The expressions of the Harish-Chandra C-functions of semisimple Lie groups Spin(n,1), SU(n,1)

Abstract

We give the recursion formula of the Harish-Chandra $C$-function with respect to the highest weight of the representations of $K$. Using this formula, we get the explicit expressions of the Harish-Chandra $C$-functions for $\mathrm{Spin}(n,1)$ and $SU(n,1)$. As an application, by using these expressions, we get the realizations of discrete series representations of $SU(n,1)$ as subquotients of nonunitary principal series representations. We shall also get the decompositions of holomorphic and antiholomorphic discrete series when restricted to $U(n-1,1)$. By using the structures of $K$-spectra of discrete series representations, we can concretely construct the invariant subspaces of the representation spaces of holomorphic and antiholomorphic discrete series.