Towards a Nonstandard Fourier Analysis in Automorphic forms: Some Results on Two Toy Examples

This thesis consists of two main parts.In the first part an integral transform between spaces of nonstandard test functions on the affine space of dimension n is constructed. The integral transform satisfies a summation formula of Poisson type, which is derived from an analogue of the Arthur--Selberg trace formula for the Lie algebra of n-by-n matrices. In the second part a proof of Conjecture 9.12 of Braverman--Kazhdan in their article "gamma-functions of representations and lifting" on the acyclicity of their l-adic gamma-sheaves over certain affine spaces is given for GL(n).