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The Brylinski Filtration for Affine Kac-Moody Algebras and Representations of \({\mathscr{W}}\)-algebras

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Abstract

We study the Brylinski filtration induced by a principal Heisenberg subalgebra of an affine Kac-Moody algebra \(\mathfrak {g}\), a notion first introduced by Slofstra. The associated graded space of this filtration on dominant weight spaces of integrable highest weight modules of \(\mathfrak {g}\) has Hilbert series coinciding with Lusztig’s t-analog of weight multiplicities. For the level 1 vacuum module L(Λ0) of affine Kac-Moody algebras of type A, we show that the Brylinski filtration may be most naturally understood in terms of representations of the corresponding \({\mathscr{W}}\)-algebra. We show that the sum of dominant weight spaces of L(Λ0) in the principal vertex operator realization forms an irreducible Verma module of \({\mathscr{W}}\) and that the Brylinski filtration is induced by the Poincaré-Birkhoff-Witt basis of this module. This explicitly determines the subspaces of the Brylinski filtration. Our basis may be viewed as the analog of Feigin-Frenkel’s basis of \({\mathscr{W}}\) for the \({\mathscr{W}}\)-action on the principal rather than on the homogeneous realization of L(Λ0).

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Acknowledgements

It is a pleasure to thank Bojko Bakalov for useful discussions and pointers to relevant literature at an early stage of this work. The authors also thank an anonymous referee whose comments helped improve the exposition. Sachin Sharma also thanks the Institute of Mathematical Sciences, where parts of this work were done, for its excellent hospitality.

Funding

Partial financial support was received from the Department of Science and Technology, Government of India through grants EMR/2016/001997 (SG) and MTR/2019/000071 (SV), from the Department of Atomic Energy, Government of India under a XII plan project (SV) and from the Indian Institute of Technology Kanpur through a faculty initiation grant (SS).

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Authors and Affiliations

  1. Department of Physics, Indian Institute of Technology Madras, Chennai, India

    Suresh Govindarajan

  2. Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, India

    Sachin S. Sharma

  3. The Institute of Mathematical Sciences, HBNI, Chennai, India

    Sankaran Viswanath

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  1. Suresh Govindarajan
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  2. Sachin S. Sharma
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  3. Sankaran Viswanath
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Correspondence to Sankaran Viswanath.

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Govindarajan, S., Sharma, S.S. & Viswanath, S. The Brylinski Filtration for Affine Kac-Moody Algebras and Representations of \({\mathscr{W}}\)-algebras. Algebr Represent Theor 26, 491–512 (2023). https://doi.org/10.1007/s10468-021-10101-6

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