12 problems

Three-term recurrence conjecture. There exist matrices $A_p$, $B_p$, and $C_p$, independent of $t$, such that

Generalized hypergeometric expression conjecture. For a given $\ell\geq 0$, the components of the functions $H(t)$ are expressible in terms of generalized hypergeometric functions…

Hook alternating property conjecture. The coefficients $A_k$ with $k$ in the range $j-i\leq k\leq j+i$ have the hook alternating property.

Matrix-valued product-expansion conjecture. The product admits a unique expansion of this form, with matrix coefficients $A_k$, for all nonnegative $n$.

Let $G$ be a semisimple linear group over a local field $F$, let $K<G$ be a maximal compact subgroup, and let $H<G$ be a closed abelian subgroup. Define … For a continuous bi-$K$-i…

Let $X$ be an irreducible symmetric space of compact type, with spectral parameter $\mu\in \Lambda^+\subset i\mathfrak{t}^*$ and spatial parameter $H\in\mathfrak{t}$. Let…

Let $n>2$, let $f:[-1,1]\to\mathbb{R}$ satisfy $f(1)=1$, and define $\widehat f:SO(n)\to\mathbb{R}$ by $\widehat f(\gamma)=f(\gamma_{11})$. Assume that $\widehat f$ defines a posit…

Spherical-function growth conjecture. If $\lambda,X\in\overline{\mathfrak{a}^+}$, then

The global spherical-function estimate. On flat Riemannian symmetric spaces with complex group $G$, one has

Let $G$ be the ambient group, $K$ a maximal compact subgroup, and $\Pi(G)_{\operatorname{sph}}$ the set of unitary irreducible spherical representations of $G$. For $g\in G$, let…

Positivity conjecture. $F_n^{\lambda,\delta}(t)>0$ for every $t\in(0,\pi]$ if $\delta\geq\lambda+1$.

Explicit lowest-spherical-function conjecture. For all $n\geq 3$,