Flatness, PI degree, and center conjectures for generalized double affine Hecke algebras

Let $H_n(u,t)$ be the generalized double affine Hecke algebra associated with an affine Dynkin diagram $D=\widetilde D_4,\widetilde E_6,\widetilde E_7$, or $\widetilde E_8$, with parameters $u=(u_{kj})$ and $t$. Set

$$q=q(u):=\prod_{k,j}u_{kj}^{-\ell/d_k}.$$

Let $Z(H_n(u,t))$ denote the center, let ${\rm e}$ be the symmetrizing idempotent, and let $\bold H_{n,\ell}(u_m,t)$ be the Hecke algebra appearing through the homomorphism $\eta_m$ given by $\eta_m(T_i)=T_i$ and $\eta_m(U)=U_m$. The flatness, PI, and representation conjecture. (i) The Gelfand–Kirillov dimension of $H_n(u,t)$ is $2n$. (ii) The algebra $H_n(u,t)$ is PI if and only if $q$ is a root of unity; if $q$ has order $N$, its PI degree is $n!(N\ell)^n$. (iii) If $q$ is a root of unity, then $H_n(u,t)$ is finitely generated over $Z(H_n(u,t))$. (iv) If $q$ is a root of unity and otherwise $(u,t)$ are generic, then $H_n(u,t)$ is Azumaya and $S(u,t):={\rm Spec}(Z(H_n(u,t)))$ is a smooth affine variety of dimension $2n$. (v) If $q=1$, then the map $Z(H_n(u,t))\to {\rm e}H_n(u,t){\rm e}$ given by $z\mapsto z{\rm e}$ is an isomorphism; in particular, ${\rm e}H_n(u,t){\rm e}$ is commutative. (vi) If $q=1$ and otherwise $(u,t)$ are generic, then every irreducible representation of $H_n(u,t)$ restricts via $\eta_m$ to the regular representation of $\bold H_{n,\ell}(u_m,t)$. These conjectures seek to extend the known properties of related rank-one and affine type-$\widetilde D_4$ algebras to the higher-rank affine cases; the paper notes that several parts are established for completions and that the general affine $\widetilde E$ cases remain difficult.