representations in characteristiq - CiteSeerX

Proceedings of Symposia in Pure Mathematics Volume 37, 1980

REPRESENTATIONS

IN CHARACTERISTIQ

LEONARD L. SCOTT’

We will be discussing here topics in three related areas: maximal subgroups, irreducible representations, and group cohomology. 1. Maximal subgroups. I will assume in this section that the finite simple groups can be classified and even that their irreducible representations in characteristicp can someday be determined. What I wish to demonstrate is that this will carry us a long way toward the determination of their maximal subgroups. It should not come as a total surprise that such a problem is tractible, since the corresponding question for complex Lie algebras and connected Lie groups was solved some time ago by E. B. Dynkin [8]. Let us review Dynkin’s solution. First of all, Dynkin treated the exceptional types separately, and I surely expect that the same will be necessary in the finite group case, requiring internal classification theory type arguments rather than representation theory. This reduced Dynkin essentially to studying the maximal subgroups of SL(,, C), O(n, C), and Sp(n, C). Next, he reduced to the case of an irreducible subgroup by simply noting that any irreducible subspace has an obvious stabilizer (when there is a form around, the irreducibility forces the subspace to be either nonsingular or totally isotropic). This had the pleasant by-product for Dynkin of also reducing to the case of a semisimple subgroup, since any connected abelian normal subgroup would have to act by scalar multiplications; in the finite groups case the maximal local subgroups associated with primes distinct from the characteristic would still have to be determined. Next Dynkin reduced the problem from the semisimple to the simple’ case (i.e., the problem of finding all maximal connected subgroups which were irreducible and simple) by using the tensor product decomposition for irreducible representations of a direct product. Any product of two or more terms would 1980 Mathematics Subject Classification. Primary 2OGO5; Secondary 20B15,2OB35,2OC20, 2OC30, 2OG10,2OJO6,2OG40, 18699, 14F05, 14L17. ‘Supported by the National Science Foundation. 2Read as “quasisimple”. Dynkin’s “simple” groups can have finite centers. In the finite case also one needs irreducible representations for central extensions as well as for the simple groups themselves. 0 American Mathematical Society1980 319

320

L. L. SCOTT

have to be contained in SL(s) x X.(t) with s + I = n and 1 < s < n; the analogous subgroup for Sp(n) is Sp(s) x O(t) and for O(n) there are two possibilities: O(s) x O(t) and Sp(s) X Sp(t). This reduction goes through in spirit for the finite case, though the list of possibilities is larger-e.g., X(s) wr S,, 3s = n. (Incidentally, Dynkin checks as he goes along that each group in his list of obvious possibilities is indeed maximal.) Also one reduces not to a quasisimple group but to an automorphism group containing it. Finally in the simple case Dynkin has available a complete classification of the simple groups in his category together with a complete determination of their irreducible representations. He also has available for each irreducible representation the knowledge of whether such a representation has a symmetric or alternating bilinear invariant, or none at all (denoted + 1, -1, or 0 in his table). This settles the question as to whether our subgroup can be put in O(n) or Sp(n) or neither, which up until now had been hanging. The analogous information will certainly be required in the finite case (even more because of the two types of quadratic forms) and will just have to be considered part of the representation theory problem. The corresponding information for hermitian forms is also required, though this would at least follow from complete knowledge of the characters. Having come this far, Dynkin now decides, in a brilliant application of the law of excluded middle, to decide which of his (theoretically classified) irreducible simple groups is not maximal (in O(n), Sp(n), or SL(n)). This makes it possible to list at least some version of the results on a single page (see the table at the end of this paper). There are just four infinite families and fourteen individual exceptions. More detailed information on the exceptions is available in longer tables [8] I have not given here. For another exposition of Dynkin’s work, see Tits [%I. Now let us assume that we have adapted Dynkin’s program to successfully find all maximal subgroups of the finite simple groups of Lie type and their containing automorphism groups, and somehow manage to treat the same problem for the finitely many sporadic groups. The question arises now as to what we do with the symmetric and alternating groups, perhaps the most interesting case of all. The answer, happily, is that we may almost be done at this point! I have listed the general forms for possible maximal subgroups in an appendix. As one might expect, the only undetermined possibilities (assuming a classification of the finite simple groups) involve a primitive embedding of an automorphism group of a simple group. If said simple group is not an altemating group of lower degree, then we would know all of its primitive permutation representations at this point. Thus we would be in a good position to inductively determine all possible exceptions to maximality, completing our classification. 2. Irreducible representations. We will be discussing here representations of groups of Lie type in their natural characteristic p. Little has been done with representations in fields of characteristic I #p, though Alperin has suggested that this theory should parallel the modular theory of the symmetric groups, which has a substantial literature [23]. In characteristic p the irreducible representations all are restrictions of rational (polynomial) representations of the ambient simply connected algebraic

REPRESENTATIONS IN CHARACTERISTIC p

321

group G over the algebraic closure k of GF(p) (Steinberg). To keep an example in mind, the group G is SL(n, k) if the original group was SL(,, p’), This is quite a serious class of examples, with the irreducible representations (or even their degrees) of SL(5,p) still unknown. (To my knowledge no one in the field has actually sat down and tried directly to construct representations. It would be a good problem (suggested originally by M. O’Nan) to try and find a general recipe for the irreducible representations of SL(,, 2) simply from a combinatorial point of view. Something like Young diagrams is what I have in mind.) Very recently (just a few weeks before this conference) George Lusztig made the first serious conjecture regarding the characters of the irreducible representations of G. To give the “character” of a representation of an algebraic group is to describe it on a maximal torus (the diagonal matrices in SL(n, k), in that example). The representation on such a torus T is a direct sum of l-dimensional representations or “weights”; from this point of view the “roots” are the representations associated with the root groups, positive roots corresponding to root groups in a fixed Bore1 subgroup B, e.g.,

)/I

in

((.

The Weyl group W = No(T)/ T obviously acts on the weights and permutes the roots. Each irreducible representation has a unique highest weight with respect to the ordering A > p iff A - lo is a sum of positive roots. The abelian group of weights carries a natural positive definite symmetric bilinear form ( , ) and the high weight h of each irreducible representation is “dominant” in the sense that (A, C.X)> 0 for each positive root (Y.In this way the dominant weights completely parameterize or label the irreducible representations of G, even though little more is known about them. To describe Lusztig’s conjecture and its context we shall go back first to the classical theory of Weyl and Kostant which successfully describes the character of the irreducible module VA with high weight A when k is replaced by the complex numbers. First of all, let p be any weight of T and regard p as a l-dimensional representation of B. The Lie algebra b of B correspondingly acts on CL,and we may consider the corresponding induced representation ZP = 91(e) Bwt,) p for the Lie algebra g of G. Here Q(g) denotes the universal enveloping algebra. The module ZP is also a rational T- (and even B-) module and is freely generated by 1 C3 p over a(n), the universal enveloping algebra for the group U- generated by the negative root groups. In the example this group is

1 I1 *

1

*

*

1

Because of this, the multiplicity of any T-weight v in ZP is just p(v - CL)= the number of ways v - p can be written as a sum of positive roots. The module Z,, is called the Vet-ma module associated with CL.The formulas of Weyl and Kostant

322

describe

L. L. SCOTT

V,, as a simple alternating

sum

[ VA]= 2,

(- l)““‘[ GA]

in the Grothendieck group of T-modules and thus give its character. Here w + A = w(A + p) - p where p is half the sum of the positive roots, and (- 1)““’ is the determinant of w in its action on the weights ( expressed in terms of the length function I(w) of reflection group theory). The proof of this result in Humphreys’ book [lo] shows quite clearly the importance of showing that the only Z,, in the same “block” as VA are of the form Z,,,, (Ha&h-Chandra)? (Each Z, is indecomposable, and even has a unique irreducible quotient module.) This gets one to the point where there must be at least some expression [VA1= ~.wEWc(w)[Z,,+]. Because of this, Humphreys and Verma pushed a corresponding block-theoretic investigation in the characteristic p case using the affine Weyl group W, (the semi-direct product of W withpZ@, the latter acting by translation on the weights, so that W, + A = We h + pZ@; here Z@ denotes the root lattice). Recently [l] H. H. Andersen (sharpening results of Humphreys, KacWeisfeiler, and Jantzen) has proved that the high weights of any two irreducible representations of G in the same block must also belong to the same orbit under the affine Weyl group W,.” This implies that there must be at least some in characteristic p expression Z wE w c( w)[ Z,+,.,] for the irreducible representation (the simplest wa$ to make sense of this in characteristic p is to think of the character of Z,, in terms of the partition function p we discussed earlier). Essentially Lusztig’s conjecture gives the coefficients c(w) for the most important weights when the prime p is large relative to the root system. I have given a precise statement in an appendix. Implicit in its philosophy are results of J. C. Jantzen [12] which allow one to obtain formulas for all weights from just a few well placed ones, also parameterized by elements of Wp. Lusztig’s conjecture is analogous to an earlier conjecture [15] of Kazhdan and Lusztig in characteristic 0 regarding irreducible modules which are the quotients of Verma modules associated with nondominant weights. The main ingredients for the characteristic 0 (resp., characteristicp) conjecture are certain polynomials P ,,,,,, defined for each pair of elements w, w’ of the Weyl group (resp., affine Weyl group), the values of these polynomials at 1 giving the requisite doubly parameterized system of coefficients (the various c(w)‘s above). The polynomials P w,w,were apparently first found in the representation theory of generic Hecke algebras, arising naturally in lifting Springer’s Weyl group representations to these algebras. Since this conference took place, they have been shown to be Poincart polynomials for a new geometric cohomology theory of Goreski,

3T~o indecomposable modules are in the same block if they can be joined by a chain of indecomposable modules with a nonzero homomorphism (either direction) between successive terms. ‘Recently S. Donkin has completed the determination of the blocks [Zs]. They are described as orbits of WP or its analogues for higher powers of p, depending on the power (plus one) of p dividing h + p.

REPRESENTATIONS IN CHARACTERISTICp

323

MacPherson, and Deligne, applied to Schubert varieties. (See Lusztig’s article in these PROCEEDINGS, where he also notes that the characteristic p conjecture implies the characteristic 0 conjecture, through a translation principle of JantZen.) Previously a Poincare polynomial interpretation in terms of group cohomology had been given by David Vogan (cf. [2] and 83), but with coefficients involving the unknown irreducible modules, and assuming a conjecture he says is equivalent. In spite of these drawbacks Vogan’s interpretation does formally imply the characteristic 0 conjecture, and conceivably could be instrumental in its proof. I have described a characteristicp analogue in $3. The theory is too young to say definitely where proofs might come from, so I will just survey some of the other approaches that have emerged so far. One really untried possibility I have already mentioned to you: directly construct representations. Another major avenue of attack is to decompose known nonirreducible representations. The main possibilities here are the Weyl modules, which may be described by a suitable reduction modp from a characteristic 0 module (k 8 z (Q u + in terms of the Kostant Z-form) and a number of results in this direction have been obtained by Jantzen. Nowadays these modules may be described as the duals of certain O-dimensional cohomology groups (H’(B, -X @ R(G)) or H’(G/B, C(-X)), where R(G) is the affine coordinate ring of G and C( -A) is the line bundle on G/B associated with A), and H. Andersen has already demonstrated the usefulness of considering the higher dimensional cohomology groups. (They are used heavily in the proof of his result on blocks cited above.) Another simple description of the Weyl module associated with X is as the universal module with high weight A, the dual of the -A) in the sense of algebraic groups. “induced” module - AJG = Morph,(G, The equivalence of all these definitions depends on the vanishing theorem first proved by George Kempf [16] and more recently by Andersen [2] and Haboush [27]. In any event the known results on the structure of these modules (as opposed to just knowing their composition factors) are very meager, the only complete results being for type A, (the group SL(2, k)), due to Carter and Cline, cf. [24]. (Carter and Cline actually give the lattice structure, though in general one might be content with well-understood filtrations.) To give the reader some appreciation of where the structure theory of these modules is today, I mention the following open problem: Let G be SL(3, k) and V its standard 3-dimensional module, with V* the dual. Describe the structure of the tensor product Sm( V) C3 Sn( V*) of symmetric powers for all integers m, n > 0. The problem

is open even with a reasonable

bound

on m and n (say m + n + 2

A. Consequently we can form a submodule Zi of an indecomposable injective Q consisting of all “sections” - plG with p < A. Next one shows that dim Extk( - hlG, Ii)

= dim Extk(Soc(

-hlG),

Ii) = multiplicity

of -hlG

5Au&d in proof. Recently S. Donkin has also found this filtration. His work will appear in Math. Z.


325

as a section of Q. One shows also this is the characteristic 0 multiplicity, which is in turn the multiplicity of the irreducible socle of Q in - AIG as a composition factor (Green). Now it might be possible to inductively determine Extb( - AIG, Z.J, using the cocycles from one answer to construct the “next largest” Zi as a maximal essential extension. 3. Cohomology. I will try to give a brief overview of some problems close to my own interests and the theory of $2. For algebraic group cohomology, it is important to study the structure of the induced modules -XIG (or equivalently their duals, the Weyl modules) and to study how the injectives are built from these (ef. the addenda to $2). The vanishing result on Exth(L,, -hlG) mentioned in the addenda is actually valid for all Ext”, n > 1, yielding a powerful dimension shift for computing algebraic group cohomology when the structure of the relevant Weyl modules is known. Results of Cline, Parshall, Scott, van der Kallen [22] indicate how to compute finite Chevalley group cohomology in terms of algebraic group cohomology for large fields. Recently Bill Dwyer, using “split buildings” constructed by Ruth Charney, has obtained some extremely promising stability results with respect to the rank of the group [7]. At the moment, his results are stated for GL,, and modules related to the standard module, but they should easily generalize. I leave the most definitive formulation as an open problem. Though there is still work to be done,6 it is now likely that entire families of finite group cohomology problems can be reduced to a finite number of cases by general methods. Turning to David Vogan’s work on the Kazhdan-Lusztig conjecture in characteristic 0, we can express his Poincare polynomial interpretation as follows, in terms of algebraic group cohomology: P,,,(q)

= x

qi

dim Ext’$“)-‘cy)-2i(

-y

. h, L( - we A))

(*)

i>O

where y, w E W and A is any dominant weight. Here we agree that any Ext group of negative degree is 0. The starting point of Vogan’s investigation in [21] seems to be the observation that (*) implies the characteristic 0 conjecture through the application of an Euler characteristic formula. The analogue of the latter in characteristicp is [L(-we A)]

= zh

T

(-l)“dimExti(-y.X,L(-w.A))[

V-,.x].

Here X is again any dominant weight and L( - w * A) is the irreducibile module with high weight - w * h, but w, y come from Wp, and we assume - we h, -y . A are both dominant. The formula is easily proved by appealing to the fact [22] that - plG 8 - v is B-acyclic for p, v dominant and replacing L( - we A) by an induced module. 6E.g., the stability results of [22] need to be treated for the twisted groups and one needs better stability theorems for growth of the characteristic p. The latter problem at least reduces, using methods of [22], to algebraic group cohomology for a module twisted by the Frobenius endomorphism. Added in proof. The stability results of [22] have now been treated for twisted groups by G. Avrunin, Trans. Amer. Math. Sot. (to appear).

326

L. L. SCOTT

PROPOSITION. Assume X is in the “bottom alcove” C, and - w * X is a dominant C, (see the Appendix; these are just the weight in the “bottom p2-alcove” hypotheses of the Lusztig conjecture). Zf (*) hoI& for ally E Wp, then so does the formula for [L _ ,++,Iconjectured by Lusztig.

This follows easily by just comparing coefficients. It would be interesting to know if other of Vogan’s results have analogues in characteristicp. It would also be interesting to have some general calculations of B-cohomology. As far as I know, complete results on H”(B, CL)with p an arbitrary weight do not exist even for n = 2. Such calculations would also be extremely helpful for specific computations in the finite Chevalley group case mentioned earlier. To complete this exposition, I would like to come back once more to maximal subgroups of finite groups. It is a theory of Bob Griess that interesting or “sporadic” 1-cocycles should lead to interesting subgroups by considering the elements of the group on which the cocycle is zero (the stabilizer of a vector in the usual extension module corresponding to the cocycle). This is supported by a number of theoretical results [19], [%I], but no one has yet made an attempt to A good starting point would be systematically look at examples. H’(SU(n, 2’) A3V) where V is the standard module, which Wayne Jones has shown to be l-dimensional for n > 7.7 The same cohomology for larger fields is 0, so that in some sense these cohomology groups are all sporadic. (The stable behavior with respect to the rank is an instance of what one should be able to prove by generalizing Dwyer’s results.) For some recent cohomology calculations, see [14] and [4]. I would like to thank H. Andersen, J. Humphreys and G. Lusztig for several conversations and my colleagues Ed Cline and Brian Parshall for numerous contributions to this lecture. Appendix: Statement of the Lwztig Conjecture (adapted from a lecture by H. Andersen). Let W be any Coxeter group and S its set of simple reflections. Let < denote the usual partial order on S in which y < w iffy has some reduced expression which is a subsequence of a reduced expression for w (well-defined, cf., Bourbaki). Let Z(w) denote the length of a reduced expression for w. We are going to inductively define some polynomials P,,,, in q for y, w E W.* We will have P,,,, = 0 unless y < w, and that the degree in q of PY,” is at most i(l(w) - Z(y) - 1). Set ~(y, w) equal to the coefficient in PY,, of this largest possible degree ~(Z(w) - Z(y) - 1) w h en the latter is a nonnegative integer. Define P,,, = 1 and PY,w = 0 if y 4 w. If y < w and P,,+, has been defined for ally with smaller w, choose s with ws < w. Put

‘The sporadic behavior actually occurs already for n = 6, where the same cohomology group is 24imensional. One regards V as n-dimensional, so that dim A’V = 20 in this case. s1 am indebted to Roger Carter for catching an error in my original description of P,,,,Y and apologize if any inaccuracies remain.


327

DEFINITION. P,,,(q)

= 4’-cp,,,ws

-q

2

+

4=py,ws

p(z, ws)q(‘(ws)-‘(“)-‘)/2p,,=.

y3

3 q2)

0

%)2

n22

3 (n+3) 4

0

(B Zn+l ) k

nzl k>l

%)k

k>l

k

o---o--o...-

In

(k+y-z)

II s=l

2k+S s(

CT& 6 0

(-I)@+1

(k+s k

)

k+4) 4

I

7

1

w3

189

1

$1

128

1

(C3)1

90

1

tc3)2

350

(4)

560

0

@)a)1

495

1

@a)2

4928

0

E6)l

351

0

@6)2

17550

0

@7)1

1539

1

(E7)2

27664

@7)3

365

@I)4

3

%)

I

1

“Reprinted from E. B. Dynkin, Maximal Transl. Ser. 2 6 (1957), p. 364.

)k

-1

-1 750

792

0%

1 -1

subgroups of the ckssical groqs, Amer. Math. See.


331

BIBLIOGRAPHY 1. H. H. Andersen, The strong linkageprinciple, J. Reine Angew. Math. 315 (1980), 53-59. The Frobenius morphism on the cohomologv of homogeneous actor burudles on G/B, preprint (Aarhus). 3. J. Ballard, Injectiw modules for restricted mueloping algebrcls, Math. Z. 163 (1978), 57-63. 4. G. Bell, On the cohomology of the finite special linear grotqs. I, II, J. Algebra 54 (1978), 216-238,239-259. 5. E. Cline, B. Parshall and L. Scott, Cohomology, hyperalgebrar and representations, J. Algebra 63 (1980), 93-123. 6. -, Induced modules and extensions of representations. I, II, Invent. Math. 47 (1978), 41-51; J. London Math. Sot. (2) 20 (1979), 403-414. 7. W. Dwyer, Twisted homological stability for general linear groups,, Ann. of Math. 111 (1980), 239-251. 8. E. B. Dynkin, The maximal subgroups of the classical groups, Amer. Math. Sot. Transl. Ser. 2 6 (1957), 245-378. 9. J. A. Green, Local&finite representations, J. Algebra 41 (1976), 137-171. 10. J. E. Hum&eys, Introduction to Lie algebras and rqwesentation theory, Grad. Texts in Math., vol. 9, Springer-Verlag, Berlin and New York, 1972. 11 . -7 OraVnaty ana’ modular representations of Chew&y groups, Lecture Notes in Math., vol. 528, Springer-Verlag, Berlin and New York, 1976. 12. J. C. Jantzen, 2% Charakterformel gewisser Darstelhmgen halbeinfacher Gruppen und Lie-Algebren, Math. Z. 140 (1974), 127-149. 13. -, Darstelhmgen halbeinfacher Gryppen und ihren Frobenius Keme, preprint (Bonn). 14. W. Jones and B. Parshall, On the I-cohomology of finite groqs of Lie type, Proc. Ccrnf. on Finite Groups (Park City, 1975), Academic Press, New York, 1976. 15. D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184. 16. G. Kempf, Linear systems on homogeneous spaces, Ann. of Math. (2) 103 (1976), 557-591. The Grothendieck-Cousin complex of an induced representation, Advances in Math. 29 (l&)~b6. 18. V. Lakshmibai, C. Musili and C. S. Seshadri, Geometry of G/B, Bull. Amer. Math. Sot. (N.S.) 1(1979), 432-433. 19. L. Scott, Permutation modules and I-co/aomology, Arch. Math. 27 (1976), 362-368. Matrices andcohomology, Ann. of Math. (2) 105 (1977), 473-492. 20. -, 21. D. Vogan, The Kazhaan-Lwztig conjectures, preprint (MIT). 22. E. Cline, B. Parshall, L. Scott and W. van der Kallen, Rational and generic cohomology, Invent. Math. 39 (1977), 143-163. 23. G. James, The representation theory of the ymmetric grotq, Lecture Notes in Math., vol. 682, Springer-Verlag, Berlin and New York, 1978. 24. E. Cline, A second look at the Weyl modules for S&, preprint (Clark). 25. R. Carter and G. Lusztig, Modular representations of finite groups of Lie type, Proc. London Math. Sot. 32 (1976), 347-384. 26. J. Tits, Sous-algebres des algebres de Lie semi-simple, Sdminaire Bourbaki 1954/1955, expod 119, Paris, 1959. 27. W. Haboush, A short proof of the Kempf vanishing theorem, Invent. Math. 56 (1980), 109-l 12. 28. S. Donkin, The blocks of a semisimple algebraic groqn, preprint (Warwick). 29. J. Towber, Young symmetry, the flag manifold, and representations of GL(n), J. Algebra 61 (1979), 414-462.

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