Related papers: Dirac series of $E_{7(-25)}$

Using the sharpened Helgason-Johnson bound, this paper classifies all the irreducible unitary representations with non-zero Dirac cohomology of $E_{7(-5)}$. As an application, we find that the cancellation between the even part and the odd…

This paper classifies all the Dirac series (that is, irreducible unitary representations having non-zero Dirac cohomology) of $E_{7(7)}$. Enhancing the Helgason-Johnson bound in 1969 for the group $E_{7(7)}$ is one key ingredient. Our…

Up to equivalence, this paper classifies all the irreducible unitary representations with non-zero Dirac cohomology for the simple Lie group $E_{6(-14)}$, which is of Hermitian symmetric type. Each FS-scattered Dirac series of $E_{6(-14)}$…

This paper classifies the Dirac series of $E_{8(-24)}$, the linear quaternionic real form of complex $E_8$. One tool for us is a further sharpening of the Helgason-Johnson bound in 1969. Our calculation continues to support Vogan's…

Up to equivalence, this paper classifies all the irreducible unitary representations with non-zero Dirac cohomology for the following simple real exceptional Lie groups: ${\rm EI}=E_{6(6)}, {\rm EIV}=E_{6(-26)}, {\rm FI}=F_{4(4)}, {\rm…

This paper classifies the equivalence classes of irreducible unitary representations with nonvanishing Dirac cohomology for complex $E_6$. This is achieved by using our finiteness result, and by improving the computing method.

In this paper, we classify all unitary representations with non-zero Dirac cohomology for complex Lie group of Type E8. This completes the classification of Dirac series for all complex simple Lie groups.

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…

This paper computes the Dirac cohomology $H_D(\pi)$ of irreducible unitary Harish-Chandra modules $\pi$ of complex classical groups viewed as real reductive groups. More precisely, unitary representations with nonzero Dirac cohomology are…

In this paper we will derive an explicit description of the genuine projective representations of the symmetric group $S_n$ using Dirac cohomology and the branching graph for the irreducible genuine projective representations of $S_n$. In…

The tempered representations of a real reductive Lie group $G$ are naturally partitioned into series associated with conjugacy classes of Cartan subgroups $H$ of $G$. We define partial Dirac cohomology, apply it for geometric construction…

By a theorem of D. Wigner, an irreducible unitary representation with non-zero $(\frak{g},K)$-cohomology has trivial infinitesimal character, and hence up to unitary equivalence, these are finite in number. We have determined the number of…

Let $G$ be a complex connected simple algebraic group with a fixed real form $\sigma$. Let $G(\mathbb{R})=G^\sigma$ be the corresponding group of real points. This paper reports a finiteness theorem for the classification of irreducible…

Let $G$ be a simple non-compact linear Lie group. Let $\pi$ be any irreducible unitary representation of $G$ with infinitesimal character $\Lambda$ whose continuous part is $\nu$. The beautiful Helgason-Jonson bound in 1969 says that the…

The author classifies Klein four symmetric pairs of holomorphic type for the non-compact Lie group of Hermitian type $\mathrm{E}_{7(-25)}$, and applies the results to branching rules.

This paper gives a criterion for the non-vanishing of the Dirac cohomology of $\mathcal{L}_S(Z)$, where $\mathcal{L}_S(\cdot)$ is the cohomological induction functor, while the inducing module $Z$ is irreducible, unitarizable, and in the…

Let $G$ be $Sp(2n, \mathbb{R})$ or $SO^*(2n)$. We compute the Dirac index of a large class of unitary representations considered by Vogan in Section 8 of [Vog84], which include all weakly fair $A_{\mathfrak{q}}(\lambda)$ modules and…

The first part (Sections 1-6) of this paper is a survey of some of the recent developments in the theory of Dirac cohomology, especially the relationship of Dirac cohomology with (g,K)-cohomology and nilpotent Lie algebra cohomology; the…

Let G be a compact connected semisimple Lie group and let H\subset G be a closed connected subgroup such that rank(G)=rank(H) and G/H is a symmetric space. Given an irreducible representation of H, we define a Dirac operator D and determine…

We study Dirac cohomology $H_D^{\mathfrak{g},\mathfrak{h}}(M)$ for modules belonging to category $\mathcal{O}$ of a finite dimensional complex semisimple Lie algebra. We prove Vogan's conjecture, a nonvanishing result for…