Vinberg's \(\theta \)-groups in positive characteristic and Kostant-Weierstrass slices
Let \(G\) be a reductive algebraic group over the algebraically closed field \(k\) and let \(\mathfrak g:={\text{Lie}}(G)\). Let \(\theta: G\to G\) be an automorphism of order \(m<\infty\) such that \({\text{char}}(k)\nmid m\) if \({\text{char}}(k)>0\). Fix a primitive \(m\)th root of unity \(\zeta\) in \(k\). Then \(\mathfrak g=\mathfrak g(0)\oplus\mathfrak g(1)\oplus\ldots\oplus\mathfrak g(m-1)\), where \(\mathfrak g(i)\) is the eigenspace of \(d\theta\) associated to the eigenvalue \(\zeta^i\). Let \(G(0)\) be the identity component of the fixed point set \(G^\theta\) of \(\theta\). Then \(G(0)\) is a reductive group and the adjoint action of \(G(0)\) stabilizes each \(\mathfrak g(i)\). For \({\text{char}}(k)=0\), in [\textit{È. B. Vinberg}, Math. USSR, Izv. 10(1976), 463--495 (1977); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 40, 488--526 (1976; Zbl 0363.20035)], some properties of the action of \(G(0)\) on \(\mathfrak g(1)\) have been established, namely: (1) any two maximal commutative subspaces of \(\mathfrak g(1)\) (called Cartan subspaces) are \(G(0)\)-conjugate and any semisimple element of \(\mathfrak g(1)\) is contained in such a subspace; (2) closed \(G(0)\)-orbits in \(\mathfrak g(1)\) are precisely \(G(0)\)-orbits of semisimple elements; (3) for any Cartan subspace \(\mathfrak c\), the restriction map is an isomorphism between polynomial \(G(0)\)-invariants on \(\mathfrak g(1)\) and polynomial \(W_{\mathfrak c}\)-invariants on \(\mathfrak c\), where \(W_{\mathfrak c}\), the little Weyl group, is the quotient of the normalizer of \(\mathfrak c\) in \(G(0)\) by its centralizer; (4) \(W_{\mathfrak c}\) is a finite linear group generated by pseudoreflections and the algebra of polynomial \(W_{\mathfrak c}\)-invariants on \(\mathfrak c\) is free. In the paper under review these results are generalized to the case of \({\text{char}}(k)>0\). In particular, for any \(G\) satisfying the so called ``standard hypotheses'' (namely, \({\text{char}}(k)\) is good; \(G'\) is simply connected; and there is a nondegenerate symmetric bilinear \(G\)-invariant form \(\mathfrak g\times \mathfrak g\to k\)) property (4) holds. This is applied to the proof of the existence of a so called Kostant--Weierstrass section in \(\mathfrak g\) for \(G\) of classical type and \({\text{char}}(k)=0\) or \(>2\). The latter result confirms for such \(G\) a long-standing general conjecture of the reviewer in characteristic zero, [see \textit{V. L. Popov}, Funct. Anal. Appl. 10, 242-244 (1976); translation from Funkts. Anal. Prilozh. 10, No. 3, 91--92 (1976; Zbl 0365.20053)].
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