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

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, F₄, G₂ 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₍ (previously only known for n 4).