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Reductive subgroups of reductive groups in nonzero characteristic. (English) Zbl 1103.20045

Let \(G\) be a (possibly nonconnected) reductive algebraic group over an algebraically closed field \(k\), and let \(N\in\mathbb{N}\). The group \(G\) acts on \(G^N\) by simultaneous conjugation, and one can form the quotient variety \(G^N/G\). The geometry of these varieties has been studied by Richardson and Vinberg.
Now let \(H\) be a (possibly nonconnected) reductive subgroup of \(G\). The inclusion of \(H^N\) in \(G^N\) gives rise to a morphism \(\psi_H^G\colon H^N/H\to G^N/G\). It is proved in the paper that the morphism \(\psi_H^G\) is finite. That result can be used to transfer information from the case \(G=\text{GL}_N(k)\), where much more is known, to arbitrary \(G\). For instance, let \(\Gamma=\{\gamma_1,\dots,\gamma_N\}\) be a finite group. The set \(R(\Gamma,G)\) of representations (that is, homomorphisms) from \(\Gamma\) to \(G\) can be viewed as a closed \(G\)-stable subvariety of \(G^N\) via the embedding \(\rho\mapsto(\rho(\gamma_1),\dots,\rho(\gamma_N))\). The author proves, as a consequence of the above result, that there are only finitely many closed conjugacy classes of representations from \(\Gamma\) to \(G\).
The fact that the morphism \(\psi_H^G\) is finite has been proved by Vinberg in the case when \(k\) has characteristic zero. The proof in the paper under review follows Vinberg’s closely, however in positive characteristic some new techniques are required.
Vinberg’s motivation was to study the ring \({\mathcal S}_N(G)\) of conjugation-invariant regular functions on \(G^N\): that is, the coordinate ring of the character variety \(G^N/G\). Given \(f\in{\mathcal S}_N(G)\) and a word \(w\) in letters \(\gamma_1,\dots,\gamma_N\), define \(f_w\in{\mathcal S}_N(G)\) by \[ f_w((g_1,\dots,g_N))=f(w(g_1,\dots,g_N)). \] Let \({\mathcal C}_N(G)\) be the \(k\)-subalgebra of \({\mathcal S}_N(G)\) generated by the \(f_w\). Then (as a consequence of a theorem of Donkin) \({\mathcal C}_N(\text{GL}_N(k))={\mathcal S}_N(\text{GL}_N(k))\), and, in particular, \({\mathcal C}_N(\text{GL}_N(k))\) is a finitely generated \(k\)-algebra. One of the main results of the paper is that \({\mathcal S}_N(G)\) is a finite module over \({\mathcal C}_N(G)\), and, \({\mathcal C}_N(G)\) is a finitely generated \(k\)-algebra.

MSC:

20G15 Linear algebraic groups over arbitrary fields
20G05 Representation theory for linear algebraic groups
14L30 Group actions on varieties or schemes (quotients)
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References:

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