  Using a combinatorial approach which avoids geometry, this paper studies the
ring structure of K_T(G/B), the T-equivariant K-theory of the (generalized)
flag variety G/B. Here the data is a complex reductive algebraic group (or
symmetrizable Kac-Moody group) G, a Borel subgroup B, and a maximal torus T,
and K_T(G/B) is the Grothendieck group of T-equivariant coherent sheaves on
G/B. We prove "Pieri-Chevalley" formulas for the products of a Schubert class
by a homogeneous line bundle (dominant or anti-dominant) and for products of a
Schubert class by a codimension 1 Schubert class. All of these Pieri-Chevalley
formulas are given in terms of the combinatorics of the Littelmann path model.
We give explicit computations of products of Schubert classes for the rank two
cases and this data allows us to make a "positivity conjecture" generalizing
the theorems of Brion and Graham, which treat the cases K(G/B) and H_T^*(G/B),
respectively.
