Affine vs Yokonuma

Question

Let $G=GL_n$. Let us start with the Hecke algebra $H_n$. It acts on K(constructible sheaves on $G/B$) by Hecke correpondences and on K(coherent sheaves on $G/B$) by Lusztig's construction [1]. Now we can extend $H_n$ by adding a commutative algebra in two ways.

In the first way we take constructible sheaves, but replace $B$ by $U$. The finite field analogue of the result is the so-called Yokonuma-Hecke algebra, the algebra of double cosets $U(F_q)\G(F_q)/U(F_q)$. It contains the group algebra of the torus $T(F_q)$ and has a surjective map to $H_n$.

In the second way we take coherent sheaves, so we can tensor by line bundles. This way we recover Lusztig's construction [1] of action of the affine Hecke algebra, which is generated by $H_n$ and a commutative subalgebra, which is the algebra of Laurent polynomials in $n$ variables, in other words the group algebra of the character group of the torus $T$.

So, we have two ways to add a commutative algebra to the Hecke algebra. Is there some kind of well-known duality between them and how to see it on the algebra level?

[1]: Lusztig, George. “Equivariant K-Theory and Representations of Hecke Algebras.” Proceedings of the American Mathematical Society, vol. 94, no. 2, 1985, pp. 337–342.

Answer 1

I am not entirely sure if this is the kind of answer you are looking for, but Chlouveraki in her thesis as well as here explains the differences in the two deformations. Generally speaking, in the finite convolution algebra you describe above we have that the additional generators coming from $T(\mathbb{F}_q)$ have finite order $d=|\mathbb{F}_q^\times|$. In particular, the Yokonuma-Hecke algebra deforms the wreath product algebra $S_n\ltimes (\mathbb{Z}/d\mathbb{Z})^n$ in a way that preserves the wreath product structure but deforms the quadratic Hecke relations.

In some sense an orthogonal deformation of these wreath products is given by the cyclotomic Hecke/Ariki-Koike algebras, which are quotients of the affine Hecke algebra. In these the Hecke relations are preserved but the wreath product structure is deformed instead. The finite Hecke algebra for $GL_n$ embeds naturally inside the Ariki-Koike algebras, but is naturally a quotient of the Yokonuma-Hecke algebras.

Additionally, you could do both deformations at the same time to arrive at cyclotomic and affine Yokonuma-Hecke algebras as in work of Chlouveraki and Poulain d'Andecy.

As to the actual question about a duality between Yokonuma-Hecke and the affine Hecke algebra, I don't know. Geometrically, one might suspect something like it based on Bezrukavnikov-Yun. Also, Weideng Cao has some purely algebraic work proving that the Yokonuma-Hecke algebra is isomorphic to a tensor product of (matrix algebras over) cyclotomic Hecke algebras.