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.

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].

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.

Representations of the principal series of a complex semisimple Lie group

1970 ◽  
Vol 4 (2) ◽  
pp. 117-121 ◽  
Author(s):  
E. A. Gutkin
Keyword(s):  
Lie Group ◽  
Principal Series ◽  
Semisimple Lie Group ◽  
Complex Semisimple

Pseudo-K�hlerian structure on domains over a complex semisimple Lie group

2002 ◽  
Vol 323 (1) ◽  
pp. 1-29
Author(s):  
Gregor Fels
Keyword(s):  
Lie Group ◽  
Semisimple Lie Group ◽  
Complex Semisimple

DISCRETE DECOMPOSABLE BRANCHING LAWS AND PROPER MOMENTUM MAPS

2012 ◽  
Vol 23 (06) ◽  
pp. 1250021 ◽  
Author(s):  
SALMA NASRIN
Keyword(s):  
Lie Group ◽  
Discrete Series ◽  
Geometric Quantization ◽  
Series Representation ◽  
Coadjoint Orbit ◽  
Irreducible Unitary Representation ◽  
Symmetric Pair ◽  
Holomorphic Discrete Series ◽  
Branching Laws ◽  
Scalar Type

Suppose an irreducible unitary representation π of a Lie group G is obtained as a geometric quantization of a coadjoint orbit [Formula: see text] in the Kirillov–Kostant–Duflo orbit philosophy. Let H be a closed subgroup of G, and we compare the following two conditions. (1) The restriction π|H is discretely decomposable in the sense of Kobayashi. (2) The momentum map [Formula: see text] is proper. In this article, we prove that (1) is equivalent to (2) when π is any holomorphic discrete series representation of scalar type of a semisimple Lie group G and (G, H) is any symmetric pair.

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].

On the irreducible representations of a complex semisimple Lie group

1970 ◽  
Vol 4 (2) ◽  
pp. 163-165 ◽  
Author(s):  
D. P. Zhelobenko
Keyword(s):  
Lie Group ◽  
Irreducible Representations ◽  
Semisimple Lie Group ◽  
Complex Semisimple
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