标题： 交错代数和仿射赫克代数对于经典$p$-adic 群的有限中心扩张及其在拟幂群中的应用 显示英文标题

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

摘要： 对于一个经典$p$-adic 仿射群的有限中心扩张$\tilde{G}$，我们考虑该群的光滑表示范畴中某个类似 Bernstein 的诱导投射生成元的自同态代数$\tilde{G}$。在 Levi 子群可分解的情况下，我们可以计算这个代数，得到类似于第一位作者之前对经典$p$-adic 群所得的结果，表明这个交织代数是带有参数的仿射 Hecke 代数与一个扭曲的有限群代数的扭曲半直积。我们也讨论了一般情况。接着，我们给出了一个应用，即对$p$-adic 超张量群的真实表示范畴的应用。利用 C. M\oe 关于 Howe 对应的结果，我们证明了这些群的 Bernstein 组件等价于经典群的单奇表示范畴的张量积。这推广了第一位作者之前的结论。它意味着$p$-adic 超张量群的真实表示范畴与相应奇特殊正交群及其纯内形变的光滑表示范畴的直和之间存在范畴等价。 显示英文摘要

摘要： 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 can 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 general case. We give then 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 odd special orthogonal group and its pure inner form.