The isomorphism conjecture for the representation variety map

Assume $q=1$ and otherwise $(u,t)$ are generic. Let $\Bbb R_{n,u,t}$ be the variety of equivalence classes of irreducible representations of $H_n(u,t)$ that restrict via $\eta_m$ to the regular representation of the relevant Hecke algebra, and let $\Bbb M_{n,u,t}$ be the variety of tuples of matrices satisfying the defining relations constructed from the invariant subspace $V^{{\bold H}_{n-1,\ell}(u_m,t)}$. The construction defines an algebraic map

$$\Phi:\Bbb R_{n,u,t}\to\Bbb M_{n,u,t}.$$

Isomorphism conjecture. The map $\Phi$ is an isomorphism of algebraic varieties. This would identify the moduli of the relevant irreducible representations with the explicitly defined matrix variety; the paper proves that the map is well-defined but leaves its bijectivity and algebraic inverse open.