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A new Weyl group action related to the quasi-classical Gelfand–Graev action

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Abstract

We construct a Weyl group action on the DKS type varieties, a certain class of varieties associated with quivers. As a result, on some special DKS type varieties, we can give a quiver theoretic explanation of the quasi-classical Gelfand–Graev action discovered by Ginzburg and Riche and studied by Ginzburg and Kazhdan recently.

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Notes

  1. \({{\,\mathrm{m}\,}}^{-1}(0)\) is isomorphic to the variety \(\Lambda _{D,V}\) defined in [19, § 3.1].

  2. Unfortunately, the symbol \(v_i\) has been used for a component of a dimension vector. The good news is that we only use \(v_i\) to denote a vector in V in this subsection. Outside this subsection, \(v_i\) always denotes a component of a dimension vector.

  3. Here and in the following, we often only define a variety (or a map between two varieties) as a set. But all such definitions can be enhanced scheme-theoretically by using suitable base changes. For details, one can see [10].

  4. Compared to [10], the moment maps here have an extra minus sign because [10] and this paper use different sign conventions for moment maps.

  5. The Weyl group action given here is a little different from [10] because we want to ensure the Weyl group action to be a left action.

  6. We choose to write the components of \(\mathbf {v}\) in a descending order, which is in conformity with the convention used in [21] but different from the convention in [7].

  7. If we use the notations of Sect. 3, the orbit equivalence class \([(\alpha _0,\beta _0,\ldots ,\alpha _{n-1},\beta _{n-1})]\) should be written as \(\pi ((\alpha _0,\beta _0,\ldots ,\alpha _{n-1},\beta _{n-1}))\). In this section, we prefer using this shorter notation.

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Acknowledgements

The author is very grateful for the helpful correspondence with Prof. Victor Ginzburg about the contents of this paper and he also would like to thank Prof. Gang Tian and Prof. Weiping Zhang for their encouragement during the preparation of this paper. Last but not least, the author thanks the anonymous referee, whose comments help to improve the quality of this paper to a great degree.

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  1. Xiangsheng Wang

    Present address: School of Mathematics, Shandong University, Jinan, 250100, Shandong, People’s Republic of China

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  1. Beijing International Center for Mathematical Research, Peking University, No. 5 Yiheyuan Road, Beijing, 100871, People’s Republic of China

    Xiangsheng Wang

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Partially supported by China Postdoctoral Science Foundation (Grant No. BX201700008), and the fundamental research funds of Shandong University (Grant No. 2020GN063).

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Wang, X. A new Weyl group action related to the quasi-classical Gelfand–Graev action. Sel. Math. New Ser. 27, 38 (2021). https://doi.org/10.1007/s00029-021-00655-0

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