A kernel formula for the action of the Weyl element in the Kirillov model of SL(2,C)SL(2,C)

We give an explicit description of the action of the Weyl element on smooth functions with compact support in the Kirillov model of non-principal series irreducible representations of SL(2,C)SL(2,C) generalizing a result of Motohashi. An important ingredient in the proof of the kernel formula is a new “classical” formula for an integral involving Bessel functions. We also give an analogous formula for GL(2,C)GL(2,C).