THE IMAGE OF THE SEGAL-BARGMANN TRANSFORM
SYMMETRIC SPACES AND GENERALIZATIONS


GESTUR OLAFSSON

Louisiana State University, Baton Rouge

The heat equation on Rn is the partial differential equation

Δu(x,t)= ∂t u(x,t)

lim t → 0+ u(x,t)= f(x)

where f is a L2-function on Rn and Δ is the usual Laplace operator on Rn . The solution is given by

u(x,t)= Ht f(x)= e tΔ f(x)= (4π t)- n/2∫ f(y)e-||x-y||2/(4t) dt =f*ht(x)

where ht(x) = (4π t)- n/2 e-||x||2/(4t) is the heat kernel on Rn . It is clear from the explicit expression for ht that the heat kernel, as well as f*ht has a holomorphic extension to Cn , the complexification of Rn . The transform f →Htf ∈O(Cn )is called the Segal-Bargmann transform. Its image is the space of holomorphic functions F : Cn →C, such that

|| F||t2:= (2πt)-n/2 ∫Cn |F(x + iy)|2 e −||y||2/2 dxdy < ∞

The Heat equation has a natural generalization to all Riemannian manifolds. The solutionis again given by the transform

u(x,t)= Htf(x)= ∫ f(y)ht(x,y)dy = e tΔ f(x)

where ht is the heat kernel, but as there is no natural complexification in general it is not clear how to realize the image in a space of holomorphic functions.

In this talk, we will discuss the Heat transform on Rn in some details to modify the concepts and ideas that are needed for Riemannian symmetric spaces of the form G/K where G is a connected semisimple Lie group and K a maximal compact subgroup. In particular, we discuss the case of K-invariant functions and how that result can be generalized to arbitrary root multiplicities on Rn . The main tools here are the spherical Fourier transform and the Abel transform.