Unramified representations of p-adic groups on spherical varieties
The description of irreducible representations of a group G can be seen as a question in harmonic analysis; namely, decomposing a suitable space of functions on G into irreducibles for the action of G x G by left and right multiplication. For a split p-adic reductive group G over a local non-archimedean field, unramified irreducible smooth representations are in bijection with semisimple conjugacy classes in the "Langlands dual" group. In the first part of this work, we generalize this description to an arbitrary spherical variety X of G as follows: Irreducible quotients of the "unramified" Bernstein component of C°°(X) (or, more generally, CC°(X, LT), where L,y is a G-linear line bundle over X) are in natural almost bijection with (a number of copies of) the quotient of a complex torus by a finite reflection group (the "little Weyl group" of X, if LT is trivial). This leads to a weak analog of results of D. Gaitsgory and D. Nadler on the Hecke module of unramified vectors, and an understanding of the phenomenon that representations "distinguished" by certain subgroups are functorial lifts. In the course of the proof, rationality properties of spherical varieties are examined and a new interpretation is given for the action, defined by F. Knop, of the Weyl group on the set of Borel orbits. In the second part of this work, we examine the dual problem of embedding an irreducible unramified representation 7r into the space C°°(X, LT); the goal here is to compute an explicit formula for the image of the unramified vector. We develop a variant of the Casselman-Shalika method, extending work of Y. Hironaka, in order to compute such a formula for the Shalika model of GLn, which is known to distinguish lifts from odd orthogonal groups.