37 problems

Let $F$ be a non-archimedean field, let $\rho_0$ be an irreducible cuspidal representation, and let $\mathbf{a}$ and $\mathbf{b}$ be multisegments whose cuspidal supports lie on th…

Generic-extension subrepresentation conjecture. The representation $\pi(C*D)$ is a subrepresentation of

Ext criterion conjecture. The representation $\pi(C)\times\pi(D)$ is irreducible if and only if there exist open subsets $U_C\subseteq C$ and $U_D\subseteq D$ such that

Rigid-component conjecture. For $C\in\mathbf{Comp}(\mathbf{d})$, the induced representation $\pi(C)\times\pi(C)$ is irreducible if and only if $C$ contains an open $G_{\mathbf{d}}$…

Tadić's preservation of unitarizability conjecture. If $\pi\in\Pi(G_n)$ is weakly real, then $\pi$ is unitary if and only if $X_{\rho_i}(\pi)$ is unitary for every $i=1,\ldots,r$.

Serre self-duality conjecture. The self-duality induced by the functor $\Theta_{\on{I}(\cG,\cP^-)^{\on{spec,glob}}}$ equals the canonical (Serre) self-duality defined on $\IndCoh$…

Compatibility conjecture. The following diagram of functors commutes:

Let $bbM$ be a Levi subgroup of $bbG$, let $bbP=bbM\ltimesbbN$ be a parabolic subgroup, let $bbrho\inbbPi_{bbC}(bbM)$, and let $bbpi$ be an irreducible subquotient of…

Let $N_1=q_1++q_w$ and $N_2=q_{w+1}++q_l$, and let $f$, $f_1$, and $f_2$ be the nilpotent elements defining the corresponding non-rectangular $W$-algebras. Let … be the affine Yang…

Let $G$ be a reductive group, and let $P=LU$ be a Levi decomposition of a parabolic subgroup of $G$, with $L$ and $U$ the corresponding Levi and unipotent subgroups. Let $H_0$ be a…

Let $C_i\in\operatorname{Comp}(\bm{d}_i)$ for $i=1,\dots,k$, let $V=V^1\oplus\cdots\oplus V^k$, and let $C\in\operatorname{Comp}(V)$. Let $\pi(C)$ denote the irreducible representa…

Let $C$ be an irreducible component of a type $A$ nilpotent variety, and let $\pi(C)$ be the corresponding irreducible representation. Call a representation square-irreducible when…

Let $C_i\in\operatorname{Comp}(\bm{d}_i)$ for $i=1,2$, and suppose that at least one of $C_1,C_2$ is rigid, meaning that it contains an open orbit. Let $\pi(C_i)$ be the correspond…

Theta-packet induction conjecture. The representation $I(\pi_0)$ is irreducible for every $\pi_0\in\Pi_{\psi_0}^\theta(G_0)$. Moreover, if $I(\pi_0)\simeq I(\pi'_0)$ for…

Let $F$ be a non-archimedean local field with finite residue field of characteristic $p$, let $C$ be a field of characteristic $c\neq p$, and let $P=MN$ be a parabolic subgroup of…

Parabolic-induction temperedness question. Does one have

Let $I_{\mathbf{P}}(\lambda)$ be an automorphic realization of a degenerate principal series similar to the Eisenstein series in the source, induced from a parabolic subgroup…

Let $\mathfrak{m}$ and $\mathfrak{m}'$ be multisegments satisfying the condition referred to in the source as … , (1) $\operatorname{SA}(\mathfrak{m},\mathfrak{m}')$ and…

Let $V$ and $V'$ be finite-dimensional graded complex vector spaces, and let $\mathfrak{C}_{\mathfrak{m}}\subset\mathfrak{X}(V)$ and…

Let $V$ and $V'$ be finite-dimensional graded complex vector spaces, and let $\mathbb{G}(V)$ and $\mathbb{G}(V')$ be their groups of grading-preserving automorphisms. For multisegm…

Deficiency-length conjecture. The GLS condition, equivalently $d=0$, is equivalent to $l=1$. Moreover, $d=1$ if and only if $l=2$, and this condition is equivalent to…

Let $F$ be a $p$-adic field, let $G$ be a classical group over $F$, and let $P$ be a proper parabolic subgroup of $G$ with Levi subgroup $M$ defined over $F$. The dual-group inclus…

Let $I^G_P(\nu,\sigma)$ be a tempered generalized principal series, where $P=MN$ is a parabolic subgroup, $\sigma$ is a unitary supercuspidal representation of $M$, and…

Let $G$ be a reductive group, let $P=MN$ be a parabolic subgroup, and let $I^G_P(\sigma)$ be the parabolic induction of an irreducible admissible representation $\sigma$ of $M$. Su…

Let $G\cdot f=\operatorname{Ind}^\mathfrak{g}_\mathfrak{l}(L\cdot f_\mathfrak{l})$ be an induced nilpotent orbit, with good gradings $\Gamma$ on $\mathfrak{g}$ and…