  Let G be the group of points of a split reductive algebraic group over a
local field k and let X=G/U where U is 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) 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. We also give an interpretation of this space in terms of
certain equivariant K-group (this was also done by G.Lusztig). 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.
