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