11 problems

Let $G$ be a reductive group with Langlands dual group $G^\vee$, and let $W$ be the Weyl group. Let $X$ be a smooth spherical $G$-variety such that $T^*X$ is hyperspherical and has…

Let $G$ be a reductive group acting on a smooth spherical variety $X$, and let $W$ be the Weyl group of $G$. Let $M^\vee$ be the dual Hamiltonian space attached to $X$ by relative…

Let $G$ be a complex reductive group, let $X$ be a smooth affine spherical $G$-variety with connected stabilizers in the open Borel orbit, and set $M=T^*(2)(X)$. Let…

Assume that $\check{M}=T^*\check{X}$, where $\check{X}$ is a $\check{G}$-spherical variety. Let $\pi$ be a tempered, everywhere unramified automorphic representation of…

Dolbeault S-duality matching conjecture. Over the $\diamondsuit$-locus, the following matching data hold under $\mathbf{S}_{\mathrm{Dol}}$: the dual of $A_{\mathrm{Dol}}(M_\rho')$…

A hyperspherical Hamiltonian action is a distinguished class of Hamiltonian boundary condition arising in the relative Langlands program. Relative Langlands conjecture. Hyperspheri…

Let $V$ be a nondegenerate quadratic space of even dimension $2n$ containing a nondegenerate line $L$, with $V=L\oplus L^{\perp}$, and let $X={\rm O}(V)/{\rm O}(L^{\perp})$. In thi…

Let $G$ be a complex reductive group and $X$ a homogeneous spherical $G$-variety. Write $\check G$ for the Langlands dual group and $\frak L X/\frak L^+G$ for the corresponding loo…

Let $(G,H)$ be a strongly tempered spherical pair over a local field $F$, and suppose that the spherical variety $G/H$ has one open Borel orbit over $F$. An $(H,\chi)$-distinguishe…

Sakellaridis–Venkatesh's conjecture. If $\pi$ is $H(F)$-distinguished, then $\pi$ should be a Langlands functorial lift from the dual group $G_X$ attached to $X$.

Let $G$ be a quasi-split group over a local field $F$, let $E/F$ be a separable quadratic extension, and let $\pi$ be an irreducible admissible representation of $G(E)$. Let…