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. It is generated by $H_n$ and a commutative subalgebra, which is the group algebra of the torus $T$.

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.

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.