Generically transitive actions on multiple flag varieties

TL;DR

This paper classifies when a semisimple algebraic group acts with finitely many orbits on multiple flag varieties, extending previous results by Popov and focusing on groups without type A simple components.

Contribution

It extends Popov's classification to a broader class of groups, identifying all triples where the group action has an open orbit or finitely many orbits on multiple flag varieties.

Findings

Identifies all triples (G, P, n) with an open G-orbit on the multiple flag variety.

Classifies all triples (G, P, n) with finitely many G-orbits on the multiple flag variety.

Extends previous classifications to groups without simple factors of type A.

Abstract

Let $G$ be a semisimple algebraic group whose decomposition into a product of simple components does not contain simple groups of type $A$, and $P\subseteq G$ be a parabolic subgroup. Extending the results of Popov [7], we enumerate all triples $(G, P, n)$ such that (a) there exists an open $G$-orbit on the multiple flag variety $G/P\times G/P\times\ldots\times G/P$ ($n$ factors), (b) the number of $G$-orbits on the multiple flag variety is finite.