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Segal-Bargmann and Weyl transforms on compact Lie groups. (English) Zbl 1203.22010
Let $K$ be a connected and compact Lie group and suppose that it is contained in its complexification $\widetilde K= K_{\bbfC}$. The authors introduce the Fock space associated with $K$ and give a simple proof of the unitarity of the Segal-Bargmann transform proved by B. C. Hall for the group $K$. First, let $\Delta_K$ be the $K$-invariant Laplacian operator on $K$. Let $\rho_t: K\to \bbfC$ be a heat kernel of $\Delta_K$, i.e., a solution of the equation $\partial\rho_t/\partial t= 1/2\Delta_K\rho_t$, $\lim_{t\to 0} \rho_t=\delta_1$, where $\delta_1$ is the delta distribution at the unit element 1 of $K$. After recalling a result of {\it B. C. Hall} [J. Funct. Anal. 122, No. 1, 103--151 (1994; Zbl 0838.22004)], the authors assert that $\rho_t(g)$ can be holomorphically extended to $\widetilde K$. Let $\mu_t$ $(t> 0)$ be a heat kernel of the Laplacian $\Delta_{\widetilde K}$ on $K_{\bbfC}$. By a theorem in the book by {\it D. W. Robinson} [Elliptic operators and Lie groups, Oxford etc.: Clarendon Press (1991; Zbl 0747.47030)] we know that $\mu_t$ is a positive function for each $t> 0$. We define $\nu_t: K\to\bbfC$ by $$\nu_t(g)= \int_K\mu_t(gk)\,dk.$$ Now, we define the Fock space $F(\widetilde K,\nu_t)$ $(t> 0)$ for the group $K$ by the space of all holomorphic functions $f:\widetilde K\to\bbfC$ such that $$\int_K|f(z)|^2\nu_t(z)\, dz< \infty,$$ where $dz$ denotes a Haar measure on $\widetilde K$. The authors prove that the Fock space $F(\widetilde K,\nu_t)$ has a reproducing kernel $E_t$ given by $E_t(g,h)= \rho_{2t}(g\overline{h^{-1}})$ for $g,h\in\widetilde K$. In fact, they prove that $$E_t(g,h)= \int_K\rho_t(k^{-1} g)\overline{\rho_t(k^{-1} h)}\,dk$$ holds for $g$, $h\in\widetilde K$. Next, they prove that there exists a linear map $C_t: L^2(K)\to F(\widetilde K,\nu_t)$ which is a unitary $K$-equivariant operator. The authors call $C_t$ the Segal-Bargmann transformation. To define the Weyl transform we put $\omega_t(g)= E_t(g,g)\nu_t(g)$ for $g\in K$ and consider the space $L^2(\widetilde K,\omega_t(g)\,dg)$. The authors define an operator $W: L^2(\widetilde K,\omega_t dg)\to F(\widetilde K,\nu_t)\otimes \overline{F(\widetilde K,\nu_t)}$, which they call the Weyl transform, and then they prove the boundedness of this operator.

MSC:
 22E30 Analysis on real and complex Lie groups 32A25 Integral representation; canonical kernels (several complex variables) 44A15 Special transforms (Legendre, Hilbert, etc.)
Full Text:
References:
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