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On the normality of the null-fiber of the moment map for \(\theta\)- and tori representations. (English) Zbl 1395.14038

Let \(G\) be a connected reductive algebraic group and let \(V\) be a finite dimensional \(G\)-representation. Then the symplectic double \(W=V \oplus V^*\) is a symplectic \(G\)-representation equipped with a \(G\)-equivariant moment map \(\mu : W \to \mathfrak{g}^*\). The scheme-theoretic fiber \(\mu^{-1}(0)\), and the corresponding symplectic reduction \(\mu^{-1}(0)/\!/G\), are expected to be normal varieties under reasonable assumptions on \((G,V)\); see M. Bulois et al. [Compos. Math. 153, No. 3, 647–666 (2017; Zbl 1391.53096)] for a more precise statement and some background.
In this paper, the author makes a new step in this direction. He considers two classes of representations: the tori representations, i.e. the case where \(G\) is a torus, and the \(\theta\)-representations, i.e. the case where \((G,V)\) is isomorphic to some \((H_0,\mathfrak{h}_1)\) where \(H\) is a connected reductive algebraic group acting on its Lie algebra \(\mathfrak{h}\), with \(\mathfrak{h}\) equipped with a \(\mathbb{Z}/m\mathbb{Z}\)-grading \(\bigoplus_{i \in \mathbb{Z}/m\mathbb{Z}} \mathfrak{h}_i\) and where \(H_0\) is the connected subgroup of \(H\) whose Lie algebra is \(\mathfrak{h}_0\).
For the tori representations, he proves that \(\mu^{-1}(0)\) is a normal variety if and only if it is irreducible (and he gives a combinatorial criterion to characterize the irreducibility). For the stable locally free \(\theta\)-representations, he proves that \(\mu^{-1}(0)\) is always a normal variety except in five exceptional cases where it is a non-normal variety.
In both cases he also proves that the symplectic reduction \(\mu^{-1}(0)/\!/G\) is always a normal variety. (This was known already for the stable locally free \(\theta\)-representations by M. Bulois et al. [Compos. Math. 153, No. 3, 647–666 (2017; Zbl 1391.53096)].) In particular, if \((G,V)\) is a visible polar representation, then this implies that \(\mu^{-1}(0)/\!/G\) is a symplectic variety isomorphic to a quotient singularity.

MSC:

14L24 Geometric invariant theory
17B20 Simple, semisimple, reductive (super)algebras
17B70 Graded Lie (super)algebras
20G05 Representation theory for linear algebraic groups
53D20 Momentum maps; symplectic reduction

Citations:

Zbl 1391.53096

Software:

SLA; GAP
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References:

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