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)

Masaaki EGUCHI, Shin KOIZUMI, and Masaichi MAMIUDA

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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

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

First available in Project Euclid: 10 June 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22E45: Representations of Lie and linear algebraic groups over real fields: analytic methods {For the purely algebraic theory, see 20G05}
Secondary: 22E30: Analysis on real and complex Lie groups [See also 33C80, 43-XX]

Harish-Chandra $C$-function infinitesimal operator discrete series Cohn's corgecture


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.

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