Searcharxiv⌕ Search

arXiv subjects

Haian He

Publications and source records attributed to Haian He.

8 recordsLinked to original sources
arXiv2019

A criterion for discrete branching laws for Klein four symmetric pairs and its application to $E_{6(-14)}$

Let $G$ be a noncompact connected simple Lie group, and $(G,G^\Gamma)$ a Klein four symmetric pair. In this paper, the author shows a necessary condition for the discrete decomposability of unitarizable simple $(\mathfrak{g},K)$-modules for Klein for symmetric pairs. Precisely, if certain conditions hold for $(G,G^\Gamma)$, there does not exist any unitarizable simple $(\mathfrak{g},K)$-module that is discretely decomposable as a $(\mathfrak{g}^\Gamma,K^\Gamma)$-module. As an application, for $G=\mathrm{E}_{6(-14)}$, the author obtains a complete classification of Klein four symmetric pairs $(G,G^\Gamma)$ with $G^\Gamma$ noncompact, such that there exists at least one nontrivial unitarizable simple $(\mathfrak{g},K)$-module that is discretely decomposable as a $(\mathfrak{g}^\Gamma,K^\Gamma)$-module and is also discretely decomposable as a $(\mathfrak{g}^\sigma,K^\sigma)$-module for some nonidentity element $\sigma\in\Gamma$.

math.RT↗
arXiv2019

Kobayashi's conjecture on associated varieties for $(\mathrm{E}_{6(-14)},\mathrm{Spin}(8,1))$

The author confirms a conjecture on associated varieties by Toshiyuki KOBAYASHI for the Klein four symmetric pair $(\mathrm{E}_{6(-14)},\mathrm{Spin}(8,1))$, which provides an alternative way to confirm the conjecture for the symmetric pair $(\mathrm{Spin}(8,2),\mathrm{Spin}(8,1))$. Also, for Klein four symmetric pairs $(G,G^\Gamma)$ with the exceptional simple Lie groups $G$ of Hermitian type, there exists a discrete series representation of $G$ which is $G^\Gamma$-admissible if and only if $(G,G^\Gamma)$ is of holomorphic type.

math.RT↗
arXiv2019

Discretely decomposable restrictions of $(\mathfrak{g},K)$-modules for Klein four symmetric pairs of exceptional Lie groups of Hermitian type

Let $(G,G^Γ)$ be a Klein four symmetric pair. The author wants to classify all the Klein four symmetric pairs $(G,G^Γ)$ such that there exists at least one nontrivial unitarizable simple $(\mathfrak{g},K)$-module $π_K$ that is discretely decomposable as a $(\mathfrak{g}^Γ,K^Γ)$-module. In this article, three assumptions will be made. Firstly, $G$ is an exceptional Lie group of Hermitian type, i.e., $G=\mathrm{E}_{6(-14)}$ or $\mathrm{E}_{7(-25)}$. Secondly, $G^Γ$ is noncompact. Thirdly, there exists an element $σ\inΓ$ corresponding to a symmetric pair of anti-holomorphic type such that $π_K$ is discretely decomposable as a $(\mathfrak{g}^σ,K^σ)$-module.

math.RT↗
arXiv2019

Dirac series for $E_{6(-14)}$

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 $E_{6(-14)}$ is realized as a composition factor of certain $A_{\mathfrak{q}}(\lambda)$ module. Along the way, we have also obtained all the fully supported irreducible unitary representations of $E_{6(-14)}$ with integral infinitesimal characters.

math.RT↗
arXiv2015

On the Reducibility of Scalar Generalized Verma Modules of Abelian Type

A parabolic subalgebra $\mathfrak{p}$ of a complex semisimple Lie algebra $\mathfrak{g}$ is called a parabolic subalgebra of abelian type if its nilpotent radical is abelian. In this paper, we provide a complete characterization of the parameters for scalar generalized Verma modules attached to parabolic subalgebras of abelian type such that the modules are reducible. The proofs use Jantzen's simplicity criterion, as well as the Enright-Howe-Wallach classification of unitary highest weight modules.

math.RT↗