Lie Groups Of Hermitian Type Research Articles

A simplified approach to the holomorphic discrete series

Expository article on semisimple Lie groups of Hermitian type and their unitary representations known as the holomorphic discrete series. The realization of the symmetric spaces associated to the groups as bounded symmetric domains is described. The representations in question are defined by holomorphic induction and realized on spaces of vector-valued holomorphic functions on the domain. A key question is whether the induction process yields a non-zero space. It is answered by Harish-Chandra’s condition, for which a complete proof is given.

On the discretely decomposable restrictions of (𝔤,K)-modules for Klein four symmetric pairs

We prove a criterion for Klein four symmetric pairs [Formula: see text] satisfying the condition that there exist infinite-dimensional simple [Formula: see text]-modules which are discretely decomposable as [Formula: see text]-modules. Making use of the criterion, we classify all the Klein four symmetric pairs satisfying the condition for the exceptional Lie groups of Hermitian type: [Formula: see text] and [Formula: see text].

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

Let [Formula: see text] be a Klein four symmetric pair. The author wants to classify all the Klein four symmetric pairs [Formula: see text] such that there exists at least one nontrivial unitarizable simple [Formula: see text]-module [Formula: see text] that is discretely decomposable as a [Formula: see text]-module. In this paper, three assumptions will be made. First, [Formula: see text] is an exceptional Lie group of Hermitian type, i.e. [Formula: see text] or [Formula: see text]. Second, [Formula: see text] is noncompact. Third, there exists an element [Formula: see text] corresponding to a symmetric pair of anti-holomorphic type such that [Formula: see text] is discretely decomposable as a [Formula: see text]-module.

The geometry of maximal components of the PSp(4, ℝ) character variety

In this paper we describe the space of maximal components of the character variety of surface group representations into PSp(4,R) and Sp(4,R). For every rank 2 real Lie group of Hermitian type, we construct a mapping class group invariant complex structure on the maximal components. For the groups PSp(4,R) and Sp(4,R), we give a mapping class group invariant parameterization of each maximal component as an explicit holomorphic fiber bundle over Teichm\"uller space. Special attention is put on the connected components which are singular, we give a precise local description of the singularities and their geometric interpretation. We also describe the quotient of the maximal components for PSp(4,R) and Sp(4,R) by the action of the mapping class group as a holomorphic submersion over the moduli space of curves. These results are proven in two steps, first we use Higgs bundles to give a non-mapping class group equivariant parameterization, then we prove an analogue of Labourie's conjecture for maximal PSp(4,R) representations.

Classification of Klein four symmetric pairs of holomorphic type for $$\mathrm {E}_{7(-25)}$$ E 7 ( - 25 )

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.

On Borel–de Siebenthal Representations

AbstractHolomorphic representations are lowest weight representations for simple Lie groups of Hermitian type and have been studied extensively. Inspired by the work of Kobayashi on discrete series for indefinite Stiefel manifolds, Gross–Wallach on quaternonic discrete series and their analytic continuation, and Ørsted–Wolf on Borel–de Siebenthal discrete series, we define and study Borel–de Siebenthal representations (also called quasi-holomorphic representations) associated with Borel–de Siebenthal root systems for simple Lie groups of non-Hermitian type.

Higgs bundles, the Toledo invariant and the Cayley correspondence

Motivated by the study of the topology of the character variety for a non-compact Lie group of Hermitian type G, we undertake a uniform approach, independent of classification theory of Lie groups, to the study of the moduli space of G-Higgs bundles over a compact Riemann surface. We give an intrinsic definition of the Toledo invariant of a G-Higgs bundle which relies on the Jordan algebra structure of the isotropy representation for groups defining a symmetric space of tube type, and prove a general Milnor–Wood type bound of this invariant when the G-Higgs bundle is semistable. Finally, we prove rigidity results when the Toledo invariant is maximal, establishing in particular a Cayley correspondence when G is of tube type, which reveals new topological invariants only seen in particular cases from the character variety viewpoint.

On weakly maximal representations of surface groups

We introduce and study a new class of representations of surface groups into Lie groups of Hermitian type, called weakly maximal representations. We prove that weakly maximal representations are discrete and injective and we describe the structure of the Zariski closure of their image. Furthermore, we prove that the set of weakly maximal representations is a closed subset of the representation variety and describe its relation to other geometrically significant subsets of the representations variety.

Maximal representations of uniform complex hyperbolic lattices

Let $\rho$ be a maximal representation of a uniform lattice $\Gamma\subset{\rm SU}(n,1)$, $n\geq 2$, in a classical Lie group of Hermitian type $H$. We prove that necessarily $H={\rm SU}(p,q)$ with $p\geq qn$ and there exists a holomorphic or antiholomorphic $\rho$-equivariant map from complex hyperbolic space to the symmetric space associated to ${\rm SU}(p,q)$. This map is moreover a totally geodesic homothetic embedding. In particular, up to a representation in a compact subgroup of ${\rm SU}(p,q)$, the representation $\rho$ extends to a representation of ${\rm SU}(n,1)$ in ${\rm SU}(p,q)$.

Duflo’s conjecture for the branching problem to the Iwasawa $$AN$$ A N -subgroup of a semisimple Lie group

The purpose of this paper is to prove Duflo’s conjecture for \((G,\pi ,\,AN)\) where \(G\) is a real connected simple Lie group of Hermitian type and \(\pi \) is a discrete series of \(G\) and \(AN\) is the maximal exponential solvable subgroup for an Iwasawa decomposition \(G=KAN\). This is essentially deduced from the following general theorem which we prove in this paper: let \(G\) be a real connected semisimple Lie group with Lie algebra \({\mathfrak {g}}\) . Then a strongly elliptic \(G\)-coadjoint orbit \({\mathcal {O}}\) is holomorphic if and only if \(\text {p}({\mathcal {O}})\) is an open \(AN\)-coadjoint orbit. Here \(\text {p} : {\mathfrak {g}}^* \longrightarrow ({\mathfrak {a}}\oplus {\mathfrak {n}})^*\) is the natural projection.

Stability of branching laws for spherical varieties and highest weight modules

If a locally finite rational representation $V$ of a connected reductive algebraic group $G$ has uniformly bounded multiplicities, the multiplicities may have good properties such as stability. Let $X$ be a quasi-affine spherical $G$-variety, and $M$ be a $(\mathbf{C}[X],G)$-module. In this paper, we show that the decomposition of $M$ as a $G$-representation can be controlled by the decomposition of the fiber $M/\mathfrak{m}(x_{0})M$ with respect to some reductive subgroup $L \subset G$ for sufficiently large parameters. As an application, we apply this result to branching laws for simple real Lie groups of Hermitian type. We show that the sufficient condition on multiplicity-freeness given by the theory of visible actions is also a necessary condition for holomorphic discrete series representations and symmetric pairs of holomorphic type. We also show that two branching laws of a holomorphic discrete series representation with respect to two symmetric pairs of holomorphic type coincide for sufficiently large parameters if two subgroups are in the same $\epsilon$-family.

Dirac cohomology, K -characters and branching laws

Inspired by work of Enright and Willenbring, we prove a generalized Littlewood’s restriction formula in terms of Dirac cohomology. Our approach is to use a character formula of irreducible unitary lowest weight modules instead of the Bernstein-Gelfand-Gelfand resolution, and the proof is much simpler. We also show that our branching formula is equivalent to the formula of Enright and Willenbring in terms of nilpotent Lie algebra cohomology. This follows from the close relationship between the Dirac cohomology and the corresponding nilpotent Lie algebra cohomology for unitary representations of semisimple Lie groups of Hermitian type, which was established by Huang, Pandzic, and Renard.

The Toledo invariant on smooth varieties of general type

We propose a definition of the Toledo invariant for representations of fundamental groups of smooth varieties of general type into semisimple Lie groups of Hermitian type. This definition allows to generalize the results known in the classical case of representations of complex hyperbolic lattices to this new setting: assuming that the rank of the target Lie group is not greater than two, we prove that the Toledo invariant satisfies a Milnor-Wood type inequality and we characterize the corresponding maximal representations.

Tight Homomorphisms and Hermitian Symmetric Spaces

We introduce the notion of tight homomorphism into a locally compact group with nonvanishing bounded cohomology and study these homomorphisms in detail when the target is a Lie group of Hermitian type. Tight homomorphisms between Lie groups of Hermitian type give rise to tight totally geodesic maps of Hermitian symmetric spaces. We show that tight maps behave in a functorial way with respect to the Shilov boundary and use this to prove a general structure theorem for tight homomorphisms. Furthermore, we classify all tight embeddings of the Poincare disk.

Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type

Let Г be a torsion-free uniform lattice of SU(m, 1), m > 1. Let G be either SU(p, 2) with p ≥ 2, \({{\rm Sp}(2,\mathbb {R})}\) or SO(p, 2) with p ≥ 3. The symmetric spaces associated to these G’s are the classical bounded symmetric domains of rank 2, with the exceptions of SO*(8)/U(4) and SO*(10)/U(5). Using the correspondence between representations of fundamental groups of Kähler manifolds and Higgs bundles we study representations of the lattice Г into G. We prove that the Toledo invariant associated to such a representation satisfies a Milnor-Wood type inequality and that in case of equality necessarily G = SU(p, 2) with p ≥ 2m and the representation is reductive, faithful, discrete, and stabilizes a copy of complex hyperbolic space (of maximal possible induced holomorphic sectional curvature) holomorphically and totally geodesically embedded in the Hermitian symmetric space SU(p, 2)/S(U(p) × U(2)), on which it acts cocompactly.

Bounded differential forms, generalized Milnor–Wood inequality and an application to deformation rigidity

We establish sufficient conditions for a cohomology class of a discrete subgroup Γ of a connected semisimple Lie group with finite center to be representable by a bounded differential form on the quotient by Γ of the associated symmetric space; furthermore if \(\rho : \Gamma\to\mathrm{PU}(1,q)\) is any representation of any discrete subgroup Γ of SU (1, p), we give an explicit closed bounded differential form on the quotient by Γ of complex hyperbolic space which is a representative for the pullback via ρ of the Kähler class of PU(1,q). If G,G′ are Lie groups of Hermitian type, we generalize to representations \(\rho : \Gamma\to G'\) of lattices Γ < G the invariant defined in [Burger, M., Iozzi, A.: Bounded cohomology and representation variates in PU (1,n). Preprint announcement, April 2000] for which we establish a Milnor–Wood type inequality. As an application we study maximal representations into PU(1, q) of lattices in SU(1,1).

The Action of the Mapping Class Group on Maximal Representations

Let Γ g be the fundamental group of a closed oriented Riemann surface Σ g , g ≥ 2, and let G be a simple Lie group of Hermitian type. The Toledo invariant defines the subset of maximal representations Repmax(Γ g , G) in the representation variety Rep(Γ g , G). Repmax(Γ g , G) is a union of connected components with similar properties as Teichmuller space $$\mathcal{T}(\Sigma_g) = {\rm Rep}_{\max}(\Gamma_g, {\rm PSL}(2,\mathbb{R}))$$ . We prove that the mapping class group $$Mod_{\Sigma_g}$$ acts properly on Repmax(Γ g , G) when $$G= {\rm Sp}(2n,\mathbb{R})$$ , SU(n,n), SO*(4n), Spin(2,n).

The Plancherel Formula for Spherical Functions with a One-Dimensional K-Type on a Simply Connected Simple Lie Group of Hermitian Type

We generalize the Harish-Chandra inversion formula for the spherical transform on a Riemannian symmetric space to homogeneous line bundles on a Hermitian symmetric space. We determine the Plancherel measure explicitly.