Orbit closures in flag varieties for the centralizer of an order-two nilpotent element : normality and resolutions for types A, B, D

TL;DR

This paper studies the geometric properties of orbit closures in flag varieties associated with nilpotent elements of order two in classical algebraic groups, establishing normality, Cohen-Macaulayness, and resolutions across various types and characteristics.

Contribution

It proves normality and Cohen-Macaulayness of orbit closures for nilpotent elements of order two in types A, B, D, extending previous results and providing explicit resolutions.

Findings

Orbit closures are normal in all cases studied.

Cohen-Macaulay property holds in characteristic zero and for type A in any characteristic.

A birational morphism involving Schubert varieties provides a resolution.

Abstract

Let G be a reductive algebraic group in classical types A, B, D and e be an element of its Lie algebra with Z its centraliser in G for the adjoint action. We suppose that e identifies with an nilpotent matrix of order two, which guarantees the number of Z-orbits in the flag variety of G is finite. For types B, D in characteristic two, we also suppose the image of e is totally isotropic. We show that any closure Y of such orbit is normal. We also prove that Y is Cohen-Macaulay with rational singularities provided that the base field is of characteristic zero, and that Cohen-Macaulayness remains in any characteristic for type A. We exhibit a birational, rational morphism onto Y involving Schubert varieties. Our work generalizes a result by N. Perrin and E. Smirnov on Springer fibers ([PS12]).