Abstract
We prove that for every Bushnell–Kutzko type that satisfies a certain rigidity assumption, the equivalence of categories between the corresponding Bernstein component and the category of modules for the Hecke algebra of the type induces a bijection between irreducible unitary representations in the two categories. Moreover, we show that every irreducible smooth G-representation contains a rigid type. This is a generalization of the unitarity criterion of Barbasch and Moy for representations with Iwahori fixed vectors.
Highlights
Let F denote a non-Archimedean local field with finite residue field
H is an associative nonunital algebra. As it is well known H acts on every complex G-representation and this induces an equivalence between the category C(G) of smooth G-representations and the category of nondegenerate H-modules
We summarize the necessary results about the Jantzen filtration and the signature character, see [43, §3], [2, §14], [4, §5] for the p-adic analogue
Summary
Let F denote a non-Archimedean local field with finite residue field. Let G be the F-points of a connected reductive algebraic group G defined over F. Let H = H(G) be the Hecke algebra of G, i.e., H(G) is the space of locally constant, compactly supported functions f : G → C endowed with the convolution product with respect with μ. As it is well known H acts on every complex G-representation and this induces an equivalence between the category C(G) of smooth (complex) G-representations and the category of nondegenerate H-modules
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