Mathematics – Differential Geometry
Scientific paper
2004-11-15
Mathematics
Differential Geometry
final version, to appear in Journ. Funct. Anal
Scientific paper
A natural one-parameter family of K\"ahler quantizations of the cotangent bundle $T^*K$ of a compact Lie group $K$, taking into account the half-form correction, was studied in \cite{FMMN}. In the present paper, it is shown that the associated Blattner-Kostant-Sternberg (BKS) pairing map is unitary and coincides with the parallel transport of the quantum connection introduced in our previous work, from the point of view of \cite{AdPW}. The BKS pairing map is a composition of (unitary) coherent state transforms of $K$, introduced in \cite{Ha1}. Continuity of the Hermitian structure on the quantum bundle, in the limit when one of the K\"ahler polarizations degenerates to the vertical real polarization, leads to the unitarity of the corresponding BKS pairing map. This is in agreement with the unitarity up to scaling (with respect to a rescaled inner product) of this pairing map, established by Hall.
Florentino Carlos
Matias Pedro
Mourão José
Nunes Joao P.
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