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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
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