The Jacobian modules of a representation of a Lie algebra and geometry of commuting varieties. (English) Zbl 0834.17003

Panyushev, Dmitrij I.

Compos. Math. 94, No. 2, 181-199 (1994).

Let \(G\) be a reductive group over an algebraically closed field of characteristic 0, and \({\mathfrak g}= \text{Lie } G\). Given a finite- dimensional rational \(G\) module \(V\) the author studies two algebraic varieties (generalized commuting varieties) associated with \(V\). The first one is the zero fiber of the moment map \[ \varphi: V\times V^*\to {\mathfrak g}^*, \qquad \varphi (v,f) (g)= f(gv). \] The second one is the zero fiber of the map \[ \psi: {\mathfrak g}\times V\to V, \qquad \psi(g, v)= gv. \] He obtains several theorems that extend classical results of the invariant theory of the adjoint representation of \(G\) to those representations of \(G\) whose invariant theory resembles that of \(\text{Ad}: G\to \text{gl} ({\mathfrak g})\). In particular, he proves that if the action \(G:V\) is stable locally free, and \(V\) has finitely many instable \(G\)-orbits, then the moment map \(\varphi\) is surjective, equidimensional, and all its fibers are irreducible reduced complete intersections.
At the end of the paper the author formulates three open problems. Problem (2) looks very challenging indeed: Prove that the algebraic variety \(\{x,y\in {\mathfrak g}\times {\mathfrak g}\mid [x,y ]=0\}\) is normal.

Reviewer: A.A.Premet (Manchester)

MSC:

17B10	Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B45	Lie algebras of linear algebraic groups

Keywords:

reductive group action; commuting variety; Jacobian module; module of covariants; open problems

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