34 problems

Let $V_n^{(\ell,r)}$ be the polynomial representation of the DAHA, and let … be the increasing sequence of subrepresentations defined by the multi-wheel condition. Kasatani's conje…

Affine Hecke coinvariant module conjecture. The $\widetilde{\mathbf H}_\kappa$-module $\widetilde{\mathcal C}(L_\kappa(\mu),L_\kappa^\dagger(\lambda))$ should be isomorphic to…

Isomorphism conjecture. The map $\Phi$ is an isomorphism of algebraic varieties. This would identify the moduli of the relevant irreducible representations with the explicitly defi…

Assume $q=1$ and otherwise the parameters $u,t$ are generic. Let $H_n(u,t)$ be the generalized double affine Hecke algebra, let $\bold H_{n,\ell}(u_m,t)$ be the Hecke algebra embed…

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 p…

Let $I^{(k,r)}_m$ be the ideals in the polynomial representation $V$ of the double affine Hecke algebra representation $mathcal{H}n^{(k,r)}$, with…

Let ${\roman k}$ be an algebraically closed field of characteristic $\ell>0$. Let $G={\roman{GL}}_n$, and retain the categories and parameters defined above, including the category…

Let $H(\underline{t};q)$ be the rank-one double affine Hecke algebra with parameters $q$ and $\underline{t}$. Here $HH^i$ denotes its $i$-th Hochschild cohomology group, and…

Deformation conjecture. The simple module $Z_\chi$ can be deformed, for each $v\neq 0$, to a simple object of $\mathcal{M}_{q,t}(\mathsf{H}_v,\mathbb{C}\boldsymbol{X})$.

Let $\theta$ be a quasisplit real form involution of $G$, and let $\check\theta$ be the dual involution of the Langlands dual group $\check G$. The stacks of fixed points are…

Let $(C,0)$ be a planar curve singularity. Let $L(q,t,a)$ be its Hilbert $L$-function and let $\mathrm{H}^{\mathrm{mot}}(q,t,a)$ be its normalized motivic superpolynomial. Cheredni…

Let $\mathrm{Hilb}^n(\mathbb{C}^2)$ denote the Hilbert scheme of $n$ points on $\mathbb{C}^2$, and let DAHA Verma modules be the corresponding representation-theoretic objects. Sta…

Partially symmetrized DAHA construction conjecture. The representation of $\mathbb{B}_{q,t}^{\mathrm{ext}}$ on $\bigoplus_{k=0}^{N}\mathbb{C}[x_1,\ldots,x_{N}]^{S_{N-k}}$ can be co…

Degeneration conjecture. The algebra $H_{D_4}(t,q)$ could be obtained as a limit of $H_{E_6}(t,q)$. Theorem 1.1 proves this conjecture by constructing the indicated functor for the…

Isomorphism conjecture. The functors $\bbV^{\lambda}_c$ and $\bfV^{\lambda}_c$ are isomorphic.

Hikami's genus 2 DAHA is obtained by gluing together the $A_1$ and $(C^\vee C_1)$ spherical DAHAs. Hikami's conjecture. His algebra can also be used to compute coloured Jones polyn…

Double-affine Kazhdan–Lusztig involution conjecture. There exists a completion of $\mathcal{H}_I$ such that the displayed right-hand side defines an algebra involution.

Let $K\subset S^3$ be a knot. Let $\widehat{K}_q(S^3\setminus K)$ be its nonsymmetric skein module, let $\widehat{K}^{\mathrm{loc}}_q(S^3\setminus K)$ be its localization, and let…

Let $\xi$ be an admissible system, let $\bbH_{\xi}$ be the associated degenerate double affine Hecke algebra, and let … be the ind-object obtained from spiral induction. For a simp…

Let $K\subset S^3$ be a knot, let $\sk_q(S^3\setminus K)$ be the Kauffman bracket skein module of its complement, and let $\mathrm{SH}_{q,t_1,t_2,1,1}$ denote the spherical double…

Let $\ddot{X}_n$ be a double affine Coxeter diagram with finite nodes labelled $1,\ldots,n$. Let $B(X_n)$ be the Artin group of the finite subdiagram, and let $\mho$ and…

Let $\dddot{X}_n$ be a double affine Coxeter diagram, let $B(X_n)$ be the Artin group associated to its finite subdiagram, and let $\mho_{01}$, $\mho_{02}$, and $\mho_{03}$ be the…

Let $M$ be a nonsymmetric skein module with localization $M^{\mathrm{loc}}$, let $\boldsymbol{\mathrm{e}}M=K_q(S^3\setminus K)$, and set $x=X+X^{-1}$. Define $U_0$ as the localized…

Let $K\subset S^3$ be a knot. Let $\hat K_q(S^3\setminus K)$ be its nonsymmetric skein module, let $\bar K_q(S^3\setminus K)$ be the quotient by the image of the skein module of th…

Let $K\subset S^3$ be any knot. Write $K_q(S^3\setminus K)$ for its Kauffman bracket skein module, and let $A_q^{\mathbb{Z}_2}$ act on these modules. Unknot submodule conjecture. F…