Crystals via the affine Grassmannian

Let G be a connected reductive group over C, and let g(boolean OR) be the L anglands dual Lie algebra. Crystals for g(boolean OR) are combinatorial obj ects that were introduced by M. Kashiwara (cf., e.g., [6]) as certain "comb inatorial skeletons" of finite-dimensional representations of g(boolean OR) . For every dominant weight lambda of g(boolean OR) Kashiwara constructed a crystal B(lambda) by considering the corresponding finite-dimensional repr esentation of the quantum group Uq(g(boolean OR)) and then specializing it to q = 0. Other (independent) constructions of B(lambda) were given by G. L usztig (cf. [9]) using the combinatorics of root systems and by P. Littelma nn (cf. [7]) using the "Littlemann path model." It was also shown in [5] th at the family of crystals B(lambda) is unique if certain reasonable conditi ons are imposed (cf. Theorem 1.1). The purpose of this paper is to give another (rather simple) construction o f the crystals B(lambda) using the geometry of the affine Grassmannian G(G) = G(K)/G(O) of the group G, where K = C((t)) is the field of Laurent power series and O = C[[t]] is the ring of Taylor series. We then check that the family B(lambda) satisfies the conditions of the uniqueness theorem from [ 5], which shows that our crystals coincide with those constructed in the re ferences above. It sould be interesting to find these isomorphisms directly (cf., however, [10]).