19 problems

van Dijk's conjecture. Every complex symmetric pair $(G,H)$ is a unitary Gelfand pair.

Aizenbud–Gourevitch's conjecture. All symmetric pairs are regular. In particular, this would imply van Dijk's conjecture, since all complex symmetric pairs are stable.

Orbit Conjecture. Let $K$ be a compact connected Lie group acting unitarily on $V$. Then $(K,H_V)$ is a Gelfand pair if and only if the action of $K$ on $V$ is multiplicity-free.

Let $E/F$ be a degree-two extension, let $G={\rm GL}_n(E)={\rm R}_{E/F}{\rm U}(n)$ in the notation of the source, and let $H={\rm U}(n)$. The pair $(G,H)$ is a supercuspidal Gelfan…

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 $j$ be a positive parameter, let $d\in\mathbb{N}$, and set $m=j-d$. Let $\lambda^{(0)},\lambda^{(1)},\ldots,\lambda^{(2j)}$ be the eigenvalues of the Berezin transform…

Let $j$ be a positive parameter, let $d\in\mathbb{N}$, and set $m=j-d$. Denote the eigenvalues of the Berezin transform $\mathcal{B}_{j,j-d}$ by…

Let $P$ be a product of simply-connected compact Lie groups, regarded as a topological analogue of a finite abelian group. Let $M$ be a fiber bundle with base $N$ and fiber $P$, an…

Let $H\subset G$ be a finite Gelfand pair, and suppose that $Aut(G)=G/Z(G)$. Then $Out_H(G)=N(H)/H$, where $N(H)$ is the normalizer of $H$ in $G$, and the preceding setup identifie…

Let $H\subset G$ be a finite Gelfand pair with dual pair $\check{H}\subset\check{G}$, and let $L$ be the common splitting field of the regular representations over $\mathbb{Q}$. Th…

The GL$(2n,q)$–GL$(n,q^2)$ conjecture. Condition $(\bigstar)$ is also satisfied for the last case (iv), namely $G=GL(2n,q)$ and $K=GL(n,q^2)$.

Equality conjecture. The polynomials $C_n(q)$ and $S_n(q)$ are identical for all $n\geq 1$. This would identify the dimension-generating polynomial arising from the multiplicity-fr…

GK-pair conjecture for good pairs. Every good symmetric pair is a GK-pair.

GK-pair conjecture. Any connected symmetric pair over $\mathbb{C}$ is a GK-pair.

Regularity conjecture. Any symmetric pair is regular.

Let $N$) be a connected, simply-connected, two-step nilpotent Lie group, let $K$ be a compact group acting by automorphisms on $N$, and assume that $(K\ltimes N,K)$ is a Gelfand pa…

Good symmetric pair conjecture. Every good symmetric pair is a GK pair.

Good-pair GK conjecture. Every good symmetric pair is a GK pair.

Regularity conjecture. Any symmetric pair is regular.