Let G be a reductive group and let ·G be its Langlands dual. We give an interpretation of the dynamical Weyl group of ·G de¯ned in  in terms of the geometry of the a±ne Grassmannian Gr of G. In this interpretation the dynamical parameters of  correspond to equivariant parameters with respect to certain natural torus acting on Gr. We also present a conjectural generalization of our results to the case of a±ne Kac-Moody groups.
We consider the multimode generalization of the normally ordered factorization formula of squeezings. This formula allows us to establish relationships between various representations of squeezed states, to calculate partial traces, mean values, and variations. The main results are expressed in terms of the matrix representation of canonical transformations which is a convenient and numerically stable mathematical tool. Explicit representations are given for the inner product and the composition of generalized multimode squeezings. Explicitly solvable evolution problems are considered.
In this paper, we consider quantum multidimensional problems solvable by using the second quantization method. A multidimensional generalization of the Bogolyubov factorization formula, which is an important particular case of the Campbell–Baker–Hausdorff formula, is established. The inner product of multidimensional squeezed states is calculated explicitly; this relationship justifies a general construction of orthonormal systems generated by linear combinations of squeezed states. A correctly defined path integral representation is derived for solutions of the Cauchy problem for the Schrödinger equation describing the dynamics of a charged particle in the superposition of orthogonal constant (E,H)-fields and a periodic electric field. We show that the evolution of squeezed states runs over compact one-dimensional matrix-valued orbits of squeezed components of the solution, and the evolution of coherent shifts is a random Markov jump process which depends on the periodic component of the potential.