**Date & Time:** Wednesday, February 19, 2014, 16:00-17:00.

**Venue:** Ramanujan Hall

**Speaker:** E. K. Narayanan , IISc Bangalore

**Title:** Bounded hypergeometric functions associated to root systems

**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. This is joint work with Angela Pasquale and Sanjoy Pusti.