A comparison of endomorphism algebras (Q6612167)

In this paper the author considers a Bernstein block \(\mathcal{R}^{[M,\sigma]_G}(G(F))\), a full subcategory of the category of smooth complex representations of \(G(F)\) , where \(F\) is a non-archimedean local field and \(G\) is a connected reductive group, which, in the Bernstein decomposition, corresponds to an inertial equivalence class, say, \(\mathcal{s}=[M,\sigma]_G\), where \(M\) is a Levi subgroup and \(\sigma\) an irreducible supecuspidal representation of \(M\). Many authors have studied a Bernstein block, using different approaches. \N\NOne approach is by using theory of types, and the other is by using a progenerator constructed in the following way. First, one restricts \(\sigma\) to the subgroup \(M^1\), the intersection of all the kernels of all the unramified characters of \(M\). Let \(\sigma_1\) be one irreducible component of that restriction. Let \(P=MU\) be a parabolic subgruop with Levi \(M\). Then the parabolically induced representation \(I_P^G(\mathrm{ind}_{M^1}^M(\sigma))\) is a progenerator of \(\mathcal{R}^{[M,\sigma]_G}(G(F))\). The author recalls the results of Solleveld about the description of the endomorphism algebra of this progenerator, which is isomorphic to the extension of an affine Hecke algebra by a twisted group algebra. \N\NOn the other hand, Morris has constructed the endomorphism algebra of the representation of \(\mathrm{ind}_K^{G(F)}(\rho)\), where \((K,\rho)\) is a depth zero type (also an extension of an affine Hecke algebra by a twisted group algebra). Under some assumptions which gurantee that the Morris's type parametrizes indeed a singleton Bernstein component, the author constructs an explicit isomorphisms between two endomorphism algebras. The paper offers a very comprehensive treatment of the problem, with four appendices whith the backrgound material.