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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 K ˜=K . 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 Δ K be the K-invariant Laplacian operator on K. Let ρ t :K be a heat kernel of Δ K , i.e., a solution of the equation ρ t /t=1/2Δ K ρ t , lim t0 ρ t =δ 1 , where δ 1 is the delta distribution at the unit element 1 of K. After recalling a result of B. C. Hall [J. Funct. Anal. 122, No. 1, 103–151 (1994; Zbl 0838.22004)], the authors assert that ρ t (g) can be holomorphically extended to K ˜. Let μ t (t>0) be a heat kernel of the Laplacian Δ K ˜ on K .

By a theorem in the book by D. W. Robinson [Elliptic operators and Lie groups, Oxford etc.: Clarendon Press (1991; Zbl 0747.47030)] we know that μ t is a positive function for each t>0. We define ν t :K by

ν t (g)= K μ t (gk)dk·

Now, we define the Fock space F(K ˜,ν t ) (t>0) for the group K by the space of all holomorphic functions f:K ˜ such that

K |f(z)| 2 ν t (z)dz<,

where dz denotes a Haar measure on K ˜. The authors prove that the Fock space F(K ˜,ν t ) has a reproducing kernel E t given by E t (g,h)=ρ 2t (gh -1 ¯) for g,hK ˜. In fact, they prove that

E t (g,h)= K ρ t (k -1 g)ρ t (k -1 h) ¯dk

holds for g, hK ˜.

Next, they prove that there exists a linear map C t :L 2 (K)F(K ˜,ν t ) which is a unitary K-equivariant operator. The authors call C t the Segal-Bargmann transformation. To define the Weyl transform we put ω t (g)=E t (g,g)ν t (g) for gK and consider the space L 2 (K ˜,ω t (g)dg). The authors define an operator W:L 2 (K ˜,ω t dg)F(K ˜,ν t )F(K ˜,ν t ) ¯, which they call the Weyl transform, and then they prove the boundedness of this operator.

MSC:
22E30Analysis on real and complex Lie groups
32A25Integral representation; canonical kernels (several complex variables)
44A15Special transforms (Legendre, Hilbert, etc.)
References:
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