Some new examples in 1-cohomology

This paper gives some new examples in the 1-cohomology theory of finite groups of Lie type, obtained from both computer calculations and the use of several theoretical results. In particular, the paper gives the first known examples of 1-cohomology groups of dimension greater than 2 for absolutely irreducible faithful modules of a finite group. The computer calculations were made originally while checking special cases of Lusztig’s conjecture on characteristic p representations of algebraic groups, and we take this opportunity to announce in print some results in that direction. (They reinforce Lusztig’s conjecture, even in a strong form suggested by Kato.)  2003 Published by Elsevier Science (USA). Observed dimensions of 1-cohomology groups for finite groups have been remarkably small. This led Guralnick and Hoffman [GH, Conjecture 2] to make the following conjecture: Conjecture 1. Let V be a finite-dimensional faithful absolutely irreducible module for a finite group G. Then dimH 1(G,V ) 2. This restated an earlier version by Guralnick [G, Conjecture 2], who had conjectured, under the same hypotheses, that “dimH 1(G,V ) , for some fixed (perhaps = 2).” In this paper we present counterexamples to the above = 2 conjecture, in the process exhibiting some methods which could illuminate the original, weaker conjecture that there was some absolute bound. While we would be pleasantly surprised if there was such a E-mail address: [email protected]. 0021-8693/03/$ – see front matter  2003 Published by Elsevier Science (USA). doi:10.1016/S0021-8693(02)00667-1 L.L. Scott / Journal of Algebra 260 (2003) 416–425 417 bound, it is not yet out of the question, and, in any case, we feel there is a lot in Guralnick’s idea that 1-cohomology groups are, generally, remarkably small. One central interest in 1-cohomology groups with irreducible coefficients V arises from their role in maximal subgroup theory [AS,S2], where the coefficients are a finite field F . For example, if F is a prime field, a maximal subgroup M is obtained from any complement to V in the semidirect product V.G (of V with the group G acting on it). Such complements, up to conjugacy, are parameterized by elements of H 1(G,V ). The irreducible module V may be viewed as absolutely irreducible by replacing F with the finite field EndFG(V ), and only a little more information is required to reduce the calculation to the case of a faithful action. Here, the best known general bounds are due to Guralnick and his collaborators [AG,G,GH], using the classification of finite simple groups. These bounds say that the dimension of H 1(G,V ) is at most a constant, currently 2/3, times the dimension of V. In this form, the constant 2/3 is sharp, but could presumably be lowered further by a more asymptotic formulation. However, the true asymptotic form of the growth rate is likely much smaller, and remains a mystery. In lectures on this work, I have been asked what were the first examples where one even had a two-dimensional 1-cohomology group, since so many cases result in a dimension of one or zero. I do not know the very first case, but there is a family of dimension two examples, for finite orthogonal groups of type D2 in characteristic 2, in the 1975 paper [CPS1]. The present examples of larger dimension were found in the context of an entirely different investigation (reported briefly here in a later section), aimed at checking special cases of the Lusztig conjecture on representations in characteristic p. This work led us to incidentally calculate many dimensions of 1-cohomology groups, as coefficients, discussed below, of particular powers of q (always the highest power, when the coefficient in question is nonzero) in certain Kazhdan–Lusztig polynomials. More precisely, the polynomials involved are the parabolic Kazhdan–Lusztig polynomials of Deodhar [D], for the case of an affine Weyl group, with the ordinary Weyl group playing the parabolic subgroup role. These may also be interpreted as certain standard Kazhdan–Lusztig polynomials Py,w(q). In the Deodhar case y and w are the longest element in their respective right cosets of the ordinary Weyl group. For a given such polynomial, the coefficient of interest is that of q(l(w)−l(y)−1)/2, which may be zero, but, when nonzero, is the coefficient of the largest power of q . Should these coefficients tend to infinity with increasing rank, a purely combinatorial issue, the methods of this paper would provide a counterexample to the weaker form of Guralnick’s conjecture. In any case, if some information on the growth rate of these coefficients could be obtained, even for the affine type A case, it would give insight into the growth properties of the above 1-cohomology groups. 1. The theory behind the examples Our first examples were established using the generic cohomology theory of [CPSK]. This is a “defining characteristic” theory for finite groups of Lie type, meaning that the representations involved have the same characteristic as that used to define the groups. The “large prime” version of the Lusztig conjecture, proved by [AJS], is also used in these first examples. Later, after a first version of this paper was written, it was realized 418 L.L. Scott / Journal of Algebra 260 (2003) 416–425 that similar examples could be obtained in cross-characteristic, also called “nondefining characteristic,” using the cross-characteristic generic cohomology theory of [CPS4], and the (much less difficult) “large prime” validity of the James conjecture, due to Geck– Gruber–Hiss, cf. [CPS4]. In either case, however, one must know validity of Lusztig’s characteristic 0 conjecture for quantum groups of type A at a root of unity, known now by either work of Kazhdan– Lusztig [KL1,KL2,KL3,KL4], and Kashiwara and Tanisaki [KT1,KT2] for all types, or in type A by the somewhat more combinatorial “LLT theory,” cf. Ariki’s exposition [Ar2] of his earlier paper [Ar] and Leclerc’s paper [LeT]. The works of Ariki and Leclerc also contain further references, and Leclerc especially cites Varagnolo and Vasserot [VV]. Curiously, both the Kashdan–Lusztig–Kashiwara–Tanisaki approach and the Ariki– Leclerc approach involve affine Lie algebras and perverse sheaves [BBD], though the perverse sheaves are on different spaces. The answers agree [Le], both expressible in terms of Kazhdan–Lusztig polynomials, through work of [LeT]. In each case—defining characteristic or cross-characteristic—the ultimate source of our examples is a coefficient in a Kazhdan–Lusztig polynomial for the affine Weyl group of type Â5. In the defining characteristic case, the underlying finite group is a PSL(6, q), with q a sufficiently large power of a sufficiently large prime p. The precise size requirement on p is unknown. If the original Lusztig conjecture is true, one may take p = 7. The unknown size of p is a feature of the “large prime” Lusztig conjecture theory, not the generic cohomology theory, which gives specific bounds. Similarly, the size of the representation characteristic must be large, with size unknown, in the cross-characteristic case, though q could be a large power of a very small prime, even the prime 2. The crosscharacteristic examples, however, appear to require representations of much larger rank groups, e.g., PGL(66, q). We will give details only in the defining characteristic case and just sketch the approach for cross-characteristic in a final section. Let n be a fixed nonnegative integer and V a fixed finite-dimensional module for a reductive algebraic group G over Fp. Then the n-cohomology groups H( G(q),V ), over the group of Fq -rational points G(q) all have the same Fp-dimension, for all sufficiently large powers q of p [CPSK]. (Parshall and Friedlander [FP, (3.2)] point out one may even take q = p, if p is—specifiably—sufficiently large.) Moreover, [CPS2], this dimension is at least the dimension of the space H( G,V ). The latter space is isomorphic to Ext(∆(0),V ) where ∆(0) is the one-dimensional trivial module, which happens also to be a Weyl module. Thus, if V = L(λ) is irreducible with high weight λ, the dimension of this Ext space has an interpretation in terms of Kazhdan–Lusztig theory [CPS3]. In fact, if λ is a regular weight in the Jantzen region, and the Lusztig conjecture holds for G, the dimension of this Ext space is a coefficient (see below) in a Kazhdan– Lusztig polynomial. (Cf. [S, p. 3], [A, (2.12)]; the result, a relatively easy property of parity conditions implied by the Lusztig conjecture, was first observed by Vogan in a category O context. A similar but more sophisticated calculation of the dimensions of spaces Ext(L(μ),L(λ)), is given in [CPS3, (3.6), (5.3), (3.9.1)]; note that the validity of the Lusztig conjecture is again assumed. The Lusztig conjecture is known by [AJS] to hold for all sufficiently large primes p, depending on the root system, though it is not known how large p must be.) If n = 1, and μ = λ the coefficient involved is that of the highest power of the indeterminant q = t2 (not to be confused with the prime power q). L.L. Scott / Journal of Algebra 260 (2003) 416–425 419 We recall that a dominant weight λ is in the Jantzen region provided (λ + ρ,α∨) p(p−h+ 2) for all positive roots α, where ρ is the sum of all fundamental weights, and h is the Coxeter number, which is n, for G= SL(n. Fq). For our purposes it is sufficient to consider modules L(λ) in the “principal block”—those with indexing weight λ of the form w0w(−ρ) − ρ, with (w0w) = (w0) + (w). These weights are regular if p h. The known results discussed above may be summarized precisely as follows: Theorem 2. (1) If λ is any dominant integral weight and n is any nonnegative integer, then we have dimH ( G,L(λ)) dimHn( G(q),L(λ)) for all sufficiently large q, depending on λ and n. (2) dimH( G,L(λ)) is the coefficient of q (w0w)− (w0)−n)/2 in the Kazhdan–Lusztig polynomial Pw0,w0w(q), if λ = w0w.(−ρ) − ρ is in the Jantzen region, and p is large enough for the Lusztig conjecture to hold for G. Deodhar [D] has shown that the Kazhdan–Lusztig polynomials required above can be computed using more tractable “parabolic” Kazhdan–Lusztig polynomials. Using Deodhar’s approach and natural recursions, Chris McDowell, a student in the University of Virginia Research Experience for Undergraduates (REU) program (and supported also by NSF), wrote a computer program in the summer of 1998 to compute the polynomials Pw0y,w0w for SL(n, Fp), n 6, with lengths additive in the expressions w0y,w0w as above. McDowell’s results are available on my web page http://www.math.virginia.edu/~lls2l. We will quote from these results below. An independent program, of Anders Buch and Niels Lauritzen, which uses similar Kazhdan–Lusztig polynomial recursions, is available at http://home.imf.au.dk/abuch/dynkin/index.html, but it has apparently not yet been tested for the SL(6, Fp) case which leads to the counterexamples. 2. The examples in defining characteristic The underlying finite groups will be of the form SL(6, q), with q a power of a prime p 7.More accurately, we will use PSL(6, q), but first we will initially apply the theory in the SL case. Momentarily, we will assume p is large enough so that the Lusztig conjecture holds for this group, but it is convenient for notation to allow p 7. (Of course all such primes are “large enough,” if the original Lusztig conjecture is true.) The Kazhdan–Lusztig polynomials Pw0y,w0w, will be denoted also by the notation Pμ,λ when λ=w0w(−ρ)− ρ =w0w.(−2ρ), and μ=w0y.(−2ρ) are regular dominant (the weights λ,μ will change with p), and we write (λ) = (w). Again, the notation (λ) depends on p, or more precisely, the p-alcove to which λ belongs. Also, this “length,” as given here in terms of w, depends on our convention for fundamental reflection generators below, but it is also expressible in terms of alcove geometry (as the number of affine hyperplanes in the geometry separating λ from the weight 0). 420 L.L. Scott / Journal of Algebra 260 (2003) 416–425 The above conventions have been chosen partly to agree with the notation used by McDowell, whom we quote below, in his computer calculations. However, there is another issue we should discuss regarding the notation for Kazhdan–Lusztig polynomials, especially as they relate to the Lusztig conjecture. Lusztig, in stating his conjecture [L], wrote dominant weights in the form −w(ρ)− ρ, for some element w of the affine Weyl group. This weight may also be written w̃(−ρ) − ρ = w̃.(−2ρ), where w → w̃ is the automorphism fixing each ordinary Weyl group element, but taking a translation to its negative. The twist by the automorphism is often ignored in the literature—incorrectly, in some cases. (I am grateful to Jens Jantzen for alerting me to this issue, and explaining that such an inaccuracy occurs in his book [J, p. 294], copied in [CPS3, (5.2)] and in other papers.) If we want to ignore the twist, we can, by changing the generating set of fundamental reflections: View the generating reflections of the affine Weyl group, commonly viewed as reflections in hyperplanes containing the walls of a dominant alcove [J,CPS3], as instead occurring in hyperplanes containing the walls of an anti-dominant alcove. This changes a Kazhdan–Lusztig polynomial Pỹ,w̃ into Py,w , and vice-versa. Consequently, when one uses these latter fundamental reflections and adjusts notation, Lusztig’s conjectured character formula [L, (4)] reads, for p h and w.–2ρ in the Jantzen region,