Perverse sheaves on affine flags and langlands dual group

Abstract

This is the first in a series of papers devoted to describing the category of sheaves on the affine flag manifold of a simple algebraic group in terms of the Langlands dual group. In the present paper we provide such a description for categories which are geometric counterparts of a maximal commutative subalgebra in the Iwahori Hecke algebra ℍ; of the anti-spherical module for ℍ; and of the space of Iwahori-invariant Whittaker functions. As a byproduct we obtain some new properties of central sheaves introduced in [G].