A motivic formalism in representation theoryshanekelly/MSJSep19.pdf · If you know more representation theory than me: 1 G is a connected reductive linear algebraic group, 2 B is

A motivic formalism in representation

theory

Shane KELLY

(joint with Jens Niklas EBERHARDT)

Slides: www.math.titech.ac.jp/∼ shanekelly/MSJSep19.pdf

20th Sep.2019, Kanazawa

Shane KELLY (joint with Jens Niklas EBERHARDT) Slides: www.math.titech.ac.jp/∼ shanekelly/MSJSep19.pdfA motivic formalism in representation theory

Overview

1 General representation theory background

2 The modular category O

3 Geometry

4 Motives

5 A toy application


An apology.


If you know more representation theory than me:

1 G is a connected reductive linear algebraic group,

2 B is a Borel subgroup,

3 T is a split maximal torus.

We always work over an algebraically closed field k = k , butG ,B ,T are defined over Z. Sometimes G (C) and G (Fp)appear in the same equation.

If you know less (or equal) representation theory than me:

1 G = SLn = {n × n matricies M s.t. detM = 1},2 B = upper triangular matricies of SLn,

3 T = diagonal matricies of SLn.


We want to study (algebraic) representations of G :

ρ : G × V → G

V a finite dimensional k-vector space,ρ(g ,−) : V → V is linear ∀ g ∈ G ,ρ(g1g2, v) = ρ(g1, ρ(g2, v)),algebraic: ρ is defined using polynomials.

Example

V = {f (x , y) = αxn + βxn−1y + · · ·+ ωyn : α, β, . . . , ω ∈ k}= homogeneous polynomials in x , y of degree n,G = SL2 acts by

[acbd ], f (x , y) 7→ f (dx − by , ay − cx)


Def. A character is a group homomorphism λ : T → k∗.

Example

Given a character λ : T → k∗, extend to B → k∗, defineO(λ) = sheaf of λ-invariant functions,

∇(λ) = Γ(G/B ,O(λ))

with the action induced by G acting on G/B .

Note: the previous slide was the case G = SL2, and

λ : [a00a−1] 7→ an

of this construction.

Definition

The ∇(λ) are called costandard representations.


Strategy: If there is a subspace W ⊆ V such thatρ(g ,W ) ⊆ W for all g ∈ G , then ρ : G×V→V is built fromG×W→W and G×(V /W )→(V /W ).

Definition

ρ is irreducible if there is no nonzero proper subrepresentations.

Theorem (Jordan-Holder)

Every representation can be built from a unique (up toreordering) tuple of irreducible representations.

Facts:1 If char.k = 0, then irreducible ⇐⇒ costandard.2 If char.k = p > 0, then irreducible 666⇐⇒ costandard.3 In general, ∀ costandard representation ∇(λ) ∃!

irreducible subrepresentation L(λ) ⊆ ∇(λ) (← definitionof L(λ)).

4 All irreducible representations are of the form L(λ).


Example

Recall,

ρn : SL2×{

homogeneousdegree n polynomials

}→{

homogeneousdegree n poly.

}[ac

bd ], f (x , y) 7→ f (dx − by , ay − cx)

Because (u + v)p = up + vp the vector space

{αxp + ωyp : α, ω ∈ k}

is fixed by SL2.So the costandard representation ρp is not irreducible (butρ0, ρ1, . . . , ρp−1 are irreducible).


Question

Given two characters λ, µ : T → k∗, how many copies of L(λ)are used to build ∇(µ)?

Definition

[∇(ν) : L(λ)] is the number of copies of L(λ) used to build∇(µ).


The modular category OConsider RepG = the category of all representations.

Problems:1 Infinitely many irreducible representations.2 No projective objects.

Replace RepG with: the modular category O.

Nice properties of O:1 Finite set of irreducibles {L(λx)}x∈W .2 Irreducibles are canonically indexed by the Weyl group

W = NGT/T . (If G = SLn, then W ∼= Symn).3 O is an abelian category with enough projectives.4 There is a “smallest” projective cover P(λ)→ L(λ) of

each irreducible L(λ) (← definition of P(λ))Shane KELLY (joint with Jens Niklas EBERHARDT) Slides: www.math.titech.ac.jp/∼ shanekelly/MSJSep19.pdfA motivic formalism in representation theory

Super sketchy definition for non-experts:

Start with two sets

∅ ⊆ N ⊆ A ⊆ { characters T → k∗}.

Objects of O = the set A of representations that can be builtout of L(λ) for λ ∈ A.

Morphisms: Designed to formally kill any representations thatcan be built out of L(λ) for λ ∈ N .


More detailed definition for experts:

-Omitted because I’m underqualified.-


Red = A; Green = A \ N .


Geometry

Question: Is there a “geometric description” of O?

Consider G/B . It has a canonical decomposition

G/B = ∪x∈WBxB/B

with BxB/B ∼= Anx for some nx ∈ Z.

Example

If G = SLn, then

SLn/B ∼= Fln = {0 ( V1 ( · · · ( Vn−1 ( kn}

[aij ] 7→ 0 ( 〈a•1〉 ( 〈a•1, a•2〉 ( 〈a•1, a•2, a•3〉 ( · · · ( kn

SLn/B = ∪x∈SymnBxB/B is the costandard decomposition ofFln.


Even more specifically,

SL2/B ∼= P1 = {∞} ∪ A1

SL3/B ∼= FL2 = A0 ∪ A1 ∪ A1 ∪ A2 ∪ A2 ∪ A3


What do we mean by “geometric representation of O”?

Given a stratified variety X = ∪s∈SXs , consider sheaves ofF -vector spaces on X which are constant on each strata.(sheaf of F -vector spaces “=” set of vector spacescontinuously parametrised by X )


On P1 = A1 ∪ {∗}, (stratified sheaf) ! (φ : V0 → V1).V0 = vector space at {∗},V1 = vector space at points of A1,φ = information about how V1 deforms to V0.

More generally, (stratified sheaf on X = ∪s∈SXs , s.t.π1(Xs) = {1}, e.g., Xs

∼= Ans ) !

Vs ; s ∈ S

φst : Vs → Vt whenever Xt ⊂ Xs .

Compatibility condition.


More generally, we can consider the bounded derived categoryof sheaves on X = ∪s∈SXs . For s ∈ S , we have intersectioncomplexes ICs .

Theorem (Soergel)

⊕i

homDb((G/B)(C),C)(ICx , ICy [i ]) ∼= homO0(P(x),P(y))

(G/B)(C) = complex variety associated to G/B .

Db((G/B)(C),C) bounded derived category of complexesof sheaves of C-vector spaces on (G/B)(C).

O0 = complex Lie algebra inspiration for O.

P(x),P(y) complex Lie algebra versions of P(µ).

Remark

The above isomorphism is used in Soergel’s proof of theKazhdan-Lusztig conjecture.


Replacing intersection complexes ICx with parity sheaves[Soergel, Juteau-Mautner-Williamson], we can get an Fp-linearversion:⊕

i

homDb((G/B)(C),Fp)(Ex ,Ey [i ]) ∼= homO(P(µx),P(µy ))

Here, O is the modular category O from above, and µx , µ arethe characters corresponding to x , y ∈ W .

Cannot replace (G/B)(C) on the left with (G/B)(Fp) because

H1et(An

Fp,Fp) 6= 0.

However, motivic cohomology gives the correct groups

HnM(An

Fp,Fp) =

{Fp n = 00 n 6= 0


Motives

Idea (Grothendieck 1960’s): there are many cohomologytheories in algebraic geometry. {

vector spaces +action of Gal(ksep/k)

}{Varieties}

&&

etale 11singular //

crystalline --

{vector spaces +Hodge structure

}{vector spaces +Frobenius linearautomorphism

}{Motives}

33

66

::

Motives should be a “universal” cohomology theory, thatcontains information about all the other ones.

Notice: targets are vector spaces + some structure.


Motives are defined by a universal property.

Like tensor product of modules.

Sometimes one can give a concrete representative of a motive,but its often better to work with them abstractly.

Like tensor product of modules.

However, for conceptual purposes:a motive is a vector space equipped with some extra structure(what structure exactly depends on the setting).Cf. Tannakian formalism.


Thanks to work of Ayoub, Cisinski-Deglise, Morel, Voevodsky,. . . , there is a good theory of relative motives (think: motivescontinuously parametrised by some base) with functorsf!, f

!, f ∗, f∗,⊗, hom.

Definition

Let H(S ,Fp) be the triangulated category of relative motivesover a variety S , with Fp-coefficients.

Remark

In [Eberhardt-K.] we construct H(S ,Fp) using relativelyelementary methods. No algebraic cycles (no intersectiontheory / moving lemmas), no model categories, no infinitycategories. Our description is possible thanks to a theorem of[Geisser-Levine] about Milnor K -theory.


Definition

A motive M ∈ H(Fp,Fp) is mixed Tate if it can be built fromthe cohomology of projective spaces (by extension, directsummand, direct sum, internal hom).

A relative motive M ∈ H(X ,Fp) is mixed Tate if it isconstant. I.e., of the form f ∗M ′ for some mixed Tate motiveM ′ ∈ H(Fp,Fp), where f : X → Fp.

Let X = ∪s∈SXs be a stratified space, is : Xs → X . A motiveM ∈ H(X ,Fp) is stratified mixed Tate if i∗s M ∈ H(Xs ,Fp) ismixed Tate, for all s ∈ S .

MTDerS(X ) := the category of stratified mixed Tate motives.


Example

Observation: Since H∗M(An,Fp) = Fp,MTDer(An) ∼= Db(Vec .SpZ)← graded vector spaces.

Theorem (Eberhardt-K.)

Let G be a semisimple simply connected split algebraic groupover Fp and G∨ the Langlands dual group. Then there is anequivalence of categories

MTDer(B∨)(G∨/B∨) ∼= Db(O2Z(G ))

between the category of stratified mixed Tate motives onG∨/B∨ and the derived evenly graded modular categoryO2Z(G ).


A toy application

For B ⊆ P ⊆ SLn a parabolic subgroup, consider

ΠP : MTDer(SLn/B)π∗→ MTDer(SLn/P)

π∗→ MTDer(SLn/B)

For any stratum is : Xs→X we have the skyscraper motiveis!Fp.

Given “nice” characters µ, ν, using ΠP and is!Fp for various Pand s, one can construct a relative motive Pµ such that for anappropriate stratum Xν we have

dim i∗νPµ = [∇(ν) : L(µ)].

(= number of copies of L(µ) used to build ∇(µ)).


Thank-you.


Documents

A motivic formalism in representation theoryshanekelly/MSJSep19.pdf · If you know more representation theory than me: 1 G is a connected reductive linear algebraic group, 2 B is