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

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…

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 $E_{7(-25)}$.

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…

Motivated by the $(\mathfrak{g},K)$-cohomology and Dirac cohomology, we determine Dirac series of $\mathrm{GL}(n,\mathbb{H})$, and show that the spin lowest $K$-type of any Dirac series, which determines the Dirac cohomology, is unique and…

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…

We determine the Dirac series of $U(n,2)$ and give related information about Dirac series: the spin lowest $K$-types are multiplicity free, and the Dirac cohomology are consistent with the Dirac index.

The unitary dual of $GL(n, \mathbb{R})$ was classified by Vogan in the 1980s. Focusing on the irreducible unitary representations of $GL(n, \mathbb{R})$ with half-integral infinitesimal characters, we find that Speh representations and the…

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 for complex $E_7$. As applications, we verify a few conjectures raised in 2011, 2019 and 2020 for this exceptional Lie group. In particular, according to Conjecture 1.1 of Barbasch and Pandzic [BP],…

The idea of using Dirac cohomology to study branching laws was initiated by Huang, Pandzi\'c and Zhu in 2013 [HPZ]. One of their results says that the Dirac cohomology of $\pi$ completely determines $\pi|_{K}$, where $\pi$ is any…

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 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…

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…

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.

We define analogues of the Casimir and Dirac operators for graded affine Hecke algebras, and establish a version of Parthasarathy's Dirac operator inequality. We then prove a version of Vogan's Conjecture for Dirac cohomology. The…

In this paper, we study the Dirac cohomology of minimal representations for all real reductive groups G. The Dirac indices of these representations are also studied when G is of equal rank, giving some counterexamples of a conjecture of…

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…

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…

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…