10 problems

Let $\mu^{\mathcal{R}(\theta)}_{a_1,a_2}$ denote a solution of the product-formula measure problem for parameters $a_1,a_2\in\mathbb{R}^N$, with $\mathcal{R}$ a finite root system…

Let $G$ be a finite reflection group acting on $\mathbb R^q$, let $k\geq0$ be a multiplicity function, let $J_k$ be the associated $G$-invariant generalized Bessel function, and le…

Let a root system be given with non-negative multiplicities, and let the associated Dunkl-type Bessel functions be defined on the underlying Weyl chamber. Positive product formula…

Existence conjecture. It is conjectured that there exists a commutative hypergroup on $C$ whose multiplicative functions include the functions $J_k(\,\cdot\,,y)$ for all…

Hypergroup conjecture. For arbitrary reflection groups and non-negative multiplicities, this convolution should make $\overline C$ into a commutative hypergroup. In the special cas…

Let $R$ be a root system with reflection group $G$, Weyl chamber $C$, and generalized Bessel function $J_k$ associated with a multiplicity function $k$. For $x,y\in\overline C$, su…

Let $N\geq 1$, let $\mathcal{R}\subset\mathbb{R}^N$ be a finite root system, and let $\theta\in\theta(\mathcal{R})$ be nonnegative. For $a_1,a_2\in\mathbb{R}^N$, consider the Dunkl…

Boundary-measure conjecture. For $h\in\{0,k_2,\ldots,k_2(n-1)\}$, the distribution $S_{k,h}$ is a positive measure supported on the boundary of $[0,1]^n$, and the intertwiner…

Let $R$ be a root system and let $k$ and $k^\tri$ be multiplicities on $R$, with $k^\tri>k>0$. Write $V_{k^\tri,k}=V_{k^\tri}\circ V_k^{-1}$ for the intertwining operator between t…

De Jeu–Rösler conjecture. If $x$ is regular and $\mathfrak{a}$ is spanned by the roots $\alpha$ whose multiplicities satisfy $k_\alpha>0$, then $\mu_x$ is absolutely continuous wit…