16 problems

Let $\bmod\text{-}\mathcal{H}_q^{l}$ be the subcategory of $\mathcal{H}_q$-modules on which $\gamma=q^l\operatorname{Id}$, and let…

Let $M_1(\bm{\lambda},\bm{\mu})$ and $N_1(\bm{\eta})$ be the Weyl and Heisenberg modules described in the source, with $\mu_M\ne0$, $\lambda_N\ne0$, and $\eta\ne0$. Let $\psi$ be a…

Let $m\in\mathbb{Z}_+$ and $n\in\mathbb{N}$ have different parities. Let $\mathcal{G}^{(m,n)}$ be the corresponding subalgebra of the planar Galilean conformal algebra…

Suppose that $\mathfrak{g}=\mathfrak{g}_{\bar 0}$, let $\zeta\in\mathfrak n_{\bar 0}^+$ be generic, and let $\lambda$ be anti-dominant. Write $L(\lambda,\zeta)$ for the correspondi…

Let $eta$ be a character of the nilpotent subalgebra, let $Uinmathcal{Z}mathfrak{g}$, and let $Ieta(U)$ be the associated Whittaker-induced module. Denote by $Psi{Ieta(U)}$ the v…

Let $mathfrak{g}$ be the Lie algebra under consideration, let $Z(mathfrak{g})$ be its center, and let $U$ be a finite-dimensional $Z(mathfrak{g})$-module. Let $Ieta(U)$ denote th…

Assume that $eta$ is a nondegenerate character, let $W$ be the Weyl group, and let $mathcal{Z}mathfrak{g}$ be the category of modules used in the Whittaker induction functor…

Fix a regular weight lambdainmathfrak{h}^ and let $X$ be the flag variety of $mathfrak{g}$. Let $Q$ be an open $B$-orbit in a $Peta$-orbit on $X$, let $mathcal{O}Q$ be an…

Duality involution conjecture. If $X\in\mathcal{N}_\eta$ and $betainmathbb{C}[Weta]$ is nonzero, then $X^{veebeta}inmathcal{N}eta$ and

Let $M(1)$ be the rank-$\ell$ Heisenberg vertex operator algebra, let $S_{\ell}$ act on it by permuting a basis $h_1,\dots,h_{\ell}$ of the underlying space, and let…

Whittaker-like-state conjecture. The state $|\mathcal{N}_{L,\xi}\rangle$ satisfies the Whittaker-like conditions referred to as. This conjecture proposes that the state created by…

Let $|1^d\rangle=[\mathcal U_G^d]$ be the fundamental class in equivariant intersection cohomology, and let $\widetilde W^{(\kappa)}_n$ denote the positive modes of the generators…

Let $\mathfrak{w}_n$ be the Witt algebra with positive nilpotent subalgebra $\mathfrak{n}_+$, and let $L(\lambda,\mu)$ be the modules constructed in Theorem 91 from parameters sati…

Let $\mathfrak{g}$ be in the setup of the preceding construction, let $\mu\in\mathfrak{h}^*$ be such that the Verma module $M(\mu)$ is simple, and let…

Let $\mathfrak{g}$ be a Lie algebra with a fixed triangular decomposition … and set $\mathfrak{n}=\mathfrak{n}_+$. Let $\lambda\in(\mathfrak{n}/[\mathfrak{n},\mathfrak{n}])^*$, let…

Let $\mathfrak{g}$ be a Lie algebra with a fixed triangular decomposition … Let $L$ be a simple Whittaker module for the Whittaker pair $(\mathfrak{g},\mathfrak{n}_+)$. Write…