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

Abstract

This paper initiates 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 such forms that were known to exist 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.