On the Schwartz Space of the Basic Affine Space

Let G be the group of points of a split reductive algebraic group G over a local field k and let X = G/U where U is the group of k-points of a maximal unipotent subgroup of G. In this paper we construct certain canonical G-invariant space S(X) (called the Schwartz space of X) of functions on X , which is an extension of the space of smooth compactly supported functions on X . We show that the space of all elements of S(X), which are invariant under the Iwahori subgroup of G coincides with space generated by the elements of the so called periodic Lusztig’s basis, introduced recently by G. Lusztig (cf. [10] and [11]). We also give an interpretation of this space in terms of certain equivariant K-group (this was also done by G.Lusztig – cf. [12]). Finally we present a global analogue of S(X), which allows us to give a somewhat untraditional treatment of the theory of principal Eisenstein series.