On the Schwartz space of the basic affine space

摘要

Let G be the group of points of a split reductive algebraic group over alocal field k and let X=G/U where U is a maximal unipotent subgroup of G. Inthis paper we construct certain canonical G-invariant space S(X) (called theSchwartz space of X) of functions on X, which is an extension of the space ofsmooth compactly supported functions on X. We show that the space of allelements of S(X) invariant under the Iwahori subgroup of G coincides with spacegenerated by the elements of the so called periodic Lusztig's basis, introducedrecently by G.Lusztig. We also give an interpretation of this space in terms ofcertain equivariant K-group (this was also done by G.Lusztig). Finally wepresent a global analogue of S(X) which allows us to give a somewhatuntraditional treatment of the theory of principal Eisenstein series.