31 problems

Let $F$ be a finite extension of ${\mathbb Q}_p$, and let $G$ be a quasi-split group over $F$. Infinitesimal-character conjecture for quasi-split groups. Does the infinitesimal-cha…

Let $G$ be a connected reductive group over a local field with Lie algebra $\frak{g}$, let $L$ be a sufficiently large extension of ${\mathbb Q}_p$ over which $G$ is split, and let…

Kudla–Rallis conservation relation. The first occurrence indices satisfy

Let $\operatorname{E}/\operatorname{F}$ be the setting of distinguished representations, and let $\pi$ be a supercuspidal representation of $\operatorname{GL}(n,\operatorname{E})$…

Let $u$ be a closed point of $U$, let $\LS_{\prescript{L}{}{G},F_u}$ be the local Langlands parameter stack, and let $\LS_{\prescript{L}{}{G},\cO_u}$ denote its unramified locus. L…

Let $F$ be a local field, let $G$ be a quasisplit reductive group over $F$, and let $B$ be a Borel subgroup with unipotent radical $N$. Choose a generic continuous character…

Let $G$ be a connected reductive algebraic group over a locally compact non-Archimedean local field $F$. Let $\lambda:W_F\to{}^LG$ be an infinitesimal parameter. For geometric para…

Categorical mod-$p$ Langlands conjecture. The Hecke $\operatorname{Ext}$-algebra is isomorphic to the $\operatorname{Ext}$-algebra of a certain line bundle on $C_\sigma$.

Assume that $G$ is quasi-split, with maximal torus $T\subset B\subset G$, and let $\phi:W_{\mathbb Q_p}\to{}^LG(\overline{\mathbb F}_\ell)$ be a toral semisimple $L$-parameter, mea…

Let ${\rm G}$ be a classical group, let $\mathfrak{Z}_{{\rm G},\overline{\mathbb{Z}}[1/p]}^{\operatorname{st}}$ denote the stable center, and let $\mathcal{EP}({\rm G})$ be the set…

Let ${\rm G}$ be a reductive group over a non-archimedean local field, let $5chat{{\rm G}}$ be its dual group, and let…

Let $({\rm G},h)$ be an inner form of a general linear group or of a classical group with $p eq 2$, and let $frak{t}$ be an endo-parameter for ${rm G}$. Let ${rm R}$ be an integral…

Square-irreducibility conjecture. $\mathrm{Z}(\mathfrak{m})$ is square-irreducible if and only if $\mathcal{C}_\mathfrak{m}^+$ contains an open $\mathrm{GL}(V)$-orbit.

Let $\ell\neq p$ be primes, let $W_{\mathbb{Q}_p}$ be the Weil group of $\mathbb{Q}_p$, and let $\mathrm{G}$ be a reductive group over $\mathbb{Q}_p$. Write…

Let $(\mathrm{G},\mathrm{G}')$ be a reductive dual pair over a non-Archimedean local field, let $\Pi'$ be an irreducible admissible representation of $\widetilde{\mathrm{G}'}$ occu…

Cauchy Harish-Chandra character correspondence conjecture. Assuming the character conjecture referred to in the source, define

Let $G={\rm GL}_n(F)$, let $X_{\check G}$ be the space of L-parameters $(\varphi,N)$ for the principal block, and let $\widetilde{\mathcal{V}}_G$ be the family of smooth $G$-repres…

Let $F$ be a non-archimedean local field, let $\mathbb{G}$ be a split reductive group over $F$, and set $G=\mathbb{G}(F)$. Let $C$ be a characteristic-zero field containing…

Non-proportionality conjecture. For any irreducible quotient $\pi$ of $\operatorname{ind}^G_K \sigma/(T_\sigma)$, the images of $f_0$ and $f_1$ in $\pi$ are not proportional.

Let $\psi:P_F\rightarrow {^{L}\mathbf G}$ be a wild parameter, and let $\mathbf G_\psi$ be the associated tamely ramified group of ${\rm GL}$-type with…

Let $\mathbf G$ be a group of ${\rm GL}$-type, let $\phi:I_F^{(\ell)}\rightarrow {^{L}\mathbf G}$ be an admissible parameter, and choose an extension…

Let $\xi:{^{L}\mathbf G'}\rightarrow {^{L}\mathbf G}$ be as above, let $\phi'$ and $\phi=\xi\circ\phi'$ be the associated admissible parameters, and let…

Let $\mathbf{G}'$ and $\mathbf{G}$ be groups of ${\rm GL}$-type with an $L$-homomorphism $\xi:{^{L}\mathbf{G}'}\rightarrow {^{L}\mathbf{G}}$. Let…

Let $G=\operatorname{GL}_n$ and let $b=1$, so that $J_1(F)=G(F)$. For a suitable pair $(f,\underline{w}_f)$, write … for the affine Deligne–Lusztig induction map, and suppose that…

Local converse conjecture. If