Symplectic Fourier–Deligne Transforms on G/U and the Algebra of Braids and Ties (2024)

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Volume 2024 Issue 13 July 2024
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Calder Morton-Ferguson

Massachusetts Institute of Technology

, 77 Massachusetts Ave., Cambridge, MA 02141, USA

Correspondence to be sent to: e-mail: caldermf@mit.edu

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International Mathematics Research Notices, Volume 2024, Issue 13, July 2024, Pages 10219–10235, https://doi.org/10.1093/imrn/rnae052

Published:

11 April 2024

Article history

Received:

02 April 2023

Revision received:

25 November 2023

Accepted:

25 November 2023

Published:

11 April 2024

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Abstract

We explicitly identify the algebra generated by symplectic Fourier–Deligne transforms (i.e., convolution with Kazhdan–Laumon sheaves) acting on the Grothendieck group of perverse sheaves on the basic affine space |$G/U$|⁠, answering a question originally raised by A. Polishchuk. We show it is isomorphic to a distinguished subalgebra, studied by I. Marin, of the generalized algebra of braids and ties (defined in Type |$A$| by F. Aicardi and J. Juyumaya and generalized to all types by Marin), providing a connection between geometric representation theory and an algebra defined in the context of knot theory. Our geometric interpretation of this algebra entails some algebraic consequences: we obtain a short and type-independent geometric proof of the braid relations for Juyumaya’s generators of the Yokonuma–Hecke algebra (previously proved case-by-case in types |$A, D, E$| by Juyumaya and separately for types |$B, C, F_{4}, G_{2}$| by Juyumaya and S. S. Kannan), a natural candidate for an analogue of a Kazhdan–Lusztig basis, and finally an explicit formula for the dimension of Marin’s algebra in Type |$A_{n}$| (previously only known for |$n \leq 4$|⁠).

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