# zbMATH — the first resource for mathematics

Relative spherical functions on $$\wp$$-adic symmetric spaces (three cases). (English) Zbl 1059.22019
From the author’s 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 $$g \mapsto g^*$$, let $$K$$ be a maximal compact of $$G$$. $$G$$ acts on the space $$\left\{x \in G\,| \,x=x^*\right\}$$ by $$g \cdot x=g\,x\,g^*$$. 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 $$F$$.

##### MSC:
 22E50 Representations of Lie and linear algebraic groups over local fields 43A85 Harmonic analysis on homogeneous spaces 43A90 Harmonic analysis and spherical functions 11F70 Representation-theoretic methods; automorphic representations over local and global fields