SIMPLE p - ADIC GROUPS , II

0.1. For any finite group Γ, a “nonabelian Fourier transform matrix” was introduced in [L1]. This is a square matrix whose rows and columns are indexed by pairs formed by an element of Γ and an irreducible representation of the centralizer of that element (both defined up to conjugation). As shown in [L2], this matrix, which is unitary with square 1, enters (for suitable Γ) in the character formulas for unipotent representations of a finite reductive group. In this paper we extend the definition of the matrix above to the case where Γ is a reductive group over C. We expect that this new matrix (for suitable Γ) relates the characters of unipotent representations of a simple p-adic group and the unipotent almost characters of that simple p-adic group [L6]. The new matrix is defined in §1. The definition depends on some finiteness results established in 1.2. Several examples are given in 1.4-1.6 and 1.12. In §2 a conjectural relation with unipotent characters of p-adic groups is stated. In §3 we consider an example arising from an odd spin group which provides some evidence for the conjecture. Since the group Γ which enters in the conjecture is described in the literature only up to isogeny, we give a more precise description for it (or at least for the derived subgroup of its identity component) in the Appendix. Notation. If G is an affine algebraic group and g ∈ G we denote by gs (resp. gu) the semisimple (resp. unipotent) part of g. Let ZG be the centre of G and let G be the identity component of G. Let Gder be the derived subgroup of G. If g ∈ G and G is a subgroup of G we set ZG′(g) = {x ∈ G; xg = gx}.