$G$-complete reducibility and saturation
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866908336899751936 |
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| author | Bate, Michael Böhm, Sören Litterick, Alastair Martin, Benjamin Roehrle, Gerhard |
| author_facet | Bate, Michael Böhm, Sören Litterick, Alastair Martin, Benjamin Roehrle, Gerhard |
| contents | Let $H \subseteq G$ be connected reductive linear algebraic groups defined over an algebraically closed field of characteristic $p> 0$. In our first main theorem we show that if a closed subgroup $K$ of $H$ is $H$-completely reducible, then it is also $G$-completely reducible in the sense of Serre, under some restrictions on $p$, generalising the known case for $G = GL(V)$. Our proof uses R.W. Richardson's notion of reductive pairs to reduce to the $GL(V)$ case. We study Serre's notion of saturation and prove that saturation behaves well with respect to products and regular subgroups. Our second main theorem shows that if $K$ is $H$-completely reducible, then the saturation of $K$ in $G$ is completely reducible in the saturation of $H$ in $G$ (which is again a connected reductive subgroup of $G$), under suitable restrictions on $p$, again generalising the known instance for $G = GL(V)$. We also study saturation of finite subgroups of Lie type in $G$. We show that saturation is compatible with standard Frobenius endomorphisms, and we use this to generalise a result due to Nori from 1987 in case $G = GL(V)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_16927 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $G$-complete reducibility and saturation Bate, Michael Böhm, Sören Litterick, Alastair Martin, Benjamin Roehrle, Gerhard Representation Theory Group Theory 20G15, 14L24 Let $H \subseteq G$ be connected reductive linear algebraic groups defined over an algebraically closed field of characteristic $p> 0$. In our first main theorem we show that if a closed subgroup $K$ of $H$ is $H$-completely reducible, then it is also $G$-completely reducible in the sense of Serre, under some restrictions on $p$, generalising the known case for $G = GL(V)$. Our proof uses R.W. Richardson's notion of reductive pairs to reduce to the $GL(V)$ case. We study Serre's notion of saturation and prove that saturation behaves well with respect to products and regular subgroups. Our second main theorem shows that if $K$ is $H$-completely reducible, then the saturation of $K$ in $G$ is completely reducible in the saturation of $H$ in $G$ (which is again a connected reductive subgroup of $G$), under suitable restrictions on $p$, again generalising the known instance for $G = GL(V)$. We also study saturation of finite subgroups of Lie type in $G$. We show that saturation is compatible with standard Frobenius endomorphisms, and we use this to generalise a result due to Nori from 1987 in case $G = GL(V)$. |
| title | $G$-complete reducibility and saturation |
| topic | Representation Theory Group Theory 20G15, 14L24 |
| url | https://arxiv.org/abs/2401.16927 |