BESSEL FUNCTIONS FOR GL2

In classical analytic number theory there are several trace formulas or summation formulas for modular forms that involve integral transformations of test functions against classical Bessel functions. Two prominent such are the Kuznetsov trace formula and the Voronoi summation formula. With the paradigm shift from classical automorphic forms to automorphic representations, one is led to ask whether the Bessel functions that arise in the classical summation formulas have a representation theoretic interpretation. We introduce Bessel functions for representations of GL(2) over a finite field first to develop their formal properties and introduce the idea that the gamma-factor that appears in local functional equations for L-functions should be the Mellin transform of a Bessel function. We then proceed to Bessel functions for representations of GL(2)(R) and explain their occurrence in the Voronoi summation formula from this point of view. We briefly discuss Bessel functions for GL(2) over a p-adic field and the relation between gamma-factors and Bessel functions in that context. We conclude with a brief discussion of Bessel functions for other groups and their application to the question of stability of gamma-factors under highly ramified twists.