Dirac Cohomology for the Cubic Dirac Operator

Dirac cohomology, elliptic representations and endoscopy

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 second part (Sections 7-12) is devoted to understanding the unitary elliptic representations and endoscopic transfer by using the techniques in Dirac cohomology. A few problems and conjectures are proposed for further investigations.

Dirac cohomology of unipotent representations of Sp(2n,R) and U(p,q)

In this paper we study the problem of computing the Dirac cohomology of the special unipotent representations of the real groups Sp(2n,R) and U(p,q).

Dirac算子、Dirac上同调与$A(m\vert n)$的单位性

Dirac Operators, Dirac Cohomology and Unitarity for $A(m\vert n)$

本文研究了$A(m\vert n)$型李超代数的Dirac算子和上同调，将其与单位性联系起来，推导了Dirac不等式，并明确给出了单位化简单超模的Dirac上同调。

三次 Dirac 算子与 colour Lie 代数的奇异 Freudenthal-de Vries 公式

Cubic Dirac operators and the strange Freudenthal-de Vries formula for colour Lie algebras

本文为 colour Lie 代数定义三次 Dirac 算子，给出从 $ε$-正交表示构造代数的充要条件，并以此证明奇异 Freudenthal-de Vries 公式与 Parthasarathy 公式，同时计算了 Dirac 算子诱导的上同调。

Dirac cohomology and character lifting

The endoscopic transfer factor is expressed as difference of characters for the even and odd parts of the spin modules, or Dirac index of the trivial representation. The lifting of tempered characters in terms of index of Dirac cohomology is calculated explicitly.

Higher Dirac cohomology of modules with generalized infinitesimal character

We modify the definition of Dirac cohomology in such a way that the standard properties of the usual Dirac cohomology, valid for modules with infinitesimal character, become valid also for modules with only generalized infinitesimal character.

Hopf-Hecke algebras, infinitesimal Cherednik algebras, and Dirac cohomology

Hopf-Hecke algebras and Barbasch-Sahi algebras were defined by the first named author (2016) in order to provide a general framework for the study of Dirac cohomology. The aim of this paper is to explore new examples of these definitions and to contribute to their classification. Hopf-Hecke algebras are distinguished by an orthogonality condition and a PBW property. The PBW property for algebras such as the ones considered here has been of great interest in the literature and we extend this discussion by further results on the classification of such deformations and by a class of hitherto unex…

Dirac cohomology for graded affine Hecke algebras

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 formulation of the conjecture depends on a uniform geometric parametrization of spin representations of Weyl groups. Finally, we apply our results to the study of unitary representations.

Dirac cohomology for the degenerate affine Hecke Clifford algebra

We define an analogue of the Dirac operator for the degenerate affine Hecke-Clifford algebra. A main result is to relate the central characters of the degenerate affine Hecke-Clifford algebra with the central characters of the Sergeev algebra via Dirac cohomology. The action of the Dirac operator on certain modules is also computed. Results in this paper could be viewed as a projective version of the Dirac cohomology of the degenerate affine Hecke algebra.

A non-vanishing criterion for Dirac cohomology

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 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 [SV]. Moreover, we find more unitary representations for w…

Unitary Representations with non-zero Dirac cohomology for complex $E_6$

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.

Dirac cohomology and Euler-Poincaré pairing for weight modules

Let $\mathfrak{g}$ be a reductive Lie algebra over $\mathbb{C}$. For any simple weight module of $\mathfrak{g}$ with finite-dimensional weight spaces, we show that its Dirac cohomology is vanished unless it is a highest weight module. This completes the calculation of Dirac cohomology for simple weight modules since the Dirac cohomology of simple highest weight modules was carried out in our previous work. We also show that the Dirac index pairing of two weight modules which have infinitesimal characters agrees with their Euler-Poincaré pairing. The analogue of this result for Harish-Chandra m…

Dirac cohomology, the projective supermodules of the symmetric group and the Vogan morphism

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 2015 Ciubotaru and He, using the extended Dirac index, showed that the characters of the projective representations of $S_n$ are related to the characters of elliptic graded modules. We derived the branching graph using Dirac theory and combinatorics relating to the cohomology of Borel varieties $\mathcal{B}_e$ of $\mathfrak{g}$ and were able to use Dirac coh…

Induction Functor in Non-commutative Equivariant Cohomology and Dirac Cohomology

The aim of this paper is to put some recent results of Huang-Pandzic (conjectured by Vogan) and Kostant on Dirac cohomology in a broader perspective.This is achieved by introducing an induction functor in the noncommutative equivariant cohomology. In this context, the results of Huang-Pandzic and Kostant are interpreted as special cases (corresponding to the manifold being a point) of more general results on noncommutative equivariant cohomology introduced by Alekseev-Meinrenken.

Dirac cohomology and simple modules of the Dunkl-Opdam subalgebra via inherited Drinfeld properties

In this paper we define a new presentation for the Dunkl-Opdam subalgebra of the rational Cherednik algebra. This shows that the Dunkl-Opdam subalgebra is a Drinfeld algebra. We use this fact to define Dirac cohomology for the DO subalgebra. We also formalise generalised graded Hecke algebras and define a Langlands classification to generalised graded Hecke algebras.

极小表示的 Dirac 上同调

Dirac cohomology of minimal representations

本文研究了所有实约化群极小表示的 Dirac 上同调，并在等秩情形下计算 Dirac 指数，给出了 Huang 2015 年猜想的反例。

Dirac 上同调与复对偶对的 $\Theta$-对应

Dirac cohomology and $Θ$-correspondence for complex dual pairs

本文研究复约化对偶对下 Dirac 上同调在 Howe 的 $\Theta$-对应中的行为，刻画了具有非零 Dirac 上同调的 Harish-Chandra 模，其 $\Theta$-提升仍保持非零，并显式计算了提升后的 Dirac 上同调。

Dirac cohomology for symplectic reflection algebras

We define uniformly the notions of Dirac operators and Dirac cohomology in the framework of the Hecke algebras introduced by Drinfeld. We generalize in this way the Dirac cohomology theory for Lusztig's graded affine Hecke algebras. We apply these constructions to the case of symplectic reflection algebras defined by Etingof-Ginzburg, particularly to rational Cherednik algebras for real or complex reflection groups with parameters t,c. As applications, we give criteria for unitarity of modules in category O and we show that the 0-fiber of the Calogero-Moser space admits a description in terms …

Symplectic Dirac cohomology and lifting of characters to metaplectic groups

We formulate the transfer factor of character lifting from orthogonal groups to symplectic groups by Adams in the framework of symplectic Dirac cohomology for the Lie superalgebras and the Rittenberg-Scheunert correspondence of representations of the Lie superalgebra $\fro\frsp(1|2n)$ and the Lie algebra $\fro(2n+1)$. This leads to formulation of a direct lifting of characters from the linear symplectic group $Sp(2n,\bbR)$ to its nonlinear covering metaplectic group $Mp(2n,\bbR)$.

三次Dirac算子与基本经典李超代数的Dirac上同调

Cubic Dirac Operators and Dirac Cohomology for Basic Classical Lie Superalgebras

本文研究了基本经典李超代数上超模的Dirac上同调，通过三次Dirac算子建立了超模的Casselman-Osborne定理超代数版本，并证明了最高权超模的Dirac上同调非平凡，同时探讨了其与Kostant上同调的关系。

Dirac cohomology of one K-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…

Cartan运动群的Dirac算子

Dirac operators for Cartan motion groups

本文引入并研究Cartan运动群的代数Dirac算子，建立其基本性质与Dirac上同调，证明Vogan猜想的强形式，明确确定Dirac系列，并解释其作为代数形变族Dirac理论零纤维的获得方式。

Unitary representations with Dirac cohomology: finiteness in the real case

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

Dirac 上同调、分支律与 Wallach 模

Dirac cohomology, branching laws and Wallach modules

本文基于 Dirac 上同调方法，针对例外李群 E_{6(-14)} 和 E_{7(-25)}，从 Dirac 上同调恢复 Wallach 模的 K-谱，推广了 2013 年 Huang 等人的分支律结果。

BGG 范畴 $\mathcal{O}$ 的 Dirac 上同调

Dirac cohomology for the BGG category $\mathcal{O}$

本文研究有限维复半单李代数范畴 $\mathcal{O}$ 中模的 Dirac 上同调，证明 Vogan 猜想，揭示其与幂零李代数上同调的联系，并验证高阶 Dirac 上同调的同调性质。

Geometric and algebraic parameterizations for Dirac cohomology of simple modules in $\mathcal{O}^\mathfrak{p}$ and their applications

In this paper, we show that the Dirac cohomology $H_{D}(L(λ))$ of a simple highest weight module $L(λ)$ in $\mathcal{O}^\mathfrak{p}$ can be parameterized by a specific set of weights: a subset $\mathcal{W}_I(λ)$ of the orbit of the Weyl group $W$ acting on $λ+ρ$. As an application, we show that any simple module in $\mathcal{O}^\mathfrak{p}$ is determined up to isomorphism by its Dirac cohomology. We describe four parameterizations of $H_D(L(λ))$ when $λ$ is regular. Two of these parameterizations are geometric in terms of a partial ordering on the dual of the Cartan subalgebra and a generali…

Unitary Representations with Dirac cohomology: a finiteness result for complex Lie groups

Let $G$ 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 tha…

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: ${\rm EI}=E_{6(6)}, {\rm EIV}=E_{6(-26)}, {\rm FI}=F_{4(4)}, {\rm FII}=F_{4(-20)}$. Along the way, we find an irreducible unitary representation of $F_{4(4)}$ 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.

Dirac operator and its cohomology for the quantum group $U_q(\mathfrak{sl}_2)$

We introduce a Dirac operator $D$ for the quantum group $U_q(\mathfrak{sl}_2)$, as an element of the tensor product of $U_q(\mathfrak{sl}_2)$ with the Clifford algebra on two generators. We study the properties of $D$, including an analogue of Vogan's conjecture. We compute the cohomology of $D$ acting on various $U_q(\mathfrak{sl}_2)$-modules.

Partial Dirac Cohomology and Tempered Representations

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 of various models of these $H$--series representations, and show how this construction fits into the framework of geometric quantization and symplectic reduction.

Drinfeld Hecke 代数的两类类 Dirac 算子

Two families of Dirac-like operators for Drinfeld's Hecke algebra

该研究为 Drinfeld Hecke 代数提出 Parthasarathy 算子与扭曲 Dirac 算子两类推广，并在 A 型情形给出具体算子族。

Classification of $A_{\mathfrak{q}}(λ)$ modules by their Dirac cohomology for type $D$, $G_2$ and $\mathfrak{sp}(2n,\mathbb{R})$

Let $G$ be a connected real reductive group with maximal compact subgroup $K$ of the same rank as $G$. In the recent paper of Huang, Pandžić and Vogan, it was shown that the admissible $Θ$--stable parabolic subalgebras $\mathfrak{q}$ of $\mathfrak{g}$ are in one-to-one correspodence with the faces of $W ρ$ intersecting the $\mathfrak{k}$--dominant Weyl chamber and that $A_{\mathfrak{q}}(0)$--modules can be classified by their Dirac cohomology in geometric terms. They described in detail the cases when $\mathfrak{g}_0$ is of type $A$, $B$, $F$ and $C$ except for $\mathfrak{g}_0 = \mathfrak{sp}(…

Archimedes域上$\mathrm{GL}(n)$的Dirac级数

Dirac series of $\mathrm{GL}(n)$ over an Archimedean field

受上同调理论启发，本文确定了$\mathrm{GL}(n,\mathbb{H})$的Dirac级数，证明了其与$\mathrm{GL}(n,\mathbb{R})$的自旋最低$K$-型均唯一且无重数，并验证了相关猜想。

A Casselman-Osborne theorem for rational Cherednik algebras

We define Lie algebra cohomology associated with the half-Dirac operators for representations of rational Cherednik algebras and show that it has property described in the Casselman-Osborne Theorem by establishing a version of the Vogan's conjecture for the half-Dirac operators. Moreover, we study the relationship between Lie algebra cohomology and Dirac cohomology in analogy of the representations for semisimple Lie algebras.

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

This paper computes the Dirac cohomology $H_D(π)$ 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 $H_D(π)$. This confirms conjectures formulated by the first named author and Pandzic in 2011.

Dirac series of $E_{7(-5)}$

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 part of the Dirac cohomology continues to happen for certain unitary representations of $E_{7(-5)}$. Assuming the infinitesimal character being integral, we further improve the Helgason-Johnson bound for $E_{7(-5)}$. This should help people to understand (part of) the unitary dual of this group.

代数族的狄拉克算子

Dirac operators for algebraic families

本文研究了与实可约李群相关的变形族的代数族狄拉克算子，证明了Vogan猜想，关联了代数族哈里-钱德罗模块的无穷小特征标与狄拉克上同调。

Dirac算子的扰动

Perturbations of Dirac Operators

本文研究了基本经典李超代数相对三次Dirac算子在颜色量子Weil代数统一框架下的扰动，提出了三种扰动类型及其不变量。

Multiplets of representations, twisted Dirac operators and Vogan's conjecture in affine setting

We extend classical results of Kostant and al. on multiplets of representations of finite-dimensional Lie algebras and on the cubic Dirac operator to the setting of affine Lie algebras and twisted affine cubic Dirac operator. We prove in this setting an analogue of Vogan's conjecture on infinitesimal characters of Harish-Chandra modules in terms of Dirac cohomology. For our calculations we use the machinery of Lie conformal and vertex algebras.

$U_q(\mathfrak{sl}_2)$ 的三次 Dirac 算子

Cubic Dirac operator for $U_q(\mathfrak{sl}_2)$

本文构造了 $\mathfrak{sl}_2$ 的 $q$-变形 Clifford 代数，定义了 $U_q(\mathfrak{sl}_2)$ 的三次 Dirac 算子 $D_q$，证明其不变性及平方的中心性，并计算了有限维模和 Verma 模上的谱与 Dirac 上同调。

$E_{7(7)}$ 的 Dirac 级数

Dirac series of $E_{7(7)}$

本文分类了 $E_{7(7)}$ 的所有 Dirac 级数，通过推广 Helgason-Johnson 界限作为关键要素，部分支持了 Vogan 的 FPP 猜想，并首次发现了旋量最低 $K$-型具有重数的 Dirac 级数。

无限维颜色李代数的Dirac算子

Dirac operators for infinite-dimensional color Lie algebras

本文构造了无限维二次Z-分次颜色李代数的三次Dirac算子和相对三次Dirac算子，建立了Parthasarathy型平方公式，并将其应用于可对称化Kac-Moody超代数，得到了Dirac不等式和Dirac核与李超代数上同调的关系。

Dirac spectrum and spectral action of SU(3)

We compute the Dirac spectrum of SU(3) for a one parameter family of Dirac operators, including the Levi-Civita, cubic, and trivial Dirac operators. We then proceed to compute the spectral action for the entire family.

Dirac operators and the Very Strange Formula for Lie superalgebras

Using a super-affine version of Kostant's cubic Dirac operator, we prove a very strange formula for quadratic finite-dimensional Lie superalgebras with a reductive even subalgebra.

Scattered representations of complex classical Lie groups

This paper studies scattered representations of $G = SO(2n+1, \mathbb{C})$, $Sp(2n, \mathbb{C})$ and $SO(2n, \mathbb{C})$, which lies in the `core' of the unitary spectrum $G$ with nonzero Dirac cohomology. 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 non-zero Dirac cohomology.