Eisenstein series on arithmetic quotients of loop groups

We construct Eisenstein series on arithmetic quotients of loop groups, give a convergence criterion, and compute their constant term in terms of the R iemann zeta function. We also give a description of certain measures which will provide in infinite dimensions, the convolution operators needed to ob tain an analytic continuation. In the last two sections we discuss the ques tion of volumes of arithmetic quotients, and we discuss various generalizat ions of our results.