Poincaré series for GL(n): Fourier expansion, Kloosterman sums, and algebreo-geometric estimates
In this paper GL(n) Poincaré series are introduced, and their (Piatetski-Shapiro-Shalika) Fourier expansions are computed. These are shown to involve certain exponential sums, which are GL(n) analogues of the classical Kloosterman sum; one such sum arises for each element of the Weyl group W of GL(n). If \(\Gamma =SL(n, {\mathbb{Z}})\) and \(\Gamma _{\infty}\) is the subgroup of upper triangular unipotents of \(\Gamma\), then these sums are given as character sums over a certain quotient space of \(\Gamma _{\infty}\setminus \Gamma\). A coordinatization map is introduced which allows one to realize this space as a subset of the integral points of an affine algebraic variety. This is used to obtain an algebro-geometric estimate for the Kloosterman sum corresponding to \(w=(I_{n-1}^ 1)\) in W (I \(= identity)\), which generalizes the GL(2) majorization of Weil.
scientific article; zbMATH DE number 3989448