23 problems

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 $i=1,\ldots,r$.

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 $G$ 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 $C_i\in\operatorname{Comp}(\bm{d}_i)$ 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 $(W,V)$ be an admissible pair. Let $\pi_V\in \Pi_{\mathrm{CW}}(\operatorname{SO}(V))$ and $\pi_W\in \Pi_{\mathrm{CW}}(\operatorname{SO}(W))$, possibly reducible, and suppose th…

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 $\mathfrak{m}$ be a multisegment, let $V$ be the graded vector space determined by its supercuspidal support, and let $\mathfrak{C}_{\mathfrak{m}}$ be the corresponding irreduc…

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…

Irreducibility-independence conjecture. The irreducibility of

Let $W$ be the Weyl group of $(\mathbf{G},\mathbf{S})$, let $I,J\subseteq\Delta$, let $\sigma$ be a locally admissible smooth representation of $L_I$ over $A$, let…

Let $F/\mathbb{Q}_p$ and $k/\mathbb{F}_p$ be finite extensions. Let $\mathbf{G}$ be a split connected reductive $F$-group with connected centre and simply connected derived subgrou…

Let $\pi$ be a unitary representation. A representation is induced by parabolic induction when it is obtained from a representation of a parabolic subgroup in this way. Unitary Dir…

Vogan–Haines compatibility conjecture. La correspondance de Langlands devrait vérifier

Let $P,P'\subset G$ be standard parabolic subgroups with standard Levi subgroups $L\subset P$ and $L'\subset P'$, and let $\pi$ and $\pi'$ be supersingular representations of…

Let $F$ be a finite extension of $\mathbb{Q}_p$, let $G$ be a connected reductive group over $F$, and let $P\subset G$ be a standard parabolic subgroup with standard Levi subgroup…

Distinction conjecture. The representation $\pi$ is distinguished if and only if, after reordering the $\Delta_i$, there is an integer $r$ between $1$ and $t/2$ such that