I.3 - Cuspidal components

Summary

A_M-finite functions

Let M be a standard Levi subgroup of G.

Remark

Let Hom(A_M,) be the set of characters of A_M, i.e. of continuous homomorphisms of A_M into. The restriction map of M to A_M defines a map

(1) We can use this to prove surjectivity by giving a description of Hom(A_M) analogous to the description of X_M (see I.1.4). As A_MM¹ is of finite index in M, the kernel of (1) is finite. It contains since A_M⊂Z_M. We will call it X_M(A_M). We obtain an isomorphism

Suppose first that k is a number field. Let (A_M) be the enveloping algebra of the (complex) Lie algebra of the real group A_M. We have

Thus (4_M) is identified, by a map which we will denote by z ↦, with the polynomial algebra over the complex space, itself isomorphic to X_M and even to X_M/X_M(A_M), since X_M(A_M) = {0}. Suppose now that k is a function field. Let (A_M) be the convolution algebra of functions with compact support on A_M. We associate with z ∊ (A_M) its Fourier-Mellin transform, which is the function on X_M/X_M(A_M) defined by

Fix a basis (a_i=1,…,n of the ℤ-module Z_M. Then (A_M) can be identified via z ↦ with the space of polynomials in the variables for i = 1,…,n.