Abstract

Let F be a non-archimedean local field with residual field of odd characteristic. Given a reductive group G defined over F, equipped with an involution denoted g↦g^*, let K be a maximal compact of G. G acts on the space by g • x = g xg^*. Let s₀ in G be fixed by the involution and let S = G • s₀ and H = Stab_G. A relative spherical function on S is a K-invariant function on S, which is an eigenfunction of the Hecke algebra of G relative to K. The problem at hand is to classify all such functions, compute them explicitly in terms of Macdonald polynomials and obtain an explicit Plancherel measure. We obtain a complete solution in three cases relevant to the theory of Automorphic Forms. Namely:

Case 1: G = GL, H = GL × GL.

Case 2: G = GL, H = GL.

Case 3: G = GL, H = GL.

E is an unramified quadratic extension of F.

Authors