On the Dirac cohomology of complex lie group representations

Dong, Chao-Ping

doi:10.1007/s00031-013-9206-0

Cited by 22 publications

(30 citation statements)

References 11 publications

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“…Here K L := K ∩ L, and K L is the pin double covering group of K L . This completely extends Theorem 6.1 of [4] to real linear groups.…”

Section: Introductionsupporting

confidence: 71%

A non-vanishing criterion for Dirac cohomology

Dong

2021

Preprint

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This paper gives a criterion for the non-vanishing of the Dirac cohomology of LS(Z), where LS(•) is the cohomological induction functor, while the inducing module Z is irreducible, unitarizable, and in the good range. As an application, we give a formula counting the number of strings in the Dirac series. Using this formula, we classify all the irreducible unitary representations of E 6(2) with non-zero Dirac cohomology. Our calculation continues to support Conjecture 5.7' of Salamanca-Riba and Vogan [19].

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“…Here K L := K ∩ L, and K L is the pin double covering group of K L . This completely extends Theorem 6.1 of [4] to real linear groups.…”

Section: Introductionsupporting

confidence: 71%

A non-vanishing criterion for Dirac cohomology

Dong

2021

Preprint

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“…Let (c) Finally, by (13) and Theorem 3.1 and Corollary 5.85 of [4], we see that the entire bottom layer of J s is K-multiplicity free. On the other hand, if H D J s is nonzero, then by (a) and Lemma 2.3 of [2], the K-type E c must lie in the bottom layer. Hence it must occur with multiplicity one.…”

Section: Proof Of Theorem 12mentioning

confidence: 94%

“…We care the most about the case when X is unitary. Motivated by the problem of classifying all the irreducible unitary representations of G with nonzero Dirac cohomology, we have introduced the spin norm and the spin lowest K-type in [2], which turn out to give the right framework. …”

Section: Preliminaries On Dirac Cohomologymentioning

confidence: 99%

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On a Conjecture of Barbasch and Pandžić

Dong

2015

Communications in Algebra

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Let G be a connected complex semisimple Lie group. Let J s be the irreducible ( , K) module with Zhelobenko parameters c /2 −s c /2 , where s ∈ W is an involution. A conjecture of Barbasch and Pandžić claims that the Dirac cohomology of any unitary J s is either zero or the trivial K-type with multiplicity 2 l 0 /2 , where l 0 is the split rank of G. We prove this conjecture for J s in the good range.

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“…Figure 9. Finally, it remains to check (6) for any u-large µ = [a, b, c, d, e, f, g] such that 0 ≤ a ≤ 22, 0 ≤ b, c, d, e ≤ 7, 1 ≤ f ≤ 8, 1 ≤ g ≤ 31. This has been carried out on a computer.…”

Section: Now In View Ofmentioning

confidence: 99%

Unitary Representations with Dirac Cohomology: Finiteness in the Real Case

Dong

2019

International Mathematics Research Notices

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Let G be a complex connected simple algebraic group with a fixed real form σ. Let G(R) = G σ be the corresponding group of real points. This paper reports a finiteness theorem for the classification of irreducible unitary Harish-Chandra modules of G(R) (up to equivalence) having non-vanishing Dirac cohomology. Moreover, we study the distribution of the spin norm along Vogan pencils for certain G(R), with particular attention paid to the unitarily small convex hull introduced by Salamanca-Riba and Vogan.2010 Mathematics Subject Classification. Primary 22E46.

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On the Dirac cohomology of complex lie group representations

Cited by 22 publications

References 11 publications

A non-vanishing criterion for Dirac cohomology

A non-vanishing criterion for Dirac cohomology

On a Conjecture of Barbasch and Pandžić

Unitary Representations with Dirac Cohomology: Finiteness in the Real Case

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