The Method of Orbits for Real Lie Groups (original) (raw)
2002, Kyungpook Math. J. , Vol. 42, No. 2, 199-272
AI-generated Abstract
Research into representations of Lie groups has evolved significantly, notably through the work of Kirillov with the orbit method. This article expands on the development of the orbit method specifically for real Lie groups, including applications for the Heisenberg and Jacobi groups. The use of geometric quantization and properties of nilpotent orbits are discussed, shedding light on the correspondence of orbits with unitary representations, while highlighting gaps in the understanding of nilpotent representations.
Sign up for access to the world's latest research.
checkGet notified about relevant papers
checkSave papers to use in your research
checkJoin the discussion with peers
checkTrack your impact
Related papers
A remark on the invariant theory of real Lie groups
Colloquium Mathematicum, 2019
We present a simple remark that assures that the invariant theory of certain real Lie groups coincides with that of the underlying affine, algebraic R-groups. In particular, this result applies to the non-compact orthogonal or symplectic Lie groups.
Matrix coefficients and coadjoint orbits of compact Lie groups
Proceedings of the American Mathematical Society, 2007
Let G be a compact Lie group. We use Weyl functional calculus (Anderson, 1969) and symplectic convexity theorems to determine the support and singular support of the operator-valued Fourier transform of the product of the j-function and the pull-back of an arbitrary unitary irreducible representation of G to the Lie algebra, strengthening and generalizing the results of Cazzaniga, 1992. We obtain as a consequence a new demonstration of the Kirillov correspondence for compact Lie groups.
Complex geometry and representations of Lie groups
Contemporary Mathematics, 2001
Go be a real form of G. Then the representation theory of the real reductive Lie group Go is intimately connected with the geometry of Go-orbits on Z. The open orbits correspond to the discrete series representations and their analytic continuations, closed orbits correspond to the principal series, and certain other orbits give the other series of tempered representations. Here I try to indicate some of that interplay between geometry and analysis, concentrating on the complex geometric aspects of the open orbits and the related representations. Contents: Part I.
1997
A version of quantum orbit method is presented for real forms of equal rank of quantum complex simple groups. A quantum moment map is constructed, based on the canonical isomorphism between a quantum Heisenberg algebra and an algebra of functions on a family of quantum G-spaces. For the series A, we construct some irreducible * -representations of Uqg which correspond to the semi-simple dressing orbits of minimal dimension in the dual Poisson Lie group. It is shown that some complimentary series representations correspond to some quantum 'tunnel' G-spaces which do not have a quasi-classical analog.
On Lie Groups and The Theory of Complex Variables
2013
Abstract. In this note we envisage the relation existing between the Lie Groups and the Theory of Complex Variables. In particular, it is shown that the dimensions of the irreducibles representations of SUpNq may be written in terms of the Eisenstein integers and an identity is built up between the imaginary parts of the dimensions of the irreducible representations of the Lie Groups SUp3q and Spp4q.
Geometric results on linear actions of reductive Lie groups for applications to homogeneous dynamics
Ergodic Theory and Dynamical Systems
Several problems in number theory when reformulated in terms of homogenous dynamics involve study of limiting distributions of translates of algebraically defined measures on orbits of reductive groups. The general non-divergence and linearization techniques, in view of Ratner’s measure classification for unipotent flows, reduce such problems to dynamical questions about linear actions of reductive groups on finite-dimensional vector spaces. This article provides general results which resolve these linear dynamical questions in terms of natural group theoretic or geometric conditions.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.