On Automorphism group of a $G$-induced variety

TL;DR

This paper investigates the automorphism groups of certain algebraic varieties induced by a semisimple algebraic group, providing explicit computations for the connected component of the automorphism group containing the identity.

Contribution

It explicitly computes the connected automorphism group component for specific G-induced varieties, advancing understanding of their symmetry structures.

Findings

Determined the automorphism group structure for particular G-induced varieties.

Identified the connected component of automorphisms containing the identity.

Enhanced understanding of symmetries in algebraic varieties associated with semisimple groups.

Abstract

Let $G$ be a connected semisimple algebraic group of adjoint type over the field $\mathbb{C}$ of complex numbers and $B$ be a Borel subgroup of $G.$ Let $F$ be an irreducible projective $B$-variety. Then consider the variety $E:=G\times^{B}F,$ which has a natural action of $G$; we call it $G$-induced variety or $(G,B)$-induced variety. In this article, we compute the connected component containing the identity automorphism of the group of all algebraic automorphisms of some particular $G$-induced varieties $E.$