13 problems

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…

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 $H(\underline{t};q)$ 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…

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 $K\subset S^3$ 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 $K\subset S^3$ 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 $\mathcal{P}_{n,m}^{\lambda}(u,q,t)$ be the stabilized superpolynomial and define its reduced version by … Cherednik's duality conjecture. For the transposed partition…

Let $\mathcal{P}_{n,m}^{\lambda,N}(q,t)$ be the DAHA superpolynomial associated with a partition $\lambda$ in rank $N$. Cherednik's stabilization conjecture. There exists a polynom…

Let $\Gamma_n$ be the object whose $(i,j)$-graded piece is denoted by ${\rm{gr}}_{i,j}(\Gamma_n)$. Fix a rational parameter $h$ with denominator $m>1$, let…

Key conjecture. The commutative algebra $K_0(Rep_q\,G)$ can be canonically identified with the algebra of $W$-invariants in this polynomial representation of DAHA. For generic $q$,…