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

✍ Scribed by Roberto Camporesi

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 310 KB
• Volume: 214
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1006/jmaa.1997.5588

✦ Synopsis

We give two equivalent analytic continuations of the Minakshisundaram᎐Pleijel Ž . zeta function z for a Riemannian symmetric space of the compact type of U r K rank one UrK. First we prove that can be written as

Ž . function for GrK the noncompact symmetric space dual to UrK , and F z is an Ž Ž . . analytic function which is given explicitly as a contour integral cf. Eq. 4.11 . To Ž . prove the above formula we use a relation first noticed by Vretare between the scalar degeneracies of the Laplacian on UrK and the Plancherel measure on Ž . GrK. The second expression we obtain for z is in terms of a series of U r K Ž . Ž . Ž Ž . . generalized Riemann zeta functions z, q cf. Eq. 5.9 . The doubly connected R case of real projective spaces is also discussed.