Conjugacy classes of n-tuples in Lie algebras and algebraic groups
A reductive algebraic group G over an algebraically closed field F of characteristic 0 acts on itself by inner automorphisms and on its Lie algebra L(G) by the adjoint representation. Using diagonal actions, G acts on \(G^ n\) and \(L(G)^ n\), and the author establishes a number of geometric properties of the orbits of G on the latter n-tuple spaces. This generalizes results of \textit{B. Kostant} [Am. J. Math. 85, 327-404 (1963; Zbl 0124.268)] and \textit{R. Steinberg} [Publ. Math., Inst. Hautes Etud. Sci. 25, 49-80 (1965; Zbl 0136.300)] for \(n=1\). For an n-tuple \(x\in G^ n\), let A(x) be the algebraic subgroup of G generated by the entries of x. The n-tuple x is semisimple (resp., unipotent) if A(x) is a linearly reductive (resp., unipotent) algebraic group. Among the author's results is that the orbit \(G\cdot x\) is closed (resp., unstable) if and only if x is semisiple (resp., unipotent). He also shows that the stabilizer \(Gx=Gy\cap Gz\), where \(x=y\cdot z\) is the Levi decomposition of x: \(x_ i=y_ iz_ i\) for \(y_ i\in L\), a Levi subgroup of A(x), and \(z_ i\in R_ u\) (A(x)). Moreover \(G\cdot y\) is the unique closed orbit in the closure of \(G\cdot x\). He also obtains a precise formula for dim \(\pi^{-1}(\pi (x))\), where \(\pi\) : \(G^ n\to G^ n/G\) is the morphism that corresponds to the inclusion homomorphism \(F[G^ n]^ G\to F[G^ n]\), and \(G^ n/G\) is the algebra \(F[G^ n]^ G\) of invariants. He characterizes the stable points of G and the smooth points of the quotient variety \(G^ n/G.\) Most results extend to the action of K on \(G^ n\) and \(L(G)^ n\), where a linearly reductive group S acts on G by automorphisms with fixed point \(K=G^ S\). Even for the case \(n=1\), the resulting theorems are new. A key tool is the Hilbert-Mumford theorem in the strengthened form established by \textit{G. Kempf} [Ann. Math., II. Ser. 108, 299-316 (1978; Zbl 0406.14031)]. For the case \(G=GL(V)\) and K the orthogonal group O(V) or the symplectic group \(S_ p(V)\), and the actions of G and K on \(L(G)^ n\), many of the results have been obtained in a more elementary fashion by \textit{C. Procesi} [Adv. Math. 19, 306-381 (1976; Zbl 0331.15021)].
scientific article; zbMATH DE number 4122229
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