On the Analytic Continuation of the Minakshisundaram–Pleijel Zeta Function for Compact Symmetric Spaces of Rank One

Abstract

We give two equivalent analytic continuations of the Minakshisundaram–Pleijel zeta function ζU/K(z) for a Riemannian symmetric space of the compact type of rank oneU/K. First we prove that ζU/Kcan be written asζU/K(z)=eiπ(z−N/2)VU/KζG/K(z)+F(z),whereN=dimU/K,VU/Kis the volume ofU/K, ζG/K(z) is the local zeta function forG/K(the noncompact symmetric space dual toU/K), andF(z) is an analytic function which is given explicitly as a contour integral (cf. Eq. (4.11)). To prove the above formula we use a relation (first noticed by Vretare) between the scalar degeneracies of the Laplacian onU/Kand the Plancherel measure onG/K. The second expression we obtain for ζU/K(z) is in terms of a series of (generalized) Riemann zeta functions ζR(z,q) (cf. Eq. (5.9)). The doubly connected case of real projective spaces is also discussed.