Green Functions Associated to Complex Reflection Groups, Ii

Green functions associated to complex reflection groups G(e, 1, n) were discussed in the author’s previous paper. In this paper, we consider the case of complex reflection groups W = G(e, p, n). Schur functions and Hall-Littlewood functions associated to W are introduced, and Green functions are described as the transition matrix between those two symmetric functions. Furthermore, it is shown that these Green functions are determined by means of Green functions associated to various G(e, 1, n). Our result involves, as a special case, a combinatorial approach to the Green functions of type Dn. 0. Introduction This paper is a continuation of [S]. In [S], Hall-Littlewood functions associated to the complex reflection group G(e, 1, n) were introduced. Green functions associated to G(e, 1, n) are defined as a solution of a certain matrix equation arising from the combinatorics of e-symbols. It was shown that such Green functions are obtained as coeffcients of the expansion of Schur functions in terms of Hall-Littlewood functions. In the case where e = 2, G(e, 1, n) coincides with the Weyl group of type Bn, and the Green function in that case coincides with the Green function associated to finite classical groups Sp2n(Fq) or SO2n+1(Fq) introduced by Deligne-Lusztig, in a geometric way. So our resut is regarded as a first step towards the combinatorial description of such Green functions, just as in the case of Green polynomials of GLn(Fq). In this paper, we take up the complex reflection group G(e, p, n), and show that a similar formalism as in the case of G(e, 1, n) works also for such groups. In the case of G(e, p, n), symmetric functions such as Schur functions, Hall-Littlewood functions, etc. appear as p-tuples of similar functions associated to the various complex reflection groups G(e, 1, n). In particular, Green functions associated to G(e, p, n) can be described in terms of Green functions associated to G(e, 1, n). In the case where e = p = 2, the group G(e, p, n) is equal to the Weyl group of type Dn. In this case our Green functions coincide with the Green functions associated to the finite groups SO2n(Fq) of split type or non-split type. So our result implies, in this case, that the Green functions of type Dn can be described completely in terms of various “Green functions” of type Bn′. However, note that the Green function of type Bn′ appearing in this context is not the Green function associated to SO2n′+1. They are the functions introduced in [S], associated to different type of symbols. 1