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

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 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

INVERSION OF HOROSPHERICAL INTEGRAL TRANSFORM ON REAL SEMISIMPLE LIE GROUPS

1992 ◽  
Vol 07 (supp01b) ◽  
pp. 1047-1071 ◽  
Author(s):  
Anton ZORICH
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Inversion Formula ◽  
Integral Transform ◽  
Regular Representation ◽  
Integral Transforms ◽  
Semisimple Lie Groups ◽  
Semisimple Lie Group ◽  
Complex Semisimple ◽  
Local Inversion

There exists the wonderful integral transform on complex semisimple Lie groups, which assigns to a function on the group the set of its integrals over "generalized horospheres" — some specific submanifolds of the Lie group. The local inversion formula for this integral transform, discovered in 50's for [Formula: see text] by Gel'fand and Graev, made it possible to decompose the regular representation on [Formula: see text] into irreducible ones. In case of real semisimple Lie group the situation becomes more complicated, and usually there is no reasonable analogous integral transform at all. Nevertheless, in the present paper we succeed to define the integral transforms on the Lorentz group and some other real semisimple Lie groups, which are in a sense analogous to the integration over "horospheres". We obtain the inversion formulas for these integral transforms.

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].

CYCLIC MODULES FOR A COMPLEX SEMISIMPLE LIE GROUP

1973 ◽  
Vol 7 (3) ◽  
pp. 497-510 ◽  
Author(s):  
D P Želobenko
Keyword(s):  
Lie Group ◽  
Semisimple Lie Group ◽  
Complex Semisimple
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