Let (G, K) be an irreducible Hermitian symmetric pair of noncompact type with G = SU (p, q), and let λ be an integral weight such that the simple highest weight module L(λ) is a Harish-Chandra (g, K)-module. We give a combinatorial algorithm for the Gelfand-Kirillov dimension of L(λ). This enables us to prove that the Gelfand-Kirillov dimension of L(λ) decreases as the integer λ + ρ, β ∨ increases, where ρ is the half sum of positive roots and β is the maximal noncompact root. Finally by the combinatorial algorithm, we obtain a description of the associated variety of L(λ).
We prove a simple formula that calculates the associated variety of a highest weight Harish-Chandra module directly from its highest weight. We also give a formula for the Gelfand-Kirillov dimension of highest weight Harish-Chandra module which is uniform across Cartan types and is valid for arbitrary infinitesimal character.
ABSTRACT. It is demonstrated that, for the recently introduced classical magnetized Kepler problems in dimension 2k + 1, the non-colliding orbits in the "external configuration space" R 2k+1 \ {0} are all conics, moreover, a conic orbit is an ellipse, a parabola, and a branch of a hyperbola according as the total energy is negative, zero, and positive. It is also demonstrated that the Lie group SO + (1, 2k + 1) × R + acts transitively on both the set of oriented elliptic orbits and the set of oriented parabolic orbits.
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