52 problems

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 $\pi$.

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

Let $({\rm G},{\rm G}')$ be an irreducible dual pair in ${\rm Sp}(W)$ with ${\rm rk}({\rm G})\leq {\rm rk}({\rm G}')$. Let ${\rm Chc}^{*}$ be the map transferring invariant distrib…

Let $V$ and $W$ be the quadratic and symplectic spaces underlying the similitude dual pair, let $R$ be the corresponding similitude group, and let $\theta_\psi(\pi)$ denote the max…

Beilinson's Hodge index conjecture. The pairing of the arithmetic theta lift is expected to satisfy

Let $k$ be a local field of characteristic zero, let $\psi$ be a nontrivial character of $k$, and let $(U,q)$ be an odd-dimensional quadratic space over $k$. Let…

Kudla–Rallis conservation relation. The first occurrence indices satisfy

Let $G$) and $G^{\prime}$ be mutual centralizers forming a reductive dual pair inside a symplectic group over a local field $F$. Let $\omega$ be an oscillator representation of the…

Let $R_s(MG_1,MG_2)$ denote the relevant stable-range representation class, let $\theta_s(MG_1,MG_2)(\pi)$ be the stable theta lift, and let $\omega^c$ be the contragredient oscill…

Li's positivity conjecture. For unitary $\pi$, the form $(,)_{\pi}$ is always positive semidefinite.

Case 7 conjecture. Then

Let $G'$ and $G$ be the members of a reductive dual pair, and let $obreakomega^infty$ denote the smooth oscillator representation. For smooth irreducible representations ${pi}'$ a…

Let $G=G(W_n)$ and let $\psi,\psi'\in\Psi^+(G_n)$ be local Arthur parameters with $\psi\geq_C\psi'$. Suppose that $\pi\in\Pi_\psi\cap\Pi_{\psi'}$. The closure-monotonicity conjectu…

The refined Adams conjecture for ABV-packets. If $\pi\in\Pi_\phi^{\mathrm{ABV}}$, then: (1) for $\alpha\gg0$, $\theta_{-\alpha}^\pm(\pi)\in\Pi_{\phi_\alpha}^{\mathrm{ABV}}$; (2) if…

Let $(G,H)$ be a dual pair, let $\pi$ be a representation of $G$ of Arthur type, and let $\psi^{\max}(\pi)$ be the maximal Arthur parameter attached to $\pi$. Let $\alpha_0$ be the…

Muić's conjecture. The big theta lift $\Theta(\pi)$ to $H(V_m)$ is irreducible, if non-zero.

Stable-range theta lift low-rank conjecture. The theta lift $\theta(\tau)$ is a low rank representation of $G$, meaning that its $U$-support contains no open $L$-orbit.

Let $n$ be such that there is a cuspidal Hecke eigenform of weight $2n+12$ for $\operatorname{SL}_{2}(\mathbb{Z})$, and let $\pi$ be the associated level-one algebraic automorphic…

Let $F$ be an archimedean local field, let $G$ be the group of type $\mathsf{G}_2$, and let $K/F$ be the quadratic algebra used to define $G'_J$. Let $\chi:K^\times\to S^1$ be a un…

Let $\pi$ be a representation of $G$ and let $\psi^{\max}(\pi)$ denote its maximal Arthur parameter. Let $d^\pm(\pi,\psi^{\max}(\pi))$ be the corresponding going-up or going-down i…

Let $G(W)$ be the group attached to the symplectic space $W$, let $G(W\oplus W)$ be the corresponding group for the orthogonal direct sum with the given form on the first copy and…

Let $F$ be the global field implicit in the automorphic setting, let $W$ be a symplectic space, let $V$ be a quadratic space, and let $sigma\subset \mathcal{A}_{cusp}({\rm Sp}(W))$…

Let $E/k$ be the relevant quadratic extension, let ${\rm U}(V)\times {\rm U}(W)$ be a dual pair in the stable range, and let $\pi\subset\mathcal{A}_{\rm cusp}({\rm U}(V))$ have A-p…

Adams's conjecture. The representation $theta_{-alpha}(pi)$ is contained in the A-packet parametrized by

Arithmetic Rallis inner product formula conjecture. The Beilinson–Bloch height pairing of the arithmetic theta lifts should give