The local zeta function for symmetric spaces of non-compact type

Abstract

The Mellin transform of the heat kernel on a non-compact symmetric space X gives rise to a zeta function ζ ( s ; x , b ) that was studied when the rank of X was 1 . In this case the special values of the zeta function and of its derivative at s = 0 , for example, are relevant for the quantum field effective potential in space-times modelled on X , or especially on a compact locally symmetric quotient Γ ∖ X , where Γ is a discrete group of isometries of X . Also the special value of ζ ( s ; x , b ) at s = − 1 2 determines the Casimir energy of such a space-time. In this paper we extend the study of ζ ( s ; x , b ) to any symmetric space X of arbitrary real rank. One of our main results is Theorem 2.1, where we show that for general X and for x ≠ 1 ¯ , ζ ( s ; x , b ) admits a continuation to an entire function. On the other hand, we show that under a mild condition, for x = 1 ̄ , ζ ( s ; 1 ̄ , b ) has a meromorphic continuation to C with at most simple poles, all lying in the set of half-integers. In case G is complex, we give a very explicit form of the meromorphic continuation and we compute special values of the zeta function and of its derivative at s = 0 and at s = − 1 2 , which give a local contribution to the Casimir energy of X . To illustrate the difficulties present in the general case, we work out explicitly the meromorphic continuation for two infinite families of higher rank groups.