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

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.