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 $g\mapsto g^{\ast }$, let $K$ be a maximal compact of $G$. $G$ acts on the space $\left\{x\in G|x=x^{\ast }\right\}$ by $g\cdot x=gx{g}^{\ast }$. Let $s_0\in G$ be fixed by the involution and let $S=G\cdot s_0$ and $H={\text{ Stab}}_G\left(s_0\right)$. 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: $G=GL\left(2n,F\right),H=GL\left(n,F\right)\times GL\left(n,F\right)$.

Case 2: $G=GL\left(m,E\right),H=GL\left(m,F\right)$.

Case 3: $G=GL\left(2n,F\right),H=GL\left(n,E\right)$.

$E$ is an unramified quadratic extension of $F$.

Milestones