187 problems

Let $F$ be a non-Archimedean local field of characteristic zero, let $l$ be a positive integer, and let $\mathrm{GL}_l(F)$. Let $\pi,\pi'$ be irreducible generic representations of…

Let $G$ be a split classical group, and let $\pi$ be an isolated representation in the unitary dual of $G$. Automorphy conjecture. The representation $\pi$ is automorphic. The conj…

Let $K/\mathbb Q$ be a finite Galois extension with Galois group $G$, and let $\chi$ be a non-trivial irreducible character of $\operatorname{Gal}(K/\mathbb Q)$. Write $\chi(1)$ fo…

Howe's uniqueness conjecture. A local theta lift from an irreducible admissible representation has a unique irreducible subquotient.

Multiplicity and packet conjecture. For every $v\in S$, there is a representation $\Pi_v^-$ of $\mathrm{G}_2(\mathbb{Q}_v)$, different from $\mathcal{L}_\alpha(\pi_v,1/10)$, such t…

Let $E/F$ be a finite extension, let $\rho:\operatorname{Gal}(E/F)\rightarrow GL_n(\mathbb{C})$ be a representation, and let $L(\rho,s)$ denote its Artin $L$-function. Langlands' f…

Let $F$ be a number field, let $n$ be a positive integer, and let $\pi$ and $\pi'$ be cuspidal automorphic representations of $\operatorname{GL}(n,\mathbb{A}_F)$. Suppose that…

Assume the setup in which $G$ is simply connected and almost simple, $S$ is a set of places with at least two elements, and $T=\emptyset$. For each positive integer $m$, let…

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 the global function field and let ${}^LG$ and ${}^LH$ be the $L$-groups of a reductive group ${\bf G}$ and a quasi-split reductive group ${\bf H}$ over $F$, respectively…

Guo–Jacquet conjecture. If $\pi_D$ is $H_D$-distinguished and $\pi$ is deduced from $\pi_D$ by the Jacquet–Langlands correspondence, then $\pi$ is $H$-distinguished.

Sakellaridis–Venkatesh conjecture. A relative discrete series representation $\pi$ in $L^2(X)$ is contained in an Arthur packet corresponding to an $X$-distinguished $A$-parameter

Let $F$ be a non-archimedean local field, let $\mathbf{G}$ be a reductive group over $F$, and let $\mathbf{X}$ be a $\mathbf{G}$-spherical $F$-variety. Write $G$ for the group of…

Let $G$ be a reductive group with maximal compact subgroup $K$, symmetric space $X=G/K$, and let $\Gamma\subset G$ be a discrete subgroup of finite covolume. Let $A$ be the split c…

Fix a distinguished character $\overline{\chi}^0$. Let $\mathcal{A}_\epsilon(\overline{\operatorname{Sp}}_{2r}(\mathbb A))$ denote the relevant discrete-spectrum representation spa…

Let $\Pi$ be a cuspidal automorphic representation of $G_0$ or $G$. It is CAP relative to a parabolic $P$ if it is weakly equivalent to constituents of an induced representation fr…

Let $F$ be a $p$-adic field of characteristic zero, let $\mathrm{GSp}(4)$ denote the group of symplectic similitudes over $F$, and let an $L$-packet of representations of…

Langlands reciprocity conjecture. The function $L(s,\sigma)$ is the $L$-function associated to an automorphic representation of $GL(n,{\mathbb A})$.

Let $F$ be a non-archimedean local field of characteristic $0$, let $G$ be the group of points of a quasi-split connected reductive $F$-group, and let $P=MU$ be a standard paraboli…

Let $G$ be as above, and let $\pi=A_{\mathfrak q}$ be a cohomological representation of $G$ such that … where $\mathfrak l'_0$ is a compact Lie algebra and $\mathfrak g_1$ is a non…

Let $G$ be a semisimple algebraic group over ${\mathbb Q}$ obtained by restriction of scalars from a group of type ${}^3D_4$ or ${}^6D_4$ over a totally real number field, and supp…

Let $G$ be a semisimple group with discrete series. Let $\pi\in\widehat G$ be a cohomological representation, and consider the question of whether there is a lattice $\Gamma$ in…

Let $G$ be a semisimple algebraic group over ${\mathbb Q}$, let $L$ be the Levi subgroup associated with a $\theta$-stable parabolic subalgebra $\mathfrak{q}$, let $\tau$ be a unit…

Let $G$ be a semisimple algebraic group over ${\mathbb Q}$, let $\mathfrak{q}$ be a $\theta$-stable parabolic subalgebra, and let $L\subset G$ be its associated Levi subgroup. Let…

Let $\mathcal O$ be a fixed closed orbit, and let $a_{n,\lambda}$ be the coefficients in the expansion associated with $\mathcal O$ and the representation parameter $\lambda$. Clos…