Classification of Irreducible Unitary Representations of Compact Simple Lie Groups. I

1966 ◽  
Vol 7 (10) ◽  
pp. 1824-1832 ◽  
Author(s):  
M. L. Mehta
Keyword(s):  
Lie Groups ◽  
Unitary Representations ◽  
Simple Lie Groups ◽  
Irreducible Unitary Representations

scholarly journals Unitary representations with Dirac cohomology: A finiteness result for complex Lie groups

2020 ◽  
Vol 32 (4) ◽  
pp. 941-964 ◽  
Author(s):  
Jian Ding ◽  
Chao-Ping Dong
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Equivalence Classes ◽  
Unitary Representations ◽  
Parabolic Subgroups ◽  
Dirac Cohomology ◽  
Finiteness Result ◽  
Irreducible Unitary Representations ◽  
Cohomological Induction ◽  
Good Range

AbstractLet G be a connected complex simple Lie group, and let {\widehat{G}^{\mathrm{d}}} be the set of all equivalence classes of irreducible unitary representations with non-vanishing Dirac cohomology. We show that {\widehat{G}^{\mathrm{d}}} consists of two parts: finitely many scattered representations, and finitely many strings of representations. Moreover, the strings of {\widehat{G}^{\mathrm{d}}} come from {\widehat{L}^{\mathrm{d}}} via cohomological induction and they are all in the good range. Here L runs over the Levi factors of proper θ-stable parabolic subgroups of G. It follows that figuring out {\widehat{G}^{\mathrm{d}}} requires a finite calculation in total. As an application, we report a complete description of {\widehat{F}_{4}^{\mathrm{d}}}.

On non-compact groups I. Classification of non-compact real simple Lie groups and groups containing the Lorentz group

1965 ◽  
Vol 287 (1411) ◽  
pp. 519-531 ◽  
Keyword(s):  
Lie Groups ◽  
Lorentz Group ◽  
Compact Groups ◽  
Bilinear Forms ◽  
Unimodular Group ◽  
Simple Lie Groups ◽  
Real Complex

The classification of all simple real Lie groups is discussed and given explicitly in the form of tables. In particular four classes of groups are identified which contain the Lorentz group. These are the groups which leave bilinear forms in real, complex and quaternionic spaces invariant, and the unimodular group in n -dimension.

scholarly journals One more method to construct the irreducible unitary representations of nipotent Lie groups

1986 ◽  
Vol 40 (1) ◽  
pp. 89-94
Author(s):  
A. A. Astaneh
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Unitary Representations ◽  
Irreducible Representations ◽  
Nilpotent Lie Group ◽  
Irreducible Unitary Representations ◽  
Simply Connected

AbstractIn this paper one more canonical method to construct the irreducible unitary representations of a connected, simply connected nilpotent Lie group is introduced. Although we used Kirillov' analysis to deduce this procedure, the method obtained differs from that of Kirillov's, in that one does not need to consider the codjoint representation of the group in the dual of its Lie algebra (in fact, neither does one need to consider the Lie algebra of the group, provided one knows certain connected subgroups and their characters). The method also differs from that of Mackey's as one only needs to induce characters to obtain all irreducible representations of the group.

On the cohomology of uniform arithmetically defined subgroups in SU*(2n)

2011 ◽  
Vol 151 (3) ◽  
pp. 421-440 ◽  
Author(s):  
JOACHIM SCHWERMER ◽  
CHRISTOPH WALDNER
Keyword(s):  
Lie Group ◽  
Symmetric Spaces ◽  
Automorphic Forms ◽  
Close Relation ◽  
Unitary Representations ◽  
Geometric Constructions ◽  
Locally Symmetric Spaces ◽  
Totally Geodesic ◽  
Irreducible Unitary Representations

AbstractWe study the cohomology of compact locally symmetric spaces attached to arithmetically defined subgroups of the real Lie group G = SU*(2n). Our focus is on constructing totally geodesic cycles which originate with reductive subgroups in G. We prove that these cycles, also called geometric cycles, are non-bounding. Thus this geometric construction yields non-vanishing (co)homology classes.In view of the interpretation of these cohomology groups in terms of automorphic forms, the existence of non-vanishing geometric cycles implies the existence of certain automorphic forms. In the case at hand, we substantiate this close relation between geometry and automorphic theory by discussing the classification of irreducible unitary representations of G with non-zero cohomology in some detail. This permits a comparison between geometric constructions and automorphic forms.

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