T(x,ξ)(X)dπ→ Tξ(G) ' g

is surjective. Indeed, this composition is given by the inclusion of bx into g (the first component),and the map

ξ 7→ Adx(ξ)− ξ(the second component). The assertion follows now from the fact that the action of Adx− Idon g/bx is surjective: indeed the regularity assumption on x means that it has no eigenvalue 1on Adx.

7.1.3. The same proof shows that every element in g can be conjugated into a given Borelsubalgebra.

Recall that an element of G is said to be semi-simple (resp., nilpotent) if its action on OG(by left translations) is semi-simple (resp., nilpotent). Equivalently, if its action in every finite-dimensional representation is semi-simple (resp., nilpotent). The latter point of view impliesthat the condition of being semi-simple (resp., nilpotent) survives under homomorphisms ofgroups.

Corollary 7.1.4. Every semi-simple (resp., nilpotent) element of G can be conjugated into T(resp., N).

Proof. By Theorem 7.1.2, we can conjugate our element into B. Now the claim is that ev-ery semi-simple (resp., nilpotent) element of a solvable group can be conjugated into a givenmaximal torus (resp., is contained in the unipotent radical).

Corollary 7.1.5. Every torus on G can be conjugated into T .

Proof. Every torus is generated (i.e., is the Zariski closure of the abstract group generated) bya semi-simple element.

28 DENNIS GAITSGORY

7.1.6. Let K be a maximal compact in G(C). Recall that

TK := K ∩B(C)

is a maximal compact subgroup of T (C) for some Cartan T ⊂ B.

Theorem 7.1.7. Every element of K can be conjugated into TK .

Proof 1. Suppose by contradiction that k ∈ K is an element that cannot be conjugated intoTK . Consider the action of k on K/TK ' G(C)/B(C) = X(C). We obtain that this actionhas no fixed points. Now, the Lefschetz fixed point theorem implies that the action of k onH•(X(C),R) has a zero trace. Now, it is easy to see that since K is connected, the traces ofall elements of K on H•(X(C),Z) are equal (if two endomorphisms of a space are homotopic,they induce the same maps on cohomology). Hence, they are all equal to zero.

Let us take k′ = 1. Then the trace in question equals χ(X(C)) = |W | 6= 0, which is acontradiction.

Alternatively, we can take k′ to be a regular element in TK . The automorphism of X(C)defined by such a k′ preserves the orientation and has isolated fixed points (the W Borels thatcontain T ). Hence, the Lefschetz number equals |W |.

Proof 2, due to Sherry Gong. We will emulate the proof of Theorem 7.1.2. Set K be the fi-bration over X(C) that attaches to every x the intersection Bx ∩ K. We have the evidentmap

π : K → K.

Since the action of K on X(C) is transitive, it is enough to show that π is surjective. Notethat this is a map between orientable compact manifolds. In the proof of Theorem 7.1.2 we sawthat π is an orientation-preserving local isomorphism on a neighborhood of a regular element,and the preimage of such an element has has |W | many preimages.

We now have the following general assertion (well-definedness of degree):

Lemma 7.1.8. Let us be given a map f : Y → Z between orientable compact manifolds.Suppose that Z contains a a point z such that f is an orientation-preserving local isomorphismon a neighborhood of z. Then f is surjective.

7.2. Real tori. Let T be a real torus; let GC denote its complexification. We denote by t theLie algebra of T and by tC its complexification, i.e., the Lie algebra of TC.

7.2.1. We have:

TC ' Gm ⊗Z

Λ

for a canonically defined lattice Λ. The real structure τ on TC corresponds to an involution θon Λ:

τ(c⊗ λ) = c⊗ θ(λ),

so that τ θ is the unique compact real structure on TC.

Write Gm(C) = C× as U(1)× R>0,×. Hence,

T (R) = (U(1)⊗Z

Λ)τ × (R>0,× ⊗Z

Λ)τ .


Note that τ acts on U(1)⊗Z

Λ by

τ(c⊗ λ) = c−1 ⊗ θ(λ)

and on (R>0,× ⊗Z

Λ)τ by

τ(c⊗ λ) = c⊗ θ(λ).

The exponential map identifies R>0,× with the additive group R, so that

R>0,× ⊗Z

Λ ' R⊗Z

Λ,

where the latter is a vector space, equipped with a linear involution θ. Thus

(R>0,× ⊗Z

Λ)τ ' (R⊗Z

Λ)θ,

where the latter is the subspace of θ-invariants.

7.2.2. We denote

A := (R>0,× ⊗Z

Λ)τ and a := (R⊗Z

Λ)τ ,

so that the exponential map identifies a and A.

KT := (U(1)⊗Z

Λ)τ ,

i.e., KT ⊂ T (R) is the maximal compact. Thus:

(7.1) T (R) = KT ×A.This is the polar decomposition of Theorem 5.3.5 for T (R).

7.2.3. We shall say that an element in T (R) is split if it belongs to the A factor in (7.1). Weshall say that a subtorus T ′ ⊂ T is split if θ|T ′ is trivial (equivalently, KT ∩ T ′(R) is finite).

Consider the corresponding decomposition

(7.2) t ' kT ⊕ a.

We shall say that an element of k is split if it belongs to the a factor in (7.2). We shall saythat a subspace v ⊂ t is split if all of its elements are split. Note that a subtorus T ′ is split ifand only its Lie algebra t′ is split.

From the above analysis it is easy to deduce the following:

Proposition 7.2.4.

(a) An element in T (R) is split if and only if the algebraic group that it generates is connected(and hence is a torus), whose Lie algebra is split.

(a’) An element in T (R) is split if and only if its action on every (algebraic) representation haspositive real eigenvalues.

(b) An element in ξ ∈ t is split if and only if the smallest subtorus T ′ ⊂ T such that ξ ∈ t′, issplit.

(b’) An element in t is split if and only if its action on every (algebraic) representation has realeigenvalues.

7.3. Split elements in a real reductive group. In this subsection we let G be a real reduc-tive group, let GC denote its complexification. We denote by g the Lie algebra of G and by gCits complexification, i.e., the Lie algebra of GC.

30 DENNIS GAITSGORY

7.3.1. Let g ∈ G be an element. We shall say that g is split if: (i) it is semi-simple, (ii) thealgebraic group that it generates is a split torus.

Let ξ ∈ g be an element. We shall say that ξ is split if it is semi-simple and smallest torusT ′ ⊂ G such that ξ ∈ t′, is split.

We shall say that a subalgebra v ⊂ g is R-diagonalizable if it is abelian and consists of splitelements.

From Proposition 7.2.4 we obtain:

Corollary 7.3.2.

(a) An element g ∈ G if and only if its action on every (algebraic) representation of G isdiagonalizable (over R) with positive eigenvalues.

(b) A subalgebra v ⊂ g is R-diagonalizable if and only if its action on every (algebraic) repre-sentation of G is simultaneously is simultaneously diagonalizable (over R).

(c) For an R-diagonalizable subalgebra v ⊂ g, the smallest subtorus Tv ⊂ G such that v ⊂ Tv,is split.

7.3.3. We now claim:

Proposition 7.3.4.

(a) Let v ⊂ g be an R-diagonalizable subalgebra. Then there exists a compatible compact formon GC such that v ⊂ p.

(b) If v is contained in p, then v is R-diagonalizable.

Proof. We will use the following statement, which can be proved by an argument similar tothat of Theorem 5.2.3:

Theorem 7.3.5. Let φ : G1 → G2 be a homomorphism of real reductive groups. Then givena compact form σ1 on G1

C compatible with G1, one can find a compact form σ2 on G2C that is

compatible with G2 and σ2 φ = φ σ1.

For point (a), let G1 be its centralizer in G, and let g′ be its Lie algebra. Then G1 is areductive group. By Theorem 7.3.5 it suffices to find a compatible compact form for G1.

We have g1 ' zg1 ⊕ (g1)′. Choose any compact form on G1. We have

p = (p ∩ zg1)⊕ (p ∩ (g1)′).

Then by Proposition 7.2.4 and (7.2), we have v ⊂ (p ∩ zg1) ⊂ p.

For point (b), consider a representation G→ GL(V ). It suffices to show that every elementof p acts by an R-diagonalizable operator on V . By Theorem 7.3.5, we can choose a positivedefinite scalar product on V such that p maps to E(V ). This implies the claim.


8.1. The maximal R-diagonalizable subalgebra.


8.1.1. We shall say that a subalgebra a ⊂ g is a maximal R-diagonalizable subalgebra if it isan R-diagonalizable subalgebra and is not properly contained in other such.

Let a be a maximal R-diagonalizable subalgebra. Let Ta be the smallest split subtorus of Gsuch that a ⊂ Lie(Ta). By Corollary 7.3.2 and the maximality assumption the above inclusionis in fact an equality

a = Lie(Ta).

This establishes a bijection between maximal R-diagonalizable subalgebras of g and maximalsplit subtori in G.

SetA := Ta(R)0;

this is the subgroup of Ta(R) as in (7.1).

8.1.2. We claim:

Proposition 8.1.3. Given a maximal R-diagonalizable subalgebra a ⊂ g and an arbitrary R-diagonalizable subalgebra v ⊂ g, there exists an element of G(R)0 that conjugates v into a.

Proof. By Proposition 7.3.4(a), we can assume that both v and a are contained in p. Choosinga generic element ξ ∈ v, its suffices to show that there exists k ∈ K such that Adk(ξ) ∈ a; bythe maximality of a, the latter is equivalent to the condition that [Adk(ξ), η] = 0 for a givengeneric element η ∈ a. Consider the function

f(k) = (Adk(ξ), η).

By compactness, choose a point k ∈ K, on which it attains a maximum. We claim thatthis point will do. Indeed, conjugating ξ by k, we can assume that k = 1. We obtain that thefunction f has a zero differential at k = 1. However, the differential is given by

k 7→ ([k, ξ], η) = (k, [ξ, η]).

We have [ξ, η] ∈ k, and (−,−) is non-degenerate on k. Hence, [ξ, η] = 0.

8.1.4. We shall say that the real group G is split if a is a Cartan subalgebra of g. One can showthat every connected complex reductive group has a unique split form (up to conjugacy).

8.2. The centralizer of a maximal R-diagonalizable subalgebra.

8.2.1. As is the case of any abelian semi-semi subalgebra in g, the centralizer of a in g is theLie algebra m of a Levi subgroup M .

Proposition 8.2.2. m ' a⊕ (m ∩ k).

Proof. Let ξ + η be an element of g that centralizes a, with ξ ∈ k and η ∈ p. It suffices to showthat ξ and η each centralize a. For a ∈ a we have

0 = [a, ξ + η] = [a, ξ] + [a, η],

where [a, ξ] ∈ p and [a, η] ∈ k. Hence, both are zero.

Corollary 8.2.3.

(a) The map A× (K ∩M)→M is the polar decomposition of Theorem 5.3.5 for M .

(b) The map A× (K ∩ ZM )→ ZM is the polar decomposition of Theorem 5.3.5 for ZM .

Corollary 8.2.4. The group M is compact modulo its center.

32 DENNIS GAITSGORY

8.3. The minimal parabolic.

8.3.1. By assumption, the adjoint action of a on g is diagonalizable. Write

g = m⊕⊕αgα,

where α ∈ a∗. Note that since σ acts as −1 on a, we have

τ(gα) = g−α.

One can show that the collection of those α that appear forms a root system.

8.3.2. In terms of this decomposition, k identifies with the direct sum of m and the span ofelements

(ξα + ξ−α), ξα ∈ gα, ξ−α ∈ g−α, τ(ξα) = ξ−α.

8.3.3. Choose an element a ∈ a such that α(a) 6= 0 for any α. Then the subspace

q := m⊕

⊕α,α(a)>0

gα,

is the Lie algebra of a parabolic subgroup P of G. (This manipulation is valid for any abeliansemi-simple subalgebra in g.)

Let n denote the Lie algebra of N(Q). We have:

n = ⊕α,α(a)>0

gα.

Proposition 8.3.4. The subgroup Q is minimal among parabolics defined over R.

Proof. The statement is equivalent to the fact that M has no proper parabolics defined over R.This follows from the fact that M/Z(M) is compact (compact real groups do not have properparabolics because they have no unipotent elements).

8.3.5. Consider the subgroup

Q0 ⊂ P (R)

equal to the preimage of A ⊂ Ta(R) ⊂ ZM (R) ⊂M(R) under the projection

Q(R)→M(R).

By construction, P 0 is an extension

1→ Nmin(R)→ Q0 → A→ 1.


Theorem 8.3.7. Any unipotent subgroup of G can be conjugated into Nmin by an element ofG(R).

Proof. Let N ′ ⊂ G be a unipotent subgroup. By Hilbert 90, it is enough to see that N ′(R) canbe conjugated into N(R). Consider the quotient G(R)/Q(R). We claim that is it enough toshow that N ′(R) has a fixed point on G(R)/Q(R). Indeed, such a point, will conjugate N ′(R)into Q(R). Now, any unipotent element of Q(R) is contained in Nmin(R). Indeed, the quotientQ(R)/Nmin(R) 'M(R) contains no unipotent elements.

To prove the existence of the fixed point, we will show that for a proper variety X and aunipotent group N ′ acting on (everything is defined over R), every connected component ofX(R) contains an N ′(R)-fixed point. The proof is obtained as in the case of algebraically closedfields:


Reduce by induction to the case when N ′ = Ga. Then the claim is that for any x ∈ X(R),the map

A1 → X, a 7→ a · xextends to a map P1 → X (by the valuative criterion). Its value on ∞ ∈ P1(R) is the desiredfixed point.

8.4. The Iwasawa decomposition.

8.4.1. We will prove:

Theorem 8.4.2. The product map

Q0 ×K → G(R)


8.4.3. For the proof we first notice that the corresponding assertion does take place at the Liealgebra level:

Proposition 8.4.4. The sum map

a⊕ n⊕ k→ g

is an isomorphism.

Proof. Follows from Sect. 8.3.2.

We can now prove Theorem 8.4.2:

Proof. We need to show that the map

(8.1) K → G(R)/Q0,

given by the action on the element 1 ∈ G(R)/Q0 is a diffeomorphism.

Since K is compact, the image is closed. We now claim that the map in question is sub-mersive. This is an assertion at the level of tangent spaces, and follows from the surjectivityof the map in Proposition 8.4.4. Hence, the image is open. Being both open and closed, it isthe union of certain connected components. However, the map (8.1) is a bijection at the levelof connected components (by Theorem 5.3.5).

Thus, we obtain that the action of K on G(R)/Q0 is transitive. It remains to see thatStabK(1) = 1. However, this is just the fact that K ∩ P 0 = 1.

Remark 8.4.5. The action of K on G(R)/Q(R) comes from an action of the complex algebraicgroup KC = GθC on the complex algebraic variety (G/Q)C. We note, however, that the latteraction is not necessarily transitive. However, the above proof shows that the orbit of the element1 ∈ (G/Q)C under this action is open.

Here is how it looks in an example G = SL2,R. We have Q = B; K = U(1). At thecomplexified level GC = SL2 and KC = Gm; (G/P )C ' P1. The action of Gm on P1 is theusual one, and the open orbit in question is P1 − 0,∞.

8.5. The Cartan decomposition.

34 DENNIS GAITSGORY

8.5.1. Consider the product map

(8.2) K ×A×K → G(R).

The Cartan decomposition is the following statement:

Theorem 8.5.2. The map (8.2) is surjective and proper (the preimage of every compact iscompact).

The meaning of this theorem is that G(R) ”looks like A modulo something compact”.

Remark 8.5.3. The map (8.2) is (obviously) not bijective. For example, the subset

K × 1 ×K ⊂ K ×A×K → G(R)

gets collapsed onto just one copy of K.

8.5.4. Proof of properness. It suffices to show that the map K ×A→ G(R)/K is proper. Thisfollows from the fact that the map A → G(R)/K is proper (by Theorem 8.4.2) and the factthat K is compact.

8.5.5. Proof of surjectivity. By the polar decomposition, the assertion is equivalent to the factthat the adjoint action of K on P defines a surjection

K ×A→ P,

or, equivalently,

K × a→ p.

I.e., we need to show that every element of p can be conjugated by means of K into a. Butwe have shown that in Proposition 8.1.3.


9.1. Admissible representations. Let G be a real reductive group. We will assume that Gis connected as an algebraic group (which does not imply that the corresponding Lie groupis connected. To save the notation we will denote by the same symbol G (rather than G(R))the corresponding Lie group. Let K ⊂ G be its maximal compact. Note that K may bedisconnected.

9.1.1. Let V be a continuous representation of G. Recall that V is acted on by the algebraMeasc(G) by convolutions.

Recall that we denote by V∞ ⊂ V the vector subspace consisting of smooth vectors. Wehave shown that it is dense. We recall also that V∞ is acted on by a larger algebra, namelyDistrc(G). The action of elements of Measc(G) on V∞ factors through the tautological mapMeasc(G)→ Distrc(G), dual to the inclusion C∞(G)→ C(G).


9.1.2. Recall also that for an irreducible representation ρ of K, we denote by

V ρ := ρ⊗HomK(V ρ, V )

the ρ-isotypic component in V . The operator Tξρ·µHaarKacts as a projector on V ρ: i.e., its

image is in V ρ, and its restriction to V ρ is the identity.

Recall the notationV K -fin :=

⊕ρ

V ρ;

this is the space of K-finite vectors V . Its tautological map into V is injective and has a denseimage, called the space of K-finite vectors.

We claim:

Lemma 9.1.3. For every ρ, the subspace V∞ ∩ V ρ is dense in V ρ.

Proof. Choose a Dirac sequence of smooth functions fn → δ1. Then for every v ∈ V ρ, theelements

Tξρ·µHaarK? Tfn·µHaarG

(v)

belong to both V ρ and V∞ and converge to v.

9.1.4. A representation V of G is said to be admissible if for every ρ, the vector space V ρ isfinite-dimensional.

A key observation is:

Proposition 9.1.5. Let V be admissible. Then V K -fin ⊂ V∞.

Proof. It is enough to show that V ρ ⊂ V∞ for every ρ. I.e., we need to show that V∞∩V ρ equalsall of V ρ. However, this subspace is dense by Lemma 9.1.3, and since V ρ is finite-dimesnional,it must be the whole thing.

9.1.6. As a result we obtain that if V is admissible, the Lie algebra g acts on V K -fin. Note thatthe actions of K and g (on all of V∞) are compatible in the following sense:

(i) For k ∈ K and η ∈ g,Tk · Tη · Tk−1 = TAdk(η);

(ii) The action of k on V∞ arising from the action of K on V (any vector smooth with respectto G is obviously smooth with respect to K) equals the restriction of the action of g along thetautological map k→ g.

9.1.7. We define the notion of (g,K)-module to be a C-vector space, equipped with an algebraic(i.e., locally finite) action of K and an action of g that are compatible in the sense that (i) and(ii) above hold.

Let KC be the complex algebraic group corresponding to K. Note that (g,K)-modules canbe equivalently defined as complex vector spaces equipped with an algebraic action of KC andan action of gC that are compatible in the similar sense.

Thus, (g,K)-modules form a purely algebraic category; we denote it by (g,K)-mod.

Let(g,K)-modadm ⊂ (g,K)-mod

denote the full subcategory of admissible (g,K)-modules. The assignment V 7→ V K -fin is afunctor

(9.1) Rep(G)adm → (g,K)-modadm.

36 DENNIS GAITSGORY

9.1.8. Let M be an admissible (g,K)-module. We can form its algebraic dual

(M∗)alg :=⊕ρ

(Mρ)∗,

which is equipped with a natural action on g.

Equivalently, (M∗)alg is the subspace of K-finite vectors in the full linear dual M∗, the latterbeing

∏ρ

(Mρ)∗.

9.2. How well is a representation approximated by its (g,K)-module?

9.2.1. Following Harish-Chandra, we shall sat that two admissible representations V1 and V2 ofG are infinitesimally equivalent if V K -fin

1 and V K -fin2 are isomorphic as (g,K)-modules.

It is (obviously) not true that if two admissible representations V1 and V2 of G are infinites-imally equivalent then they are isomorphic. For example, take G = K and V1 = L2(K) andV2 = C(K).

The above example may not be too interesting since the representations V1 and V2 arehighly reducible. We will bring more interesting examples when we introduce principal seriesrepresentations. But in case, the phenomenon has to with the fact that a given (g,K)-modulecan be topologized in many different ways so that its completion carries an action of G.

9.2.2. Nonetheless, we have:

Theorem 9.2.3.

(a) Let V1 and V2 be two admissible representations of G. Let S : V1 → V2 be a continuous mapof the underlying vector spaces. Assume that S sends V K -fin

1 to V K -fin2 and that the resulting

map V K -fin1 → V K -fin

2 is compatible with (g,K)-module structures. Then the initial S was amap of G-representations.

(b) For V ∈ Rep(G)adm and M := V K -fin, for every (g,K)-submodule M1 ⊂ M , its closureM1⊂V is a G-subrepresentation.

(c) The assignments

(V1 ⊂ V ) 7→ (V1)K -fin ⊂M and M1 7→M1 ⊂ V

define mutually inverse bijections between the set of closed G-subrepresentations of V and theset of (g,K)-submodules of M .

Corollary 9.2.4. An admissible representation V is irreducible (i.e., contains no closed G-invariant subspaces) if and only if V K -fin is irreducible as a (g,K)-module.

It is of course not true that every G-representation is admissible (take an infinite directsum of copies of the same representation). But one can ask: is it true that irreducible G-representations are admissible. This was conjectured by Harish-Chandra, but many years later,W. Soergel produced a counterexample. However, we have the following theorem (to be provednext week):

Theorem 9.2.5. Every unitary irreducible representation of G is admissible.

In Corollary 10.1.6 we will see that a unitary irreducible representation of G is completelydetermined by its space of K-finite vectors.

Theorem 9.2.5 has a converse (also due to Harish-Chandra):


Theorem 9.2.6. Let M be an irreducible (g,K)-module equipped with an invariant Hermitianpositive-definite form. Then the Hilbert space completion V of M carries a unique unitaryG-representation such that V K -fin = M as (g,K)-modules.

In the above theorem a form (−,−) is called invariant if

(k ·m1, k ·m2) = (m1,m2), ∀k ∈ K and (ξ ·m1,m2) = −(m1, ξ ·m2), ∀ξ ∈ g.

As a formal corollary, one obtains:

Corollary 9.2.7. Let M be an admissible (g,K)-module equipped with an invariant Hermitianpositive-definite form. Then the Hilbert space completion V of M carries a unique unitaryG-representation such that V K -fin = M as (g,K)-modules.

Proof. By admissibility, M is an orthogonal direct sum of irreducible (g,K)-modules.

9.2.8. The proof of Theorem 9.2.3 is based on the following:

Proposition 9.2.9. Let V be an admissible representation of V , and let v be an element ofV K -fin. Then for every η ∈ V ∗, the function on G

g 7→ η(g(v))

is real analytic.

Let us prove Theorem 9.2.3 using this proposition:

Proof. For point (a), it is enough to show that for v1 ∈ (V1)K -fin we have

Tg S(v1) = S Tg(v1).

For that it suffices to show that for any η ∈ V ∗2 , we have

η(Tg S(v1)) = η(S Tg(v1)).

Both sides are analytic functions in g. Hence, it is enough to show that all of their derivativesat 1 ∈ G are equal, and that they agree on at least one point on each connected component ofG.

The first condition follows from the compatibility with the action of U(g). The secondcondition follows from the compatibility with the action of K.

For point (b), we recall that by the Hahn-Banach theorem, the closure of a subspace M1 ⊂ Vequals ((M1)⊥)⊥, where (M1)⊥ ⊂ V ∗ is the annihilator of M1. Hence, it suffices to show thatfor any η ∈ (M1)⊥ we have g · η ∈ (M1)⊥, i.e., for every v1 ∈ M1 we have η(g(v1)) = 0. Thisfollows by analyticity in the same way as above.

To prove point (c), we note that for a subrepresentation V1 ⊂ V , the subspace V K -fin1 is

dense in V1, so its closure is all of V1.

Vice versa, for a sudmodule M1 ⊂ M , it suffices to show that for every ρ, the image ofTξρ·µHaarK

on M1 lies in Mρ1 . However, Tξρ·µHaarK

(M1) ⊂ Mρ1 , and the assertion follows by

continuity.

9.3. Proof of analyticity. The goal of this subsection is to prove Proposition 9.2.9. With norestriction of generality, we can assume that v ∈ V ρ for some ρ ∈ Irrep(K).

38 DENNIS GAITSGORY

9.3.1. Let D be a differential operator of order n on a differentialble (resp., real analytic)manifold X. Let σn(D) be its symbol, i.e., the image of D under the projection

Diff≤n(X)→ Diff≤n(X)/Diff≤n−1(X) ' SymnC∞(X)(Vect(X)).

We can regard σn(D) as a function on T ∗(X), which is homogenous of degree n along thefibers.

Recall that D is said to be elliptic if σn(D)(η) > 0 for all 0 6= η ∈ T ∗x (X).

We have the following basic result (elliptic regularity):

Theorem 9.3.2. Let φ be an element of Distr(X) (where Distr(X) is the topological dual ofC∞c (X)). If D(φ) = 0 (i.e., φ(D(f)) = 0 for all f ∈ C∞c (X)). Then:

(a) The distribution φ is smooth. I.e., it given by a smooth function (times a smooth measureon X).

(b) If X is real analytic, and the coefficients of D are real analytic, then φ is real analytic. I.e.,it is given by an analytic function (times an analytic measure on X).

9.3.3. We will apply Theorem 9.3.2(b) to X = G and φ being the (continuous) function (timesµHaarG) given by g 7→ η(g(v)). Thus, our goal is to find an elliptic operator D with analyticcoefficients that annihilates this function.

We will define D as a left-invariant differential operator corresponding to a certain elementu ∈ U(g)≤n. The analyticity is then guaranteed by the construction.

The ellipticity will amount to checking that the image σn(u) of u in

U(g)≤n/U(g)≤n−1 ' Symn(g)

satisfies σn(u)(η) > 0 for any 0 6= η ∈ g∗.

9.3.4. Write g = g′ ⊕ zg. Let (−,−) be a bilinear form on g, which is the Killing form on g′

and a form on zg that is positive-definite on the split part of zg (i.e., zg ∩ p = zg ∩ a) andnegative-definite on the compact part of zg (i.e., zg ∩ k).

Then (−,−) is the direct sum of a positive-definite form on p and a negative-definite formon k.

9.3.5. Let Cg ∈ Z(g) ⊂ U(g) be the Casimir element corresponding to (−,−). I.e., if ei is anorthonormal basis in g with respect to (−,−), then

Cg =∑i

(ei)2.

It is an elementary fact that Cg indeed belongs to Z(g) and is independent of the choice ofa basis.

Consider also the element Ck ∈ U(k) ⊂ U(g). Set

u = Cg − 2 · Ck.

Since Z(g) commutes with the action of K, the action of u on V∞ preserves V ρ. Since V ρ

is finite-dimensional, we can find a monic polynomial p (of some degree n) such that p(Tu)annihilates V ρ. Set

u = p(u).

By construction u annihilates v, and hence annihilates our function φ. It remains to checkthe ellipticity. Take n = 2d.We have σn(u) = (σ2(u))d. Hence, it suffices to show that

σ2(u) ∈ Sym2(g)


is elliptic.

Choose an orthonormal bases for g to be of the form e′i∪e′′j , where e′i is an orthonormalbasis for k, and e′′j is an orthonormal basis for p. Then

σ2(u) =∑j

(e′′j )2 −∑i

(e′′i )2.

I.e., σ2(u) is the quadratic form corresponding to the bilinear form

(−,−)|p − (−,−)|k,and the result follows from the fact that the latter is positive-definite.

9.4. (g,K)-modules vs g-modules.

9.4.1. Consider the forgetful functor

(g,K)-mod→ g-mod.

We can factor it as(g,K)-mod→ (g,K0)-mod→ g-mod,

where K0 is the neutral connected component of K.

Lemma 9.4.2. The functor (g,K0)-mod→ g-mod is fully faithful. Its essential image is stablewith respect to taking submodules.

Proof. The action of g on a vector space determines that of k, and if the latter extends to anaction of K0, it does so in a unique way (in this case we say that the action of k integrates tothat of K0). This happens of and only if the following happens:

(i) The action of the derived Lie algebra k′ is locally finite;

(ii) the resulting action of the simply-connected cover of K ′0 factors through that of K ′0.

(iii) The abelian algebra zk acts semi-simply with characters given by characters of the torus(ZK0)0.

(iv) The two resulting actions of ZK′0 ∩ (ZK0)0 agree.

It is clear that these conditions are stable with respect to taking submodules.

Corollary 9.4.3. If M ∈ (g,K0)-mod is such that the underlying g-module is irreducible, thenM itself is irreducible.

9.4.4. Recall that an object M of an abelian category A is said to be finitely generated if forevery ascending chain

M1 ⊂M2 ⊂ ... ⊂Mwith ∪

iMi = M , we have Mi = M for some i.

We say that A is Noetherian if a subobject of a finitely generated object is finitely generated.If A = A-mod for an associative algebra A, this property is equivalent to A being (left)-Noetherian. In particular, this is the case for A = g-mod, since U(g) is Noetherian (this followsfrom the fact that gr(U(g)) ' Sym(g) is Noetherian).

We say that A is Artinian, if every finitely generated object has finite length.

Lemma 9.4.5. The forgetful functor (g,K)-mod → g-mod sends finitely generated objects tofinitely generated objects.

40 DENNIS GAITSGORY

Proof. By Lemma 9.4.2, it suffices to prove the corresponding fact for the restriction functor(g,K)-mod→ (g,K0)-mod.

Let M be a finitely generated (g,K)-module, and let

M1 ⊂M2 ⊂ ... ⊂M, ∪iMi = M

be a chain of (g,K)-submodules. Pick representatives k ∈ K for each element of π0(K). Theneach

M ′i =∑k∈K

k · (Mi)

is a (g,K)-submodule of M . Hence, M ′i = M for some i. Pick j large enough so that k · (Mi) ⊂Mj . Then Mj = M .

Corollary 9.4.6. The category (g,K)-mod is Noetherian.


Proposition 9.4.8. For an irreducible (g,K)-module, the underlying g-module is a direct sumof finitely many irreducibles.

Proof. Let M be an irreducible (g,K)-module. By Lemma 9.4.2, it suffices to show that M isa direct sum of finitely many irreducibles (g,K0)-modules.

Let M ′ ⊂M be a maximal (g,K0)-submodule with M/M ′ 6= 0 (such always exists by Zorn’slemma). Thus, N = M/M ′ is a non-zero irreducible (g,K0)-module. Pick a representativek ∈ K for each element of π0(K). Consider

M ′′ := ∩kk(M ′) ⊂M.

Then M ′′ is a proper (g,K)-submodule of M , and hence equals zero. Hence, the map

M → ⊕k

(M ′)k

is injective, where (M ′)k denotes M ′ with the action twisted by the automorphism k. Thus,M , when viewed as a (g,K0)-module is a submodule of a semi-simple module, and hence issemi-simple.


10.1. Schur’s lemma for (g,K)-modules.

10.1.1. Here is version of Schur’s lemma for Lie algebras:

Theorem 10.1.2. Let M be an irreducible g-module. Then the map C → Endg(M) is anisomorphism.

Proof. Let S be an endomorphism of M . Suppose that it is not a scalar. Consider the map

C[s]→ Endg(M), s 7→ S.

We claim that it is injective. Indeed, if it was not, it would contain an element of the form∏i

(s− ai)ni in its kernel, which would mean that one of operators S− ai · Id) is non-invertible.

Since M is irreducible, this would mean that S = ai · Id, contradicting the assumption.


Thus, the above map extends to an (automatically injective) map from the fraction fieldC(s) of C[s] to Endg(M). Note, however, that C(s), when viewed as a vector space over Chas dimensional at least continuum: indeed, the elements 1

s−a are all linearly independent.

However, we claim that Endg(M) is countable-dimensional. Indeed, for any non-zero vectorm ∈M , evaluation on m defines an injective map

Endg(M)→M,

while M itself is countable-dimensional, being the quotient of U(g) by Ann(m).

10.1.3. Note that the same proof applies for M being an irreducible object in the category(g,K)-mod. We will soon see that any irreducible (g,K)-module is admissible (Theorem 10.3.3).Now, for admissible (g,K)-modules, one can give a simpler proof of Schur:

Proposition 10.1.4. Any endomorphism of an irreducible admissible (g,K)-module is a scalar.

Proof. It is enough to show that any endomorphism S of an admissible (g,K)-module M hasa non-zero eigenspace. Pick ρ such that Mρ is non-zero. Then S preserves Mρ, and since thelatter is finite-dimensional, the assertion follows.

10.1.5. Here is an application of Schur’s lemma. This is a statement complementary to Theo-rem 9.2.5. It says that a unitary irreducible representation is fully determined by the space ofits K-finite vectors, viewed as a (g,K)-module:

Corollary 10.1.6. Let V1 and V2 be two irreducible unitary representations that are infinites-imally equivalent. Then they are isomorphic.

Proof. First off, by Theorem 9.2.5, V1 and V2 are admissible, so we can talk about the corre-sponding (g,K)-modules.

Set Mi := (Vi)K -fin. We can recover Vi as the Hilbert space completion of Mi equipped with

the induced Hermitian form.

Note that if M is equipped with a (g,K)-invariant Hermitian form then we have a canonicalidentification

M ' (M†)alg,

where † means “take dual+complex conjugate”.

Fix an isomorphism S : M1 'M2. It does not necessarily respect the given inner form, butit does so up to multiplication by a (positive) scalar. Indeed, consider the diagram

M1S−−−−→ M2

∼y y∼

((M1)†)alg S†←−−−− ((M2)†)alg.

The composite automorphism of M1 respects the (g,K)-action, and hence by Schur’s lemmaand Corollary 9.2.4, it is given by multiplication by a scalar.

Dividing the initial φ by the square root of this scalar, we can thus assume that φ respectsthe Hermitian structures. Thus, it defines an isomorphism of Hilbert spaces V1 → V2. Now theresult follows from Theorem 9.2.3(a).

10.2. Action of the center of U(g).

42 DENNIS GAITSGORY

10.2.1. Denote by Z(g) the center of the associative algebra U(g). We can identify Z(g) withinvariants in U(g) with respect to the adjoint action of g (or G itself). It is known to be “large”:

Consider the filtration on Z(g), induced by the PBW filtration on U(g).

Lemma 10.2.2. The map from grn(Z(g)) := Z(g)≤n/Z(g)≤n−1 to the space of adg-invariantsin

grn(U(g)) := U(g)≤n/U(g)≤n−1 ' Symn(g)

is an isomorphism.

Proof. Follows from complete reducibility of algebraic g-representations: the functor of g-invariants is exact, so sends cokernels to cokernels.

Hence, gr(Z(g)) identifies as a commutative algebra with Sym(g)g, and the latter identifieswith Sym(h)W , and is known to be isomorphic to be a polynomial algebra of rank equal tothe rank of g (i.e., dim(h)). Lifting the generators, we can therefore find a (non-canonical)isomorphism between Z(g) with the same polynomial algebra.

However, such isomorphism can be made canonical: this is the Harish-Chandra isomorphism

(10.1) Z(g) ' Sym(h)W ,

to be discussed later.

10.2.3. We claim:

Proposition 10.2.4. Let M be an admissible (g,K)-module. Then the action of Z(g) is locallyfinite. I.e., M splits as a direct sum M ' ⊕

χ∈Spec(Z(g))Mχ, such that Z(g) acts on each Mχ be

a generalized character χ.

Proof. Since the action of G (and hence K) commutes with that of Z(g), we obtain that Z(g)preserves each K-isotypic componnent Mρ. Since the latter are finite-dimensional, the assertionfollows.

10.2.5. For a given element χ ∈ Spec(Z(g)), let (g,K)-modχ be the full subcategory of(g,K)-mod consisting of modules, on which Z(g) acts with a generalized character χ, i.e., forevery m ∈M there exists a power n such that the ideal (ker(χ))n ⊂ Z(g) annihilates m.

Note that by Schur’s lemma, every irreducible object in (g,K)-mod belongs to (g,K)-modχ,for a uniquely defined χ.

We have:

Theorem 10.2.6. The category (g,K)-modχ has only finitely many isomorphism classes ofirreducible objects.

Remark 10.2.7. This is a rather deep theorem. Although it is completely algebraic, the initialproof by Harish-Chandra was very indirect and used analysis. Later in the semester we willsupply a proof using the localization theory of Beilinson-Bernstein.

Assuming the above theorem, we obtain:

Theorem 10.2.8. For an object M ∈ (g,K)-modχ the following conditions are equivalent:

(i) M is finitely generated;

(ii) M is of finite length;

(iii) M is admissible.


Proof. The implication (ii) ⇒ (i) is evident. Let us prove that (iii) implies (ii). Let M be anadmissible object of (g,K)-modχ, and let

0 = M0 ⊂M1 ⊂M2... ⊂Mn = M

be a chain of submodules. We will effectively bound the integer n.

Let Lα be the irreducible objects of (g,K)-modχ; by the second statement in Theorem 10.2.6,there are finitely many of them. For each α, pick ρα ∈ Irrep(K) so that Lραα 6= 0. Let ρ = ⊕

αρα.

For each i = 1, ..., n there exists an index α so that Lα is a subquotient of Mi/Mi1 . Hence

dim(HomK(ρ,Mi/Mi1)) ≥ 1.

Hence,

dim(HomK(ρ,M)) ≥ n.

Let us show that (i) implies (iii). This follows from the next assertion (of independentinterest):

Proposition 10.2.9. Let M be a finitely generated (g,K)-module. Then for any ρ, the isotypiccomponent Mρ is finitely generated over Z(g).

Proof of Proposition 10.2.9. Let M be a finitely generated (g,K)-module. Then it receives asurjection from a (g,K)-module of the form U(g) ⊗

U(k)ρ′ for ρ′ a finite-dimensional representation

of K. Hence, it is enough to show that

HomK(ρ, U(g) ⊗U(k)

ρ′)

is finitely generated over the center. The latter assertion is enough to check at the associatedgraded level. I.e., it is enough to check that

(Sym(g/k)⊗Hom(ρ, ρ′))K

is finitely generated as a module over Sym(g)G.

Let us identify g with its dual via the Killing form. Denote W := Hom(ρ, ρ′); this is afinite-dimensional K-representation. Thus, we need to show that

(Op ⊗W )K

is finitely-generated over OGg , where OGg maps to Op via the restriction map.

By Sect. 8.5.5, the K-orbit of a is dense in p; henve the restriction map under a ⊂ p definesan injection

(Op ⊗W )K → Oa ⊗W.Now the assertion follows from the fact that Oa is finite as a module over g//G. Indeed, a

is a closed subvariety in h, and g//G identifies with h//W , and the assertion follows from thefact that h is finite over h//W .

10.3. Some consequences about properties of the category (g,K)-mod.

44 DENNIS GAITSGORY

10.3.1. The implication (i) ⇒ (ii) in Theorem 10.2.8, we obtain:

Corollary 10.3.2. The category M ∈ (g,K)-modχ is Artinian.

Combining the implication (ii)⇒ (iii) in Theorem 10.2.8 with Theorem 10.1.2 we also obtain:

Theorem 10.3.3. Every irreducible (g,K)-module is admissible.

We note that the proof of Theorem 10.3.3 does not use the (more difficult) theorem Theo-rem 10.2.6.

10.3.4. Combining the above results, we obtain:

Corollary 10.3.5. For a (g,K)-module the following conditions are equivalent:

(i) M is finitely generated and admissible;

(ii) M is finitely generated and its support over Spec(Z(g)) is finite;

(iii) M is admissible and its support over Spec(Z(g)) is finite;

(iv) M is of finite length.

We will refer to (g,K)-modules satisfying the equivalent conditions of Corollary 10.3.5 asHarish-Chandra modules.

10.4. Admissibility of unitary representations. In this subsection we will begin the proofof Theorem 9.2.5, which says that an irreducible unitary representation of G is admissible. Wewill prove a sharper result:

Theorem 10.4.1. Let V be an irreducible unitary representation of G. Then for any ρ ∈Irrep(K), we have:

dim(V ρ) ≤ dim(ρ)2.

10.4.2. Let V be a topological vector soace. Recall that among the various topologies that existon the vector space End(V ) of continuous endomorphisms of V , there exists what is called thestrong topology:

The fundamental system of neighborhoods of 0 in V is given by finite collections

((v1, U1), ..., (vn, Un)),

where vi ∈ V and Ui are neighborhoods of zero in V . The corresponding neighborhood in Vconsists of those S such that

S(vi) ∈ Ui, ∀i = 1, ..., n.

Let A be an associative algebra acting on V , i.e., we have a homomorphism A → End(V ).We shall say that V is strongly irreducible if the image of A in End(V ) is dense in the strongtopology.

10.4.3. Theorem 10.4.1 follows from the following two statements:

Theorem 10.4.4. Let V be an irreducible unitary representation of G. Then V , viewed asacted on by Measc(G), is strongly irreducible.

Theorem 10.4.5. Let V be a representation of G, which is strongly irreducible when viewedas acted on by Measc(G). Then for any ρ ∈ Irrep(K), we have:

dim(V ρ) ≤ dim(ρ)2.



10.5.1. Let H be a Hilbert space. Let A be a subalgebra in End(H) closed under the operationS 7→ S†. Let A be the strong closure of A in End(H). Let Ac ⊂ End(H) be the commutant ofA, i.e.,

Ac = S′ ∈ End(V ), S′ S = S S′ for all S ∈ A.

We will use the following result of von Neumann:

Theorem 10.5.2. The inclusion A ⊂ (Ac)c is an equality.

Corollary 10.5.3. If the inclusion C → Ac is an equality, then the action of A on V is stronglyirreducible.

10.5.4. Let A be the image of Measc(G) in End(V ). The operation

f(g) 7→ f(g−1)

defines an involution on C(G), and hence on Measc(G), denoted φ 7→ φ†. We have

Tφ† = (Tφ)†.

Thus, we obtain that Theorem 10.4.4 follows from the next result, which can be viewed as aHilbert space version of Schur’s lemma:

Theorem 10.5.5. Let V be an irreducible unitary representation of G. Then any endomor-phism of V is a scalar.

10.5.6. For the proof of Theorem 10.5.5, we will use the following weak form of the spectraltheorem. Let H be a Hilbert space, and S be a continuous self-adjoint endomorphism of H.Then for any λ ∈ R there exists an idempotent πλ of H with the following properties:

(i) Any endomorphism of H that commutes with S commutes also with πλ;

(ii) For any v ∈ Im(πλ) we have (S(v), v) ≤ λ(v, v);

(iii) For any v ∈ ker(πλ) we have (S(v), v) ≥ λ(v, v).

10.5.7. Proof of Theorem 10.5.5. Let S be an endomorphism of V . First off, we can assumethat S is self-adjoint. Indeed, if S is an endomorphism, then so is S†. Then

S + S† and i(S − S†)

are self-adjoint, and if we can prove that each of them is a scalar, then so is the initial S.

Foe every λ, let πλ be the corresponding idempotent of V . By (i), it commutes with theaction of G. Hence, by the irreducibility we either have Im(πλ) = V or πλ = 0.

Let λ0 be the infimum of all λ such that

(S(v), v) ≤ λ(v, v);

it exists because S is bounded (and non-zero, which we can assume). Then from (ii) we obtainthat for all λ < λ0, πλ = 0. Hence, by (iii) we obtain that

(S(v), v) ≥ λ(v, v), ∀v ∈ V.

Thus, we obtain that (S(v), v) = λ0(v, v), i.e., S = λ0 · Id.


46 DENNIS GAITSGORY

10.6.1. For ρ ∈ Irrep consider the corresponding projector ξρ on V ρ. Set

Aρ = ξρ ·Measc(G) · ξρ.

This is a subalgebra in Measc(G) and it acts on V ρ. It is easy to see that if the action ofMeasc(G) on V is strongly irreducible, then the action of Aρ on V ρ is strongly irreducible.

Theorem 10.4.5 follows from the combination of the following two statements:

Proposition 10.6.2. There exists a family of finite-dimensional representations πρ of Aρ, suchthat:

(i) Each π is of dimension ≤ n for n = dim(ρ)2;

(ii) For every element a ∈ Aρ there exists a π such that the action of a in π is non-zero.

Proposition 10.6.3. Let A be an associative algebra equipped with a family of finite-dimensional modules satisfying conditions (i) and (ii) from Proposition 10.6.2. Then if V is atopological vector spaced equipped with a strongly irreducible A-action, then dim(V ) ≤ n.

In the rest of this subsection we will prove Theorem 10.6.3.

10.6.4. For an associative algebra A and a positive integer r consider the map

Pr : A⊗r → A, (a1, ..., ar) 7→∑σ∈Σr

(−1)σ · aσ(1) · ... · aσ(n).

Suppose that A = End(Cn). It is clear that Pr = 0 for r ≥ n2. Let r(n) be the minimalinteger such that Pr(n) vanishes. It is easy to see that the function

n 7→ r(n)

is strictly increasing.

Remark 10.6.5. The theorem of Amitzur-Levitzky says that r = 2n.

10.6.6. Let A be as in Proposition 10.6.3 it follows that Pr(n) vanishes on A. Let now V be atopological vector space equipped with a strongly irreducible action of A. By density, we obtainthat Pr(n) vanishes also on End(V ).

Assume for the sake of contradiction that dim(V ) > n. Then we can split V as a direct sum

V = Cn+1 ⊕ V ′,where V ′ is some topological vector space. Then End(V ) contains a subalgebra isomorphicto End(Cn+1). However, Pr(n) is non-zero on End(Cn+1), since r(n + 1) > r(n). This is acontradiction.

10.7. Proof of Proposition 10.6.2.

10.7.1. Let π be the set of all irreducible finite-dimensional (hence, algebraic) representationsof G. Since G is linear (and so admits a faithful finite-dimensional representation), it is clearthat for any φ ∈ Measc(G), there exists a π on which it acts non-trivially.

Take πρ := πρ, the ρ-isotypic component in π. We obtain that this family satisfies condition(ii). Hence, it remains to prove the following:

Theorem 10.7.2. For every irreducible finite-dimensional representation π of G, the dimensionof the space πρ is ≤ dim(ρ)2.

The rest of this subsection is devoted to the proof of this theorem.


10.7.3. Consider the corresponding complex groups GC and KC. Let X be the flag variety ofGC. We will use the following fundamental result (the Bott-Borel-Weil theorem):

Theorem 10.7.4. Every irreducible representation of GC can be realized as the space of regularsections of a G-equivariant line L bundle on X.

10.7.5. Recall that the Iwasawe decomposition said that K acted transitively on G/P . Thisimplies that the orbit of the unit point under the action of KC on (G/P )C is open. Since Mis compact modulo its center, we obtain that the action of KC on X also has an open orbit;denote it X0.

The map

Γ(X,L) → Γ(X0,L)

is an injective map of KC-representations.

It remains to show that for any ρ,

dim(HomKC(ρ,Γ(X0,L))) ≤ dim(ρ).

However, this is true for any KC-homogeneous space equipped with an equivariant linebundle. Indeed, Γ(X0,L) is a subrepresentation inside the regular representation Reg(KC) ofKC (i.e., the space of regular functions on KC), and

HomKC(ρ,Reg(KC)) ' ρ∗,

by Frobenius reciprocity.


11.1. More on the notion of infinitesimal equivalence.

11.1.1. Let V1 and V2 be two admissible G-representations, and let S : M1 → M2 be a mapbetween the underlying (g,K)-modules. It is (obviously) not true that one can continuouslyextend S to a map from V1 to V2. But one can do so “up to a hut”:

Proposition 11.1.2. There exists a (canonically defined) admissible G-representations V1,2

equipped with maps

V1S1←− V1,2

S2→ V2,

such that S1 induces an isomorphism M1,2 →M1, and such that the resulting map

M1S1' M1,2

S2−→M2

is the original S.

Proof. Consider the graph of S, which is a map

M1 →M1 ⊕M2

, and compose it with M1 ⊕M2 → V1 ⊕ V2.

Let V1,2 be the closure of the image of M1 in V1 ⊕ V2. By Theorem 9.2.3, V1,2 is a G-subrepresentation of V1 ⊕ V2, with the required properties.

48 DENNIS GAITSGORY

11.1.3. Let HG := Distrc(G)K×K -fin be the subspace of Distrc(G) that consists of distributionsthat are K-finite with respect to both left and right translations. We have

Distrc(G)K×K -fin = ⊕ρ1,ρ2

ξρ1 ?Distrc(G) ? ξρ2 ,

where the notation ξρ is as in Sect. 1.4.3 (it acts as a projector on the ρ-isotypic component).

The subspace H(G) is closed under convolutions, so it is a subalgebra in Distrc(G). Note,however, that it is non-unital.

If V is an admissible G-representation, we have a canonically defined action of H(G) onV K -fin. From Proposition 11.1.2 we obtain:

Corollary 11.1.4. Let V1 and V2 be two admissible G-representations, and let S : M1 → M2

be a map between the underlying (g,K)-modules. Then S intertwines the actions of H(G) onM1 and M2.

Proof. The assertion is obvious when S comes from a map V1 → V2. Now, Proposition 11.1.2reduces us to this situation.

In particular:

Corollary 11.1.5. The action of H(G) on V K -fin depends only on the class if infinitesimalequivalence of V .

11.1.6. Let V be an admissible G-representation, and let V ∗ be its dual. Note that

(V ∗)K -fin ' (V K -fin)∗,alg.

Denote M := V K -fin. Thus, for m ∈ M and m∗ ∈ M∗,alg, we obtain the matrix coefficientfunction MCV,m⊗m∗

g 7→ 〈m∗, g ·m,which is a C∞-function on G

From Corollary 11.1.5 we obtain:

Corollary 11.1.7. The function MCV,m⊗m∗ only depends on the infinitesimal equivalence classof V .

11.2. Proof of Theorem 9.2.6. :

11.2.1. Let M be an irreducible admissible (g,K)-module, equipped with a (g,K)-invariantHermitian form. We want to show that there exists unitary representation of G such that Mis its space of K-finite vectors.

The proof is based on the following proposition:

Proposition 11.2.2. There exists a Banach realization V on M , such that

(m,m) ≤ ||m||2.

Proof. We will use the fact that any irreducible (g,K)-module admits a Banach realization (to

br proved later). Let V1 be such a realization, and let V †1 be its complex-conjugate dual. Wehave

M ' V K -fin1 and M†,alg ' (V †1 )K -fin.

However, the Hermitian form on M gives rise to an identification M ' M†,alg. Thus, weobtain two embeddings

ı : M → V1 and ı† : M → V †1 .


Let V be the closure of the image of M under the diagonal embedding

M →M ⊕M ı⊕ı†−→ V1 ⊕ V †1 .

Then V is a G-subrepresentation of V1 ⊕ V †1 , by Theorem 9.2.3. It is easy to see that itsatisfies the requirements since

(m,m) ≤ ||ı(m)|| · ||ı†(m)|| ≤ (||ı(m)||+ ||ı†(m)||)2.

11.2.3. Let V be a Banach representation, supplied by Proposition 11.2.2. By definition, theHermitian form (−,−) extends continuously to V . We claim that it is G-invariant.

This would imply the assertion of Theorem 9.2.6 as the desired unitary representation canbe defined as the completion of V with respect to (−,−).

11.2.4. To prove the invariance, it suffices to show that for m1,m2 ∈M , the function

f(g) = (g ·m1,m2)− (m1, g−1 ·m2) = 0.

Note that (m1,−) and (−,m2) are continuous functionals on V . Now, by Proposition 9.2.9,it suffices to show that all the derivatives of f at 1 ∈ G vanish. However, this follows from theg-invariance of (−,−) on M .

11.3. An algebra that controls (some) (g,K)-modules.

11.3.1. Let ρ be an irreducible K-representation. Set

Mρ := U(g) ⊗U(k)

ρ.

This is a (g,K)-module, where we make K act by

k · (u⊗m) = Adk(u)⊗m.

It is is easy to see that

Hom(g,K)-mod(Mρ,M) 'Mρ.

Set

Aρ := (End(g,K)-mod(Mρ))op.

11.3.2. We have a pair of mutually adjoint functors

Φ : Aρ-mod : (g,K)-mod : Ψ,

where Ψ sends M ∈ (g,K)-mod to Hom(g,K)-mod(Mρ,M), viewed as a module over Aρ, and Φsends Q ∈ Aρ-mod to

Mρ ⊗AρQ.

The functor Ψ is exact, and the functor Φ is only right exact.

Lemma 11.3.3. The unit of the adjunction

Id→ Ψ Φ

is an isomorphism.

50 DENNIS GAITSGORY

Proof. For Q ∈ Aρ-mod, since the functor M 7→Mρ is exact, we have

(Mρ ⊗AρQ)ρ ' (Mρ)

ρ ⊗AρQ ' Aρ ⊗

AρQ ' Q.

Corollary 11.3.4. The functor Φ is fully faithful.


Proposition 11.3.6.

(a) If M is an irreducible (g,K)-module with Mρ 6= 0, then Ψ(M) is an irreducible Aρ-module.

(b) If Q is an irreducible Aρ-module, then Φ(Q) has a unique irreducible quotient, to be denotedMQ. The composite map

Q→ Ψ Φ(Q)→ Ψ(MQ)

is an isomorphism.

Proof. For point (a), let Q be a submodule of Ψ(M). By adjunction, we have a non-zero mapΦ(Q)→M . Since M is irreducible, the above map is surjective. Since Ψ is exact, the map

Q ' Ψ Φ(Q)→M

is surjective. Hence Q is all of Ψ(M).

For point (b), by Zorn’s lemma, Φ(Q) admits some irreducible quotient, denote it M . Byadjunction, we have a non-zero map

Q→ Ψ(M).

However, by point (a), Ψ(M) is irreducible, so the above map is an isomorphism.

Suppose now that Φ(Q) admits a surjective map to a direct sum M1 ⊕M2. We claim thatone of these modules must be zero. Indeed, applying Ψ, we obtain a (still surjective) map

Q ' Ψ Φ(Q)→ Ψ(M1)⊕Ψ(M2).

But Q is irreducible, so one of these maps, say Q → Ψ(M2) must be zero. However, byadjunction, this means that the initial map Φ(Q)→M2 was zero.

11.3.7. Let M be the Levi subgroup of the minimal parabolic Q of G. Denote KM := K ∩M .This is the maximal compact subgroup in M . Recall that

M = KM ×A.

In the next lecture we will construct an injective algebra homomorphism

(11.1) (Aρ)op → EndKM (ρ)⊗ U(a).

11.4. Admissibility of irreducible (g,K)-modules. Our goal on this subsection is to giveanother proof Theorem 10.3.3, by a method that we will employ again.

11.4.1. Let us decompose ρ into irreducible KM -modules:

ρ ' ⊕iπ⊕nii .

Let n = max(ni). We will show that any irreducible representation of Aρ is finite-dimensionalof dimension ≤ n. This would prove Theorem 10.3.3 in view of Proposition 11.3.6.


11.4.2. Recall the notations of Sect. 10.6.4. Since a is commutative, the polynomial Pr(n)

vanishes on EndKM (ρ) ⊗ U(a). The existence of the homomorphism (11.1) implies that Pr(n)

vanishes on Aρ.

We now claim:

Proposition 11.4.3. Let A be an associative algebra such that Pr(n) vanishes on A. Then anyirreducible representation of A is finite-dimensional of dimension ≤ n.

Proof. We claim that if M is an A module that contains n + 1 linearly independent vectors,then A has a subquotient isomorphic to Matn+1,n+1. This would be a contradiction since Pr(n)

is non-zero on Matn+1,n+1.

The existence of a subquotient follows from Burnside’s theorem: if M is an irreducibleA-module, and m1, ...,mk ∈ M linearly independent vectors, and m′1, ...,m

′k some k-tuple of

vectors, then there exists an element a ∈ A such that

a ·mi = m′i, i = 1, ..., k.

11.5. Construction of the homomorphism (11.1).

11.5.1. Consider the tensor product

Fρ := U(m) ⊗U(q)

Mρ,

which we can also think of as

C ⊗U(n)

Mρ.

It is acted on by m via the left action on the first factor. Moreover, it is easy to see as inSect. 11.3.1 that Fρ is naturally an (m,KM )-module.

In addition, Fρ carries a commuting action of (Aρ)op via the second factor.

11.5.2. Using the g = a⊕ n⊕ k decomposition, we can write

Fρ := U(m) ⊗U(q)

U(g) ⊗U(k)

ρ ' U(m) ⊗U(q)

U(q) ⊗U(kM )

ρ ' U(m) ⊗U(kM )

ρ

as an (m,KM )-module. So, this is the object of the same nature as Mρ but for the pair (m,KM )and the KM -representation ρ.

Further, writing U(m) (as an algebra) as

U(m) ' U(a)⊗ U(kM ),

we obtain that Fρ is isomorphic to

U(a)⊗ ρ,as an (m,KM )-module, where m ' a ⊕ kM acts via the action of a on the first factor and kMon the second factor.

11.5.3. Thus, we obtain that the action of (Aρ)op on Fρ gives rise to a map

(Aρ)op → End(m,K)-mod(Fρ) ' U(a)⊗ EndKM (ρ).

This is the desired map (11.1). Let us now prove that it is injective.

52 DENNIS GAITSGORY

11.5.4. Consider the algebra U(g)K of AdK-invariants in U(g). We claim that it naturally actson Mρ on the right by (g,K)-module endomorphisms. Indeed, this action is given by

(u⊗ v) · u′ 7→ u · u′ ⊗ v.

Hence, we obtain an algebra map

U(g)K → Aρ.

Let Iρ be the kernel of the action map U(k) → End(ρ). This is a two-sided ideal in U(k).Consider the left ideal U(g) · Iρ ⊂ U(g).

Note, however, that the intersection

Jρ := U(g)K ∩ U(g) · Iρis a two-sided ideal in U(g)K .

It is easy to see that the above map U(g)K → Aρ factors through a map

(11.2) U(g)K/Jρ → Aρ.

Proposition 11.5.5.

(a) The map (11.2) is an isomorphism.

(b) The composite map

(U(g)K/Jρ)op → (Aρ)

op → U(a)⊗ EndKM (ρ)

is injective.

Clearly, Proposition 11.5.5 implies that the map (11.1) is injective.

Proof of Proposition 11.5.5. Note that all objects in sight carry a filtration, induced by thecanonical filtration on U(g). To prove the proposition, it suffices to show that both (a) and (b)hold at the associated graded level.

We have:

gr(U(g)K/Jρ) ' (gr(U(g) ⊗U(k)

U(k)/Iρ))K ' (gr(U(g) ⊗

U(k)End(ρ)))K ' (Sym(g/k)⊗ End(ρ))K

and

grAρ ' gr(Hom(ρ, U(g) ⊗U(k)

ρ)K) ' (gr(Hom(ρ, U(g) ⊗U(k)

ρ))K ' (Sym(g/k)⊗ End(ρ))K ,

and it is easy to see that under these identifications, the associated graded of (11.2) is theidentity map on (Sym(g/k)⊗ End(ρ))K . This proves point (a).

Similarly, the associated graded of the map (11.1) is the map

(11.3) (Sym(g/k)⊗ End(ρ))K → (Sym(a)⊗ End(ρ))KM ,

where the map g/k→ a isg/k ' q/kM → m/kM ' a.

Thus, it remains to see that (11.3) is injective. Identifying g with its dual via the Killingform and denoting W := End(ρ), interpret the above map as

(Op ⊗W )K → (Oa ⊗W )KM ,

given by restriction along a → p.

Now, the assertion follows from the fact that under the adjoint action of K on p, the orbitof a is dense in p, see Sect. 8.5.5.


12. Week 6, Day 2 (Thurs, March 2)

12.1. Induced representations.

12.1.1. Let Q′ be a parabolic subgroup of G; let N ′ denote its unipotent radical, and let M ′

denote its Levi quotient. Let W be a representation of M ′, which we regard as a representationof Q′.

We define the induced representation IGQ′(W ) as follows: its space consists of continuousfunctions

f : G→W

that satisfy

f(g · q) = q−1 · f(g), g ∈ G, q ∈ Q′.Equivalently, we can identify this space with the space of functions f : G/N ′ → W that

satisfy

f(g ·m) = m−1 · f(g), g ∈ G, m ∈M ′,since Q′ acts on W via M ′.

Frobenius reciprocity implies:

Lemma 12.1.2. For a G-representation V , the space of continuous maps of G-representationsV → IGQ′(W ) identifies with the space of maps of Q′-representations V →W .

12.1.3. Let us identify what IGQ′(W ) looks like as a K-representation.

From the fact that G = K ·Q′ and K ∩Q′ = K ∩M ′ =: KM ′ is the maxmal compact in M ′,we obtain that the restriction of IGQ′(W ) identifies with the space of functions

f : K →W such that f(k ·m) = m−1 · f(k), k ∈ G, m ∈ KM ′ ,

i.e., we have a canonical isomorphism

(12.1) ResGK IGQ′ ' IKKM′ ResM′

KM′

This point of view lets us endow IGQ′(W ) with a variety of different topologies, and pass tothe corresponding completions. For example if W is finite-dimensional, we can consider the Lptopology on (12.1). It is easy to see that the action of G will still be continuous (but it will notpreserve the Lp norm; only the action of K does).

12.1.4. We claim:

Proposition 12.1.5. Suppose that W is admissible. Then IGQ′(W ) is admissible.

Proof. By (12.1) For a finite-dimensional representation ρ of K, we have

HomK(ρ, IGQ′(W )) ' HomKM′ (ρ,W ).

12.2. Induction for (g,K)-modules.

54 DENNIS GAITSGORY

12.2.1. We will now describe the algebraic counterpart of the induction construction. It willuse D-modules. Consider the category (m′,KM ′)-mod. We have a functor

rGM ′ : (g,K)-mod→ (m′,KM ′)-mod, M 7→Mn′ ,

where the subscript n′ means taking coinvariants with respect to the Lie algebra n′. The resultcarries an action of (m′,KM ) because M ′ normalizes n′.

We can also rewrite rGM ′ as

M 7→ U(m′) ⊗U(q′)

M.

It is called the Jacquet functor.

12.2.2. The functor rGM ′ admits a right adjoint by general nonsense. We will denote it by iGQ′ .

We observe:

Proposition 12.2.3. The natural transformations

Res(g,K)K iGQ′ → IndKKM′ Res

(m′,KM′ )KM′

and

coInd(m′,KM′ )KM′

ResKKM′ → rGQ′ coInd(g,K)K ,

induced by the tautological natural transformation

ResKKM′ Res(g,K)K → Res

(m′,KM′ )KM′

rGM ′

are isomorphisms.

In other words, whatever the functor iGQ′ happens to be, we know what it does at the level ofK-representations: this is just induction from KM ′ to K, i.e., the following diagram commutes:

(g,K)-modiGQ′←−−−− (m′,KM ′)-mod

Res(g,K)K

y yRes(m′,K

M′ )KM′

K-mod ←−−−− KM ′ -mod.

Note this is an algebraic counterpart of (12.1).

Proof. It suffices to prove the second isomorphism, as the first one is obtained by passing toright adjoints.

For the second isomorphism, we note that the functor coInd(g,K)K is given by

ρ 7→ U(g) ⊗U(k)

ρ

Using the fact that

k⊕ q′ g and q′ ∩ k = kM ′ ,

we obtain:

U(m′) ⊗U(q′)

U(g) ⊗U(k)

ρ ' U(m′) ⊗U(q′)

U(q′) ⊗U(kM′ )

ρ ' U(m′) ⊗U(k)

ρ,

as desired.

Corollary 12.2.4. The functor iGQ′ preserves admissibility.


12.2.5. Let W be an admissible M ′-representation. Let L denote the underlying (m′,KM ′)-module. We claim:

Proposition 12.2.6. The (g,K)-module underlying IGQ′(W ) identifies canonically with iGQ′(L).

Proof. We first construct a map of (g,K)-modules

(12.2) (IGQ′(W ))K -fin → iGQ′(L).

By the definition of iGQ′(L), the datum of such a map is equivalent to that of a map of

(q′,KM ′)-modules

(12.3) (IGQ′(W ))K -fin → L.

By Frobenius reciprocity (i.e., Lemma 12.1.2) we have a map of Q′-representations

IGQ′(W )→W.

Under this map the subspace IGQ′(W ))K -fin gets sent to WKM′ -fin = L. This defines the

desired map (12.3). It is easy to see that it indeed respect the (q′,KM ′)-action.

Thus, by adjunction we also obtain the map (12.2). In order to see that this map is anisomorphism, it suffices to show that it is such at the level of K-representations.

We have:Res

(g,K)K ((IGQ′(W ))K -fin) ' (ResGK(IGQ′(W )))K -fin,

which by (12.1) identifies with

(IKKM′ ResM′

KM′(W ))K -fin,

and the latter is easily seen to identify with IndKK′M (L).

Now,

Res(g,K)K (iGQ′(L)) ' IndKK′M (L)

by Proposition 12.2.3.

By unwinding the definitions, it is easy to see that the resulting map from IndKK′M (L) to itself

is the identity.

12.3. Explicit description of the induction functor on (g,K)-modules. We will nowdescribe the functor iGQ′ explicitly. In doing so we will used modules over algebraic differentialoperators, a.k.a., D-modules.

For the duration of this subsection we be in the context of algebraic geometry, so will writeG, K, etc, for their complexifications.

12.3.1. Recall that the K-orbit of 1 in X ′ := G/Q′ is open; denote it by X ′0. This is an affinescheme. Denote

X ′ := G/N ′;

this is a M ′-torsor over X ′. Let X ′0 be the preimage of X ′0 in X ′0. This is also an affine scheme.

The scheme X ′ carries an action of G (in particular K) by left translations, and a commutingaction of M ′ by right translations.

Consider the ring of differential operators D(X ′0). The above action of G on X ′ gives rise toan algebra homomorphism

ιl : U(g)→ D(X ′) ⊂ D(X ′0)

56 DENNIS GAITSGORY

and the right action of M ′ gives rise to an algebra homomorphism

ιr : U(m′)→ D(X ′) ⊂ D(X ′0).

The images of these two algebras commute with each other. In particular, every D-module

on X ′0 carries a left action of g and a right-action of m.

12.3.2. We let Y be the closed subscheme of X ′0 equal to the orbit of 1 under the K-action;note Y is isomorphic to K as K ∩N ′ = 1.

We will now consider a particular D-module on X ′0, denoted it by δY,X′0, to be thought of as

the δ-distribution2 on Y inside X ′0.

Explicitly, δY,X′0is obtained by quotienting D(X ′0) by the left ideal gererated by:

(i) IY ⊂ OX′0⊂ D(X ′0), where IY is the ideal of Y in X ′0;

(ii) ιl(k) ⊂ D(X ′0).

Note that since Y is invariant under left translations by K, the elements from ιl(k) normalizethe ideal IY .

12.3.3. We claim that the left action of g on δY,X′0extends to a structure of (g,K)-module, and

the above action of m′ extends to a structure of (m′,KM ′)-module.

Indeed, the action of K on X ′0 defines an action of K on D(X ′0). The derivative of this actionis given by

ξ ∈ k, d ∈ D(X ′0) 7→ ιl(ξ) · d− d · ιl(ξ).

This action of K preserves the ideal defining δY,X′0. Now, if ξ ∈ k, then the element ιl(ξ)

belongs to this ideal and so the above action of ξ ∈ k on δY,X′0is given by left multiplication by

ιl(ξ).

Similarly, the right action of M ′ on X ′0 defines an action of M ′ on D(X ′0). The derivative ofthis action is given by

ξ ∈ m′, d ∈ D(X ′0) 7→ ιr(ξ) · d− d · ιr(ξ).

The induced action ofKM ′ preserves the ideal defining δY,X′0. Further, we claim that elements

of the form ιr(ξ) for ξ ∈ kM ′ belong to this ideal. Indeed, this follows from the fact that therestriction of the vector field ιr(ξ) to Y is of the form∑

i

fi · ιl(ξi), fi ∈ OY , ξi ∈ kM ′ .

Hence, the above action of ξ ∈ kM ′ on δY,X′0is given by left multiplication by ιr(ξ).

2In the D-module language, it is the direct image under Y → X′0 of the D-module OY tensored with the line

(Λtop(k/kM′ ))⊗−1.


12.3.4. In what follows we will need the following construction. Let M1 and M2 be a right anda left (g,K)-modules respectively. In this case we can form the vector space3

M1

K⊗g/k

M2 := (M1 ⊗U(g)

M2)K .

Note that when g = k, the projection

(M1 ⊗M2)K →M1

K⊗kM2

is an isomorphism (this can be seen by identifying K-invariants with K-coinvariants).

So, we should think of the functor

M1,M2 7→M1

K⊗g/k

M2

as the operation of taking invariants in the K-direction and coinvariants with respect to g/k.

For example, if

M1(ρ) = coInd(g,K)K := ρ ⊗

U(k)U(g),

we have

(12.4) M1

K⊗g/k

M2 ' (ρ⊗M2)K .

Remark 12.3.5. In the particular case when G is compact modulo its center (a situation thatwill be of particular interest for us, because we will take G = M , the Levi of the minimal

parabolic), the operationK⊗g/k

can be described in simpler terms. Namely,

M1

K⊗g/k

M2 ' (M1 ⊗U(a)

M2)K .

12.3.6. We take the module δY,X′0, and given L ∈ (m′,KM ′)-mod we form

′iGQ′(L) := δY,X′0

KM′⊗

m′/kM′L.

The action of g via ιl makes it into an object of (g,K)-mod. We will prove:

Theorem 12.3.7. There is a canonical isomorphism of functors

iGQ′ ' ′iGQ′ , (m′,KM ′)-mod→ (g,K)-mod.


3The definition given below is fine when K is reductive, and when we are interested in the non-derivedfunctor. In general, more care is needed.

58 DENNIS GAITSGORY

12.4.1. We first claim that there exists a canonical isomorphism of functors

(12.5) Res(g,K)K ′iGQ′ → IndKKM′ Res

(m′,KM′ )KM′

,

i.e., that at the level of K-representations, ′iGQ′ does the right thing.

Indeed, we claim that δY,X′0when regarded as a K-module with respect to the left action

and a (m′,KM ′)-module with respect to the right action, identifies canonically with

OY ⊗U(kM′ )

U(m).

This would imply the isomorphism (12.5) in view of (12.4), since Y ' K as a K × KM ′ -scheme.

To establish the desired description of δY,X′0we note that there is a canonical map

OY → δY,X′0as K ×KM ′ -modules. By adjunction, this gives rise to a map

OY ⊗U(kM′ )

U(m)→ δY,X′0

as K-modules and (m′,KM ′)-modules. To check that this map is an isomorphism, it is enoughto do so at the associate graded level.

The latter, however, is easily seen to be the identity map on

OY ⊗ Sym(m′/kM ′).

12.4.2. We next consider the case of Q′ = G. We claim that ′iGG is indeed isomorphic to theidentity functor on (g,K)-mod. I.e., we claim:

Proposition 12.4.3. The functor

M 7→ δK,GK⊗g/k

M

is isomorphic to the identity functor on (g,K)-mod.

Proof. For M ∈ (g,K)-mod, the action of K on M gives rise to a map

M → OK ⊗M.

Its image lies in (OK ⊗M)K , where we consider the diagonal action of K comprised fromthe given action on M and the action of K on OK by right translations.

Composing, we obtain a map

M → (OK ⊗M)K → (δK,G ⊗M)K → δK,GK⊗g/k

M.

It is straightforward to check that the above map is a map of (g,K)-modules. This definesa natural transformation

Id→ ′iGG.

To check that it is an isomorphism, it suffices to show that the induced natural transformation

Res(g,K)K → Res

(g,K)K ′iGG

is an isomorphism.

However, it is straightforward to check that the above map is the isomorphism of (12.5) inthe particular case of Q′ = G.


12.4.4. Next we construct a natural transformation

(12.6) rGQ′ ′iGQ′ → Id .

In view of Proposition 12.4.3, this amounts to constructing a map of (q′,KM ′)-modules

δY,X′0→ δKM′ ,M ′

that respects the (q′,KM ′)-action on the left and the (m′,KM ′)-action on the right.

Consider M ′ as a subscheme in X ′0 (i.e., the fiber over 1 of G/N ′ → G/Q′) and consider thetensor product

OM ′ ⊗OX′0

δY,X′0.

This is a D-module on M ′. The projection

δY,X′0→ OM ′ ⊗

OX′0

δY,X′0

respects the (q′,KM ′)-action on the left and the (m′,KM ′)-action on the right.

Thus, it suffices to construct an isomorphism of D-modules on M ′

δKM′ ,M ′ ' OM ′ ⊗OX′0

δY,X′0.

We construct a map

D(M)→ OM ′ ⊗OX′0

δY,X′0

by sending the generator to 1 ∈ OM ′ . It is easy to see that this map factors via a map

(12.7) δKM′ ,M ′ → OM ′ ⊗OX′0

δY,X′0.

To show that the latter map is an isomorphism, we do it at the associated graded level.However, it is easy to see that gr of both sides identifies naturally with

OKM′ ⊗ Sym(m/kM ′)

so that (12.7) induces the identity map.

12.4.5. By adjunction, the map (12.6) constructed above, gives rise to a natural transformation

′iGQ′ → iGQ′ .

We claim that this natural transformation is an isomorphism. To prove this, it is sufficientto show that the induced natural transformation

Res(g,K)K ′iGQ′ → Res

(g,K)K iGQ′

is an isomorphism.

However, by (12.5), the left-hand side identifies with IndKKM′ Res(m′,KM′ )KM′

, and the right-hand

side identifies with the same by Proposition 12.2.3.

Now, by unwinding the constructions, we see that the resulting natural transformation from

IndKKM′ Res(m′,KM′ )KM′

to itself is the identity map.

60 DENNIS GAITSGORY

13. Week 7, Day 1 (Tue., March 7)

13.1. Harish-Chandra homomorphism.

13.1.1. Let Q′ be a parabolic in G. Consider the tensor product

F := U(m′) ⊗U(p′)

U(g),

as a right g-module and a left m′-module.

We claim:

Proposition 13.1.2. The action of Z(g) ⊂ U(g) on F factors through a uniquely defined map

φ : Z(g)→ Z(m′)

and the action of Z(m′) ⊂ U(m). Furthermore, φ is injective and Z(m′) is finite as a Z(g)-module.

Remark 13.1.3. The above map is most commonly used when Q′ = B, and so m′ = h. It isusually referred to as the Harish-Chandra homomorphism.

Proof of Proposition 13.1.2. An element z ∈ Z(g) defines an endomorphism of F as a g-moduleand as a m′-module. We claim that for any endomorphism S of F as an g-module, the imageS(1) of the element 1 ∈ F lies in

U(m′) ' U(m′) ⊗U(p′)

U(p′) ⊂ U(m′) ⊗U(p′)

(U(p′)⊗ U(n′−)) ' U(m′) ⊗U(p′)

U(g) = F.

Indeed, S(1) is invariant under the adjoint action of zM ′ ⊂ m′, but the subspace of suchelements in U(n′−) equals C.

Hence 1 · z = φ(z) · 1 for a well-defined element φ(z) ∈ U(m). For u ∈ U(m′) we have

u · φ(z) = u · z = z · u = φ(z) · u,(the second equality since z ∈ Z(g)), as elements in U(m′) ⊂ F . Hence, φ(z) ∈ Z(m′). Asimilar argument shows that φ is an algebra homomorphism.

To show that φ injective and Z(m′) is finite as a Z(g)-module it is enough to do so at theassociated graded level. By transitivity, it is enough to consider the case of Q′ = B. Thecorresponding map is

Sym(g)G → Sym(g/n)T ' Sym(h).

Identifying g with its dual via the Killing form, the latter map is the restriction map

Sym(g)G → Sym(h),

which is injective because the orbit if h in g under the adjoint action is dense.

13.1.4. Consider the functor rGQ′ . We claim:

Lemma 13.1.5. For M ∈ (g,K)-mod, the action of Z(g) on rGQ′(M) induced by the Z(g)-action

on M equals the action obtained from φ : Z(g)→ Z(m′) and the action of Z(m′) on rGQ′(M) as

an m′-module.

Proof. Follows from the fact that the functor g-mod→ m′-mod underlying rGQ′ is given by

M 7→ F ⊗U(g)

M.


By adjunction, we obtain:

Corollary 13.1.6. For L ∈ (m′,KM ′)-mod, the action of Z(g) on iGQ′(L) as a (g,K)-module

equals the action induced by the action of Z(g) on L via φ.

13.1.7. We claim:

Proposition 13.1.8.

(a) The functor rGM ′ sends finitely generated objects in (g,K)-mod to finitely generated objectsin (m,M ′)-mod.

(b) The functor rGM ′ sends objects of finite length in (g,K)-mod to objects of finite length in(m,M ′)-mod.

Proof. Point (a) follows from the second isomorphism in Proposition 12.2.3. Point (b) followsfrom point (a) and Corollary 10.3.5.

Remark 13.1.9. The proof of Proposition 13.1.8 used Corollary 10.3.5, and that in turned usedthe (non-trivial) Theorem 10.2.6, applied to the reductive group M ′. However, when Q′ is theminimal parabolic Q, so that the Levi M ′ = M is compact modulo its center, the conclusionof Theorem 10.2.6 is evident.

So, we obtain that the corresponding functor rGQ sends (g,K)-modules of finite length to

finite-dimensional (m,KM )-modules.

The proof amounts to the fact that for M ∈ (g,K)-mod of finite length, the (m,KM )-modulerGM (M) is finitely generated and has finite support over Z(g). This implies that its support overZ(m) is also finite. The latter means that its support in Spec(U(a)) is finite and that it hasonly finitely many KM -isotypics components.

13.2. The subquotient theorem.

13.2.1. In this subsection we will prove the following fundamental result, known as the Sub-quotient Theorem:

Theorem 13.2.2. Every irreducible (g,K)-module can be realized as a subquotient of iGP (L),where P is the minimal parabolic and L is a finite-dimensional (m,KM )-module.

The particular significance of this theorem is explained by the following corollary:

Theorem 13.2.3. Every irreducible (g,K)-module can be realized as the (g,K)-module under-lying an admissible representation of G on a Banach (and even Hilbert) space.

Let us see how Theorem 13.2.2 implies Theorem 13.2.3:

Proof of Theorem 13.2.3. Let us realize a given irreducible (g,K)-module M as a subquotientof iGQ(L). By Proposition 12.2.6 we can realize iGQ(L) as the space of K-finite vectors in IGQ (L),

which is a Banach representation of G (and also in the L2-version of IGQ (L), which is a Hilbert

(but not unitary!) representation of G).

Hence, by Theorem 9.2.3, M can be realized as a subquotient of IGQ (L) (or its L2 version).

Remark 13.2.4. Note that Theorem 13.2.2 does not explicitly mention the fact that the (g,K)-modules iGQ(L) have a finite length. The latter is true, but is a hard theorem, essentiallyequivalent to Theorem 10.2.6, see below.

62 DENNIS GAITSGORY

13.2.5. Let us make the following observation on the structure of the theory:

Proposition 13.2.6. The following assertions are logically equivalent:

(i) For an irreducible (m,KM )-module L, the (g,K)-module iGQ(L) is of finite length.

(i’) For a finite-dimensional (m,KM )-module L, the (g,K)-module iGQ(L) is finitely generated.

(ii) The assertion of Theorem 10.2.6 holds, i.e., for every character χ of Z(g) there are onlyfinitely many classes of irreducible objects in (g,K)-modχ.

Proof. Let us show that (i) implies (ii). By Theorem 13.2.2, we know that every irreducible(g,K)-module can be realized as a subquotient of iGQ(L) for some irreducible (m,KM )-module

L. By Corollary 13.1.6, if L is irreducible, then Z(g) acts on iGQ(L) by a single character,

obtained via the Harish-Chandra homomorphism φ : Z(g)→ Z(g) from the character by whichZ(m) acts on L.

Hence, it suffices to show that, given χ, there are only finitely many irreducible (m,KM )-modules L, on which Z(g) acts by χ. Since the map Spec(Z(m)) → Spec(Z(g)) is finite, itsuffices to show that for a given character χ′ of Z(m), there are at most finitely many irreducible(m,KM )-modules L, on which Z(m) acts by χ′. In fact, there is at most one such module:

Te datum of χ′ determines the action of a, and it is known that finite-dimensional represen-tations of a compact group (in our case KM ) are distinguished by the action of the center4 ofthe universal enveloping algebra.

Let us show that (ii) implies (i). We have seen that Theorem 10.2.6 implies Corollary 10.3.5.In particular, in order to prove that iGQ(L) is of finite length it is sufficient to it is admissible

(which follows from Corollary 12.2.4) and that its support over Z(g) is finite (which followsfrom Corollary 13.1.6).

Clearly (i) implies (i’). For the inverse implication we will use the fact (to be proved nexttime) that the algebraic dual of iGQ(L) is isomorphic to iGQ(L∗), where L∗ is the dual of L (up

to a ρ-shift). If... ⊂Mi ⊂Mi+1 ⊂ ...

is an infinite chain of subobjects of iGQ(L) (with non-zero subquotients), consider the corre-sponding chain

... ⊂ (Mi+1)⊥ ⊂ (Mi)⊥ ⊂ ...

in iGQ(L∗). Assuming (i’) and using the fact that the category (g,K)-mod is Noetherian, weobtain that both these chains stabilize on the right. This is a contradiction.

13.3. Proof of the subquotient theorem.

13.3.1. Let ρ be an irreducible representation of K. Recall the algebra Aρ from Sect. 11.3. Weclaim that it suffices to show that every irreducible Aρ-module can be realized as a subquotientof Ψ(iGQ(L)) = HomK(ρ, iGQ(L)) for some finite-dimensional representation L of M .

Indeed, let M be an irreducible (g,K)-module. Let ρ be such that Mρ 6= 0. Denote Q :=Ψ(M). Let L be such that Q can be realized as a subquotient of Ψ(iGQ(L)). I.e., we havesubmodules

Q1 ⊂ Q2 ⊂ Ψ(iGQ(L))

and an isomorphism Q ' Q2/Q1.

4If we just consider the action of the Casimir, for every eigenvalue there will be at most finitely manyirreducibles on which it acts with this eigenvalue.


Consider the corresponding maps

Φ(Q1)→ Φ(Q2)→ iGQ(L).

We claim that the composition

Φ(Q2)→ iGQ(L)→ coker(Φ(Q1)→ iGQ(L))

is non-zero. Indeed, if it were, then applying the functor Ψ we would obtain that the composition

Ψ Φ(Q2)→ Ψ(iGQ(L))→ coker(Ψ Φ(Q1)→ Ψ(iGQ(L)))

is zero. However, since Ψ Φ ' Id, the latter map identifies with

Q2 → Ψ(iGQ(L))→ coker(Q1 → Ψ(iGQ(L))),

which was non-zero by assumption.

Therefore, since Φ is right exact, we obtain that Φ(Q) surjects onto a subquotient, (denoteit M′) of iGQ(L). We claim that in this case M is a quotient of M′. Indeed, this follows from the

fact that Φ(Q) has a unique irreducible quotient (Proposition 11.3.6(b)).

13.3.2. Let us now write down explicitly the functor Ψ iGQ. (This will be a lot easier than

writing down iGQ itself).

By definition, the functor Ψ iGQ is the right adjoint of the functor rGQ Φ. Now, the latterfunctor sends

Q 7→ U(m) ⊗U(p)

U(g) ⊗U(k)

ρ ⊗Aρ

Q ' Fρ ⊗Aρ

Q,

where we regard Fρ as an (m,KM )-module equipped with a commuting right action of Aρ.Note that Fρ, veiwed as an (m,KM )-module identifies with rGQ(U(g) ⊗

U(k)ρ).

Hence, the functor Ψ iGQ is given by

L 7→ Hom(m,KM )-mod(Fρ,L).

Recall that Fρ, when viewed as an (m,KM )-module, isomorphic to

U(m) ⊗U(kM )

ρ.

So, the composition of Ψ iGQ with the forgetful functor Aρ-mod→ Vect is just

L 7→ HomKM (ρ,L).

We also note:

Lemma 13.3.3. The module Fρ are finitely generated over Z(g).

Proof. Note that by Lemma 13.1.5 the Z(g)-action on Fρ, equals the action obtained fromφ : Z(g) → Z(m) and the Z(m) action on Fρ as a (m,KM )-module. Clearly, Fρ is finitelygenerated as a module over U(a), and hence over all of Z(m). Now the assertion follows fromthe fact that Z(m) is finitely generated as a module over Z(g).

64 DENNIS GAITSGORY

13.3.4. Since the (right) action of Aρ on Fρ is faithful by Sect. 11.5, (and Fρ is finitely generatedover U(a)), we can find an embedding

(13.1) Aρ → F⊕kρ

as right Aρ-modules, for some k.

Let Q be an irreducible (automatically, finite-dimensional) module over Aρ; consider its lineardual Q∗ as a right Aρ-module; it is irreducible. In particular, it can be realized as a quotient ofAρ. It suffices to show that Q can be realized as a subquotient of Hom(m,KM )-mod(F⊕kρ ,L) forsome L.

Let χ be the character of Z(g) through which it acts on Q. Then Z(g) acts by the samecharacter on Q∗. Let Iχ ⊂ Z(g) be the corresponding ideal. Recall that by Lemma 13.3.3, Fρis finitely generated as a Z(g)-module. It follows from Artin-Rees applied to the embedding(13.1) that Aρ/Iχ is a subquotient of F ′ := F⊕kρ /Inχ for some n.

This F ′ is an (m,KM )-module that carries a commuting right action of Aρ. It suffices to showthat Q can be realized as a subquotient of Hom(m,KM )-mod(F ′,L) for some finite-dimensional(m,KM )-module L.

Note that F ′ is finite-dimensional as a vector space. Take L = F ′ ⊗ (F ′)∗, considered as amodule over (m,KM ) via the action on the first factor. We claim that it does the job.

Indeed, the evaluation map F ′ ⊗ (F ′)∗ → C defines a surjection

Hom(m,M)(F′,L)→ (F ′)∗

as left Aρ-modules. However, (F ′)∗ contains Q as a subquotient, by construction.

14. Week 7, Day 2 (Tue., March 9)

14.1. Normalized induction and duality on induced representations.

14.1.1. Consider the adjoint action of q′ on g/q′, and let us consider the top exterior power ofthis action. We obtain an (algebraic) character of the group Q′, i.e., a homomorphism

Q′ → Gm,which automatically factors via a homomorphism

(14.1) M ′ → Gm.In its turn, the homomorphism (14.1) factors through a homomorphism

(14.2) M ′/[M ′,M ′]→ Gm.Note that M ′/[M ′,M ′] is a connected torus defined over R. Therefore it (rather, the group

of its R-points) can be written as

KM ′/[M ′,M ′] ×AM ′/[M ′,M ′].(Note that when Q′ is the minimal parabolic Q, then AM ′/[M ′,M ′] ' A.)

Since KM ′/[M ′,M ′] is connected, (14.2) factors through a homomorphism

(14.3) AM ′/[M ′,M ′] → R>0,×.

We denote the latter by −2ρQ′ . By a slight abuse of notation, we will denote by the samesymbol the composite homomorphism

M ′(R)→M ′/[M ′,M ′](R)→ AM ′/[M ′,M ′] → R>0,×

and also the corresponding characters of Lie algebras.


Explicitly, let us write n′ as a sum of aM ′/[M ′,M ′]-eigenspaces

n′ '⊕α′

(n′)α′ .

Then

−2ρQ′ = −∑α′

α′ · dim((n′)α′).

14.1.2. Let us once and for all choose a trivialization of the line Λtop(g/q−). The key observationis the following:

Proposition 14.1.3. There exists a canonically defined G-invariant functional∫G/Q′

: IGQ′(−2ρQ′)→ C.

Proof. By definition IGQ′(−2ρQ′) is the space of scalar-valued functions on G that satisfy

f(g · p) = (−2ρQ′)(p−1) · f(g).

We claim that this vector space identifies canonically with the space of top continuous degreedifferential forms on G/Q′. This follows from the fact that the action of Q′ on the top exteriorpower of Te(G/Q

′) ' g/q′ is given by −2ρQ′ .

Now the sought-for functional is given by integration.

14.1.4. The homomorphism (14.3) has a well-defined square root, which we denote by −ρQ′ .We denote the normalized induction functor

In,GQ′ : Rep(M ′)→ Rep(G′)

by

In,GQ′ (W ) := IGQ′(W ⊗ (−ρQ′)).

Note that In,GQ′ (W ) and IGQ′(W ) are the same as K-representations. This is because −ρQ′ istrivial on KM ′ .

14.1.5. Let W be a representation of M and let W ∗ be its dual. From Proposition 14.1.3 weobtain that there exists a canonically defined G-invariant pairing

(14.4) In,GQ′ (W )× In,GQ′ (W ∗)→ C, f1, f2 7→∫G/Q′

〈f1, f2〉.

We clam:

Proposition 14.1.6. Let W is admissible. Then the induced pairing

(In,GQ′ (W ))K -fin ⊗ (In,GQ′ (W ∗))K -fin → C

is perfect, i.e., identifies (In,GQ′ (W ∗))K -fin with

((In,GQ′ (W ))K -fin)∗,alg.

66 DENNIS GAITSGORY

Proof. It is enough to verify the statement at the level of K-representations. In the latter case,the pairing in question is given by

IndKKM (L)⊗ IndKKM (L∗,alg)→ IndKKM (C)→ C,

where the latter map is the K-invariant (!) integration over K/KM ' G/P , or which is thesame as unique splitting of the map

C → IndKKM (C)

as K-representations.

Now, for a given ρ ∈ Irrep(K), we have

(IndKKM (L))ρ ' ρ⊗HomKM (ρ,L)

and similarly

(IndKKM (L∗,alg))ρ∗' ρ∗ ⊗HomKM (ρ∗,L∗,alg),

and our pairing is the perfect pairing induced by

ρ⊗ ρ∗ → C

and

HomKM (ρ,L)⊗HomKM (ρ,L∗,alg)→ C.

14.1.7. Let in,GQ denote the functor (m′,KM ′)-mod→ (g,K)-mod given by

in,GQ′ (W ) := iGQ′(W ⊗ (−ρQ′)).

Note that in,GQ′ (W ) and iGQ′(W ) are the same as K-representations.

Combining Proposition 14.1.6 with Theorem 14.3.6 (see below) and Proposition 11.1.2, weobtain:

Corollary 14.1.8. Let L be an admissible (m′,KM ′)-module. Then there exists a canonical

isomorphism between (in,GQ (L))∗,alg and in,GQ (L∗,alg).

Remark 14.1.9. The assertion of Corollary 14.1.8 is purely algebraic. It amounts to the factthat the canonical K-invariant functional

iGQ′(−2ρQ′) ' OK/KM → C

is G-invariant. However, I was not able to give an algebraic proof of this fact: I needed toappeal to Proposition 14.1.6 that involves integration over the real manifold G/Q′.

14.1.10. Let us return to the situation of Sect. 14.1.5. Suppose that W is unitary.

In this case, we have a G-invariant sesquilinear pairing

(14.5) In,GQ′ (W )× In,GQ′ (W )→ C,∫G/Q′

(f1, f2).

Interpreting the topologigal space underlying In,GQ′ (W ) as the space of continuous functions

f : K →W, f(k ·m) = m−1 · f(k), m ∈ KM ,

we obtain that (14.5) equips In,GQ′ (W ) positive-definite scalar product, which is continuous in

the original topology on In,GQ′ (W ).


Let L2In,GQ′ (W ) denote the completion of In,GQ′ (W ) with respect to the resulting L2-norm.

We obtain that L2In,GQ′ (W ) is a unitary representation of G. Thus, normalized induction mapsunitary representations to unitary representations.

14.2. Casselman’s submodule theorem.

14.2.1. We will prove the following:

Theorem 14.2.2. Every irreducible (g,K)-module can be realized as a submodule of someiGQ(L) for an irreducible finite-dimensional (m,KM )-module L.

By adjunction, this theorem is equivalent to:

Theorem 14.2.3. Let M be a finitely generated admissible (g,K)-module. Then the space Mn

of n-coinvariants is non-zero.

There exist two algebraic proofs of this theorem. One by O. Gabber and another by Beilinson-Bernstein via localization theory. We will present an analytic proof. It is based on consideringthe growth of matrix coefficients of K-finite vectors in a realization of our (g,K)-module.

14.2.4. Let a be a vector in a. Let V be a G-representation on a Banach space.

Lemma 14.2.5. There exists a real number λ such that for all v ∈ V and v∗ ∈ V ∗ there existsa constant C so that we have

|〈v∗, exp(t · a) · v〉| ≤ C · exp(t · λ), t ∈ R≥0.

Proof. Follows from the fact that the operator Texp(a) is bounded, and we take λ it be itsnorm.

14.2.6. We shall say that a is regular if all of its eigenvalues on n are non-zero. We shall saythat a is strictly positive if all of its eigenvalues on n are strictly positive.

We have the following key assertion:

Theorem 14.2.7. Let a ∈ a be regular. Let V be admissible, and let v and v∗ be K-finite suchthat 〈v∗, exp(t · a) · v〉 is not identically equal to zero. Then the set of λ ∈ R such that

|〈v∗, exp(t · a) · v〉| ≤ C · exp(t · λ), t ∈ R≥0 for some C

is bounded below.

In other words, this theorem says that the matrix coefficient functions 〈v∗, exp(t · a) · v〉cannot decay faster than all exponents.

14.2.8. Let us deduce Theorem 14.2.3 from Theorem 14.2.7. We will need the following assertionof independent interest, proved below.

Proposition 14.2.9. Any admissible finitely generated (g,K)-module is finitely generated overU(n).

Proof of Theorem 14.2.2. Pick a ∈ a to be strictly negative. We pick a realization V of M. Fixv∗ ∈ (V ∗)K -fin. For v ∈ V K -fin such that 〈v∗, exp(t · a) · v〉 is not identically equal to zero, letλv be the infimum of those real numbers that

|〈v∗, exp(t · a) · v〉| ≤ C · exp(t · λ) for some C.

By Lemma 14.2.5 and Theorem 14.2.7, this infimum is well-defined.

68 DENNIS GAITSGORY

We claim that if v = u · v′ for u an element of U(n) with eigenvalue µ with respect to a(remember, it is negative), then

λv = λv′ + µ.

Indeed, this is just the fact that

exp(t · a) · v = exp(t · µ) · exp(t · a) · v′.

Hence, since M is finitely generated over U(n), the function

v 7→ λv

attains a maximum. Let v0 ∈M be a vector on which this maximum is attained. By the above,it cannot be of the form u · v′ for u in the augmentation ideal of U(n). I.e., v0 projects tonon-zero in Mn.

Proof of Proposition 14.2.9. It is enough to show that any object of (g,K)-mod of the form

U(g) ⊗U(k)

ρ,

where ρ is a finite-dimensional representation of K, is finitely generated over U(n)⊗ Z(g).

I.e., it suffices to show that U(g) is finitely generated as a module over U(n)⊗U(k)op⊗Z(g).The latter is suffices to do at the associated graded level. I.e., it suffices to show that Sym(g)is finitely generated as a module over Sym(n)⊗ Sym(k)⊗ Sym(g)G.

Since everything is positively graded, it suffices to show that Sym(g/n) is finitely generatedover Sym(k)⊗ Sym(g)G. However, the action of Sym(g)G on Sym(g/n) factors through Sym(a)(which is finite as a Z(g)-module), and the assertion follows.

14.3. Further realization results.

14.3.1. We will now consider the category (q,KM )-mod. It contains as a full subcategory(m,KM )-mod; it consists of those objects L on which n acts trivially.

We have the pair of adjoint functors

Res(g,K)(q,KM ) : (g,K)-mod (q,KM )-mod : iGQ.

The precomposition of iGQ with (m,KM )-mod → (q,KM )-mod is the functor that we earlier

denoted iGQ. The proof of Proposition 12.2.3 applies and we have an isomorphism

Res(g,K)K iGQ ' IndKKM Res

(q,KM )KM

.

14.3.2. We have a similar situation at the level of group representations:

ResGQ : Rep(G) Rep(P ) : IGQ .

As in Proposition 12.2.6, for a KM -admissible (q,KM )-module W , we have

(IGQ (W ))K -fin ' iGQ(WKM -fin).


14.3.3. Inside the category (q,KM )-mod we single out the full subcategory (q,KM )-modn -nilp

consisting of objects on which the Lie algebra n acts locally nilpotently.

Writing m as kM ⊕ a, we obtain that an object of (q,KM )-modn -nilp can be thought of as analgebraic representation of the group

ker(Q→M → TA),

equipped with an action of a that commutes with KM and interacts with the N -action accordingto the adjoint action of a on N .

It is easy to see that any finite-dimensional object of (q,KM )-mod belongs in fact to

(q,KM )-modn -nilp. Indeed, the action of n is necessarily nilpotent because a has non-trivialadjoint eigenvalues on n.

Moreover, it is easy to see that restriction defines an equivalence from the category of finite-dimensional representations of the (real Lie group) Q to that of finite-dimensional objects in(q,KM )-mod.

14.3.4. We will prove:

Theorem 14.3.5. Let M be a (g,K)-module of finite length. Then there exists a finite-dimensional object L ∈ (q,KM )-mod such that M embeds into iGQ(L).

As a consequence, as in we obtain (as in the case of Theorem 13.2.3):

Theorem 14.3.6. And (g,K)-module of finite length can be realized as K-finite vectors in aBanach (or even Hilbert) representation of G.

14.4. Proof of Theorem 14.3.5. We will replace K by its connected component–this doesnot change the results of the preceding sections.

14.4.1. For an integer k, consider M/nk ·M as a (q,KM ∩K0)-module. According to Proposi-tion 14.2.9, it is finite-dimensional. We will prove that there exists an integer k, such that themap

M→ iGQ(M/nk ·M),

(obtained by adjunction from the tautological map Res(g,K0)(q,KM∩K0)(M)→M/nk ·M) is injective.

14.4.2. Let n be a nilpotent Lie algebra. We consider the following functor on the category ofn-modules:

M 7→ M := lim←k

M/nk ·M.

By induction on the degree of nilpotence, from the Artin-Rees lemma we deduce:

Lemma 14.4.3. The above functor is exact when restricted to the subcategory of finitely gen-erated n-modules.

Let M be a (g,K0)-module of finite length. We claim:

Corollary 14.4.4. The map M 7→ M is injective.

Proof. Lemma 14.4.3 and Proposition 14.2.9 reduce the assertion to the case when M is irre-ducible, which is what we will now assume.

We now note that M also naturally acquires an g-action so that the natural map M → M

is a map of g-modules (this is true for any g-module M). This follows from the fact that there

70 DENNIS GAITSGORY

exists an integer k such that for any ξ ∈ g, if we take k times its bracket with elements from n,we get zero. For such k, the action of g is well-defined as a map

M/nk′+k ·M→M/nk

′·M.

Recall also that if M is irreducible as a (h,K0)-module, then it is such as a g-module. Hence,

it suffices to show that the map M→ M is non-zero. However, we know that the composition

M→ M→M/n ·Mis non-zero. Indeed, this is equivalent to the statement of Theorem 14.2.2.

14.4.5. Consider now the map of (g,K0)-modules

M→ lim←k

iGQ(M/nk ·M).

It is injective because its composition with

lim←k

iGQ(M/nk ·M)→ lim←k

M/nk ·M = M

is injective by Corollary 14.4.4.

Let M′k denote the kernel of the map M→ iGQ(M/nk ·M). We have

M′k+1 ⊂M′k ⊂ ...Since M was assumed of finite length, this chain stabilizes. Hence its stable value M′ lies in

the kernel of all the maps M→ iGQ(M/nk ·M). But then M′ lies in the kernel of the map

M→ lim←k

iGQ(M/nk ·M).

Hence M′ = 0.

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NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS …people.math.harvard.edu/~gaitsgde/RepRealRed_2017/Notes.pdf · NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS MATH 224, SPRING