The Affine Closure of T^*(SL_n/U)

TL;DR

This paper investigates the geometric structure of the affine closure of the cotangent bundle of SL_n/U, revealing symplectic singularities and specific isomorphisms in the case n=3, connecting to nilpotent orbits and Weyl group actions.

Contribution

It establishes that the affine closure of T^*(SL_n/U) has symplectic singularities and, for n=3, identifies it with the closure of a minimal nilpotent orbit in so(8,C), linking group actions.

Findings

Affine closure of T^*(SL_n/U) has symplectic singularities.

For n=3, it is isomorphic to the closure of the minimal nilpotent orbit in so(8,C).

Weyl group actions correspond to triality automorphisms in this setting.

Abstract

We show that the affine closure of T^*(SL_n/U) has symplectic singularities, in the sense of Beauville. In the special case n=3, we show that the affine closure of T^*(SL_3/U) is isomorphic to the closure of the minimal nilpotent adjoint orbit in so(8,C). Moreover, the quasi-classical Gelfand-Graev action of the Weyl group W on the affine closure of T^*(SL_3/U) can be identified with the restriction to the closure of the minimal nilpotent adjoint orbit of the triality action on so(8,C).