According to Kirillov's idea, the irreducible unitary representations of a Lie group G roughly correspond to the coadjoint orbits (?).In the forward direction one applies the methods of geometric quantization to p...According to Kirillov's idea, the irreducible unitary representations of a Lie group G roughly correspond to the coadjoint orbits (?).In the forward direction one applies the methods of geometric quantization to produce a representation, and in the reverse direction one computes a transform of the character of a representation, to obtain a coadjoint orbit. The method of orbits in the representations of Lie groups suggests the detailed study of coadjoint orbits of a Lie group G in the space (?)~* dual to the Lie algebra (?) of G. In this paper, two primary goals are achieved: one is to completely classify the smooth coadjoint orbits of Virasoro group for nonzero central charge c; the other is to find representatives for coadjoint orbits. These questions have been considered previously by Segal, Kirillov, and Witten, but their results are not quite complete. To accomplish this, the authors start by describing the coadjoint action of D-the Lie group of all orientation preserving diffeomorphisms on the circle S^1, and its central extension (?), then the authors will give a complete classification of smooth coadjoint orbits. In fact, they can be parameterized by a subspace Of conjugacy classes of (?)(1,1). Finally, the authors will show how to find representatives f coadjoint orbits by analyzing the vector fields stabilizing the orbits, and describe the amazing connection between the characteristic (trace) of conjugacy classes of (?)(1, 1) and that of vector fields stabilizing orbits.展开更多
In a previous paper,the author and his collaborator studied the problem of lifting Hamil-tonian group actions on symplectic varieties and Lagrangian subvarieties to their graded deformation quantizations and apply the...In a previous paper,the author and his collaborator studied the problem of lifting Hamil-tonian group actions on symplectic varieties and Lagrangian subvarieties to their graded deformation quantizations and apply the general results to coadjoint orbit method for semisimple Lie groups.Only even quantizations were considered there.In this paper,these results are generalized to the case of general quantizations with arbitrary periods.The key step is to introduce an enhanced version of the(truncated)period map defined by Bezrukavnikov and Kaledin for quantizations of any smooth sym-plectic variety X,with values in the space of Picard Lie algebroid over X.As an application,we study quantizations of nilpotent orbits of real semisimple groups satisfying certain codimension condition.展开更多
We study the L^p-Fourier transform for a special class of exponential Lie groups, the strong ＊-regular exponential Lie groups. Moreover, we provide an estimate of its norm using the orbit method.
文摘According to Kirillov's idea, the irreducible unitary representations of a Lie group G roughly correspond to the coadjoint orbits (?).In the forward direction one applies the methods of geometric quantization to produce a representation, and in the reverse direction one computes a transform of the character of a representation, to obtain a coadjoint orbit. The method of orbits in the representations of Lie groups suggests the detailed study of coadjoint orbits of a Lie group G in the space (?)~* dual to the Lie algebra (?) of G. In this paper, two primary goals are achieved: one is to completely classify the smooth coadjoint orbits of Virasoro group for nonzero central charge c; the other is to find representatives for coadjoint orbits. These questions have been considered previously by Segal, Kirillov, and Witten, but their results are not quite complete. To accomplish this, the authors start by describing the coadjoint action of D-the Lie group of all orientation preserving diffeomorphisms on the circle S^1, and its central extension (?), then the authors will give a complete classification of smooth coadjoint orbits. In fact, they can be parameterized by a subspace Of conjugacy classes of (?)(1,1). Finally, the authors will show how to find representatives f coadjoint orbits by analyzing the vector fields stabilizing the orbits, and describe the amazing connection between the characteristic (trace) of conjugacy classes of (?)(1, 1) and that of vector fields stabilizing orbits.
基金Supported by China NSFC grants(Grant Nos.12001453 and 12131018)Fundamental Research Funds for the Central Universities(Grant Nos.20720200067 and 20720200071)。
文摘In a previous paper,the author and his collaborator studied the problem of lifting Hamil-tonian group actions on symplectic varieties and Lagrangian subvarieties to their graded deformation quantizations and apply the general results to coadjoint orbit method for semisimple Lie groups.Only even quantizations were considered there.In this paper,these results are generalized to the case of general quantizations with arbitrary periods.The key step is to introduce an enhanced version of the(truncated)period map defined by Bezrukavnikov and Kaledin for quantizations of any smooth sym-plectic variety X,with values in the space of Picard Lie algebroid over X.As an application,we study quantizations of nilpotent orbits of real semisimple groups satisfying certain codimension condition.
文摘We study the L^p-Fourier transform for a special class of exponential Lie groups, the strong ＊-regular exponential Lie groups. Moreover, we provide an estimate of its norm using the orbit method.
