Young Tableaux, Canonical Bases, and the Gindikin-karpelevich Formula

A combinatorial description of the crystal B(∞) for finitedimensional simple Lie algebras in terms of certain Young tableaux was developed by J. Hong and H. Lee. We establish an explicit bijection between these Young tableaux and canonical bases indexed by Lusztig’s parametrization, and obtain a combinatorial rule for expressing the Gindikin-Karpelevich formula as a sum over the set of Young tableaux. 0. Introduction The Gindikin-Karpelevich formula is a p-adic integration formula proved by Langlands in [18]. He named it the Gindikin-Karpelevich formula after a similar formula originally stated by Gindikin and Karpelevich [5] in the case of real reductive groups. The formula also appears in Macdonald’s work [25] on p-adic groups and affine Hecke algebras. Let G be a split semisimple algebraic group over a p-adic field F with ring of integers oF , and suppose the residue field oF /πoF of F has size t, where π is a generator of the unique maximal ideal in oF . Choose a maximal torus T of G contained in a Borel subgroup B with unipotent radical N , and let N− be the opposite group to N . We have B = TN . The group G(F ) has a decomposition G(F ) = B(F )K, where K = G(oF ) is a maximal compact subgroup of G(F ). Fix an unramified character τ : T (F ) −→ C, and define a function f : G(F ) −→ C by f(bk) = (δτ)(b), b ∈ B(F ), k ∈ K, where δ : B(F ) −→ R>0 is the modular character of B and τ is extended to B(F ) to be trivial on N(F ). The function f is called the standard spherical vector corresponding to τ . Let G be the Langlands dual of G with the dual torus T. The set of coroots of G is identified with the set of roots of G and will be denoted by Φ. Received April 9, 2013. 2010 Mathematics Subject Classification. Primary 17B37; Secondary 05E10.