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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.

22E30Analysis on real and complex Lie groups
32A25Integral representation; canonical kernels (several complex variables)
44A15Special transforms (Legendre, Hilbert, etc.)
Full Text: DOI
[1] Berger C.A., Coburn L.A.: Heat flow and Berezin--Toeplitz estimates. Am. J. Math. 116, 563--590 (1994) · Zbl 0839.46018 · doi:10.2307/2374991
[2] Duistermaat J.J., Kolk J.A.C.: Lie Groups. Springer, Berlin (2000) · Zbl 0955.22001
[3] Hall B.C.: The Segal--Bargmann coherent state transform for compact Lie groups. J. Funct. Anal. 122, 103--151 (1994) · Zbl 0838.22004 · doi:10.1006/jfan.1994.1064
[4] Hall B.C.: The inverse Segal--Bargmann transform for compact Lie groups. J. Funct. Anal. 143(1), 98--116 (1997) · Zbl 0869.22005 · doi:10.1006/jfan.1996.2954
[5] Hall B.C.: Phase space bounds for quantum mechanics on a compact Lie group. Comm. Math. Phys. 184(1), 233--250 (1997) · Zbl 0869.22013 · doi:10.1007/s002200050059
[6] Hall B.C.: Harmonic analysis with respect to heat kernel measure. Bull. Am. Math. Soc. 38, 43--78 (2000) · Zbl 0971.22008 · doi:10.1090/S0273-0979-00-00886-7
[7] Hall B.C., Lewkeeratiyutkul W.: Holomorphic Sobolev spaces and the generalized Segal--Bargmann transform. J. Funct. Anal. 217(1), 192--220 (2004) · Zbl 1077.46025 · doi:10.1016/j.jfa.2004.03.018
[8] Hijab O.: Hermite functions on compact Lie groups I. J. Funct. Anal. 125, 480--492 (1994) · Zbl 0836.43016 · doi:10.1006/jfan.1994.1134
[9] Neeb K.-H.: Holomorphy and Convexity in Lie Theory. de Gruyter, Berlin (2000) · Zbl 0964.22004
[10] Ólafsson G., Ørsted B. et al.: Generalizations of the Bargmann transform. In: Doebner, H.-D. (eds) Lie Theory and its Applications in Physics, World Scientific, Singapore (1996)
[11] Ørsted B., Zhang G.: Weyl quantization and tensor products of Fock and Bergman spaces. Indiana Univ. Math. J. 43, 551--582 (1994) · Zbl 0805.46053 · doi:10.1512/iumj.1994.43.43023
[12] Robinson D.W.: Elliptic Operators and Lie Groups. Clarendon Press, Oxford (1991) · Zbl 0747.47030
[13] Stenzel M.B.: The Segal--Bargmann transform on a symmetric space of conpact type. J. Funct. Anal. 165, 44--58 (1999) · Zbl 0929.22007 · doi:10.1006/jfan.1999.3396
[14] Stein, E.M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III (1993) · Zbl 0821.42001