Hecke algebras with parameters and representations of a classical \(p\)-adic group: preservation of tempered spectrum
Let \(G\) be a reductive \(p\)-adic group. Let \(M\) be a Levi subgroup of \(G\) and \(\sigma\) an irreducible cuspidal representation of \(M\). Fix an irreducible component \(\sigma_1\) of the restriction of \(\sigma\) to \(M^1\) and let \(\pi_1\) denote the parabolically induced representation \(i^G_P(\text{ind}^M_{M^1}(\sigma_1))\). Let \(\mathcal H\) denote the algebra \(\text{End}_G(\pi_1)\). In the Bernstein decomposition of \(Rep(G)\) the factor determined by \((M,\sigma)\) is equivalent to the category of all right \(\mathcal H\)-modules. The algebras \(\text{End}_G(\pi_1)\) have been computed explicitly by Heiermann when \(G\) is orthogonal or symplectic or an inner form of some \(\text{GL}_n\), cf. \textit{V. Heiermann} [Sel. Math., New Ser. 17, No. 3, 713--756 (2011; Zbl 1246.22021)] and [Int. Math. Res. Not. 2010, No. 9, 1607--1623 (2010; Zbl 1195.22012)]. In particular, \(\text{End}_G(\pi_1)\) is a semi-direct product of a Hecke algebra with parameters and the algebra of a finite group. In the present article, the author proves that the equivalence \(V\mapsto \text{Hom}_G(\pi_1,V)\) from \(Rep_{(M,\sigma)}(G)\) to the category of \(\mathcal H\)-modules preserves the tempered spectrum and the square integrable (mod centre of G) representations.
scientific article; zbMATH DE number 6166677
scientific article