Split subvarieties of group embeddings

TL;DR

This paper investigates the structure of certain subvarieties within group embeddings, demonstrating that all compatibly split subvarieties are generalized Richardson varieties, extending previous results in Frobenius splitting theory.

Contribution

It extends the classification of compatibly split subvarieties to generalised projected Richardson varieties in complete G-embeddings.

Findings

All compatibly split subvarieties are generalized projected Richardson varieties.

The result generalizes previous work by Knutson, Lam, and Speyer.

Provides a new understanding of Frobenius splitting in group embeddings.

Abstract

Let G be a connected reductive group and X an equivariant compactifiction of G. In X, we study generalised and opposite generalised Schubert varieties, their intersections called generalised Richardson varieties and projected generalised Richardson varieties. Any complete G-embedding has a canonical Frobenius splitting and we prove that the compatibly split subvarieties are the generalised projected Richardson varieties extending a result of Knutson, Lam and Speyer to the situation.