### Constant term of smooth $H_\psi$-spherical functions on a reductive $p$-adic group

Patrick Delorme
2009 Transactions of the American Mathematical Society
Let ψ be a smooth character of a closed subgroup, H, of a reductive p-adic group G. If P is a parabolic subgroup of G such that P H is open in G, we define the constant term of every smooth function on G which transforms by ψ under the right action of G. The example of mixed models is given: it includes symmetric spaces and Whittaker models. In this case a notion of cuspidal function is defined and studied. It leads to finiteness theorems. Example 1 (Symmetric spaces). The group H is the fixed
more » ... oup H is the fixed point set in G of a rational involution σ defined over F of the group G, P is a σ-parabolic subgroup of G, i.e. P and σ(P ) are opposite, and F is of characteristic different from 2. For the purpose of induction, we do not limit ourselves to ψ trivial. Notice that G itself appears as a symmetric space for G × G, where σ(x, y) = (y, x). This will be referred as "the group case". Example 2 (Whittaker models). The group H is the unipotent radical, U 0 , of a minimal parabolic subgroup P 0 of G, P is a parabolic subgroup of G which contains P 0 , and ψ is a nondegenerate smooth character of U 0 (see Definition 5.1). Example 3 (Mixed models). Let Q be a parabolic subgroup of G, with Levi subgroup L and unipotent radical U Q . Let H be the fixed point set of a rational involution, σ , of L. If σ is not trivial, we assume that F is of characteristic different from 2. We take H = H U Q and ψ is nondegenerate (see Definition 5.1). We assume that P − Q is open and that P − ∩ L is a σ -parabolic subgroup of L. We denote by δ P the modulus function of P . Let V be a smooth G-module, V P its normalized Jacquet module along P . Assuming (1.1) and denoting by M the common Levi subgroup of P and P − , our first result (cf. Theorem 3.4) is the definition of a natural linear map from the space of H ψ -fixed linear forms on V to the space of (M ∩H) ψ P -fixed linear forms on V P , where ψ P is equal to the product of the restrictions to M ∩ H of ψ and δ