On the decomposition numbers of the finite general linear groups

Abstract

Let G = GL,,(q), q a prime power, and let r be an odd prime not dividing q. Let s be a semisimple element of G of order prime to r and assume that r divides qdeg(X\) 1 for all elementary divisors A of s. Relating representations of certain Hecke algebras over symmetric groups with those of G, we derive a full classification of all modular irreducible modules in the r-block Bh of G with semisimple part s. The decomposition matrix D of Bs may be partly described in terms of the decomposition matrices of the symmetric groups corresponding to the Hecke algebras above. Moreover D is lower unitriangular. This applies in particular to all r-blocks of G if r divides q 1. Thus, in this case, the r-decomposition matrix of G is lower unitriangular. Introduction. The modular representation theory of finite groups of Lie type defined over a field of characteristic p is naturally divided into the cases of equal characteristic r = p and unequal characteristic r / p. This paper begins the study of decomposition numbers in the unequal characteristic case of the finite general linear groups when r > 2. Let G be the full linear group of degree n over GF(q) and 2 /= r be a prime not dividing q. Let s c G be semisimple and consider the geometric conjugacy class (s)G of irreducible characters corresponding to the G-conjugacy class of s. Using basic properties of the Deligne-Lusztig operators introduced in [7] we will determine a parabolic subgroup P = Ps of G and a cuspidal irreducible character of X of P such that (s)G is just the set of constituents of XG, the induced character. Let { R, R, K } be an r-modular system for G and M a KP-module affording X. Then End KG(M ) is called the Hecke algebra of M (compare [5]) and has been calculated in a more general context by R. B. Howlett and G. I. Lehrer in [12]. It turns out that End KG(M )_ K[W], the Hecke algebra of W, where W = WJ denotes the Weyl group of CG(s) (the centralizer of s in G) (compare [1, 2, 4, 13]). In particular, K[W] has a basis { Tw, Iw c W } such that the structure constants of the multiplication are all contained in R c K. So let R[W] be the R-order in K[W] generated by wTIw EW} and R[W] = R ?R R[W]. Choosing a special RP-lattice S in M, and using the classification of the r-blocks of RG given by P. Fong and B. Srinivasan in Received by the editors July 25, 1983 and, in revised form, October 18, 1983, May 23, 1984 and September 1, 1984. 1980 Mathematics Subject Classification. Primary 20C20; Secondary 20G40. 'This work was supported by the Deutsche Forschungsgemeinschaft. ?1985 American Mathematical Society 0002-9947/85 $1.00 + $.25 per page