Contraction of the Actions of Reductive Algebraic Groups

Abstract

The angular dependence of the anomalous transmission of polarised Mossbauer radiation through a perfect single-crystal platelet of 57FeB0, was studied. Rocking curves of the Laue reflected and of the Laue transmitted beams were simultaneously recorded for the pure nuclear reflection (001). Striking asymmetries were observed between the transmitted and the reflected intensities for the two polarisation components. The asymmetries were caused by the strong anisotropy of the nuclear resonance scattering. The asymmetry between the total intensities (transmitted plus reflected beam) for the two polarisation components was related to a strongly asymmetric division of the incident radiation into the two wavefields built up inside the crystal. The asymmetry between the intensities of the transmitted and reflected beams revealed a strongly asymmetric composition of the wavefield experiencing anomalous low resonance absorption. The observed anomalous transmission of such a strongly asymmetric wavefield uniquely demonstrated the origin of the suppression of the incoherent reaction channels, i.e. the cancellation of the total amplitude for the individual nuclear excitation (the Kagan-Afanasev rule) rather than the cancellation of the total amplitude of the radiation field at the nuclear position.

Toric degenerations of spherical varieties

“…Returning to an arbitrary G-scheme X with limit X 0 , many geometric properties hold for X if and only if they hold for X 0 , as can be shown along the lines of [Pop87, §6] (see also [Gro97,§18]). We record three such properties that will be of use in the sequel.…”

Convexity and Thimm’s Trick

“…Given a Hamiltonian G-manifold (M, ω, Φ), Hilgert-Martens-Manon define a symplectic contraction map Φ M which is a continuous, surjective, and proper map from M to a singular space M sc called the symplectic contraction of M . M sc is the symplectic analogue of the horospherical contraction/degeneration of a reductive group action due to Popov and Vinberg [48]. M sc is equipped with an action of G × T that is Hamiltonian in the appropriate sense for these singular spaces.…”

On deformation method in invariant theory

“…5 (see Section 4). One may observe that the second part of Theorem (0.1) may be deduced from the results in [26]. However, I think that the present approach is more natural and certainly allows to give more short and transparent proofs.…”

“…A detailed development of this method together with numerous applications has been given by V.L. Popov [26] (see [9] for the results in arbitrary characteristic). Various aspects are also touched on in [6], [28], [14].…”

“…Suppose Y = G and the G-action on G is that by right multiplication. Then k[XxG}°^ ^ k[X} and we recover in this way the G-invariant filtration in k[X} and the description of gr^ k[X} given in [26], th. 5 (see Section 4).…”