On subfield symmetric spaces over a finite field
Let \(\mathbf G\) be a connected reductive linear algebraic group defined over a finite field \({\mathbf F}_{q'}\). We put \(G = {\mathbf G}({\mathbf F}_ q)\), \(q = q'{}^ 2\). Then the \(q'\)-th power Frobenius map \(\sigma: {\mathbf G}\to {\mathbf G}\) induces on \(G\) an involutory automorphism \(\tau: g\to^ \tau g\) with the fixed point set \(G_ \tau = {\mathbf G}({\mathbf F}_{q'})\). We are concerned with the irreducible representations of the Hecke algebra \(H(G,G_ \tau)\), or, almost equivalently, with the zonal spherical functions on the subfield symmetric space \(G/G_ \tau\). In the present paper, we take up the following problem: (A) Classify the irreducible representations of \(H(G,G_ \tau)\), and determine their dimensions. Since the classification of the irreducible representations of \(G\) is well-understood by works of G. Lusztig, we can reduce problem (A) to the following one: \((\text{A}')\) For each irreducible character \(\chi\) of \(G\) determine the multiplicity \(m_ \tau(\chi) = \langle 1^ G_{G_ \tau i},\chi\rangle\) with which \(\chi\) appears in the induced character \(1^ G_{G_ \tau}\).
connected reductive linear algebraic group
scientific article; zbMATH DE number 91623