Formal Dimension for Semisimple Symmetric Spaces

Abstract

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′〉 dμ_G(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.