Spinoriality of orthogonal representations of reductive groups

TL;DR

This paper provides a straightforward criterion to determine when an orthogonal representation of a connected reductive group over a characteristic zero field lifts to the spin group, based on the highest weights of its irreducible components.

Contribution

It introduces a simple, weight-based criterion for spin-liftability of orthogonal representations of reductive groups, simplifying previous complex methods.

Findings

Criterion based on highest weights for spin-liftability

Applicable to all orthogonal representations of reductive groups

Simplifies the analysis of spin structures in representation theory

Abstract

Let G be a connected reductive group over a field of characteristic zero, and consider an orthogonal representation of G. We give a simple criterion for whether the representation lifts to the spin group, in terms of the highest weights of the irreducible constituents of the representation.

Equations

$$|\nu {|}^2=\sum _{\alpha \in R}⟨\alpha ,\nu {⟩}^2\in 2\mathbb{Z}.$$

$$p(\underline{\nu })=\frac{1}{2}\gcd \left(|{\nu }_1{|}^2,\dots ,|{\nu }_r{|}^2\right).$$

$$p(\underline{\nu })\cdot \frac{\mathrm{tr}(C,V)}{\dim \mathfrak{g}}$$

Full text

1. 1 Introduction

1. Introduction

Let $G$ be a connected reductive group over a field $F$ of characteristic [math]. Let $(\varphi,V)$ be a representation of $G$, which in this paper always means a finite-dimensional $F$-representation of $G$. Suppose that $V$ is orthogonal, i.e., carries a symmetric nondegenerate bilinear form preserved by $\varphi$. Thus $\varphi$ is a morphism from $G$ to $\operatorname{SO}(V)$. Write $\rho:\operatorname{Spin}(V)\to\operatorname{SO}(V)$ for the usual isogeny ([Veldkamp]). Following [Bou.Lie.7-9], we say that $\varphi$ is spinorial when it lifts to $\operatorname{Spin}(V)$, i.e., provided there exists a morphism $\hat{\varphi}:G\to\operatorname{Spin}(V)$ so that $\varphi=\rho\circ\hat{\varphi}$. We call $\varphi$ aspinorial otherwise.

By an argument in Section LABEL:reduction.closed, we may assume that $F$ is algebraically closed, which we do for the rest of this introduction. Let $T$ be a maximal torus of $G$. Write $\pi_{1}(G)$ for the fundamental group of $G$ (the cocharacter group of $T$ modulo the subgroup $Q(T)$ generated by coroots), and $T_{V}$ for a maximal torus of $\operatorname{SO}(V)$ containing $\varphi(T)$. Then $\varphi$ induces a homomorphism $\varphi_{*}:\pi_{1}(G)\to\pi_{1}(\operatorname{SO}(V))\cong\mathbb{Z}/2\mathbb{Z}$, and $\varphi$ is spinorial iff $\varphi_{*}$ is trivial. If we take a set of cocharacters $\underline{\nu}=\{\nu_{1},\ldots,\nu_{r}\}$ whose images generate $\pi_{1}(G)$, then $\varphi$ is spinorial iff each cocharacter $\varphi_{*}\nu_{i}$ of $T_{V}$ lifts to $\operatorname{Spin}(V)$. (See Section LABEL:s3.)

Write $\mathfrak{g}$ for the Lie algebra of $G$, and $X^{*}(T)$ for the character group of $T$. Suppose $(\varphi,V)$ is an orthogonal representation of $G$. Write $C$ for the Casimir element associated to the Killing form. Given a cocharacter $\nu$ of $T$, put

[TABLE]

We introduce the integer

[TABLE]

Theorem 1**.**

Suppose that $\mathfrak{g}$ is simple and let $\varphi$ be an orthogonal representation of $G$. Then $\varphi$ is spinorial iff the integer

[TABLE]

*is even. *

Alternatively, this can be reformulated in terms of the Dynkin index ‘$\operatorname{dyn}(\varphi)$’ of $\varphi$ and the dual Coxeter number $\check{h}$ of $\mathfrak{g}$. (We recall these integers in Section LABEL:s7.)