國立清華大學 數學系

Abstract:

Let g be a complex semisimple Lie algebra. Let F be a finite-dimensional g-module with weights μ_1,..., μ_k, and X be an arbitrary g-module with infinitesimal character χ_λ. Kostant proved that an infinitesimal character which occurs in the tensor product of X⊗F is necessarily of the form χ_{λ+μ_i} (i=1,…,k). However, it is a difficult question whether a nonzero submodule with infinitesimal character χ_{λ+μ_i} indeed occurs in X⊗F. Assume X is a Harish-Chandra module with infinitesimal character χ_λ. We prove a criterion when a nonzero submodule with infinitesimal character χ_{λ+μ_i } occurs in X⊗F by using Dirac cohomology.