The second degree cohomology of finite orthogonal groups, II

Abstract

Let V be a finite-dimensional vector space over GF(q), where q is odd. Suppose that dim V= 2m + 1, m = 1,2,..., and that V is equipped with a nondegenerate quadratic form Q. Let R(2m + 1, q)-or just Sa, when m and q are clear from the context--denote the commutator subgroup of the general orthogonal group of this quadratic form; that is, l2(2m + 1, q) is a Chevalley group of type B(m, q). In this paper we compute the second degree cohomology of n on its standard module V. In the smallest case (dim V= 3), H*(fi, V) is zero for q = 3, and it is of dimension 1 over GF(q), otherwise [lo]. Hence we will assume from now on that dim Va 5. Our main result is that, under these assumptions, H*(J2, V) is always zero, except for the group n(7,3), where we obtain an upper bound of 1 for the dimension of the second degree cohomology group over GF(3). The group H*(0(7, 3), V) is already known to be nonzero [5], so that it constitutes the only exception. The general outline of the proof is to calculate H*(G, V), where G is the maximal parabolic subgroup of R which is the stabilizer of a singular point. Since the restriction map from n to G induces an injection on the cohomology level, H*(R, V) will be trivial provided that H*(G, V) is. Except for the cases of q = 3, dim V= 5, 7 or 9, it is fairly straightforward to obtain the result that H’(G, V) is zero, by studying certain exact cohomology sequences obtained from the E,-terms of the Hochschild-Serre spectral sequence. In the two cases of q = 3, dim V= 5 or 9, where H*(G, V) is nonzero, we obtain explicit formulas for the nontrivial cocycle classes on the 3Sylow subgroup of the orthogonal group which is contained in the maximal parabolic subgroup G. We then show that these cocycle classes are not stable ones, hence in these two cases, H*(R, V) must be trivial again. 88 0021-8693/80/110088-22$02.00/O