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{Twisted Poincar\'e Series and Geometric Factorization of Affine Weyl Groups
We study the relation between the alternating product of twisted Poincar'e
series of parabolic subgroups of affine Weyl groups and hyperbolic stabilizers
of geodesic tubes. Moreover, from such relation, we find a length-preserving
factorization of affine Weyl groups of type and
Two-dimensional Noncommutative atom Gas with Anandan interaction
Landau like quantization of the Anandan system in a special electromagnetic
field is studied. Unlike the cases of the AC system and the HMW system, the
torques of the system on the magnetic dipole and the electric dipole don't
vanish. By constructing Heisenberg algebra, the Landau analog levels and
eigenstates on commutative space, NC space and NC phase space are obtained
respectively. By using the coherent state method, some statistical properties
of such free atom gas are studied and the expressions of some thermodynamic
quantities related to revolution direction are obtained. Two particular cases
of temperature are discussed and the more simple expressions of the free energy
on the three spaces are obtained. We give the relation between the value of
and revolution direction clearly, and find Landau like levels of the
Anandan system are invariant and the levels between the AC system and the HMW
system are interchanged each other under Maxwell dual transformations on the
three spaces. The two sets of eigenstates labelled by can be related
by a supersymmetry transformation on commutative space, but the phenomenon
don't occur on NC situation. We emphasize that some results relevant to Anandan
interaction are suitable for the cases of AC interaction and HMW interaction
under special conditions.Comment: Latex, 10 page
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