Bounded hypergeometric functions associated to root systems.

A natural extension of Harish-Chandra's theory of spherical
functions on Riemannian symmetric spaces of non-compact type was introduced
by Heckman and Opdam in the late eighties. In this theory, the symmetric
space $G/K$ is replaced with a triple $(\mathfrak{a}, \Sigma, m)$ where
$\mathfrak{a}$ is a Euclidean vector space with an inner product, $\Sigma$ a root system in $\mathfrak{a}^{*}$ and $m$ a multiplicity function on $\Sigma.$ Associated to this triple, there is a family of commuting differential operators (which coincide with left $G$-invariant differential operators on $G/K$ when the triple is geometric) which admit joint eigenfunctions called hypergeometric functions (these functions coincide with Harish-Chandra's spherical functions in the geometric case). We study these functions and characterize the bounded hypergeometric functions, thus establishing an analogue of the celebrated theorem of Helgason and Johnson. This is joint work with Angela Pasquale and Sanjoy Pusti.