A study of Kostant-Kumar modules via Littelmann paths[HBNI Th190]

Abstract

Let g be a symmetrizable Kac-Moody Lie algebra. For each dominant integral weight λ of g, let V λ denote the corresponding irreducible integrable highest weight g-module and let v λ be a highest weight vector in V λ . Given dominant integral weights λ, μ and an element w of the Weyl group of g, the Kostant-Kumar (KK) module K(λ, w, μ) is the cyclic g-submodule of V λ ⊗ V μ generated by v λ ⊗ v wμ , where v wμ is a nonzero vector in the one-dimensional weight space of weight wμ in V μ . Littelmann has given a path model for the tensor product V λ ⊗ V μ . We give, in the spirit of Littelmann, a path model for Kostant-Kumar modules in terms of Lakshmibai-Seshadri (LS) paths. Littelmann’s path model gives a generalized Littlewood-Richardson rule for decomposing tensor products into irreducibles. An analogous rule for Kostant-Kumar modules was given by Joseph under the hypothesis that the Kac-Moody algebra is symmetric. We extend Joseph result to finite type Lie algebras and use this rule to study Parthasarathy-Ranga Rao-Varadarajan (PRV) components and generalized PRV components in Kostant-Kumar modules. At the end, we discuss Kostant-Kumar modules for the finite dimensional Lie algebras g of type A. In this case, it is well known that the semistandard Young tableaux are very useful to study representations theory. We gave a procedure to associate a permutation w(T ) to semistandard Young tableau T . Permuatation w(T ) corresponds to the right key of T introduced by Lascoux-Schützenberger. It is well known that Littlewood-Richardson (LR) tableaux count multiplicities of irreducible modules in the tensor product. Given a LR tableaux S of type μ, we can easily associate a semi standard Young tableau T of shape μ. We associate a permutation w(S) to LR tableau S, by simply defining w(S) := w(T ). Then Littlewood-Richardson tableaux S such that w(S) ≤ w count multiplicities of irreducible modules in the KK module K(λ, w, μ).