On the structure of certain subalgebras of a universal enveloping algebra

Abstract

The representation theory of a semisimple group G, from an algebraic point of view, reduces to determining the finite dimensional representation of the centralizer U k {U^\mathfrak {k}} of the maximal compact subgroup K of G in the universal enveloping algebra U of the Lie algebra g \mathfrak {g} of G. The theory of spherical representations has been determined in this way since by a result of Harish-Chandra U k {U^\mathfrak {k}} modulo a suitable ideal I is isomorphic to the ring of Weyl group W invariants U ( a ) W U{(\mathfrak {a})^W} in a suitable polynomial ring U ( a ) U(\mathfrak {a}) . To deal with the general case one must determine the image of U k {U^\mathfrak {k}} in U ( k ) ⊗ U ( a ) U(\mathfrak {k}) \otimes U(\mathfrak {a}) , where k \mathfrak {k} is the Lie algebra of K. We prove that if W is replaced by the Kunze-Stein intertwining operators W ~ \tilde W then U k {U^\mathfrak {k}} suitably localized and completed is indeed isomorphic to U ( k ) ⊗ U ( a ) W ~ U(\mathfrak {k}) \otimes U{(\mathfrak {a})^{\tilde W}} suitably localized and completed.