59 problems

Let $V$ be a linear representation of a complex reductive algebraic group $G$ and let … be the associated moment map. Set $V:=\mu^{-1}(0)$. For a generic…

Let $(\mathfrak g,\mathfrak k)$ be a symmetric pair listed in Table 1 in Section. Let the relevant Nakajima varieties carry an involution $\sigma$, and let the associated Steinberg…

The stronger factor-sequence conjecture. Under these hypotheses, $(\dots,X',Y',\dots)$ is also a factor sequence for $\gamma$.

Let $\mathfrak g$ be the Lie algebra used in the quiver-variety construction, let $\mathbf w^1$ and $\mathbf w^2$ be the two framing data, let $\mathfrak L(\mathbf w^i)$ be the cor…

Fix a quiver of finite type, take $m=(1,\dots,1)$ and $\lambda=0$, and let $d,v$ be fixed. Write $M_{m,0}(d,v)$ and $M_{0,0}(d,v)$ for the corresponding quiver varieties. Connected…

Let $M_{0,\lambda}(d,v)$ be the quiver variety in question, and set … Let $(L_d)_0$ denote the $0$-weight space of the Kac–Moody algebra associated to the quiver, for the represent…

Let $I'=I\setminus\{p\}$ index the irreducible components $Y_k$ of the exceptional variety $\mathfrak{L}=\mathfrak{L}_m$, and let $\widetilde E_k(m)$ be the characteristic function…

Let $\mathbf{v}$ be a dimension vector, let $\GL_{\mathbf{v}}$ be the corresponding product of general linear groups, and let $\mathbf{N}_{\mathbf{v}}$ be the associated representa…

Let $\mathcal{L}(n,r)$ be the specified closed projective subvariety of the Gieseker space, let $K_{\mathcal{T}}(\mathcal{L}(n,r))$ denote its equivariant $K$-theory, and let the c…

Let $\mathfrak M_{(0,-2i\zeta_{\mathbb{R}})}$ be the indicated quiver-variety moduli space, let $p_0$ be a fixed point of the $\mathbb{C}^{\star}$-action, and let $W^1(p_0)$ be the…

Let $(\mathbf{Q},\mathbf{W},\mathbf{v})$ be a quiver with potential and dimension vector, with associated quiver variety $X$ and superpotential, and let $Z$ be its critical locus;…

Let $(\mathbf{Q},\mathbf{W},\mathbf{v})$ be a quiver with potential and dimension vector, with associated gauge group $\mathsf{G}$, representation space $\mathsf{N}$, stability con…

Level-$(0,-|\boldsymbol w|)$ geometric action conjecture. The tensor product representation

Level-$(0,-1)$ geometric action conjecture. The representation $\mathcal F^{(0,-1)(k)}_u$ of $U_{t_1,t_2,p}(\mathfrak{gl}_{N,tor})$ is equivalent to an expected level-$(0,-1)$ geom…

Elliptic quantum group and geometric representation conjectures. (1) Vertex operators of $U_{q,p}(\widehat{\mathfrak g})$ or $U_{t_1,t_2,p}(\mathfrak g_{tor})$ defined as intertwin…

Let $X$ be a type A Nakajima quiver variety with its natural torus action, let $p$ be a $\mathsf{T}$-fixed point, and let $\boldsymbol{a}\to 0$ denote the non-equivariant limit. Qu…

Let $\root\in\Sig$ be a dimension vector of Type $\II$, and let the associated quiver variety be the variety determined by this dimension vector. Type II uniqueness conjecture. If…

Let $N(\I(\D[4],n))$, $N(\I(\D[m],1))$, $N(\I(\D[m],n)')$, $N(\I(\D[m],0)')$, $N(\I(\E[6,7],n))$, and $N(\I(m))$ denote the numbers of projective crepant resolutions for the indica…

Let $\mathcal{M}^{!}=\overline{\mathcal{W}}^{\lambda}_{\mu}$ be the 3d mirror dual of a zero-dimensional quiver variety, with unique torus fixed point $p^{!}$, and set…

Let $Q$ be a quiver without loops, let $\mathcal{M}=\mathcal{M}_{Q,\theta}(\mathsf{v},\mathsf{w})$ be a zero-dimensional quiver variety with $\theta=(1,1,\ldots,1)$, and let…

Homological Hikita conjecture. There is an isomorphism of graded $\mathbb C[(\mathfrak t_\bv/S_\bv)\times\ff]\otimes\mathbb C[\hbar]$-algebras

K-theoretic Hikita conjecture. There is an isomorphism of algebras over $\mathbb C[(T_{\bf{v}} / S_\bv) \times F] \otimes \mathbb C[q^{\pm 1}]$

Let \tilde{\frac?} be the tensor product variety associated with dominant weights $\lambda_1,\lambda_2$, and let $\omega^1,\omega^2$ be the corresponding framing data, with…

Let $Y$ be the additive character variety associated to a reductive group $G$, a genus $g$, and a tuple of $G$-orbits in the Lie algebra, under the additive genericity assumptions.…

Let $(Q,\mathbf v,\mathbf w)$ be a $T^{\sigma}_{[1^n]}[\SU(n)]$ quiver, and let $\mathbb W=\mathfrak S_n$ be its Namikawa–Weyl group. Let…