The description of irreducible representations of a group G can be seen as a problem in harmonic analysis; namely, decomposing a suitable space of functions on G into irreducibles for the action of GG 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. We generalize this description to an arbitrary spherical variety X of G as follows. Irreducible unramified quotients of the space Cc∞M are in natural almost bijection with a number of copies of AX*/WX, the quotient of a complex torus by the little Weyl group ofX. This leads to a description of the Hecke module of unramified vectors (a weak analog of geometric results of Gaitsgory and Nadler), 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 Knop, of the Weyl group on the set of Borel orbits.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Distinguished representations
- Spherical varieties