20 problems

Let $G$ be an almost algebraic group, let $H$ be a closed algebraic subgroup, and let $\mathfrak{g}$ and $\mathfrak{h}$ be their Lie algebras, with duals $\mathfrak{g}^*$ and…

Kobayashi's conjecture. The multiplicity $m_{\tau}(\pi|_{G'})$ is finite for every $\tau\in\hat{G}'$. This is a finite-multiplicity assertion for discretely decomposable restrictio…

Finiteness conjecture. There are at most finitely many discrete spectra in the branching law $\varpi^{p,q}|_{G'}$.

Let $F$ be a non-archimedean local field. Let $\pi_1$ and $\pi_2$ be Arthur type representations of $\operatorname{GL}_n(F)$ and $\operatorname{GL}_{n-1}(F)$, respectively. Say tha…

Projectivity conjecture. The restriction $\pi|_H$ is projective if and only if $\pi$ is supercuspidal.

Let $pi in \mathrm{Irr}(G_{n+1})$ and $pi' in \mathrm{Irr}(G_n)$. A pair $(pi,pi')$ is relevant if there exist multisegments $mathfrak m,mathfrak n$ such that…

Spherical branching conjecture. The period integral for spherical functions of finite-dimensional representations is nonzero if and only if the $H'$-spherical representation $V'$ o…

Uniqueness and multiplicity-one conjecture. If

Let $G_n=\mathrm{GL}_n(F)$, and let $\pi_M$ and $\pi_N$ be Arthur type representations of $G_{n+1}$ and $G_n$, respectively. For $k\geq 1$, write $\pi_M^{[k]}$ and…

Let $G=G(V)$ and $H=G(W)$ be the classical groups in the restriction problem, let $(M_A,N_A)$ be A-parameters with associated L-parameters $(M,N)$, and let $\Pi_M$ and $\Pi_N$ be t…

Let $k$ be a local field, let $M_A$ and $N_A$ be A-parameters of ${\rm GL}_n(k)$ and ${\rm GL}_{n-1}(k)$, respectively, and let $\pi_M$ and $\pi_N$ be the associated irreducible re…

Arthur–Vogan packet branching conjecture. Suppose that $\underline G$ and $\underline{G}^1$ are a relevant pair. Then the restriction and branching behavior predicted for the corre…

Let $\mathfrak m$ be a multisegment for $n+1$, and let $\zeta(\mathfrak m)$ denote the associated Zelevinsky induced module. For the restriction to $G_n$, a Bernstein component is…

Uniform boundedness conjecture. If $\pi$ is an irreducible highest weight representation of $G$ or if $\pi$ is a minimal representation of $G$, then the uniform boundedness propert…

Let $G$ be a real reductive group, let $G'$ be a reductive subgroup, and choose a maximal compact subgroup $K$ of $G$ such that $K'=G'\cap K$ is a maximal compact subgroup of $G'$.…

Januszewski's conjecture. In the setting of Kobayashi's conjecture, $m_{\tau}(\pi|_{G'})$ grows at most polynomially in the norm of the infinitesimal character of $\tau$, in the se…

Let $F$ be a non-Archimedean local field. Let $\pi_2$ be a discrete series representation of ${\rm SO}_n(F)$, and let $\pi_1$ be an irreducible admissible, say tempered, representa…

Kobayashi's multiplicity-growth conjecture. This dimension is finite and grows at most polynomially in the norm of the infinitesimal character of $\tau$. The statement is presented…

Interlacing-pattern conjecture. There exists a bijective correspondence between $I(a_.,b_.)$ and discrete series representations $(\pi_1,\pi_2)$ of relevant pairs $(G,H)$ with thes…

Let $H$ be the connected fixed-point set of an involution, with ${\mathfrak g}={\mathfrak h}\oplus{\mathfrak s}$. Let $A_{\mathfrak q}(\lambda)$ be a $K\cap H$-admissible represent…