The local zeta function for symmetric spaces of the non-compact type, con T. Godoy y F. Williams, Journal of Geometry and Physics, V. 61, 125-136, 2011

GODOY TOMÁS, MIATELLO ROBERTO JORGE, WILLIAMS, FLOYD

JOURNAL OF GEOMETRY AND PHYSICS

ELSEVIER SCIENCE BV

Año: 2011 vol. 61 p. 125 - 136

0393-0440

 The Mellin transform of the heat kernel on a non-compact symmetric space X gives rise to a zeta function  $\zeta(s;x,b)$    that has been studied when the rank of X is $1$. In this case the special values  of this 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 $\Gamma\ X$, where $\Gamma$ is a discrete group of isometries of  X .  Also the special value of $\zeta(s;x,b)$  at $s=-\frac{1}{2}$ determines the Casimir energy of such an space-time.   In this paper we extend the study of $\zeta(s;x,b)$ to any symmetric space X of arbitrary real rank.  In one of our main results   we show that for general X and for $x\neq \overline{1}$,    $\zeta(s;x,b)$ admits a continuation to an entire function. On the other hand, we show that, under a mild condition, for $ x = \bar 1$, $\zeta(s;\bar 1,b)$ 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  (for all values of x).  We compute special values of the zeta function and of its derivative at $s=0$ and at $s=-\frac{1}{2}$, 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.