On commutator subspaces of GF(pa)SL2(pa)-modules

Abstract

Let p a prime, X = SL 2 (p a ) and K = GF(p a ). For S a Sylow p-subgroup of X, we explicitly describe the structure of the commutator series of S on each of the irreducible KX-modules.Of the finite rank 1 Lie type groups, SL 2 (p a ), the 2-dimensional special linear group over the field of p a elements is, in many respects, pre-eminent. It appears frequently as a subquotient in minimal parabolic subgroups of groups of Lie type in characteristic p and, on a wider canvas, often arises in classification problems. One reason why SL 2 (p a ) is encountered in such problems is that its natural 2-dimensional GF(p a )-module is a failure of factorization module-see for example [1], [3], [4]. Other modules for SL 2 (p a ) are also important by virtue of their appearance as chief factors in the unipotent radical of rank 1 parabolic subgroups. Specifically, we see (Galois twists of) V 1 , V 2 and V 3 (see below for the definitions of the V i ) in the Chevalley groups PSL 3 (p a ), PSp 4 (p a ), and G 2 (p a ) respectively, while V 1 ⊗ V σ 1 (σ 2 = 1) and V 1 ⊗ V σ 1 ⊗ V σ 2 1 (σ 3 = 1) are to be found in the twisted Lie type groups U 4 (p 2a ) and 3 D 4 (p 3a ) respectively. Put X = SL 2 (p a ) where p is a prime and a ∈ N. Let K = GF(p a ) and Aut(K) = σ with σ the standard Frobenius automorphism which, for k ∈ K, maps k to k p . In particular we note that |Aut(K)| = a. Let P = K[x, y] be the polynomial ring in two variables. Then P admits a natural action of X and of course with respect to this action the subspaces V n (n a non-negative integer) consisting of homogeneous polynomials of fixed degree n are KX-submodules of P . For 0 n p − 1, the modules V n are all irreducible and comprise the set F of so-called fundamental modules for KX. According to [2] K is a splitting field for X and we obtain all the irreducible KX-modules from the members of F by using the Steinberg Tensor Product Theorem. Hence an arbitrary irreducible KX-module V is given bywhere each W i ∈ F. In particular, there are p a irreducible KX-modules.Mathematics Subject Classification (2000): 20D08.