21 problems

Let $\psi\in\Psi({}^{\vee}G^{\Gamma})$ be an Arthur parameter, let $\varphi_\psi$ be its associated Langlands parameter, and let $\Pi(G)_\varphi$ denote the corresponding $L$-packe…

Let $G$ be a real reductive group with maximal compact subgroup $K$, and let $G_0=K\ltimes\mathfrak{s}$ be its Cartan motion group. Write $\Pi_{\mathrm{adm}}(G)$ and…

Let $G$ be a real reductive group and let $\pi$ be an irreducible Casselman–Wallach representation of $G$. Let $\operatorname{WF}_{\mathrm{tr}}(\pi)$ denote its analytic wavefront…

Let $G$ be a connected reductive algebraic group defined over $\mathbb{R}$), let $W_{\mathbb{R}}$ be the Weil group of $\mathbb{R}$, and let ${}^{\vee}G^{\Gamma}$ be the correspond…

Let $G\subset \widetilde{G}$ be reductive real linear groups satisfying the hypothesis of the main theorem, and let $I_{{\mathfrak{g}},\tilde{{\mathfrak{g}}}}$ denote the largest i…

FPP conjecture. One of the following holds: (1) the infinitesimal character of $\pi$ belongs to the fundamental parallelepiped; or (2) $\pi$ is cohomologically induced, in the weak…

Let $G$ be a real reductive Lie group and let $V$ be an irreducible representation of $G$ with real infinitesimal character. The representation $V$ carries a canonical Hodge filtra…

Unipotent and complementary-series conjecture. The representations in $\widehat{G}$ can be obtained from one of the following:

Salamanca-Riba–Vogan's convex-hull conjecture. If $X$ is unitary, then $\Lambda$ must lie in

Salamanca-Riba–Vogan's unitarity conjecture. The above bijection preserves unitarity.

Let $G_{\mathbb R}$ be the real points of a connected reductive algebraic group, and let $\mathsf{Cov}^{i\mathbb R}(G_{\mathbb R})$ denote the real co-adjoint covers of…

Let $G$ be an equal-rank real reductive group with maximal compact subgroup $K$, and let $\widetilde K$ be the spin double cover of $K$. For an irreducible unitary…

Let $\Gamma$ be a finitely generated group. For a real reductive group $G$ with maximal compact subgroup $K$, call $\Gamma$ $G$-flawed if $\mathfrak{X}_{\Gamma}(G)$ strongly deform…

Extension conjecture. Every such $T$ extends uniquely to an element of $\mathcal{C}^{p}(G)'$ if and only if $T$ can be expressed as a finite sum of derivatives of a class of functi…

A hd-compactification of a Lie group is a compactification satisfying the properties described in the paper: inversion extends smoothly, right-invariant vector fields extend as smo…

Let $G_\mathbb{R}$ be a real reductive Lie group, with complexification $G$ and maximal compact subgroup $K$ as in the source. Let $\mathcal{O}_\mathfrak{p}$ be a nilpotent $K$-orb…

Let $G_\mathbb{R}$ be a noncompact real reductive Lie group. Barbasch–Vogan conjecture. Unipotent representations should be the building blocks of the unitary dual of…

Low-rank reduction conjecture. An irreducible admissible representation $\pi$ of $\operatorname{GL}_n({\mathbb R})$ occurs as a Jordan–Hölder factor of $Ps(\underline{\pi})$ if and…

Let $G$ be a semisimple Lie group, let $K$ be a maximal compact subgroup, and let $G_0=K\ltimes\mathfrak{p}$ be the Cartan motion group associated with the Cartan decomposition of…

Let $G$ be a reductive group, let $\delta_0$ be the specified automorphism, and let $\xi$ be a strong involution for $G$ with dominant regular integral infinitesimal cocharacter…

Cartan-subspace quotient conjecture. The map $\pi$ sets up a one-to-one correspondence between the $G_{\mathbf R}$-conjugacy classes of Cartan subspaces of $V_{\mathbf R}$ and the…