Nonzero Dirac Cohomology Research Articles

Dirac series of E7(−25)

By further sharpening the Helgason-Johnson bound in 1969, this paper classifies all the irreducible unitary representations with non-zero Dirac cohomology of the Hermitian symmetric real form E7(−25).

Dirac series of [formula omitted

Using the sharpened Helgason–Johnson bound, this paper classifies all the irreducible unitary representations with non-zero Dirac cohomology of E7(−5). As an application, we find that the cancellation between the even part and the odd part of the Dirac cohomology continues to happen for certain unitary representations of E7(−5). Assuming the infinitesimal character being integral, we further improve the Helgason–Johnson bound for E7(−5). This should help people to understand (part of) the unitary dual of this group.

A Non-vanishing Criterion for Dirac Cohomology

This paper gives a criterion for the non-vanishing of the Dirac cohomology of \({\mathscr{L}}_{S}(Z)\), where \({\mathscr{L}}_{S}(\cdot )\) 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 E6(2) with non-zero Dirac cohomology. Our calculation continues to support Conjecture 5.7’ of Salamanca-Riba and Vogan (Ann. Math., 148(3), 1067–1133 1998). Moreover, we find more unitary representations for which cancellation happens between the even part and the odd part of their Dirac cohomology.

Dirac Series of GL(n, ℝ)

Abstract The unitary dual of $GL(n, {\mathbb {R}})$ was classified by Vogan in the 1980s. In particular, the Speh representations and the special unipotent representations are the building blocks of the unitary dual with half-integral infinitesimal characters. In this paper, we classify all irreducible unitary $({\mathfrak {g}}, K)$-modules with non-zero Dirac cohomology for $GL(n, {\mathbb {R}})$, as well as a formula for (one of) their spin-lowest $K$-types. Moreover, analogous to the $GL(n,{\mathbb {C}})$ case given in [12], we count the number of the FS-scattered representations of $GL(n, {\mathbb {R}})$.

Dirac index of some unitary representations of [formula omitted] and SO⁎(2n)

Let G be Sp(2n,R) or SO⁎(2n). We compute the Dirac index of a large class of unitary representations considered by Vogan in Section 8 of [33], which include all weakly fair Aq(λ) modules and (weakly) unipotent representations of G as two extreme cases. We conjecture that these representations exhaust all unitary representations of G with nonzero Dirac cohomology. In general, for certain irreducible unitary module of an equal rank group, we clarify the link between the possible cancellations in its Dirac index, and the parities of its spin-lowest K-types.

Dirac series for complex classical Lie groups: A multiplicity-one theorem

This paper computes the Dirac cohomology HD(π) of irreducible unitary Harish-Chandra modules π of complex classical groups viewed as real reductive groups. More precisely, unitary representations with nonzero Dirac cohomology are shown to be unitarily induced from unipotent representations. When nonzero, there is a unique, multiplicity free K-type in π contributing to HD(π). This confirms conjectures formulated by the first named author and Pandžić in 2011.

Dirac series for E6(−14)

Up to equivalence, this paper classifies all the irreducible unitary representations with non-zero Dirac cohomology for the simple Lie group E6(−14), which is of Hermitian symmetric type. Each FS-scattered Dirac series representation of E6(−14) is realized as a composition factor of certain Aq(λ) module. Along the way, we have also obtained all the fully supported irreducible unitary representations of E6(−14) with integral infinitesimal characters.

Scattered Representations of Complex Classical Lie Groups

Abstract As a continuation of [9], this paper studies scattered representations of $SO(2n+1, {{\mathbb{C}}})$, $Sp(2n, {{\mathbb{C}}})$, and $SO(2n, {{\mathbb{C}}})$. We describe the Zhelobenko parameters of these representations, count their cardinality, and determine their spin-lowest $K$-types. We also disprove a conjecture raised in 2015 asserting that the unitary dual can be obtained via parabolic induction from irreducible unitary representations with nonzero Dirac cohomology.

On the Dirac series of U(p, q)

This paper computes the Dirac index of all the weakly fair $$A_{{\mathfrak {q}}}(\lambda )$$ modules of U(p, q). We find counter-examples to a conjecture of Vogan on the unitary dual of U(p, q), which was phrased by Trapa in 2001. However, we still believe that any irreducible unitary representation of U(p, q) with non-zero Dirac cohomology must be a weakly fair $$A_{{\mathfrak {q}}}(\lambda )$$ module.

Dirac series for some real exceptional Lie groups

Up to equivalence, this paper classifies all the irreducible unitary representations with non-zero Dirac cohomology for the following simple real exceptional Lie groups: EI=E6(6),EIV=E6(−26),FI=F4(4),FII=F4(−20). Along the way, we find an irreducible unitary representation of F4(4) whose Dirac index vanishes, while its Dirac cohomology is non-zero. This disproves a conjecture raised in 2015 asserting that there should be no cancellation between the even part and the odd part of the Dirac cohomology.

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 {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 {E_{6}} .

A vanishing theorem for Dirac cohomology of standard modules

This paper studies the Dirac cohomology of standard modules in the setting of graded Hecke algebras with geometric parameters. We prove that the Dirac cohomology of a standard module vanishes if and only if the module is not twisted-elliptic tempered. The proof makes use of two deep results. One is some structural information from the generalized Springer correspondence obtained by S. Kato and Lusztig. Another one is a computation of the Dirac cohomology of tempered modules by Barbasch–Ciubotaru–Trapa and Ciubotaru.We apply our result to compute the Dirac cohomology of ladder representations for type An. For each of such representations with non-zero Dirac cohomology, we associate to a canonical Weyl group representation. We use the Dirac cohomology to conclude that such representations appear with multiplicity one.

Dirac cohomology of cohomologically induced modules for reductive Lie groups

We extend the setting and a proof of the Vogan's conjecture on Dirac cohomology to a possibly disconnected real reductive Lie group $G$ in the Harish-Chandra class. We show that the Dirac cohomology of cohomologically induced module ${\mathcal L}_S(Z)$ is completely determined by the Dirac cohomology of the inducing module $Z$. More precisely, we prove that if $Z$ is weakly good then the Dirac cohomology $H_D({\mathcal L}_S(Z))$ is equal to ${\mathcal L}^{\widetilde{K}}_S(H_D(Z)\otimes\Bbb{C}_{-\rho(\frak{p}\cap\frak{u})})$. An immediate application is a classification of tempered irreducible unitary representations with nonzero Dirac cohomology.

Dirac cohomology of one-$W$-type representations

The smooth hermitian representations of a split reductive p-adic group whose restriction to a maximal hyperspecial compact subgroup contain a single K-type with Iwahori fixed vectors have been studied in [D. Barbasch, A. Moy, Classification of one K-type representations, Trans. Amer. Math. Soc. 351 (1999), no. 10, 4245-4261] in the more general setting of modules for graded affine Hecke algebras with parameters. We show that every such one K-type module has nonzero Dirac cohomology (in the sense of [D. Barbasch, D. Ciubotaru, P. Trapa, The Dirac operator for graded affine Hecke algebras, arXiv:1006.3822]), and use Dirac operator techniques to determine the semisimple part of the Langlands parameter for these modules, thus completing their classification.

On the Dirac cohomology of complex lie group representations

This note studies the problem of classifying all the irreducible unitary representations with nonzero Dirac cohomology for a complex Lie group G. We reduce it to the classification of spherical ones with nonzero Dirac cohomology on the Levi level. Then in the spherical unitary dual, by computing spin norm and utilizing Vogan pencil, we show how to further reduce the classification to fairly few candidate representations.

Dirac cohomology and translation functors

We study the relationship between the Dirac cohomology of a (g,K)-module X and the Dirac cohomology of a Jantzen–Zuckerman translate of X. More precisely, we show that if X is unitary, and if some submodule X′ of a translate of X has nonzero Dirac cohomology, then X has nonzero Dirac cohomology. We also show that the space of harmonic spinors (i.e., the kernel of the Dirac operator) related to X′ embeds into a certain product of harmonic spinors for X and harmonic spinors for the finite-dimensional module used to define the translation. This generalizes, with a simpler proof, results of Mehdi and Parthasarathy (2008) [MP1] and Mehdi and Parthasarathy (2010) [MP2].

Dirac cohomology and the bottom layer K-types

Let G be a connected real reductive Lie group with a maximal compact subgroup K corresponding to a Cartan involution Θ of G. Let q = l ⊕ u be a θ-stable parabolic subalgebra of the complexified Lie algebra g of G, where θ = dΘ. Let L be the centralizer of q in G. We show that, under certain dominance assumptions, cohomological induction with respect to q takes irreducible unitary (l, L∩K)-modules with nonzero Dirac cohomology to irreducible unitary (g,K)-modules which also have nonzero Dirac cohomology.

Dirac cohomology of unitary representations of equal rank exceptional groups

In this paper, we consider the unitary representations of equal rank exceptional groups of type E with a regular lambda-lowest K-type and classify those unitary representations with the nonzero Dirac cohomology.

Unitary representations of classical Lie groups of equal rank with nonzero Dirac cohomology

In this paper, we consider unitary representations of classical groups of equal rank (rank G = rank K) except type CI with regular lambda-lowest K-type and get the necessary and sufficient condition such that those unitary representations considered have nonzero Dirac cohomology.

Dirac cohomology, unitary representations and a proof of a conjecture of Vogan

Let G G be a connected semisimple Lie group with finite center. Let K K be the maximal compact subgroup of G G corresponding to a fixed Cartan involution θ \theta . We prove a conjecture of Vogan which says that if the Dirac cohomology of an irreducible unitary ( g , K ) (\mathfrak {g},K) -module X X contains a K K -type with highest weight γ \gamma , then X X has infinitesimal character γ + ρ c \gamma +\rho _{c} . Here ρ c \rho _{c} is the half sum of the compact positive roots. As an application of the main result we classify irreducible unitary ( g , K ) (\mathfrak {g},K) -modules X X with non-zero Dirac cohomology, provided X X has a strongly regular infinitesimal character. We also mention a generalization to the setting of Kostant’s cubic Dirac operator.