345 problems

Prasad's conjecture. The representation $\pi$ is $H$-distinguished if and only if both the restriction of $\phi$ to $W_E$ is trivial and $\pi$ is generic. Under the first condition…

Howe duality conjecture. If $\Theta(\pi)$ is non-zero, then it admits a unique irreducible quotient, denoted by $\theta(\pi)$ and called the small theta lift of $\pi$.

Let $G$ be a connected reductive $p$-adic group and let $\psi:W_F”\to{}^LG$ be an Arthur parameter. Write $\phi_\psi$ for the associated Langlands parameter, and let…

Let $\lambda$ be a Hodge type, viewed as an element of the character lattice of $G$, and let $\tau$ be a tame inertial type. Let $V(\lambda-\eta_{\mathbb{Q}_p})$ denote the restric…

Let $G=Sp_4(L)$, where $L$ is as in the paper. For $bin G(L)$, let … be the $\sigma$-centralizer of $b$, and let $\operatorname{def}(b)$ be the difference between the $F$-rank of…

Self-similarity conjecture. Assume that $n\geq 2$ and $p\geq d(n^2-1)$. Then no open subgroup of $SL_1^{ne}(D)$ is self-similar.

Let $G$ be a quasi-split reductive group over a non-Archimedean local field $F$, let $G'$ be an inner form, and let enhanced L-parameters for $G'$ be defined as pairs…

Let $F/F_{0}$ be a quadratic extension of $p$-adic fields, let $G=\mathrm{GL}_{n}(F)$), and let $\sigma$ be the non-trivial automorphism of $F/F_{0}$. Let $\tau$ be a unitary invol…

Dat's torsion conjecture. The cokernel of this induction map is $\widetilde w_G$-torsion. Dat proved this conjecture for $G=\operatorname{GL}_n(F)$. The source explains that the co…

Let $T\subset G$ be a maximal torus split over $F$, with root datum $(X(T),\Phi(T),X^{\vee}(T),\Phi^{\vee}(T))$, and let $G^\sphat$ be the dual group. Define … and let…

Let $F/F_0$ be the unramified quadratic extension and let $G(\mathbb{W})\times G(\mathbb{V})$ act on the formal scheme $\mathcal{N}_{m,n}$. For a regular element…

Let $G$ be a reductive group over a local field, let $\mu\in X_*(T)_+$ be a dominant cocharacter, and let $\tau$ represent the unique basic $\sigma$-conjugacy class such that…

Depth-preservation conjecture. For every $\pi\in\Pi(\phi)$,

Higher Deligne–Lusztig representations are representations associated with parahoric subgroups of $p$-adic groups, and compact induction produces representations of the ambient $p$…

Let $F$ be a non-Archimedean local field, let $G$ be the reductive group under consideration, and let $f^{EP}$ be the Euler–Poincaré function with respect to a Haar measure $dg$ on…

Let $F$ be a non-Archimedean locally compact field, let $D$ be a non-split quaternion algebra with centre $F$, and fix an integer $n>1$. Set $G={\rm GL}_n(D)$, equipped with an inv…

Canonical integral models conjecture. For every such Shimura datum and quasi-parahoric model, there exists a system of canonical integral models…

Let $k$) be an algebraic closure of $\a0\mathbb F_p$, let $W(k)$ be its ring of Witt vectors with Frobenius automorphism $\sigma$, and put $L=W(k)[1/p]$. Let $G$ be a connected red…

Howe's conjecture. For every compact subset $Omega\subset G(F)$ and every open subgroup $K\subset G(F)$, the restriction of $J(Omega)$ to $C_c(G(F)/K)$ is finite-dimensional.

Let $G$ be the $p$-adic group under consideration and let $R$ be a field of characteristic different from $p$. A cuspidal irreducible $R$-representation of $G$ is an irreducible re…

Let $F/\mathbb{Q}_p$ be a finite extension with $p\neq 2$, let $G$ be a connected semisimple linear algebraic group split over $F$, and let $\mathcal{D}_F(G)$ denote the $G^\sphat$…

Let $F$ be a non-archimedean local field, let $\mathbb{G}$ and $\mathbb{H}$ be algebraic groups over $F$, with $\mathbb{H}$ quasi-split, and write $G=\mathbb{G}(F)$ and…

Kottwitz–Rapoport conjecture. The nonemptiness pattern of $X(\mu,b)$ is given by Mazur's inequality.

Aubert–Baum–Plymen–Solleveld conjecture. The set of irreducible objects in a Bernstein block has the structure of a particular twisted extended quotient.

Let $\Pi$ be a cuspidal support, let $\mathcal{O}(\Pi)$ denote the corresponding Zelevinsky line, and let $\mathbf{a}$ and $\mathbf{b}$ be multisegments belonging to the same…