A LIMIT FORMULA FOR EVEN NILPOTENT ORBITS

2008 ◽  
Vol 19 (02) ◽  
pp. 223-236
Author(s):  
MLADEN BOŽIČEVIĆ
Keyword(s):  
Lie Group ◽  
Coadjoint Orbits ◽  
Nilpotent Orbits ◽  
Semisimple Lie Group ◽  
Real Form ◽  
Parabolic Subalgebra ◽  
Limit Formula ◽  
Canonical Measure ◽  
Complex Semisimple

Let Gℝ be a real form of a complex, semisimple Lie group G. Assume [Formula: see text] is an even nilpotent coadjoint Gℝ-orbit. We prove a limit formula, expressing the canonical measure on [Formula: see text] as a limit of canonical measures on semisimple coadjoint orbits, where the parameter of orbits varies over the negative chamber defined by the parabolic subalgebra associated with [Formula: see text].

scholarly journals Invariant measures on nilpotent orbits associated with holomorphic discrete series

2021 ◽  
Vol 25 (24) ◽  
pp. 732-747
Author(s):  
Mladen Božičević
Keyword(s):  
Lie Group ◽  
Invariant Measures ◽  
Discrete Series ◽  
Coadjoint Orbits ◽  
Semisimple Lie Group ◽  
Real Form ◽  
Holomorphic Discrete Series ◽  
Content Type ◽  
Wave Front Set ◽  
Complex Semisimple

Let G R G_\mathbb R be a real form of a complex, semisimple Lie group G G . Assume G R G_\mathbb R has holomorphic discrete series. Let W \mathcal W be a nilpotent coadjoint G R G_\mathbb R -orbit contained in the wave front set of a holomorphic discrete series. We prove a limit formula, expressing the canonical measure on W \mathcal W as a limit of canonical measures on semisimple coadjoint orbits, where the parameter of orbits varies over the positive chamber defined by the Borel subalgebra associated with holomorphic discrete series.

scholarly journals ON THE EXACTNESS OF KOSTANT–KIRILLOV FORM AND THE SECOND COHOMOLOGY OF NILPOTENT ORBITS

2012 ◽  
Vol 23 (08) ◽  
pp. 1250086 ◽  
Author(s):  
INDRANIL BISWAS ◽  
PRALAY CHATTERJEE
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Nilpotent Orbits ◽  
Semisimple Lie Group ◽  
Simple Lie Algebra ◽  
Adjoint Orbits ◽  
Adjoint Orbit

We give a criterion for the Kostant–Kirillov form on an adjoint orbit in a real semisimple Lie group to be exact. We explicitly compute the second cohomology of all the nilpotent adjoint orbits in every complex simple Lie algebra.

Corwin–Greenleaf multiplicity functions for complex semisimple symmetric spaces

2015 ◽  
Vol 26 (05) ◽  
pp. 1550039
Author(s):  
Salma Nasrin
Keyword(s):  
Symmetric Space ◽  
Lie Algebras ◽  
Lie Group ◽  
Symmetric Spaces ◽  
Coadjoint Orbit ◽  
Natural Projection ◽  
Real Form ◽  
Single Orbit ◽  
Complex Semisimple ◽  
Complex Symmetric

Let Gℂ be a complex simple Lie group, GU a compact real form, and [Formula: see text] the natural projection between the dual of the Lie algebras. We prove that, for any coadjoint orbit [Formula: see text] of GU, the intersection of [Formula: see text] with a coadjoint orbit [Formula: see text] of Gℂ is either an empty set or a single orbit of GU if [Formula: see text] is isomorphic to a complex symmetric space.

scholarly journals Satake diagrams and real structures on spherical varieties

2015 ◽  
Vol 26 (12) ◽  
pp. 1550103 ◽  
Author(s):  
Dmitri Akhiezer
Keyword(s):  
Lie Group ◽  
Dynkin Diagram ◽  
Real Structure ◽  
Semisimple Lie Group ◽  
Spherical Varieties ◽  
Spherical Subgroup ◽  
Complex Semisimple ◽  
Uniqueness And Existence ◽  
Definition Of

With each antiholomorphic involution [Formula: see text] of a connected complex semisimple Lie group [Formula: see text] we associate an automorphism [Formula: see text] of its Dynkin diagram. The definition of [Formula: see text] is given in terms of the Satake diagram of [Formula: see text]. Let [Formula: see text] be a self-normalizing spherical subgroup. If [Formula: see text] then we prove the uniqueness and existence of a [Formula: see text]-equivariant real structure on [Formula: see text] and on the wonderful completion of [Formula: see text].

scholarly journals ON HOMOGENEOUS CR MANIFOLDS AND THEIR CR ALGEBRAS

2006 ◽  
Vol 03 (05n06) ◽  
pp. 1199-1214
Author(s):  
ANDREA ALTOMANI ◽  
COSTANTINO MEDORI
Keyword(s):  
Lie Group ◽  
Levi Form ◽  
Flag Manifold ◽  
Semisimple Lie Group ◽  
Real Form ◽  
Cr Manifolds ◽  
Geometrical Properties ◽  
Homogeneous Cr Manifolds

In this paper we show some results on homogeneous CR manifolds, proved by introducing their associated CR algebras. In particular, we give different notions of nondegeneracy (generalizing the usual notion for the Levi form) which correspond to geometrical properties for the corresponding manifolds. We also give distinguished equivariant CR fibrations for homogeneous CR manifolds. In the second part of the paper we apply these results to minimal orbits for the action of a real form of a semisimple Lie group Ĝ on a flag manifold Ĝ/Q.

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