Compositio Mathematica

Research Article

On the unramified spectrum of spherical varieties over p-adic fields

Yiannis Sakellaridisa1

a1 School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel (email:


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 G×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. We generalize this description to an arbitrary spherical variety X of G as follows. Irreducible unramified quotients of the space $C_c^\infty (X)$ are in natural ‘almost bijection’ with a number of copies of AX*/WX, the quotient of a complex torus by the ‘little Weyl group’ of X. 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.

(Received June 26 2006)

(Accepted December 24 2007)

2000 Mathematics subject classification

  • primary 22E50;
  • secondary 11F85;
  • 14M17


  • distinguished representations;
  • spherical varieties