Split covers for certain representations of classical groups

Abstract

Let R(G) denote the category of smooth representations of a p-adic group. Bernstein has constructed an indexing set B(G) such that R(G) decomposes into a direct sum over s 2 B(G) of full subcategories R(G) known as Bernstein subcategories. Bushnell and Kutzko have developed a method to study the representations contained in a given subcategory. One attempts to associate to that subcategory a smooth irreducible representation (⌧,W ) of a compact open subgroup J < G. If the functor V 7! HomJ(W,V ) is an equivalence of categories from R(G) ! H(G, ⌧) mod we call (J, ⌧) a type. Given a Levi subgroup L < G and a type (JL, ⌧L) for a subcategory of representations on L, Bushnell and Kutzko further show that one can construct a type on G that “lies over” (JL, ⌧L) by constructing an object known as a cover. In particular, a cover implements induction of H(L, ⌧L)-modules in a manner compatible with parabolic induction of L-representations. In this thesis I construct a cover for certain representations of the Siegel Levi subgroup of Sp(2k) over an archimedean local field of characteristic zero. In particular, the representations I consider are twisted by highly ramified characters. This compliments work of Bushnell, Goldberg, and Stevens on covers in the self-dual case. My construction is quite concrete, and I also show that the cover I construct has a useful property known as splitness. In fact, I prove a fairly general theorem characterizing when covers are split.