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Journal of Lie Theory 31 (2021), No. 1, 265--286
Copyright Heldermann Verlag 2021

Spinorial Representations of Orthogonal Groups

Jyotirmoy Ganguly
The Institute of Mathematical Sciences, Chennai 600113, Tamil-Nadu, India

Rohit Joshi
Pune 411004, Maharashtra, India


Let $G$ be a real compact Lie group, such that $G=G^0\rtimes C_2$, with $G^0$ simple. Here $G^0$ is the connected component of $G$ containing the identity and $C_2$ is the cyclic group of order $2$. We give criteria for whether an orthogonal representation $\pi\colon G\to \text{O}(V)$ lifts to $\text{Pin}(V)$ in terms of the highest weights of $\pi$ and also in terms of character values. From these criteria we compute the first and second Stiefel-Whitney classes of the representations of the orthogonal groups.

Keywords: Orthogonal group, spinorial representation, Stiefel-Whitney class, highest weight.

MSC: 22E41, 22E47, 57R20.

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