25 problems

Let $X_{\la}$ be the affine Grassmannian Schubert variety associated with a dominant coweight $\la$, let $\mathcal A(\la)$ be the union of Schubert varieties in the affine flag var…

Reuman's conjecture. If $x$ lies in the shrunken Weyl chambers, then $X_x(b)\ne\emptyset$ if and only if

Coherence conjecture. Provided that $\mu$ is a sum of minuscule coweights for $G_{\rm ad}$,

Derived Koszul duality conjecture. The category of perverse sheaves on the affine flag variety of a semi-simple group, smooth along the Schubert stratification, is derived Koszul e…

Let $\fg$ be the Lie algebra under consideration, with index set $I$, and let $\fF^\lambda_{\fg,\fb}$ be the corresponding fiber space with open stratum…

Let $G$ be a reductive group with Langlands dual $G^\vee$, let $C$ be the underlying curve, and let $D=\{c_i\}$ be the parabolic divisor. Let $(E,\varphi,Q)$ be a smooth fixed poin…

Geometric Jordan-type conjecture. For all $i,j\in\mathbb{Z}$, the partitions in the growth diagram of $w$ should be interpretable as the Jordan types of

Jordan-type conjecture. There is an open set $\mathcal{N}\subseteq\mathfrak{n}_F\cap\mathfrak{n}_{F'}$ such that, for every $er\in\mathcal{N}$,

Geometric realization conjecture. The growth diagrams constructed in the paper should admit a natural geometric realization in terms of relative positions of affine flags, analogou…

Periodic Kazhdan–Lusztig stalk conjecture. The stalks of simple objects in the category of semiinfinite sheaves are computed by the periodic Kazhdan–Lusztig polynomials $p_{y,w}$.

Let $\Fl_{\calG}$ be an affine flag variety, and let $\Fl_{\calG,\leq w}$ be its Schubert subvariety indexed by a double coset $w$ of the Iwahori--Weyl group. Its perfection is den…

Let $F$) be a nonarchimedean local field, let $G$ be a quasi-split connected reductive group over $F$, and let $K$ be a special parahoric subgroup. For…

Let $\lambda$ and $\tau$ be parameters as in the paper, with combinatorial genericity meaning that the relevant weights are sufficiently deep in the base alcove. The main theorem g…

Let $V_1$ and $V_2$ be $N$-dimensional vector spaces, let $\bI$ be an Iwahori subgroup, and let $\bFfl_i$ be the variety of complete flags in $V_i$. Define the dg-scheme with zero…

The local model conjecture. There exists a projective and flat scheme … over … which supports an action of ${\mathcal G}\otimes_{\mathbb Z_p}{\mathcal O}_E$ and is such that: (i) t…

Open-and-closed immersion conjecture. The maps $\iota^0$ and $\iota^\pm$ are always open and closed immersions.

Let $\mathfrak g$ be a simple Lie algebra over $\mathbb C$, let $h$ be its Coxeter number, let $t\geq1$ be relatively prime to $h$, and let $N_t$ be the topologically nilpotent reg…

Let $G$ be a simply connected, simple, complex Lie group, let $B\subset G$ be a Borel subgroup, and let ${\mathcal F\ell}_{G}$ be the associated affine flag manifold. Write…

Assume that $\mathcal G_\eta$ is semisimple, simply connected, and absolutely simple, and that $\mathcal G_{\eta_x}$ splits over a tamely ramified extension of $K_x$ for every…

Let $G$ be as in the paper, let $\mathcal{F}\ell_G$ be its affine flag variety, and let $I^0$ be the pro-unipotent radical of the Iwahori subgroup $I$. Write…

Basic-class comparison conjecture. There exists $N_b\in\mathbb Z_{\ge 0}$ such that, for every $x\in\widetilde W$ with $\ell(x)\ge N_b$,

P-alcove conjecture. $X_x(b)\ne\emptyset$ if and only if, for every semistandard $P=MN$ for which $x\mathbf a$ is a $P$-alcove, $b$ is $\sigma$-conjugate to an element $b'\in M(L)$…

Let $\bD_{\mathrm{ren}}(\fD(\Fl^{\operatorname{aff}}_G)_\crit\operatorname{mod})$ carry the old and new t-structures, and let $J_\cla$ be the convolution objects indexed by…

Let $G$ be the reductive group in the affine flag construction, let $\bD^b(\fD(\bFl^{\operatorname{aff}}_G)_\crit\operatorname{mod})$ be the derived category of critically twisted…

Let $C$ be a smooth projective curve and let $\mathcal{G}$ be a semisimple Bruhat–Tits group scheme on $C$. The moduli stack $\operatorname{Bun}_{\mathcal{G}}$ parametrizes…