The asymptotic expansion of the zeta-function of a compact semisimple Lie group

Abstract

If G G is a connected, simply connected, semisimple Lie group with metric given by the negative of the Killing form and zeta-function Z ( t ) Z(t) , then \[ Z ( t ) = Vol ⁡ G ( 4 π t ) dim ⁡ G / 2 exp ⁡ | δ | 2 t + exponentially small error as t ↓ 0. Z(t) = \frac {{\operatorname {Vol} G}} {{{{(4\pi t)}^{\dim G/2}}}}\exp |\delta {|^2}t + {\text {exponentially small error as }}t \downarrow 0. \]