Non-spherical Poincaré series, cusp forms and L-functions for GL(3)

Non-spherical Poincaré series, cusp forms and L-functions for GL(3)

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Jack Buttcane, University of Buffalo
IAS Room S-101

The analytic theory of Poincaré series and Maass cusp forms and their L-functions for SL(3,Z) has, so far, been limited to the spherical Maass forms, i.e. elements of a spectral basis for L^2(SL(3,Z)\PSL(3,R)/SO(3,R)).  I will describe the Maass cusp forms of L^2(SL(3,Z)\PSL(3,R)) which are minimal with respect to the action of the Lie algebra and give a (relatively) simple method for constructing Kuznetsov-type trace formulas by considering Fourier coefficients of certain Poincaré series.  In recent work with Valentin Blomer, we have extended our proof of spectral-aspect subconvexity for L-functions of SL(3,Z) Maass forms to the non-spherical case, and I will discuss the structure of that proof, as well.