arXiv research

We study the isotropy representation of real flag manifolds associated to simple Lie algebras that are split real forms of complex simple Lie algebras. For each Dynkin diagram the invariant irreducible subspaces for the compact part of the isotropy subgroup are described. Contrary to the complex flag manifolds the deco…

Unique submaximal symmetry found for certain parabolic geometries.

Generic Hitchin representations avoid hyperplanes in Lie algebras.

The infinitesimal symmetry algebra of any Cartan geometry has maximum dimension realized by the flat model, but often this dimension drops significantly when considering non-flat geometries, so a gap phenomenon arises. For general (regular, normal) parabolic geometries of type (G,P), we use Tanaka theory to derive a un…

Study shows infinite volumes of moduli spaces for certain groups.

We prove a global Birkhoff decomposition for almost split real forms of loop groups, when an underlying finite dimensional Lie group is compact. Among applications, this shows that the dressing action - by the whole subgroup of loops which extend holomorphically to the exterior disc - on the $U$-hierarchy of the ZS-AKN…

The study classifies transverse spheres in flag manifolds and finds new examples.

We develop a new geometric method of understanding principal G-Higgs bundles through their spectral data, for G a real form of a complex Lie group. In particular, we consider the case of G a split real form, as well as G = SL(2,R), U(p,p), SU(p,p), and Sp(2p,2p). Further, we give some applications of our results, and d…

We introduce a class of Higgs bundles called cyclic which lie in the Hitchin component of representations of a compact Riemann surface into the split real form of a simple Lie group. We then prove that such a Higgs bundle is equivalent to a certain class of solutions to the affine Toda equations. We further explain thi…

We study the Hitchin component in the space of representations of the fundamental group of a Riemann surface into a split real simple Lie group in the rank 2 case. We prove that such representations are described by a conformal structure and class of Higgs bundle we call cyclic and we show cyclic Higgs bundles correspo…

Through Cayley and Langlands type correspondences, we give a geometric description of the moduli spaces of real orthogonal and symplectic Higgs bundles of any signature in the regular fibres of the Hitchin fibration. As applications of our methods, we complete the concrete abelianization of real slices corresponding to…

Introduces $Θ$-positivity in Lie groups, generalizing Lusztig's positivity.

We study the existence of invariant Einstein metrics on real flag manifolds associated to simple and non-compact split real forms of complex classical Lie algebras whose isotropy representation decomposes into two or three irreducible sub-representations. In this situation, one can have equivalent sub-modules, leading …

In this work we study the existence of invariant almost complex structures on real flag manifolds associated to split real forms of complex simple Lie algebras. We show that, contrary to the complex case where the invariant almost complex structures are well known, some real flag manifolds do not admit such structures.…

New submersion proves complex-valued harmonic map existence.

In this paper we describe all the nilradicals of parabolic subalgebras of split real simple Lie algebras admitting symplectic structures. The main tools used to obtain this list are Kostant's description of the highest weight vectors (hwv) of the cohomology of these nilradicals and some necessary conditions obtained fo…

We introduce $Θ$-positivity, a new notion of positivity in real semisimple Lie groups. The notion of $Θ$-positivity generalizes at the same time Lusztig's total positivity in split real Lie groups as well as well known concepts of positivity in Lie groups of Hermitian type. We show that there are two other families of …

We study hamiltonian actions of compact groups in the presence of compatible involutions. We show that the lagrangian fixed point set on the symplectically reduced space is isomorphic to the disjoint union of the involutively reduced spaces corresponding to involutions on the group strongly inner to the given one. Our …

The first part of this paper surveys several characterizations of Teichmüller space as a subset of the space of representation of the fundamental group of a surface into PSL(2,R). Special emphasis is put on (bounded) cohomological invariants which generalize when PSL(2,R) is replaced by a Lie group of Hermitian type. T…

Real del Pezzo surfaces split real lines into elliptic and hyperbolic types.

Let $M$ be a compact connected special flat affine manifold without boundary equipped with a Gauduchon metric $g$ and a covariant constant volume form. Let $G$ be either a connected reductive complex linear algebraic group or the real locus of a split real form of a complex reductive group. We prove that a flat princip…

Study on $G_2$-type flag manifolds, focusing on invariant metrics and Ricci flow.

We give the complete classification of all sub-Riemannian model spaces with both step and rank three. They will be divided into three families based on their nilpotentization. Each family will depend on a different number of parameters, making the result crucially different from the known case of step two model spaces.…

Local equivalence found between maximally symmetric rolling and flat Cartan distributions.

For a semisimple real Lie group $G$, we study topological properties of moduli spaces of polystable parabolic $G$-Higgs bundles over a Riemann surface with a divisor of finitely many distinct points. For a split real form of a complex simple Lie group, we compute the dimension of apparent parabolic Teichm{ü}ller compon…

The study finds non-uniform lattices with thin Hitchin representations in specific Lie groups.

New minimal surface theory disproves a conjecture in symmetric spaces.

Study of conformal limits for special opers in Lie groups.

We study parabolic G-Higgs bundles over a compact Riemann surface with fixed punctures, when G is a real reductive Lie group, and establish a correspondence between these objects and representations of the fundamental group of the punctured surface in G with arbitrary holonomy around the punctures. Three interesting fe…

Given a generic 2-plane field on a 5-dimensional manifold we consider its (3,2)-signature conformal metric [g] as defined in math.DG/0406400. Every conformal class [g] obtained in this way has very special conformal holonomy: it must be contained in the split-real-form of the exceptional group G_2. In this note we show…

The paper finds dense subgroups in certain Lie groups.

Local equivalence shown between specific distributions and flat Cartan distribution.

The holonomy of the ambient metrics of Nurowski's conformal structures associated to generic real-analytic 2-plane fields on 5-manifolds is investigated. It is shown that the holonomy is always contained in the split real form G_2 of the exceptional Lie group, and is equal to G_2 for an open dense set of 2-plane fields…

Minimal orbits of semi-simple Lie groups are studied and related to invariant subspaces.

Cataclysm deformations study Anosov representations and their convergence.

Establishes geometric properties of elements in the positive semigroup of a general real semisimple Lie group.

Cataclysm deformations study Anosov representations, leading to new formulas and non-open sets.

New classification of conformal structures with maximal $G_2$ symmetry.

Associated to the problem of rolling one surface along another there is a five-manifold M with a rank two distribution. If the two surfaces are spheres then M is the product of the rotation group SO_3 with the two-sphere and its distribution enjoys an obvious symmetry group; the product of two SO_3's, one for each sphe…

We find two different families of $Sp(2,R)$ symmetric $G_2$ structures in seven dimensions. These are $G_2$ structures with $G_2$ being the split real form of the simple exceptional complex Lie group $G_2$. The first family has $τ_2\equiv 0$, while the second family has $τ_1\equivτ_2\equiv 0$. The families are differen…

In 1992, Hitchin used his theory of Higgs bundles to construct an important family of representations of the fundamental group of a closed, oriented surface of genus at least two into the split real form of a complex adjoint simple Lie group. These Hitchin representations comprise a component of the space of conjugacy …

Harmonic metrics are established for SO0(n,n)-Higgs bundles on non-compact hyperbolic surfaces.

The paper studies holomorphic curves in a pseudo-Riemannian space and their moduli space.

Researchers parametrize spaces of positive representations for Lie groups.

Starting from the observation that a flying saucer is a nonholonomic mechanical system whose 5-dimensional configuration space is a contact manifold, we show how to enrich this space with a number of geometric structures by imposing further nonlinear restrictions on the saucer's velocity. These restrictions define cert…

Introduces Fock bundles for studying surface group character varieties.

Classifies states of four rebits using group theory.

Classifies multiply-transitive (2,3,5)-distributions using modern Cartan geometry.