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}}}.

scholarly journals Unitary representations with non-zero Dirac cohomology for complex E 6

2019 ◽  
Vol 31 (1) ◽  
pp. 69-82 ◽  
Author(s):  
Chao-Ping Dong
Keyword(s):  
Equivalence Classes ◽  
Unitary Representations ◽  
Dirac Cohomology ◽  
Finiteness Result ◽  
Unitary Dual ◽  
Irreducible Unitary Representations ◽  
Computing Method

Abstract This paper classifies the equivalence classes of irreducible unitary representations with non-vanishing Dirac cohomology for complex {E_{6}} . This is achieved by using our finiteness result, and by improving the computing method. According to a conjecture of Barbasch and Pandžić, our classification should also be helpful for understanding the entire unitary dual of complex {E_{6}} .

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.

scholarly journals Cylindrical representations of some infinite dimensional nuclear Lie groups

1992 ◽  
Vol 46 (2) ◽  
pp. 295-310 ◽  
Author(s):  
Jean Marion
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Additive Group ◽  
Test Functions ◽  
Necessary And Sufficient Condition ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Irreducible Unitary Representations ◽  
Direct Product Group ◽  
Necessary And Sufficient

Let Γ.𝒜 be the semi-direct product group of a nuclear Lie group Γ with the additive group 𝒜 of a real nuclear vector space. We give an explicit description of all the continuous representations of Γ.𝒜 the restriction of which to 𝒜 is a cyclic unitary representation, and a necessary and sufficient condition for the unitarity of such cylindrical representations is stated. This general result is successfully used to obtain irreducible unitary representations of the nuclear Lie groups of Riemannian motions, and, in the setting of the theory of multiplicative distributions initiated by I.M. Gelfand, it is proved that for any connected real finite dimensional Lie groupGand for any strictly positive integerkthere exist non located and non trivially decomposable representations of orderkof the nuclear Lie group(M;G) of all theG-valued test-functions on a given paracompact manifoldM.

LEFT INVARIANT COMPLEX STRUCTURES ON NILPOTENT SIMPLY CONNECTED INDECOMPOSABLE 6-DIMENSIONAL REAL LIE GROUPS

2007 ◽  
Vol 17 (01) ◽  
pp. 115-139 ◽  
Author(s):  
L. MAGNIN
Keyword(s):  
Automorphism Group ◽  
Normal Form ◽  
Lie Groups ◽  
Lie Algebras ◽  
Lie Group ◽  
Normal Forms ◽  
Equivalence Classes ◽  
Complex Structures ◽  
Simply Connected ◽  
Connected Lie Group

Integrable complex structures on indecomposable 6-dimensional nilpotent real Lie algebras have been computed in a previous paper, along with normal forms for representatives of the various equivalence classes under the action of the automorphism group. Here we go to the connected simply connected Lie group G0 associated to such a Lie algebra 𝔤. For each normal form J of integrable complex structures on 𝔤, we consider the left invariant complex manifold G = (G0, J) associated to G0 and J. We explicitly compute a global holomorphic chart for G and we write down the multiplication in that chart.

scholarly journals On Square-Integrable Representations of A Lie Group of 4-Dimensional Standard Filiform Lie Algebra

CAUCHY ◽  
2020 ◽  
Vol 6 (2) ◽  
pp. 84
Author(s):  
Edi Kurniadi
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Unitary Representations ◽  
Identity Operator ◽  
Orbit Method ◽  
Irreducible Unitary Representations ◽  
Dimension 2 ◽  
Filiform Lie Group ◽  
Filiform Lie Algebra

<p class="Abstract">In this paper, we study irreducible unitary representations of a real standard filiform Lie group with dimension equals 4 with respect to its basis. To find this representations we apply the orbit method introduced by Kirillov. The corresponding orbit of this representation is genereric orbits of dimension 2. Furthermore, we show that obtained representation of this group is square-integrable. Moreover, in such case , we shall consider its Duflo-Moore operator as multiple of scalar  identity operator. In our case  that scalar is equal to one.</p>

Singular Integrals on Ultraspherical Series

1972 ◽  
Vol 24 (1) ◽  
pp. 60-71
Author(s):  
Charles F. Dunkl
Keyword(s):  
Compact Group ◽  
Harmonic Analysis ◽  
Singular Integral ◽  
Singular Integrals ◽  
Integral Operators ◽  
Equivalence Classes ◽  
Unitary Representations ◽  
Irreducible Unitary Representations ◽  
Ultraspherical Series ◽  
Zonal Functions

One of the main uses of harmonic analysis on the sphere is to discover new theorems about series of ultraspherical (Gegenbauer) polynomials. In this paper, we will construct singular integral operators from scalar functions on the sphere to vector functions. These operators when restricted to zonal functions give Lp-bounded (1 < p < ∞ ) operators on ultraspherical series.We will use [7, Chapter 9] as our main reference. Let G denote a compact group, with identity e, and Ĝ its dual, the set of equivalence classes of continuous irreducible unitary representations of G.

Weak Containment and Induced Representations of Groups

1962 ◽  
Vol 14 ◽  
pp. 237-268 ◽  
Author(s):  
J. M. G. Fell
Keyword(s):  
Dual Space ◽  
Locally Compact Group ◽  
Equivalence Classes ◽  
Unitary Representations ◽  
Unitary Equivalence ◽  
Locally Compact ◽  
Induced Representations ◽  
Weak Containment ◽  
Irreducible Unitary Representations ◽  
Special Case

Let G be a locally compact group and G† its dual space, that is, the set of all unitary equivalence classes of irreducible unitary representations of G. An important tool for investigating the group algebra of G is the so-called hull-kernel topology of G†, which is discussed in (3) as a special case of the relation of weak containment. The question arises: Given a group G, how do we determine G† and its topology? For many groups G, Mackey's theory of induced representations permits us to catalogue all the elements of G†. One suspects that by suitably supplementing this theory it should be possible to obtain the topology of G† at the same time. It is the purpose of this paper to explore this possibility. Unfortunately, we are not able to complete the programme at present.

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