A vector-valued version of Kostant's separation of variables theorem (Representation theory and various problems in algebra, analysis, and geometry)

Abstract

Let G be a connected reductive algebraic group defined over mathbb{C} and mathfrak{g} its Lie algebra. Let (pi_{mu}, V_{mu}) be a minuscule representation of G. The space mathcal{P}(mathfrak{g})otimes V_{mu} of V_{mu}-valued polynomials on mathfrak{g} is naturally a module of a commutative algebra mathcal{I}{mu} containing mathcal{P}(mathfrak{g})^{G} (A. A. Kirillov's family algebra). In this report, we define the space mathcal{H}_{mu} of V_{mu}|-valued harmonic polynomials and show the separation of variables formula "mathcal{P}(mathfrak{g})otimes V_{mu}=mathcal{I}_{mu}otimes mathcal{H}_{ mu}" as a generalization of Kostant's well-known result.