On the analytic continuation of the Minakshisundaram-Pleijel zeta function for compact symmetric spaces of rank one

[摘要] We give two equivalent analytic continuations of the Minakshisundaram-Pleijel zeta function zeta(U/K)(z) for a Riemannian symmetric space of the compact type of rank one U/K. First we prove that zeta(U/K) can be written as zeta(U/K)(z) = e(i pi(z-N/2))V(U/K)zeta(G/K)(z) + F(z), where N = dim U/K, V-U/K is the volume of U/K, zeta(G/K)(z) is the local zeta function for G/K (the noncompact symmetric space dual to U/K), and F(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 on U/K and the Plancherel measure on G/K. The second expression we obtain for zeta(U/K)(z) is in terms of a series of (generalized) Riemann zeta functions zeta(R)(z, q) (cf. Eq. (5.9)). The doubly connected case of real projective spaces is also discussed. (C) 1997 Academic Press.

[发布日期] 1997-10-15