# Parabolic subgroups and Automorphism groups of Schubert varieties

## TL;DR

This paper investigates the automorphism groups of Schubert varieties in complex simple algebraic groups, showing that each proper parabolic subgroup corresponds to automorphisms of a specific Schubert variety.

## Contribution

It establishes a correspondence between proper parabolic subgroups and automorphism groups of certain Schubert varieties in complex simple algebraic groups.

## Key findings

- Each proper parabolic subgroup is the automorphism group of some Schubert variety.
- Automorphism groups of Schubert varieties can be characterized by elements of the Weyl group.
- The result links algebraic group structure with geometric automorphisms of Schubert varieties.

## Abstract

Let $G$ be a simple algebraic group of adjoint type over the field $\mathbb{C}$ of complex numbers, $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ Let $w$ be an element of the Weyl group $W$ and $X(w)$ be the Schubert variety in $G/B$ corresponding to $w$. In this article we show that given any parabolic subgroup $P$ of $G$ containing $B$ properly, there is an element $w\in W$ such that $P$ is the connected component, containing the identity element of the group of all algebraic automorphisms of $X(w).$