On Orbits in Double Flag Varieties for Symmetric Pairs

Let G be a connected, simply connected semisimple algebraic group over the complex number field, and let K be the fixed point subgroup of an involutive automorphism of G so that (G, K) is a symmetric pair. We take parabolic subgroups P of G and Q of K, respectively, and consider the product of partial flag varieties G/P and K/Q with diagonal K-action, which we call a double flag variety for a symmetric pair. It is said to be of finite type if there are only finitely many K-orbits on it. In this paper, we give a parametrization of K-orbits on G/P × K/Q in terms of quotient spaces of unipotent groups without assuming the finiteness of orbits. If one of P ? G or Q ? K is a Borel subgroup, the finiteness of orbits is closely related to spherical actions. In such cases, we give a complete classification of double flag varieties of finite type, namely, we obtain classifications of K-spherical flag varieties G/P and G-spherical homogeneous spaces G/Q.     

High resolution off-magic-angle spinning NMR for cholesteric liquid crystals

High resolution NMR of cholesteric liquid crystals is realized by off-magic-angle spinning (OMAS). Using the average potential theory, it is shown that the pitch axes of a cholesteric liquid crystal with a positive (negative) magnetic susceptibility anisotropy are aligned along the spinning axis with an OMAS angle larger (smaller) than the magic angle without any distortion of the helical structure, which is untwisted or distorted by the static magnetic field in some static samples. Hence, the line broadening due to the anisotropies of chemical shifts and/or quadrupole couplings is removed, while information of the anisotropies in cholesteric alignments can be obtained from the line positions. A detailed theoretical analysis of effects of spinning frequency and molecular diffusion along the pitch axis on the linewidth is given, predicting that the resolution is improved greatly at a spinning frequency much higher than the rotational diffusion coefficient. These theoretical results are verified by ¹³C OMAS experiments on a cholesteric liquid crystal of p-ethoxybenzyl-p-[(S)-2- methylbutyl]aniline (EBMBA) and cholesteric mixtures of p-methoxybenzylidene-p-n-butylaniline (MBBA) and cholesteryl chloride (CC).