Intersection cohomology of Vinberg-Popov varieties

TL;DR

This paper develops a recursive method to compute the intersection cohomology of Vinberg-Popov varieties, especially for the group SL_n, linking algebraic geometry, symplectic geometry, and representation theory.

Contribution

It introduces a new recursive procedure for calculating intersection cohomology of Vinberg-Popov varieties, advancing understanding of their geometric and algebraic structure.

Findings

Recursive computation method for intersection cohomology

Explicit results for SL_n cases

Connection to symplectic implosion and algebraic group actions

Abstract

The Vinberg-Popov variety of a simply connected reductive algebraic group $G$ is a singular affine variety that contains the basic affine space $G/U$ as a Zariski open subset. It is defined as the spectrum of the ring of functions on $G/U$, and can also be identified with the universal symplectic implosion for the maximal compact subgroup of $G$. We provide a recursive procedure for computing the intersection cohomology of this variety, with an emphasis on the case where $G = \operatorname{SL}_n$.