Monoid Embeddings of Symmetric Varieties

TL;DR

This paper characterizes antiinvolutions on algebraic groups and explores their extension to monoids, applying these results to classify Borel orbits in certain symmetric variety compactifications.

Contribution

It provides a criterion for extending antiinvolutions to monoids and describes Borel orbit parametrizations in symmetric variety embeddings.

Findings

Criteria for antiinvolution extension to monoids

Classification of Borel orbits in symmetric variety compactifications

Extension of Springer’s involution work to new settings

Abstract

We determine when an antiinvolution on an adjoint semisimple linear algebraic group extends to an antiinvolution on a $J$-irreducible monoid. Using this information, we study a special class of compactifications of symmetric varieties. Extending the work of Springer on involutions, we describe the parametrizing sets of Borel orbits in these special embeddings.