Speaker: E.K. Narayanan
Date: September 12, 2013
Affiliation: IISc., Bangalore

Abstract: 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. <p> This is joint work with Angela Pasquale and Sanj