20 problems

Let $(\pi,V)$ be an irreducible supercuspidal representation of $G$, and let ${\mathscr{S}}$ be the $G$-equivariant Schneider–Stuhler sheaf on the building $X$ associated with…

Let $F$ be a nonarchimedean local field, let $\boldsymbol{\mathrm{G}}$ be a connected reductive group defined over $F$, and let $G=\boldsymbol{\mathrm{G}}(F)$ with center $Z$. Let…

Let $G_1,\ldots,G_p$ be twisted Levi subgroups, let $\phi_{(i)}:\fa_i\to\bC$ be refined leading terms, let $X_{(i)}\in\check{\fa}_i$ be their duals, and let $A'$ and the central su…

Let $G$ be a connected simple algebraic group, let $\hg$ be the associated affine Kac–Moody algebra at critical level, and let $(J,\tphi)$ be a $K$-type for the loop group obtained…

Supercuspidal wavefront-set conjecture. One has

Fix a reductive group $G$, a rigid inner twist, a Whittaker datum $\mathfrak w$, and the refined local Langlands reciprocity map $\operatorname{rec}_{\mathfrak w}$. An enhanced $L$…

Let $F_0$ be a local field, let $G$ be a reductive group over $F_0$, let $\pi_0\in\Pi(G/F_0)$ be any supercuspidal representation, and let $\Pi_0\subset\Pi(G/F_0)$ be the packet co…

Let $F$ be a local field, let $G$ be a reductive group over $F$, and call a supercuspidal representation pure incorrigible when it is pure and remains supercuspidal in the base-cha…

Let $G$ be quasisplit over $E$, fix a Whittaker datum, and assume that the connected split center of $G$ is trivial. Let $\varphi$ be an elliptic $L$-parameter, let $S_\varphi$ be…

Let $F$ be a non-archimedean local field, let $G$ be a reductive group that splits over a tame extension of $F$, and assume that the residue characteristic is sufficiently large fo…

Unitary-group supercuspidality conjecture. The packet $\Pi_{\phi}$ consists of supercuspidal representations if and only if $\phi$ is discrete and its restriction to…

Let $G$ be a reductive group and let $\mathcal{M}(G)$ denote the representations of $G$. Let $\pi\in\mathcal{M}(G)$ be irreducible and supercuspidal. A subgroup $K$ of $G$ is compa…

KST character-decay conjecture. The following assertions hold:

Let $G=\mathbf{G}(F)$ be the group of rational points of a connected reductive algebraic group over a non-archimedean local field, and let $\pi$ be a supercuspidal representation o…

Let $G/F$ be a connected reductive group, and let $|W|$ be the cardinality of the Weyl group of a maximal torus in $G\times\overline{F}$. Let $\mathfrak{p}$ be a prime such that…

Cuspidal Langlands parameter conjecture. Le $L$-paquet $cPi_{cvarphi}(G)$ contient des représentations supercuspidales de $G$ si et seulement si $cvarphi$ est un paramètre de Langl…

Let $F$ be a locally compact non-archimedean local field, let ${\mathrm{G}}_n={\mathrm{GL}}_n(F)$, and let $\psi$ be a nontrivial additive character of $F$. Assume that $\pi_1$ and…

Let $F$ be a non-Archimedean local field, let $N$ be a positive integer, and let $\bar{\pi}$ be a supercuspidal representation of $\mathbf{SL}_N(F)$. Choose a supercuspidal represe…

Let $G(F)$ be the group under consideration, let $T(F)$ be the relevant torus, and let ${}^w\Theta_{\chi}$ be the character formula associated with a character $\chi$ and an elemen…

Let $PGSp_6$ be the split adjoint group, and let $\operatorname{Irr}_g^\circ(PGSp_6)$ denote the set of isomorphism classes of its generic supercuspidal irreducible representations…