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 $B, C, F_4, G_2$ 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 $n \leq 4$).