# Semilinear automorphisms of reductive algebraic groups

## TL;DR

This paper investigates the structure of semilinear automorphisms of reductive algebraic groups over fields, establishing conditions for splitting of automorphism sequences and providing examples where automorphism groups do not decompose straightforwardly.

## Contribution

It characterizes the automorphism groups of reductive algebraic groups over fields, especially quasi-split cases, and explores the splitting conditions of associated exact sequences.

## Key findings

- Aut_G(k) is isomorphic to Aut_{R(G)}(k) for quasi-split G
- The exact sequence of automorphisms splits under specific conditions
- Examples of groups with non-decomposable automorphism groups

## Abstract

Let $ G $ be a connected reductive algebraic group over a field $ k $. We study the group of semilinear automorphisms Aut($ G\to $Spec $k$) consisting of algebraic automorphisms of $ G $ over automorphisms of $ k $. We focus on the exact sequence $ 1\to $Aut $G\to $Aut ($ G\to $Spec $k$)$\to $Aut$_{G}(k)\to 1 $. When $G$ is quasi-split, we show that Aut$_{G}(k)$ is isomorphic to Aut$_{\mathcal{R}(G)}(k)$, where $\mathcal{R}(G)$ denotes the scheme of based root datum of $G$. Furthermore, the exact sequence $ 1\to $Aut $G\to $Aut ($ G\to $Spec $k$)$\to $Aut$_{G}(k)\to 1 $ splits if and only if the exact sequence $ 1\to \text{Aut }\mathcal{R}(G) \to \text{Aut }(\mathcal{R}(G) \to \text{Spec } k)\to \text{Aut}_{\mathcal{R}(G)}(k)\to 1 $ splits. As a corollary, we get many examples of algebraic groups $ G $ over $ k $ whose group of abstract automorphisms does not decompose as the semidirect product of $ \text{Aut } G $ with $\text{Aut}_G(k) $. We also study the same questions for inner forms of SL$_n $ over a local field.