Degenerate Whittaker functions for $\Sp_n(\R)$

Abstract

In this paper, we construct Whittaker functions with exponential growth for the degenerate principal series of the symplectic group of genus n induced from the Siegel parabolic subgroup. This is achieved by explicitly constructing a certain Goodman–Wallach operator which yields an intertwining map from the degenerate principal series to the space of Whittaker functions, and by evaluating it on weight- l standard sections. We define a differential operator on such Whittaker functions which can be viewed as generalization of the ξ -operator on harmonic Maass forms for \SL2(\R) .