9 problems

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…

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.

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

van Dijk's GK conjecture. If $F=\mathbb{C}$, any connected symmetric pair is a GK pair.

van Dijk's conjecture. Any connected symmetric pair $(G,H)$ over $\mathbb{C}$ such that $G/H$ is connected is a Gelfand pair.