## Journal of the Mathematical Society of Japan

### 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 $Spin(n, 1)$ 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.

#### Article information

Source
J. Math. Soc. Japan, Volume 51, Number 4 (1999), 955-985.

Dates
First available in Project Euclid: 10 June 2008

https://projecteuclid.org/euclid.jmsj/1213107830

Digital Object Identifier
doi:10.2969/jmsj/05140955

Mathematical Reviews number (MathSciNet)
MR1705256

Zentralblatt MATH identifier
0937.22007

#### Citation

EGUCHI, Masaaki; KOIZUMI, Shin; MAMIUDA, Masaichi. The expressions of the Harish-Chandra $C$ -functions of semisimple Lie groups $Spin(n, 1)$ , $SU(n,1)$. J. Math. Soc. Japan 51 (1999), no. 4, 955--985. doi:10.2969/jmsj/05140955. https://projecteuclid.org/euclid.jmsj/1213107830