Zeta functions and their asymptotic expansions for compact symmetric spaces of rank one

Abstract

We assemble the basic facts required to discuss c~-functions of compact symmetric spaces from the representation-theoretic viewpoint, in principle, everything here in w 1 is contained in Garth Warner's book [6], and we refer to Warner [6] and Helgason [4] for the original sources (of which Caftan [3] is the principal one). Fix a compact riemannian symmetric space M and let G be the largest connected group of isometries. Thus G is a compact connected Lie group with an involutive automorphism a, and M = G/K where K is an open subgroup of G~= {geG :a(g) =g}, and the riemannian metric on M derives from a positive definite invariant bilinear form on the Lie algebra of G. (~ denotes the set of all equivalence classes In] of irreducible unitary representations n of G. Given In], V~ denotes the (finite dimensional complex Hilbert) space on which n represents G. A class [n]e(~ is of class 1 relative to K if there exists