The Cohen-Macaulay property of the category of $ ({\frak g}, K) $-modules

Abstract

Let \( ({\frak g}, K) \) be a Harish-Chandra pair. In this paper we prove that if P and P' are two projective \( ({\frak g}, K) \)-modules, then Hom(P, P') is a Cohen-Macaulay module over the algebra \( {\cal Z}({\frak g}, K) \) of K-invariant elements in the center of \( U({\frak g}) \). This fact implies that the category of \( ({\frak g}, K) \)-modules is locally equivalent to the category of modules over a Cohen-Macaulay algebra, where by a Cohen-Macaulay algebra we mean an associative algebra that is a free finitely generated module over a polynomial subalgebra of its center.