Abstract

abstract:

The Kuznetsov and Petersson trace formulae for ${\rm GL}(2)$ forms may collectively be derived from Poincar\'e series in the space of Maass forms with weight. Having already developed the spherical spectral Kuznetsov formula for ${\rm GL}(3)$, the goal of this series of papers is to derive the spectral Kuznetsov formulae for non-spherical Maass forms and use them to produce the corresponding Weyl laws. Aside from general interest in new types of automorphic forms, this is a necessary step in the development of a theory of exponential sums on ${\rm GL}(3)$. We take the opportunity to demonstrate a sort of minimal method for developing Kuznetsov-type formulae, and produce auxillary results in the form of generalizations of Stade's formula and Kontorovich-Lebedev inversion. This first paper is limited to the non-spherical principal series forms as there are some significant technical details associated with the generalized principal series forms, which will be handled in a separate paper. The best analog of this type of form on ${\rm GL}(2)$ is the forms of weight one which sometimes occur on congruence subgroups.

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