Exceptional Isogenies and the Classifying Spaces of Simple Lie Groups

Annals of Mathematics

Exceptional Isogenies and the Classifying Spaces of Simple Lie GroupsAuthor(s): Eric M. FriedlanderSource: Annals of Mathematics, Second Series, Vol. 101, No. 3 (May, 1975), pp. 510-520Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1970938 .

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Exceptional isogenies and the classifying spaces of simple Lie groups

By ERIC M. FRIEDLANDER*

In this paper, we exhibit a homotopy equivalence

(BS02&+ 1)112 - (BSPn.)12 where ( ) denotes localization away from the prime 2. We also provide homotopy equivalences

(BG2)113 - (BG2)113 t (BF4)112 - (BF4)112

for the classifying spaces of the exceptional simple Lie groups G2 and F4 which correspond to non-trivial symmetries of the associated Dynkin diagrams. The same methods that provide these homotopy equivalences also yield another proof of a theorem of C. Wilkerson asserting the existence of homotopy equivalences for any connected reductive complex Lie group G(C),

T: (BG(C))11 --* (BG(C))11v

whose action on rational cohomology is multiplication by the appropriate power of p ([11]).

These constructions establish the existence of the only allowable maps between localizations (away from finitely many primes) of classifying spaces of simple Lie groups of equal rank which are not classifying maps of homo- morphisms ([1]).

Our method is to utilize exceptional isogenies existing between algebraic groups defined over fields;of positive characteristic. The key technical tool is the existence of the delooping (classifying space) of the etale homotopy type of an algebraic group of any characteristic. With the aid of Sullivan's "localized arithmetic square in homotopy theory" ([9]), we prove that the delooping of an isogeny of simple algebraic groups in characteristic p provides a map on the classifying spaces of the associated complex simple Lie groups localized away from p.

We are greatly indebted to J. F. Adams who conjectured the existence of the homotopy equivalences here considered, and who passed on to us the suggestion of J-P. Serre and D. Quillen that these equivalences should relate

* Partially supported by the N.S.F. and the U.S.-France Exchange of Scientists program.



EXCEPTIONAL ISOGENIES 511

to finite characteristic algebraic geometry. Furthermore, we appreciate valuable discussions with Spencer Bloch and Clarence Wilkerson.

1. Purely inseparable isogenies

We begin by recalling the definition of a purely inseparable isogeny of algebraic groups.

Definition 1.1. Let G' and Gk be connected algebraic groups defined over a field. A surjective homomorphism with finite kernel

f: G k- Gk

is said to be a purely inseparable isogeny if the induced map on rational functions fields, f*: K(G,) K(G9), is a purely inseparable field extension

([3, No. 181). If X' - X is a finite, surjective, purely inseparable map of algebraic

varieties defined over a field of characteristic p, then any etale map U' ) X' descends uniquely to an etale map U-s X, with U x x X' isomorphic to U' over X'. This is a consequence of the fact that if p: X -+ X is the total pth

power map (i.e., the "absolute Frobenius") and if U-s X is an etale map, then p*(Us X) = Us X (i.e., U x x X- U is the total pth power map p: Usf U).

The following proposition is now immediate.

PROPOSITION 1.2. Let f: G' - Gk be a purely inseparable isogeny of connected algebraic groups defined over k. Then the induced map on etale homotopy types ([2])

fet: (Gk)et 4 (Gk)et

is an isomorphism.

For any algebraic group Gk over a field k, let W{Gk}ret denote the "rigid etale homotopy type" of the simplicial variety

W{Gk}: ...G G 2 4Gk 4 Spec/c.

As explained in [6], W{Gk}rel is a good model for the classifying space of (Gk)et.

Another construction of classifying spaces which is given in [8] would prob- ably work equally well for our purposes.

An easy consequence of the definition of W{Gk}ret is the following.

COROLLARY 1.3. Let f: G' - Gk be a purely inseparable isogeny of connected algebraic groups. Then the induced map

f: W{Gk}ret > W{Gk}ret

is an isomorphism of pro-objects of bi-simplicial sets.



512 ERIC FRIEDLANDER

To relate the homotopy types of the classifying spaces of complex algebraic groups to characteristic p algebraic geometry, we provide the following proposition (cf. [6], [8]). Our proof depends on the existence of a reductive group scheme Gz over Z (to be viewed as a "nice family" of algebraic groups defined over the rationals and over each finite field) such that the rational points of G, constitute a given reductive, complex Lie group ([4]). More details of this proof can be found in [6].

PROPOSITION 1.4. Let G(C) be a reductive Lie group given as the complex points of the complex algebraic group G,. Let Gz be the reductive "Chevalley group scheme" defined over Z with Gz x z C = G. Let p be a prime, let k denote the algebraic closure of the prime field Fp, and let R C be a chosen embedding of the Witt vectors of k into C.

Then there exist natural weak homotopy equivalences

(JW{Gk}ret)" > ( V{GR}ret)" (JW{Gc}ret)" < (BG(C))

where ( )' denotes pro-finite completion away from p. Consequently, the embedding R C determines a map

0: BG(C)jIp ? lim ( WIGk~ret)" 9

with O' a homotopy equivalence.

Proof. We observe that the structure map w: GR Spec R may be factored as

GR ' GRI UR CGR/BR > Spec R

with - proper and smooth, with 8 a geometric fibration whose fibres are tori, and with a a geometric fibration whose fibres are prime-to-p l-aspherical and acyclic ([5]); that is, BR is a maximal solvable algebraic subgroup of GR and UR is the maximal unipotent algebraic subgroup of BR.

For any integer 1 prime to p, R'aZ/1 = 0 for t > 0 and a,,Z/1 = Z/l on GR/UR. Furthermore, RtScZ/1 is locally constant and satisfies base change for all t ? 0; and so R/* (R'l* Z/1) is locally constant and satisfies base change for all s, t > 0. With the aid of Leray spectral sequences, we conclude that R"'7*Z/l is locally constant and satisfies base change. This enables us to conclude that the maps

(Gk)et ?(GR)et < (Gc)et induce isomorphisms in Z/l cohomology, as does the canonical map G(C) (Gc)et. Consequently, a comparison of spectral sequences enables us to conclude that the maps




W{G,} ret > W{GR}ret < W{Gc}ret < BG(C)

induce isomorphisms in Z/l cohomology. Since each of these pro-spaces is simply connected, the Whitehead theorem now implies the proposition.

To relate pro-finite completion away from a prime p to localization away from p, we use D. Sullivan's "interaction" functor which associates to any map of "genotypes" a map of spaces unique up to "visible homotopy" ([10]) (i.e., the equivalence relation between maps f, g: Xlp - Y11p which are homotopic when restricted to finite subcomplexes of X). In more classical Sullivan language, we employ the "localized arithmetic fibre square of homotopy theory" ([9]). In our notation, (X)' denotes a pro-object in the homotopy category of CW complexes and lim (X)' denotes the categorical inverse limit of (X)f.

The uniqueness assertion of Lemma 1.5 is essentially due to C. Wilkerson.

LEMMA 1.5. Let p be a prime, let ( )' denote the pro-finite completion away from p, and let A1,/ = lim (Z)N Q. Let X be a nilpotent space of finite type, let X(0) X ? A,/, denote the formal completion of X(0), and let lim (X)^ X ? A,/, denote the composition of localization at 0 and the natural lim (Z)' homotopy equivalence (lim (X)^)(O) X (0 Allp.

If Xand Y are two nilpotent spaces of finite type, then to give a map $: (X)llp (Y)11p whose pro-finite prime-to-p completion is a given map I',: lim (X)' lim (Y)^, it suffices to fit (J' in a homotopy commutative

diagram

X(o) - X (D Allp ' lim (X)^

(1.5.1) T) 01

Y(0)- ) Y (DA,/p 11m(Y)

Furthermore, $D is a homotopy equivalence if and only if (O, $D', and (D' are homotopy equivalences.

Finally, if G'(C) and G(C) are complex connected, reductive Lie groups, there exists at most one map D: BG'(C)1ll p BG(C)11p inducing a given map (D": lim (BG'(C)) lim (BG(C))^-

Proof. The existence of (D: (X)ll p (Y)llp for (1.5.1) (and uniqueness up to "visible" homotopy) is Sullivan's basic observation about genotypes ([10]). The fact that 1D is a homotopy equivalence if and only if (DO, $D', (D' are homotopy equivalences is a simple consequence of the fact that ($D, (Doy (Do. VI) determines a map of homotopy cartesian squares; one merely applies the




observation that a map of fibre triples of connected spaces which is a weak homotopy equivalence on two of the base, fibre, and total space is a weak homotopy equivalence on the third as well.

To prove the last assertion of the lemma, we recall that BG'(C)0 and BG(C)O are products of K(Q, n)'s. Thus, a map $O: BG'(C)0 - BG(C)O is equivalent to a map $D*: H*(BG(C), Q) - H*(BG'(C), Q) which is determined by the map D'^: lim H*(BG(C), Z/l1) ? Q lim H*(BG'(C), Z/l1) ? Q. More- over, since H%(BG'(C), Z) and wu(BG(C)) are infinite only for i even, H'(S(BG'(C)11p), wi(BG(C)11j)) is finite for all i > 0 (where S(BG'(C),1p) denotes the suspension of BG'(C)11,). Consequently, maps f, g: BG'(C)11, - BG(Q, which are homotopic when restricted to every finite subcomplex of BG'(C) ("visibly homotopic") are homotopic. Thus, if $, $': BG'(C), - BG(C)l, satisfy (VN = $PN, then (D. = $. Sullivan's theory then implies that (D and $' are visibly homotopic, so that (D = V.

We now provide our general theorem asserting the existence of maps of localized classifying spaces arising from maps of algebraic groups of finite characteristic algebraic geometry.

THEOREM 1.6. Let f: G' - Gk be a purely inseparable isogeny of connected reductive algebraic groups over the algebraic closure k of the prime field F,. There exists a homotopy equivalence between the localizations away from p of the classifying spaces of the associated complex Lie groups

(D: BG'(C)l, p > BG (Q~, p

which fits in a homotopy commutative square

BG'(Q11, -( , BG(CQj~p

(1.6.1) 10 {o

lim ( W{Gk}ret)' -* lim ( W{Gk}ret)

where 0 is induced by pro-finite completion and the homotopy equivalences as in Proposition 1.4, and where ( )' denotes pro-finite prime-to-p completion. Moreover, such a (D is canonically determined by a choice of an embedding of the Witt vectors of k into C (which determines 0).

Proof. We first consider the special case of any isogeny of tori. If T' and T are tori of rank r, then

Homag. grps. ( Tk, Tk) Mrxr(Z)

where the isomorphism is determined once isomorphisms




Tk > (GL1,J x and Tk - (GL1 k)xr

are chosen. Consequently, f: Tk -Tk determines f: TR > TR fitting in a commutative diagram of group schemes

(1.6.2) 4 4n Tk - TR < TC

where R denotes the Witt vectors of k. Then the homotopy commutativity of (1.6.1) with $D = B9 follows from (1.6.2) by the naturality of our delooping construction (W{ })ret and the map BT(C) (W{IT})ret.

More generally, let G' and G be connected reductive groups with maximal tori T' and T. A homomorphism f: GI Gk induces f,: Tko Tk; and an identification of Weyl groups

W- = N(Tk((k))/ Tk(k) - N(Tk(1))I Tk(1) = W

with respect to which ft is equivariant. If we identify the Weyl groups of Gk, GR, and GC, then the associated commutative diagram of tori (1.6.2) consists of equivariant maps. In particular, 9t: Tcf - Tc is equivariant under the respective actions of W' and W. Since H*(BG(C), Q) is naturally identified as H*(BT(C), Q)TvY qc determines an isomorphism $D*: H*(BG(C), Q) H*(BG'(C), Q). Since BG'(C)0o, and BG(C)(o, are products of K(Q, n)'s, the map (D* is induced by a unique homotopy equivalence

(o0: BG'(C)(o) BG(C)(o) Let $2: BG'(C) ?z A1,/ BG(C) ?z A,/, denote the homotopy equivalence

induced by (D. via formal completion. By Lemma 1.5, to complete the proof of the theorem it suffices to verify the commutativity of the following diagram

BG'(C)(o) - BG'(C) $& Al/, < lim (BG'(C))- IDo { 1-l.f.0

BG(C)(o) ->BG(C) &i Aqp < lim (BG(Q)).

The left hand square commutes by construction. Since BG(C) ?z A1/, is a product of K(Q, n)'s, we may verify the commutativity of the right hand square by checking the commutativity of the induced square

H*(BG'(C) & A, Q) { H*(lim (BG'(C))f Q)

(T'O'\)* T'?-1 f ?)* IT*/BG(C) A, Q\ ) HT*(lim /B(Q Q)~r\^




This square imbeds (with each corner mapping injectively) in the square

H *(B T'(C) (&z Allp, Q) , H*(lim (BT'(C))^ Q)

H* (BT(C) ($) Aq~, Q) , H*(lim (BT(Q))^ Q)

Finally, the last square commutes because both vertical maps are induced by the map $D,: BT'(C) BT(C); this is true by construction for (PDt)* and by our previous argument for the special case of a map of tori for (6-1 ft)*.

In exhibiting $d: (BG'(C))11l, (B(C)),1, in the proof of Theorem 1.6, we actually proved that PD is compatible with the classifying map of an explicit homomorphism of maximal tori. We emphasize this observation in the following corollary.

COROLLARY 1.7. Let the hypotheses and notation be as in Theorem 1.6. If ft: T, -* Tk denotes the restriction of f to a maximal torus of G', and if It: T'(C) T(C) denotes the lifting of ft then D fits in a homotopy commutative square

BT1(C)_ BT(C)

(1.7.) 1 1 (BGf(Q)j)1/PD (BG(Q))l/P

whose vertical maps are induced by the inclusions

T'(C) - G'(C), T(C) - G(C) .

Moreover, At determines the "same map" between the euclidean root spaces of G'(C) and G(C) as does ft between the euclidean root spaces of G' and Gk.

Proof. By construction of $D and Sullivan's theorem concerning the uniqueness of maps induced by a map of genotypes ([10]), (1.7.1) commutes up to "visible homotopy". By Lemma 1.5, (1.7.1) must then be homotopy commutative.

Using the diagram (1.6.2), we may identify the free abelian groups X(Tk) and X(T(C)) of rational characters of Tk and T(C). This identification is equivariant with respect to respective actions of the Weyl groups and sends a fundamental system of roots of Gk to a fundamental system of roots of G(C). Moreover, this identification respects the inner product structures on the euclidean spaces X(Tk) (? R and X( T(C)) (? R. Under this identification, ft and -a clearly determine the same map,

X(T'(C)) ? R - X(T(C)) R .




2. Applications

In this section, we apply Theorem 1.6 to each of the "exceptional isogenies". These are isogenies which exist between simple algebraic groups defined over the algebraic closure k of some finite field but which do not correspond to an isogeny of associated complex simple Lie groups.

Our first theorem is an immediate application of Theorem 1.6 applied to the complex simple Lie groups SO2,,,1(C) and Sp2,,(C), and stated in terms of their compact forms. This theorem can be viewed as a "delooping" of a theorem of B. Harris ([7]). Since BSO2.+1 and BSpn have very diff erent Z/2Z cohomology, localization away from 2 is essential before one can obtain a homotopy equivalence as given below.

We apologize for the notational confusion involving the symplectic group. We recall that Sp2n (k) ci GL2,(I) preserves a non-degenerate skew-symmetric bilinear form on the vector space k1'2 , whereas SpN c Sp2n (C) is the compact form of Sp2n (C) (which may be viewed as the automorphism group of quaternionic n-space with the quaternionic inner product structure).

THEOREM 2.1. Let k be the algebraic closure of the prime field F2. Let f: SO2.+l,k SP2.,k be the exceptional isogeny determined by restricting an automorphism of the quadratic module (kIc2n+1; xJx"+l + ... + XnX2n + X2n+1) to the non-degenerate symplectic space k 2,,+'/k e2,,+1. Then there exists a homotopy equivalence

02: (B S02n+1l)l/2 (B SPn)1/2


(B S02n+ 1)1/2 >- + (BSp1/2

I I lim (J/V{SO2n+1,k}ret)N -f lim (WV{SP2nk}ret)'

where ( )^ denotes pro-finite prime-to-2 completion. Moreover, D is canonically determined by a choice of embedding Witt (k) = R C (which determines 0).

The following two theorems exhibit self homotopy equivalences of the localized classifying spaces of two of the exceptional simple Lie groups. Since these self equivalences determine non-trivial symmetries of the Dynkin diagrams which do not preserve the weights, they are not the classifying maps of automorphisms of the groups themselves.

THEOREM 2.2. Let G2 denote the exceptional simple complex Lie group




of dimension 14 and rank 2. There exists a homotopy equivalence

AtD: (BG2)113 ) (BG2)113


BT'(G2) > BT(G2)

(BG2)113 > (BG2)113 where 0: T'(G2) T(G2) is an isogeny of maximal tori of G2 sending a short root to a long root and a long root to 3 times a short root.

Proof. Let f: (G2)k - (G2)k be the exceptional isogeny of algebraic groups over k, the algebraic closure of the prime field F3 ([3, No. 21]). Let (D: (BG2)113

(BG2)113 be the map provided by Theorem 1.6; thus $D is compatible with

O-1.f.O: lim (BG2)' - lim (BG2)' and $D,: BT'(G2) - BT(G2),

the classifying map of 0t: T'(G2) T(G2) "lifting" f,: T'(G2)k T(G2)k. Since 0 and f, induce the same map on root spaces by Corollary 1.7, the theorem follows from the fact that f, sends a short root to a long root and a long root to three times a short root.

The same argument applies to the exceptional simple complex Lie group F4, since there exists an exceptional isogeny in characteristic 2, f: (Fj)k (F4)k, where k is the algebraic closure of the prime field with 2 elements. This isogeny has the effect of reflecting the Dynkin diagram of (F4)k, sending short roots to long roots and long roots to 2 times short roots ([3, No. 24]).

THEOREM 2.3. Let F4 denote the exceptional simple complex Lie group of dimension 52 and rank 4. There exists a homotopy equivalence

$D: (BF4)112 > (BF4)112


BT'(F4) > BT(F4)

1$@ 1 (BF4)112 > (BF4)112

where 0: T'(F4) T(F4) is an isogeny of maximal tori of F4 sending short roots to long roots and long roots to 2 times short roots.

The only other exceptional isogenies of simple algebraic groups are powers of the geometric Frobenius, UP1: Gku Gk, where k is a field of characteristic p. We recall that 0zP is defined on the coordinate ring level as the




map of k-algebras

.** *, t .]/(f1, *, fm) > klt, * * f](fl m)

sending each tj to ti (so that UP1 is well-defined provided that each fi has coefficients in Fr).

These geometric Frobenius maps together with Theorem 1.6 give a new proof of a theorem of C. Wilkerson ([11]).

THEOREM 2.4. Let G(C) be any complex, connected, reductive Lie group and let Gk be the corresponding reductive algebraic group over the algebraic closure k of the prime field Fp. Then there exists a homotopy equivalence (determined by a choice of embedding Witt (k) - C)

T: (BG(C))1,/p ,(B(G(-))+p

such that (i) T fits in a homotopy commutative square

(BG(Q)j1/P > (B (Q) j/ p lo0

lim ( V{Gk}ret)' ii lim ( VV{Gk}ret)'

where a Gk G, Gk is the geometric Frobenius and ( )^ denotes pro-finite completion prime-to-p;

(ii)I* =p- H n(B(G(C)), Q) H f(B(G(C)), Q). Proof. Since Gk is obtained by reducing the equations modulo p of an

integral group scheme Gz, we conclude that the geometric Frobenius 5: Gk Gk is well-defined. By Theorem 1.6, there exists a unique homotopy equivalence P: (BG)11,- (BG)11I satisfying (i). Moreover, as constructed in Theorem 1.6, P*: H*(BG(C), Q) - H*(BG(C), Q) is the restriction of the map (B*)*: H*(BT(C), Q) - H*(BT(C), Q) where er: T(C) - T(C) is the product of pth power maps p: C* C*. Since H*(BC*, Q) is a polynomial algebra on a generator in dimension 2 and since p*: H1(C*, Z) H1(C*, Z) is multiplication by p,

p*: H2 (BT(C), Q) - H2 (BT(C), Q) is multiplication by pfl

Finally, we observe that the inverse of the map exhibited in Theorem 2.4 induces a map on prime-to-p pro-finite completions which has a natural algebro-geometric interpretation.

COROLLARY 2.5. Let P: (BG(C))11/p (BG(C))11/ be the homotopy equivalence satisfying (i) and (ii) of Theorem 2.4. Then -1 fits in a homotopy commutative square




W-1 (BG(Q)j1lP> (BG(Q)jllp

lo t?~~~j lim ( W{Gkjret) lim ( W{Gkjret)_

where v: Gk -Gk is the map defined on coordinate rings kit1, *, tn]/(fi, * *, fin) - k1t, *, * t]/(fl, *, fi)

by sending each tj to itself and sending any a in k to ca1.

Proof. Recall that the total pth power map (defined by sending any ele- ment of the coordinate ring to its pth power) is the identity on W{Gk}ret- Thus, the proof is an immediate consequence of the fact APED a = a . OP equals the total pth power map.

PRINCETON UNIVERSITY

REFERENCES

[1] J. F. ADAMS and Z. MAHMUD, to appear. [2] M. ARTIN and B. MAZUR, Etale Homotopy, Lecture notes in mathematics 100, Springer,

1969. [3] C. CHEVALLEY, Seminaire C. Chevalley 1956-1958. [4] M. DEMAZURE and A. GROTHENDIECK, Schemas en Groupes III, Lecture notes in mathe-

matics 153, Springer, 1970. [5] E. FRIEDLANDER, The etale homotopy theory of a geometric fibration, Manuscripta

Mathematica 10 (1973), 209-244. [6] , Computations of K-theories of finite fields, to appear. [7] B. HARRIS, On the homotopy groups of the classical groups, Ann. of Math. 74 (1961),

407-413. [8] R. HOOBLER, Etale homotopy theory of algebraic groups, pre-print. [9] D. SULLIVAN, Geometric Topology, Part I, M.I.T. Notes, 1970. [10] , Genetics of homotopy theory and the Adams conjecture, Ann. of Math. 100

(1974), 1-79. [11] C. WILKERSON, Self-maps of classifying spaces, in Localization in Group Theory and

Homotopy Theory, Lecture notes in mathematics 418, Springer, 1974.

(Received October 15, 1974)



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Exceptional Isogenies and the Classifying Spaces of Simple Lie Groups