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.