Kuznetsov, Petersson and Weyl on GL(3), II: The generalized principal series forms

Abstract

This paper develops the study by analytic methods of the generalized principal series Maass forms on GL(3). These forms occur as an infinite sequence of one-parameter families in the two-parameter spectrum of GL(3) Maass forms, analogous to the relationship between the holomorphic modular forms and the spherical Maass cusp forms on GL(2). We develop a Kuznetsov trace formula attached to these forms at each weight and use it to prove an arithmetically-weighted Weyl law, demonstrating the existence of forms which are not self-dual. Previously, the only full level, generalized principal series forms that were known to exist on GL(3) were the self-dual forms arising from symmetric-squares of GL(2) forms. The Kuznetsov formula developed here should take the place of the GL(2) Petersson trace formula for theorems “in the weight aspect”. As before, the construction involves evaluating the Archimedian local zeta integral for the Rankin–Selberg convolution and proving a form of Kontorovich–Lebedev inversion.