The Complex-Time Segal-Bargmann Transform

We introduce a new form of the Segal–Bargmann transform for a Lie group $K$ of compact type. We show that the heat kernel $({\rho }_t(x){)}_{t>0,x\in K}$ has a space-time analytic continuation to a holomorphic function $$({\rho }_{\mathbb{C}}(\tau ,z){)}_{\mathrm{R}\mathrm{e}\tau >0,z\in K_{\mathbb{C}}}$$ where $K_{\mathbb{C}}$ is the complexification of $K$. The new transform is defined by the integral $$(B_{\tau }f)(z)={\int }_K{\rho }_{\mathbb{C}}(\tau ,z{k}^{-1})f(k)dk,z\in K_{\mathbb{C}}\mathrm{.}$$ If $s>0$ and $\tau \in \mathbb{D}(s,s)$ (the disk of radius $s$ centered at $s$), this integral defines a holomorphic function on $K_{\mathbb{C}}$ for each $f\in L^2(K,{\rho }_s)$. We construct a heat kernel density ${\mu }_{s,\tau }$ on $K_{\mathbb{C}}$ such that, for all $s,\tau$ as above, $B_{s,\tau }:=B_{\tau }{\mathrm{\mid }}_{L^2(K,{\rho }_s)}$ is an isometric isomorphism from $L^2(K,{\rho }_s)$ onto the space of holomorphic functions in $L^2(K_{\mathbb{C}},{\mu }_{s,\tau })$. When $\tau =t=s$, the transform $B_{t,t}$ coincides with the one introduced by the second author for compact groups and extended by the first author to groups of compact type. When $\tau=t\in (0,2s)$,thetransform$B_{s,t}$ coincides with the one introduced by the first two authors.