54 problems

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 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…

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.

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

The coefficient conjecture. For $k<d$, the coefficient of $t^k$ in $\mathcal P_{\mathbf p}(\mathbf n)$ is $0$; for $k>d$, the coefficient depends on the choice of $\mathbf n$.

The cluster-monomial coefficient conjecture. The coefficient of $t^d$ in $\mathcal P_{\mathbf p}(\mathbf n)$ is independent of the choice of $\mathbf n$ representing $Y$, and defin…

k-Schur multiplication positivity conjecture. The structure constants satisfy

Let $\operatorname{Gr}$ be the affine Grassmannian, and let affine Schur functions $\tilde{F}_\lambda$ be indexed by the corresponding Grassmannian affine permutations. Morse–Shimo…

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

Let $G$ be a connected reductive group, let $\check G$ be its Langlands dual group, and let $\Lambda_G$ be the union of conormal bundles to the $G({\mathbf O})$-orbits in the affin…

Let $G$ be a connected reductive group, let $G({\mathbf O})$ act on the affine Grassmannian $\operatorname{Gr}_G$, and let $\mathbb G_m$ act by loop rotation. Write $\check G$ for…

Iwahori–Whittaker–Iwahori-monodromic equivalence conjecture. There exists an equivalence between the mixed versions of $D_{\mathcal{IW}}$ and the category of Iwahori-monodromic she…

Let $\CF$ be a $\bG[[z]]$-equivariant perverse sheaf on the affine Grassmannian $\CG$. The natural projection … has target identified with … The algebra $\CA=U(\fn^L_+)$ acts on th…

Let $\CF$ be a $\bG[[z]]$-equivariant perverse sheaf on the affine Grassmannian $\CG$. For each $\beta\in Y$, let $T_{-\beta}$ denote the corresponding semiinfinite orbit, and cons…

Quasimap coefficient conjecture. There is an isomorphism of $\fg^L$-modules

Tilting conjecture. In the $\bG[[z]]$-equivariant case, this resulting $\fg^L$-module is tilting. This predicts that the geometric $\fg^L$-action on the global cohomology of quasim…

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 the affine Grassmannian slice carry its category of perverse coherent sheaves and equivariant K-theory, and let the corresponding quantum cluster algebra be equipped with Qin's…

Let ${}^\imath\overline{\mathcal{W}}_\mu^\lambda$ be an affine Grassmannian islice, let $\overline{C}_\mu^{\underline{\lambda}}$ be the indicated subvariety, and let…

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…

Dolbeault-homology conjecture for Dirichlet boundary conditions. The algebra of local operators on the Dirichlet boundary condition is

Let $T$ be a nilpotent matrix in Jordan canonical form, let ${\rm Gr}(\ell,n)^T$ be the variety of $\ell$-dimensional $T$-fixed subspaces, and let the shuffle equations be the line…

Let the symplectic $k$-Schur functions be the functions defined by Lam, Schilling, and Shimozono for the affine Grassmannian of a symplectic group. Seelinger's conjecture. The symp…