A weighted Plancherel formula II. The case of the ball
Studia Mathematica, Tome 102 (1992) no. 2, pp. 103-120
The group SU(1,d) acts naturally on the Hilbert space $L²(B dμ_α) (α > -1)$, where B is the unit ball of $ℂ^d$ and $dμ_α$ the weighted measure $(1-|z|²)^α dm(z)$. It is proved that the irreducible decomposition of the space has finitely many discrete parts and a continuous part. Each discrete part corresponds to a zero of the generalized Harish-Chandra c-function in the lower half plane. The discrete parts are studied via invariant Cauchy-Riemann operators. The representations on the discrete parts are equivalent to actions on some holomorphic tensor fields.
Keywords:
Plancherel formula, Harish-Chandra c-function, reproducing kernel, orthogonal polynomial, invariant Cauchy-Riemann operator
@article{10_4064_sm_102_2_103_120,
author = {Genkai Zhang},
title = {A weighted {Plancherel} formula {II.} {The} case of the ball},
journal = {Studia Mathematica},
pages = {103--120},
year = {1992},
volume = {102},
number = {2},
doi = {10.4064/sm-102-2-103-120},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/sm-102-2-103-120/}
}
Genkai Zhang. A weighted Plancherel formula II. The case of the ball. Studia Mathematica, Tome 102 (1992) no. 2, pp. 103-120. doi: 10.4064/sm-102-2-103-120
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