On a t-exactness property of the Harish-Chandra transform

TL;DR

This paper proves the t-exactness of the Harish-Chandra transform using hyperbolic localization, extending known results to broader sheaf-theoretic contexts and arbitrary monodromy.

Contribution

It provides a new, simplified proof of the t-exactness of the Harish-Chandra transform via hyperbolic localization, applicable to general sheaf-theoretic settings.

Findings

Identifies nearby cycles with Radon and Harish-Chandra functors composition

Extends t-exactness results to arbitrary monodromy

Simplifies proof of the transform's exactness

Abstract

Using hyperbolic localization, we identify the nearby cycles along the Vinberg degeneration with the composition of Radon and Harish-Chandra functors, both considered for the category of character sheaves. This provides a new, simple proof of the exactness of this composition, extending previously known results to arbitrary monodromy and more general sheaf-theoretic set-ups.