Intertwining Algebras and Affine Hecke Algebras for Finite Central Extensions of Classical $p$-adic Groups with Application to Metaplectic Groups

Abstract

For a finite central extension $\tilde{G}$ of a classical $p$-adic reductive group, we consider the endomorphism algebra of some induced projective generator \`a la Bernstein of the category of smooth representations of $\tilde{G}$. In the case where the Levi subgroups decompose, we compute this algebra to get a result similar to the one previously obtained by the first author for classical $p$-adic groups, showing that this intertwining algebra is a twisted semi-direct product of an affine Hecke algebra with parameters by a twisted finite group algebra. We discuss also the case with non-decomposed Levi subgroups. We then give an application to the category of genuine representations of a $p$-adic metaplectic group. Using results of C. M\oe glin relative to the Howe correspondence, we show that the Bernstein components of these groups are equivalent to tensor products of categories of unipotent representations of classical groups. This generalizes a previous result of the first author. It implies an equivalence of categories between the category of genuine representations of the $p$-adic metaplectic group and the direct sums of those of smooth representations of the corresponding split odd special orthogonal group and its pure inner form.