On the component group of a real reductive group
Abstract
For a connected linear algebraic group $G$ defined over $\mathbb{R}$, we compute the component group $\pi_0G(\mathbb{R})$ of the real Lie group $G(\mathbb{R})$ in terms of a maximal split torus $T_{\text{s}}\subseteq G$. In particular, we recover a theorem of Matsumoto (1964) that each connected component of $G(\mathbb{R})$ intersects $T_{\text{s}}(\mathbb{R})$. We provide explicit elements of $T_{\text{s}}(\mathbb{R})$ which represent all connected components of $G(\mathbb{R})$. The computation is based on structure results for real loci of algebraic groups and on methods of Galois cohomology.
- Publication:
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arXiv e-prints
- Pub Date:
- March 2022
- DOI:
- 10.48550/arXiv.2203.14024
- arXiv:
- arXiv:2203.14024
- Bibcode:
- 2022arXiv220314024T
- Keywords:
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- Mathematics - Group Theory;
- Mathematics - Algebraic Geometry;
- Mathematics - Differential Geometry;
- 20G20;
- 22E15;
- 11E72
- E-Print:
- 11 pages, typos corrected, examples added, references updated