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 $$(\rho_{\mathbb{C}}(\tau,z))_{\mathrm{Re}\,\tau>0,z\in K_{\mathbb{C}}}$$ where $K_{\mathbb{C}}$ is the complexification of $K$. The new transform is defined by the integral $$(B_{\tau}f)(z)=\int_{K}\rho_{\mathbb{C}}(\tau,zk^{-1})f(k)\,dk,\quad z\in K_{\mathbb{C}}.$$ 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, $B_{s,\tau}:=B_{\tau}|_{L^{2}(K,\rho_{s})}$ 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 $\tau=t\in (0,2s)$, the transform $B_{s,t}$ coincides with the one introduced by the first two authors.