25 problems

Let $G$ be a semisimple algebraic group with Borel subgroup $B$, Weyl group $W$, and dominant weights $\lambda,\mu\in P^+$. For $v,w\in W$, let $V_v(\lambda)$ and $V_w(\mu)$ be Dem…

Let $B$ be a perfect crystal, let $\{w^{(k)}\}$ be a sequence of Weyl group elements, and let $\lambda=l\Lambda_0$ be the level-$l$ dominant integral weight for which the sequence…

Let $\mathfrak{g}=C^{(1)}_n$, let $B$ be the perfect crystal considered for this case, and set $\lambda=l\Lambda_i$ with $i\neq 0,n$. For the associated sequence $\{w^{(k)}\}$ and…

Let $\widehat{\mathfrak{g}}$ be an affine Kac–Moody algebra, let $\Lambda=m\Lambda_0+\lambda$ be a dominant integral weight, and let $V(\Lambda)$ and $V(\lambda)$ denote the corres…

Let $D(m,\lambda^\vee)$ be a Demazure module equipped with the current-algebra structure described in the preceding corollary. Quantum affine lift conjecture. The corresponding sta…

Let $D_q(m,\lambda)$ be a quantum Demazure module whose crystal graph is a subgraph of the crystal graph of the corresponding irreducible $U_q(\widehat{\mathfrak{g}})$-representati…

Let ${\mathfrak g}$ be the simple Lie algebra under consideration, let $\lambda$ be a dominant weight, and let $D(1,\lambda)$ and $W^1(\lambda)$ denote respectively the level-one D…

Quantum Demazure module conjecture. The module $V_{-\lambda^{\vee},q}(l\Lambda_0)$ can be endowed with the structure of a $U'_q(\widehat{\mathfrak{g}})$-module admitting a crystal…

Kirillov–Reshetikhin crystal conjecture. More generally, the crystal of $KR(lc_k^{\vee}\omega_k)$ is conjectured to be isomorphic to the crystal of a Demazure module after omitting…

Let $\widetilde{\mathbf{R}}_{\lambda,i}^{\ell}$ be the set appearing in the description of the level-$\ell$ Pieri formula, and let Theorem(1) denote the corresponding assertion for…

Cancellation-free identity. There exists some $m\leq l\leq n$ such that

Saturation conjecture. For every $w\in W$ and every dominant $\lambda$, $\operatorname{char}(V_\lambda^w)$ is saturated.

Lattice-point conjecture. If $G$ is of type $E_6$, $E_7$, or $E_8$, then

Existence conjecture. For every $\mu\in P$ and $k\in\mathbb{N}$, there exists a $1$-admissible element of $\mathcal{P}(\mu,k)$. This would give a uniform minimal admissibility para…

Let $M_1,\dots,M_k$ be the modules considered in the construction of the global fusion product, let $R(M_1,\dots,M_k)$ denote that global fusion product, let…

Let $\mathfrak{g}$ be a simple Lie algebra of type $D_n$, let $\lambda,\mu\in P^+$, and let $V(\lambda,\mu)$ be the graded $\mathfrak{g}[t]$-module generated by $v_{\lambda,\mu}$ w…

Let $\Lambda$ be a level-one fundamental weight, let $b,c\in P$, let $m\in\mathbf Q$, and let $p>0$. Let $D_b$ and $D_c$ be thick Demazure modules, let $\mathring{\mathbb D}_b$ and…

Let $P$ be the weight lattice, let $D_c$ and $\mathbb D_c$ be the thick and local Demazure modules, and define … with $\deg X_{i,j}=1$. Let $\widetilde{\mathfrak b}^-$ be the relev…

Extension duality conjecture. The same Ext-vanishing and one-dimensionality statement holds for type $E_8$ as well. The conjecture extends the categorical duality between global…

Uniqueness conjecture in the rigid case. Under the rigid-case condition, $\overline{v}_{\overline{\textbf{l}}}$ is the unique vector obtained from a trail in $T^-_s(\textbf{a})$.

Absence of false trails conjecture. The functions associated with the trails in $T^+_s(\textbf{a})$ should have $T^+_s(\textbf{a})$ equal to the $\mathbb Z$ convex hull of the $S$-…

Demazure–PBW conjecture. In the setting of the cited Demazure-module description, one has

Fix a dominant integral weight $\Lambda$ and a sequence $(w^{(k)})$ in $W^{\mathrm{aff}}$ such that $l(w^{(k)}) \to \infty$. Let…

Let $V(\lambda)$ be an irreducible highest-weight module, let $W$ be the Weyl group, and let $V_w(\lambda)\hookrightarrow V(\lambda)$ be a Demazure module for $w\in W$. For special…

The discussion concerns the connection between $q$-deformed Whittaker functions, specializations of Macdonald polynomials, and Demazure modules, established above for…