Varieties of nilpotent elements for simple Lie algebras. I: Good primes.
Let \(G\) be a simple algebraic group over \(k=\mathbb C\) or \(\overline {\mathbb F}_p\) where \(p\) is good, \(\mathfrak g = \operatorname{Lie} G\), \(\rho \) an irreducible representation of minimal dimension or the adjoint representation. In the paper the description of a variety of nilpotent elements \(N_{r,\rho }=\{ x\in \mathfrak g: \rho (x)^r=0 \}\) is given in terms of orbit closures (with respect to \(G\)). The approach to the problem is based on ideas of \textit{D. K. Nakano} and \textit{T. Tanisaki} [On the realization of orbit closures as support varieties, preprint (2003), J. Pure Appl. Algebra 206, No. 1--2, 66--82 (2006; Zbl 1156.14326)] who have shown how to use orbit closure in \(\mathfrak g\mathfrak l(V)\) after intersecting with \(\mathfrak g\). The case of classical groups is considered in detail. For exceptional groups the authors provide tables listing in Bala-Carter notations the orbit closures which describe nilpotent varieties \(N_{r, \rho }(\mathfrak g)\). All non-Richardson orbits are indicated. In particular, when \(r=p=\text{char}\,k\) the results of the paper recover the calculations of the restricted nullcone \(N_1(\mathfrak g)=N_{p, \rho }(\mathfrak g)\) performed by \textit{J. F. Carlson, Z. Lin, D. K. Nakano} and \textit{B. J. Parshall} [Contemp. Math. 325, 51--75 (2003; Zbl 1059.17013)]. Note that \(N_1(\mathfrak g)\) plays a crucial role in the cohomology theory of restricted Lie algebras. The authors present an alternate method to find \(N_{r, \rho }\) when \(k=\mathbb C, ~\mathfrak g\) is classical, and \(\rho \) is the adjoint representation using Jacobson-Morozov theorem.
scientific article; zbMATH DE number 2136744
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