University of Georgia

Mathematics Department Colloquium

Brian Hall

University of California, San Diego

January 22, 1999

Holomorphic methods in harmonic analysis

Abstract: The Segal-Bargmann transform for $\mathbb{R}^{n}$ (also called the coherent state transform) is a standard and important tool in harmonic analysis and mathematical physics. Consider a function $f$ on $\mathbb{R}^{n},$ and the Fourier transform of $f,$ which is another function on $\mathbb{R}^{n}.$ The Segal-Bargmann transform combines information about \textit{both} $f$ and its Fourier transform into a \textit{single} function on $\mathbb{R}^{2n}.$ The Segal-Bargmann transform of any function $f$ on $\mathbb{R}^{n}$ is a \textit{holomorphic} function on $\mathbb{C}% ^{n}=\mathbb{R}^{2n}.$ I will give an introduction to the Segal-Bargmann transform for $\mathbb{R}% ^{n}$ and then describe a generalization of this transform in which $\mathbb{R}^{n}$ is replaced by an arbitrary compact Lie group $K$. The simplest example is the compact Lie group $K=$ SU$\left( 2\right) ,$ which is just the 3-sphere $S^{3}.$ The key to passing from $\mathbb{R}^{n}$ to $K$ is to replace the Gaussians that appear in the standard transform by their natural geometrical analogs: heat kernels. I will keep the presentation elementary, so most of the talk should be accessible to a general mathematical audience.