Nearby Cycle Sheaves for Stable Polar Representations

TL;DR

This paper computes the Fourier transform of nearby cycle sheaves for stable polar representations, extending previous results and aiming to advance the theory of character sheaves in graded Lie algebras.

Contribution

It generalizes earlier work by computing Fourier transforms of nearby cycle sheaves for a broad class of stable polar representations.

Findings

Fourier transform of nearby cycle sheaves explicitly computed

Partial generalization of previous results in the field

Application to character sheaves for graded Lie algebras

Abstract

Let G|V, G connected, reductive over C, be a stable polar representation in the sense of [DK], satisfying some mild additional hypotheses. Given a G-equivariant rank one local system L on the general fiber of the quotient map f : V --> V/G, we compute the Fourier transform of the corresponding nearby cycle sheaf P on the zero-fiber of f. This provides a partial generalization of the results of [Gr1] and [GVX1]. Our main intended application is to the theory of character sheaves for graded Lie algebras over C.