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Deformations of symplectic singularities and orbit method for semisimple Lie algebras

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Abstract

We classify filtered quantizations of conical symplectic singularities and use this to show that all filtered quantizations of symplectic quotient singularities are spherical symplectic reflection algebras of Etingof and Ginzburg. We further apply our classification and a classification of filtered Poisson deformations obtained by Namikawa to establish a version of the Orbit method for semisimple Lie algebras. Namely, we produce a natural map from the set of coadjoint orbits of a semisimple algebraic group to the set of primitive ideals in the universal enveloping algebra. We show that the map is injective for classical Lie algebras and conjecture that in that case the image consists of the primitive ideals corresponding to one-dimensional representations of W-algebras. Along the way, we get several new results on the Lusztig-Spaltenstein induction for coadjoint orbits.

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Notes

  1. This conjecture has been recently proved in [50].

  2. In the forthcoming paper [36] we prove the full conjecture using the same method.

  3. As we have mentioned in Introduction, this has been established in [50].

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Acknowledgements

This paper would have never appeared without help from Pavel Etingof and Dmitry Kaledin. I would like to thank them as well as Yoshinori Namikawa, Sasha Premet, and David Vogan for stimulating discussions. Finally, I would like to thank the referees for numerous comments that helped me to improve the exposition. I am very happy to dedicate the paper to Sasha Premet on his 60th birthday, this paper, as well as much of my other work, is inspired by his fascinating results. The paper was partially supported by the NSF under grants DMS-1161584, DMS-1501558. This work has also been funded by the Russian Academic Excellence Project ’5-100’.

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  1. Department of Mathematics, Yale University, New Haven, CT, 06511, USA

    Ivan Losev

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Dedicated to Sasha Premet, on his 60th birthday, with admiration.

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Losev, I. Deformations of symplectic singularities and orbit method for semisimple Lie algebras. Sel. Math. New Ser. 28, 30 (2022). https://doi.org/10.1007/s00029-021-00754-y

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