We give the recursion formula of the Harish-Chandra -function with respect to the highest weight of the representations of . Using this formula, we get the explicit expressions of the Harish-Chandra -functions for and . As an application, by using these expressions, we get the realizations of discrete series representations of as subquotients of nonunitary principal series representations. We shall also get the decompositions of holomorphic and antiholomorphic discrete series when restricted to . By using the structures of -spectra of discrete series representations, we can concretely construct the invariant subspaces of the representation spaces of holomorphic and antiholomorphic discrete series.
Masaaki EGUCHI. Shin KOIZUMI. Masaichi MAMIUDA. "The expressions of the Harish-Chandra -functions of semisimple Lie groups , ." J. Math. Soc. Japan 51 (4) 955 - 985, October, 1999. https://doi.org/10.2969/jmsj/05140955