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Abstract:

Many of the constructions of analytic number theory generalize from the classical GL(2) objects (like modular/Maass forms, Kloosterman sums, and L-functions) to GL(n), and studying these new objects involves special functions on GL(n) (like Bessel functions, zonal spherical functions, and Whittaker functions).  I'll discuss several of these new functions and their roles in constructing Kuznetsov-type trace formulas.

The past few years, I've been on a quest to find integral representations for the Bessel functions occurring in the GL(n) Kuznetsov formula.  We have these for GL(2), GL(3) and now GL(4) and I think I know how to construct them for GL(n).  I'll discuss the spectral families on GL(n) and their corresponding Kuznetsov formulas.  Finally, I'll talk about the integral representations we want for the Bessel functions and how (I hope) they lead to applications like Weyl laws, moments and subconvexity of L-functions.