22 problems

Genericity conjecture. A pure $L$-packet $\Pi^{\text{pure}}_\phi(G)$ is generic if and only if $\phi$ is an open parameter for $G$.

Let $G=\mathrm{GSp}_4$, let $\pi=\otimes_v\pi_v$ be an irreducible, unitary, cuspidal, generic automorphic representation of $G(\mathbb A)$ with trivial central character, and let…

Let $F$ be a non-Archimedean local field, let $G=\operatorname{GL}_{2n}(F)$, and let $P_{n,n}=MN$ be the standard parabolic subgroup with Levi factor…

Let $F$ be a nonarchimedean local field with ring of integers $\mathfrak{o}$, and let $\mathfrak{o}_l=\mathfrak{o}/(varpi^l)$. Let $L$ and $E$ be unramified extensions of $F$ of de…

Whittaker-space isomorphism conjecture. For every $W$-orbit $\mathcal{O}\subset\mathscr{X}_{Q,n}$, the embedding is an isomorphism

Let $E/F$ be a quadratic extension, let $G$ be a reductive group over $F$, and let $N_0$ be the unipotent subgroup associated with a fixed Whittaker datum. Let $\pi$ be an irreduci…

Let $r>1$ be odd, let $F$ be a global field, and let $\Theta_{r-1}^{(r)}$ be the theta representation of the $r$-fold metaplectic cover of $Sp_{r-1}({\mathbb A})$. Put…

Let $\tilde{\varphi}_0:X_{Q,n}/X_{Q,n}^{sc}\to X/X_G^{sc}$ be well-defined, let $\mathfrak{w}=(B,U,\psi)$ be Whittaker data, and let…

Let $\overline{G}_0$ be a saturated cover, let $\mathscr{X}_{Q,n}$ be the relevant lattice, and let $\mathscr{X}_{0,Q,n}$ be the Weyl-stable sublattice. Let $W$ act by the usual We…

Let $\overline{G}$ be an $n$-fold persistent covering group. Let $\Theta(\overline{G},\chi)=\pi_{\Gamma^-}$ be the theta representation associated to an unramified character $\chi$…

In the situation of Theorem, let $F_G(K_{E,\chi, \mathbb H})$ be the complex on $\operatorname{Bun}_{\mathbb G}$ constructed there, and let $\Whit$ denote the first Whittaker coeff…

Whittaker vanishing conjecture. If $J$ satisfies this dimension equation, then $J$ is zero for every choice of data.

Let $x$ be a point of the curve, let $\Whit_x^\kappa$ be the metaplectic Whittaker category at $x$, and let $\mathcal F_{x,\mu}$ denote the Whittaker object indexed by a dominant c…

Let $\rho$ be the quadratic form on the coweight lattice associated with the metaplectic data. For $\bar c$ the residue-field point and $\bar\theta$ the image of $\theta$ in the ad…

Arthur-component-group conjecture. For every $\phi\in\Psi^t(G)$,

Classical-group descent conjecture. If $\pi$ is an irreducible constituent of $\sigma^{\psi_N}(\{\pi_1,\dots,\pi_k\})$, then

Arthur-parameter conjecture. For every elliptic Arthur parameter $\rho$ of Ramanujan type,

Let $G$ be a connected reductive group, let $\operatorname{Bun}_G$ be its moduli stack of bundles, and let $\operatorname{Whit}^{\operatorname{ext}}(G,G)$ be the extended Whittaker…

Whittaker and theta-lifting conjecture. (i) $\mathcal K_E$ is a perverse sheaf, each of whose $\mathbb Z/2\mathbb Z$-parity pieces is irreducible, and

Square-root conjecture. For any $d\geq 0$, there is a complex ${\cal S}_E^d\in\operatorname{D}(\operatorname{RCov}^d)$ equipped with an isomorphism

Let $G$ be a reductive group with Langlands dual group $G$, and let $\kappa$ and $check\kappa$ be invariant forms on their Lie algebras such that, for the mutually dual Cartan subal…

Let $G$ be a reductive group, $G$ its Langlands dual group, and let $W^c(Gr_G):=D^cmod(Gr_G)^{N((t)),\chi}$ denote the twisted Whittaker category, where $c$ is the twisting paramete…