Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-09T08:32:59.882Z Has data issue: false hasContentIssue false

Formal Dimension for Semisimple Symmetric Spaces

Published online by Cambridge University Press:  04 December 2007

Bernard Krötz
Affiliation:
The Ohio State University, Department of Mathematics, 231 West 18th Avenue, Columbus, OH 43202, U.S.A. E-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

If G is a semisimple Lie group and (π, $\cal H$) an irreducible unitary representation of G with square integrable matrix coefficients, then there exists a number d(π) such that ($\forall$v, v′, w, w′ ∈ $\cal H$) 1/d(π) 〈v, v′〉 〈w′, w〉 = ∫G 〈π(g).v,w$\overline$ 〈π(g).v′.w′〉 dμG(g). The constant d(π) is called the formal dimension of (π, $\cal H$) and was computed by Harish-Chandra in [HC56, 66]. If now H\G is a semisimple symmetric space and (π, $\cal H$) an irreducible H-spherical unitary (π, $\cal H$) belonging to the holomorphic discrete series of H\G, then one can define a formal dimension d(π) in an analogous manner. In this paper we compute d(π) for these classes of representations.

Type
Research Article
Copyright
© 2001 Kluwer Academic Publishers