16 problems

Derivative conjecture. The representation $\rho_1(\tau)$ is $(rk,1)$. If $F$ is non-archimedean, then for every $c>1$ the highest derivative of $\rho_c(\tau)$ is

Let $G$ be a reductive group over a local field $F$, and let $\overline{G}^{(n)}$ be a Brylinski–Deligne cover of $G$. Write $\operatorname{Irr}_{\rm gen}(\overline{G}^{(n)})$ for…

Let $G$ be the underlying group, let $\textnormal{\textsf{Basic}}_G$ be the set of basic data, and for each $\beta\in\textnormal{\textsf{Basic}}_G$ let $\widetilde G_\beta$ be the…

Let $F$ be the local field and let $\widetilde G$ be the cover under consideration, with associated dual group data $\widetilde H$. Write $\Pi(\widetilde G)$ for the relevant set o…

Maximal Whittaker-orbit conjecture. One has $\mathcal{O}^{r,n}\leqslant\mathcal{O}$ for every $\mathcal{O}\in\mathcal{N}_{\rm Wh}^{\rm max}(\pi)$.

Let $\overline{G}$ be a covering group with subgroup $\overline{G}_0$, let $\overline{G}^\dag=\overline{Z(G)}\cdot\overline{G}_0$, and let $\phi_\pi$ be an $L$-parameter of…

Let $G$ be a group, and let $\sigma_n(G)$ denote its normalizer covering number, the least number of proper normalizers whose union is $G$, when such a covering exists. Normalizer…

Intertwining-operator conjecture. For all $1<l\leq r$, $M(\zeta,w_{(k^{l})})$ is well defined at $\zeta=\zeta^{(l)}$ and has irreducible image. For all $c\geq1$,…

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 $\chi$…

Let $\mathcal{P}_W(T)$ and $\mathcal{P}_{\rm exc}(T)$ be the two Poincaré series associated with an $n$-fold cover of a connected reductive group $G$. In the examples discussed, bo…

Let $F$ be the relevant number field, let $\nu$ be a place of $F$, and let $\tau_\nu$ be the local representation used to form the induced representations. For $l\geq 2$, write…

Let $(\widetilde{\operatorname{GL}}_r^{(n)},\widetilde{\operatorname{GL}}_R^{(n)})$ be a fundamental pair with $n\mid(\mathbf{p}\cdot n_\alpha)$. Let…

Let $(\widetilde{\operatorname{GL}}_r,\widetilde{\operatorname{GL}}_R)$ be a fundamental pair of $n$-fold covering groups satisfying $n\mid(\mathbf{p}\cdot n_\alpha)$. Let $\pi$ be…

Let $k$ and $c$ be positive integers, set $b=mc$, and let $\tau^{(m)}$ be a genuine irreducible cuspidal automorphic representation of $GL_k^{(m)}({\mathbb A})$. For…

Let $F$ be a local field. Let ${\boldsymbol{\Pi}}_\epsilon(\boldsymbol{\mathrm{\tilde G}} / F)$ denote the set of equivalence classes of irreducible admissible $\epsilon$-genuine r…

Let $G$ be a group with finite covering number … sigma(G)$is the smallest number of proper subgroups needed to cover$G$. Call$G$…$sigma(G)<$sigma(G/N) … sigma$-elementary grou…