Theta functions and arithmetic quotients of loop groups

Abstract

In this paper we observe that isomorphism classes of certain metrized vector bundles over P 1 Z − {0, ∞} can be parameterized by arithmetic quotients of loop groups. We construct an asymptotic version of theta functions, which are defined on these quotients. Then we prove the convergence and extend the theta functions to loop symplectic groups. We interpret them as sections of line bundles over an infinite dimensional torus, discuss the relations with loop Heisenberg groups, and give an asymptotic multiplication formula.