14 problems

Let $\psi$ be an Arthur parameter, let $\varphi_{\psi}$ be its associated Langlands parameter, and let $\Pi(\varphi_{\psi})$ be the $L$-packet associated to $\varphi_{\psi}$. A rep…

Let $k$ be a number field, let $\mathrm{G}$ be a reductive group over $k$, and let $\pi=\bigotimes_v\pi_v$ be an automorphic representation in the discrete spectrum of…

Let $\pi=\pi(\lambda,s,{{\mathcal O}}^\vee_{\lambda,s},{{\mathcal L}}^\vee)$ be an irreducible representation in the tempered $L$-packet…

Let $G$ be a reductive group over the global function field $K$, let $[\Pi]$ be a cuspidal automorphic representation of $G([\mathbf A_K])$ with coefficients in $\bar{\mathbb Q}$,…

Asymptotic Schur orthogonality relations. There exist $\mathbf{d}(V)\in{\mathbb Z}_{\ge 0}$ and $\mathbf{f}(V)\in{\mathbb R}_{>0}$ such that, for all $v_1,v_2,v_3,v_4\in V$,

Kazhdan–Yom Din's c-temperedness conjecture. Temperedness implies c-temperedness for irreducible unitary representations of semisimple groups over local fields.

Parabolic-induction temperedness question. Does one have

Let $\mathfrak h$ be a complex semisimple Lie algebra and $V$ an orthogonal representation of $\mathfrak h$ over $\mathbb C$. AGS equivalence conjecture. One has … This is the comp…

Let $\mathfrak h$ be a real semisimple Lie algebra and $V$ an orthogonal representation of $\mathfrak h$. AGS conjecture. If $V$ has AGS in $\mathfrak h$, then … The conjecture giv…

Let $G$ be a real, algebraic semisimple group and let $H\subset G$ be a real, algebraic reductive subgroup satisfying the condition in part (a) of Theorem 4.1 of Benoist and Kobaya…

Local Langlands conjecture. There is a one-to-one correspondence

S-group conjecture. The internal structure of the tempered L-packet $\Pi_\phi$ is controlled by the $S$-group $S_\phi$.

Point-independence conjecture. The subcategory $\Dmod^x_{\on{temp}}(\Bun_G)\subset\Dmod(\Bun_G)$ is independent of the choice of the point $x\in X$. This is deduced in the source a…

Global temperedness conjecture. If $\pi_v$ is tempered for almost all places, or even for infinitely many places, then $\pi$ is tempered; equivalently, $\pi_v$ is tempered for ever…