18 problems

The geometric interpretation conjecture. The function $\chi_1$ is well-defined and

Freeness conjecture. The stalks of $\mathcal I_!(\lambda,\mathbb Z)$ are free.

Let $G$ be a semisimple group, let $I\subset L^+G$ be an Iwahori subgroup, and let $Gr$ be the affine Grassmannian. Let $P_I(Gr)$ be the category of $I$-equivariant perverse sheave…

Let $\widetilde{M}(r)$ be the rank-$r$ moduli space, with matrices $(B_1,B_2,I,J)$, and let $\widetilde{M}(r)^{\mathbb{C}^\times}$ be the fixed locus of the $\mathbb{C}^\times$-act…

Let $Q=({\mathbb I},\Omega)$ be an ADE quiver, let $(V_i;W_i)_{i\in{\mathbb I}}$ be a representation of its framed double quiver, and impose the stated Satake-diagram symmetry and…

Let $0<k\le d<n$ be integers and let $\beta\in[0,k+1]$. Let $\mathcal{V}\subset A(k,n)$ be a set of $k$-planes in $\mathbb{R}^n$, with … Here, $A(k,n)$ denotes the affine Grassmann…

Let $G$ be the group under consideration and let $\lambda$ be a dominant coweight. Let $\overline{\mathcal{Y}^\lambda}\subset\operatorname{Gr}_G$ be the moduli scheme of principal…

Let $G$) be a simple and simply-connected complex Lie group with Borel subgroup $B$, maximal torus $T$, Weyl group $W$, affine Weyl group $W_{\mathrm{af}}=W\ltimes Q^\vee$, and cor…

Let $G$ be a complex semisimple group and let $\mu\leq\lambda$ be dominant coweights. Write $\mathcal{G}r_\mu^\lambda=\mathcal{G}r_\mu\cap\overline{\mathcal{G}r^\lambda}$ for the t…

Let $boldsymbol{G}^ atur$ be a complex connected reductive group Langlands-dual to $boldsymbol{G}$, let $\beta$ be a field of coefficients, and let…

Good-characteristic parity conjecture. If $p$ is a good prime for ${\mathbf G}$, then the standard sheaves with coefficients in a field of characteristic $p$ are $*$-parity.

Let $\Gr^{\overline{\lambda}}_\mu$ be the affine-Grassmannian slice in $\Gr_\mu$, and let $J^\lambda_\mu$ be the Poisson ideal of $\mathcal O(\Gr_\mu)$ generated by…

Let $\Gr^{\overline{\lambda}}_0$ be the slice at $\mu=0$ inside $\Gr_0=G[[t^{-1}]]_1$, and let $J^\lambda_0$ be the Poisson ideal of $\mathcal O(\Gr_0)$ generated by the coefficien…

Let $G$ be the reductive group and let $\Grlmbar$ be the affine-Grassmannian slice under consideration. A symplectic resolution is a resolution of singularities whose pullback of t…

Let $\mathcal{K}_{\mathrm{Gr},m}$ be the geometric categorical $\mathfrak{sl}_\infty$ 2-representation, let $\mathcal{K}^{\pm}_{\mathrm{Gr},m}$ denote its bounded-sided homotopy ca…

Let $Y({\underline{i}})$ be the affine-Grassmannian variety associated with ${\underline{i}}=({\0},1^k,{\u2})$ for some $k\in\mathbb N$, and let … be the projection forgetting the…

Lam's conjecture. Every $G_w$ is a finite alternating linear combination of the $G^{(k)}_\lambda$; every $G^{(k)}_\lambda$ is an alternating integer linear combination of the…

Lam's conjecture. The following hold: each $g^{(k)}_\lambda$ is a finite sum of $k$-Schur functions with positive integer coefficients;…