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Covers for self-dual supercuspidal representations of the Siegel Levi subgroup of classical \(p\)-adic groups. (English) Zbl 1133.22010

Let \(F\) be a locally compact non-archimedean field, and let \(G\) be a reductive group over \(F.\) Then the Bernstein decomposition
\[ {\mathfrak R}(G)=\prod_s{\mathfrak R}^s(G) \]
describes the decomposition of the category \(\mathfrak{R}(G)\) of smooth representations of \(G\) in terms of the subcategories \(\mathfrak{R}^s(G)\) of smooth representations with cuspidal support in the given inertial class \(s.\) The Bushnell-Kutzko theory of types proposes to describe \(\mathfrak{R}^s(G)\) (or, more precisely, \(\prod_{s\in A}\mathfrak{R}^s(G)\) over a certain finite set \(A\)) through its equivalence to the category of right modules over convolution \(\rho\)-spherical algebras \(\mathcal{H}(G,\rho).\) Here \(\rho\) is an irreducible representation of a compact subgroup \(J\) of \(G,\) and \((J,\rho)\) is called an \(s\)-type.
In an earlier work C. J. Bushnell and P. C. Kutzko [The admissible dual of \(GL(N)\) via compact open subgroups. Ann. Math. Studies. 129. (Princeton, NJ): Princeton University Press (1993; Zbl 0787.22016)] use a cover of a type for a general reductive group, viewed as a Siegel Levi subgroup for the classical group. The construction of cover has a complication due to the anisotropic part (the authors consider also the non-split quasi-split classical groups), and they use Glauberman’s correspondence, among other ingredients, to achieve it (Theorem 3.14). Another major result of the present paper is a description of the related Hecke algebra of that type. The authors give a description of it as a convolution algebra (Theorem 3.19).

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields

Citations:

Zbl 0787.22016
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