Spectral Spectral Rigidity and Geometric Quantization of Coadjoint Orbits
Abstract
Coadjoint orbits provide a fundamental link between symplectic geometry and the representation theory of Lie groups, as formalized by the orbit method of Kirillov. In this paper, we investigate the spectral properties of the Casimir operator in relation to the geometry of coadjoint orbits and their quantization.
We establish a spectral rigidity phenomenon: for compact semisimple Lie groups, the eigenvalue of the Casimir operator determines the corresponding coadjoint orbit and the associated irreducible representation. This rigidity is shown to be independent of the choice of G-invariant metric on the orbit, highlighting the intrinsic algebraic nature of the Casimir operator.
We provide explicit computations in the case of SU(2), where coadjoint orbits are 2-spheres, and analyze the relationship between the Casimir operator and the Laplace-Beltrami operator under metric variations. We further extend the discussion to real semisimple Lie groups, where rigidity persists in a weaker form through the infinitesimal character and the Harish-Chandra isomorphism.
Our results clarify the role of the Casimir operator as a bridge between geometry, spectral theory, and geometric quantization.
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References
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