23 problems

Miller–Schmid's algorithmic conjecture. The formula obtained by their algorithm satisfies all properties (i)–(v) of the birational-map theorem: it is smooth and of maximal rank on…

Let $\pi$ be the representation under consideration, with central character $\omega_\pi$ of conductor exponent $a(\omega_\pi)$, and let $n_1$ and $m_1$ be the parameters used in th…

Let ${\bfit\gamma}=(\gamma_1,\ldots,\gamma_{N-1})$ be real parameters, let $\psi_{\bfit\gamma}(\bfit x)$ be the Whittaker functions of the quantum periodic Toda chain, and let…

There is a metaplectic spherical Whittaker function, denoted $\widetilde{\mathcal{W}}^{m} = \widetilde{\mathcal{W}}^{m}(\mathbf{y}; g;v)$, and components…

Gray's U-turn lattice models associate a partition function to metaplectic covers of the symplectic group. Gray's conjecture. The partition function is the spherical Whittaker func…

Buttcane's conjecture. Let $g$ be holomorphic with rapid decay on an open tube domain in $\mathfrak{a}^*_{\mathbb{C}}$ containing $\mathfrak{a}^*$, let $t\in G(\mathbb{R})$, let…

Let $l>-\frac{1}{2}$, and let $z_l(k)$ be the real root of $\mathcal M_{k,l}$ defined near $k=l+\frac{3}{2}$ by $z_l(l+\frac{3}{2})=1+2l$. The conjecture is that its maximal interv…

In the type $B$ six-vertex model with reflecting boundary, let the wavefunctions be the partition functions obtained using the boundary $K$-matrix introduced by Brubaker, Bump, Chi…

Critical-resonance conjecture. If

Boundary-polynomial conjecture.

McNamara's conjecture.

Let $n$ be odd, let $r$ be a positive integer, and let $Z_{\lambda}(\mathbf{z};\mathbf{c})$ be the partition function of the symplectic-ice model with partition boundary data deter…

Let $\rho$ denote the action referred to in the source, let $\funct{Wh}$ be the space of Whittaker functionals, and let $\funct{Wh}_{\widetilde{K}}$ be its subspace associated with…

Let $G$ be a semisimple $p$-adic group with Weyl group $W$, let $x\leq w$, and let $S(x,w)$ be the set of positive roots $\alpha$ such that $x\leq ws_\alpha<w$. For the transition…

Weyl group multiple Dirichlet series are series in several complex variables associated with root systems, whose functional equations form a group isomorphic to the Weyl group. Fou…

Let ${\mathrm{Gr}}_{m,N}$ be the Grassmannian, and let $\Psi^{(m,N)}_{\underline{\lambda}}(x_1,\ldots,x_N)$ be the parabolic Whittaker function defined as a matrix element in a pri…

Grassmannian quantum-cohomology conjecture. $(m,\ell+m)$-Whittaker functions are relevant to a description of $S^1\times U_{\ell+m}$-equivariant quantum cohomology of…

Grassmannian Whittaker–Gromov–Witten conjecture. $(m,\ell+m)$-parabolic Whittaker functions describe equivariant Gromov–Witten invariants of $\operatorname{Gr}(m,\ell+m)$, thus gen…

Parabolic Whittaker–Gromov–Witten conjecture. A $P$-parabolic $\mathfrak{gl}_{\ell+m}$-Whittaker function describes $S^1\times U_{\ell+m}$-equivariant Gromov–Witten invariants of…

Markov-operator conjecture. The operator $\mathbb K_\mu$ is a Markov operator in general. This is known in types $A_1$ and $A_2$, and follows from integral representations there; t…

Stade's conjecture. For $\nu\in U$,

Unimodular Whittaker-limit conjecture. This limit exists. Moreover, if $X$ and $\lambda$ are real with respect to the anti-involution $f(x)\mapsto f(\overline{x}^{\,\varsigma})$, w…

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