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

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}} .

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

Forum Mathematicum ◽

10.1515/forum-2019-0295 ◽

2020 ◽

Vol 32 (4) ◽

pp. 941-964 ◽

Cited By ~ 1

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}}}.

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Singular Integrals on Ultraspherical Series

Canadian Journal of Mathematics ◽

10.4153/cjm-1972-008-1 ◽

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.

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Weak Containment and Induced Representations of Groups

Canadian Journal of Mathematics ◽

10.4153/cjm-1962-016-6 ◽

1962 ◽

Vol 14 ◽

pp. 237-268 ◽

Cited By ~ 82

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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Description of unitary representations of the group of infinite 𝑝-adic integer matrices

Representation Theory of the American Mathematical Society ◽

10.1090/ert/577 ◽

2021 ◽

Vol 25 (21) ◽

pp. 606-643

Author(s):

Yury Neretin

Keyword(s):

Irreducible Representation ◽

Unitary Representations ◽

Irreducible Representations ◽

Infinite Matrices ◽

Natural Topology ◽

Content Type ◽

Residue Ring ◽

Finite Dimensional ◽

Irreducible Unitary Representations

We classify irreducible unitary representations of the group of all infinite matrices over a p p -adic field ( p ≠ 2 p\ne 2 ) with integer elements equipped with a natural topology. Any irreducible representation passes through a group G L GL of infinite matrices over a residue ring modulo p k p^k . Irreducible representations of the latter group are induced from finite-dimensional representations of certain open subgroups.

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MULTIPLICITY-FREE DECOMPOSITIONS OF THE MINIMAL REPRESENTATION OF THE INDEFINITE ORTHOGONAL GROUP

International Journal of Mathematics ◽

10.1142/s0129167x08005084 ◽

2008 ◽

Vol 19 (10) ◽

pp. 1187-1201 ◽

Cited By ~ 2

Author(s):

MASAYASU MORIWAKI

Keyword(s):

Unitary Representation ◽

Orthogonal Group ◽

Irreducible Unitary Representation ◽

Unitary Representations ◽

Minimal Representation ◽

Infinite Dimensional ◽

Irreducible Unitary Representations ◽

Direct Product Group ◽

Multiplicity Free ◽

Kirillov Dimension

Kazhdan, Kostant, Binegar–Zierau and Kobayashi–Ørsted constructed a distinguished infinite-dimensional irreducible unitary representation π of the indefinite orthogonal group G = O(2p, 2q) for p, q ≥ 1 with p + q > 2, which has the smallest Gelfand–Kirillov dimension 2p + 2q - 3 among all infinite-dimensional irreducible unitary representations of G and hence is called the minimal representation. We consider, for which subgroup G′ of G, the restriction π|G′ is multiplicity-free. We prove that the restriction of π to any subgroup containing the direct product group U(p1) × U(p2) × U(q) for p1, p2 ≥ 1 with p1 + p2 = p is multiplicity-free, whereas the restriction to U(p1) × U(p2) × U(q1) × U(q2) for q1, q2 ≥ 1 with q1 + q2 = q has infinite multiplicities.

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