# Orthogonal multiple flag varieties of finite type II : even degree case

## TL;DR

This paper classifies certain geometric objects called multiple flag varieties associated with split orthogonal groups over infinite fields, focusing on cases where the degree is even, and determines when these varieties have finitely many orbits.

## Contribution

It provides a classification of multiple flag varieties of finite type for split orthogonal groups in the even degree case, extending previous understanding of orbit structures.

## Key findings

- Identifies conditions for finite orbit types in even degree cases
- Classifies all such multiple flag varieties over infinite fields
- Extends classification results to new algebraic group cases

## Abstract

Let $G$ be the split orthogonal group of degree $2n$ over an arbitrary infinite field $\mathbb{F}$ of chararcteristic not $2$. In this paper, we classify multiple flag varieties $G/P_1\times\cdots\times G/P_k$ of finite type. Here a multiple flag variety is said to be of finite type if it has a finite number of $G$-orbits with respect to the diagonal action of $G$.