Complex Semisimple Lie Group Research Articles

Commutative avatars of representations of semisimple Lie groups

Here we announce the construction and properties of a big commutative subalgebra of the Kirillov algebra attached to a finite dimensional irreducible representation of a complex semisimple Lie group. They are commutative finite flat algebras over the cohomology of the classifying space of the group. They are isomorphic with the equivariant intersection cohomology of affine Schubert varieties, endowing the latter with a new ring structure. Study of the finer aspects of the structure of the big algebras will also furnish the stalks of the intersection cohomology with ring structure, thus ringifying Lusztig's q-weight multiplicity polynomials i.e., certain affine Kazhdan-Lusztig polynomials.

On normalizers of maximal tori in classical Lie groups

In general, the normalizer N G ( H G ) N_G(H_G) of a maximal torus H G H_G in a semisimple complex Lie group G G does not admit a presentation as a semidirect product of H G H_G and the corresponding Weyl group W G W_G . Meanwhile, such splitting exists for classical groups corresponding to the root systems A ℓ A_\ell , B ℓ B_\ell , D ℓ D_\ell . For the remaining classical groups corresponding to the root systems C ℓ C_\ell , there still exists an embedding of the Tits extension W G T W^T_G of W G W_G into the normalizer N G ( H G ) N_G(H_G) . Here an explicit unified construction is proposed for the lifts of the Weyl groups into normalizers of maximal tori for general linear and orthogonal Lie groups via embedding into the general linear groups. For the symplectic group S p 2 ℓ \mathrm {Sp}_{2\ell } , an explanation of the impossibility of embedding W S p 2 ℓ W_{\mathrm {Sp}_{2\ell }} into the normalizer N S p 2 ℓ ( H S p 2 ℓ ) N_{\mathrm {Sp}_{2\ell }}(H_{\mathrm {Sp}_{2\ell }}) is suggested. Also, explicit formulas are computed for the adjoint action of the lifts of the Weyl groups on g = L i e ( G ) \mathfrak {g}=\mathrm {Lie}(G) . Finally, examples of Weyl groups lifts are given for the groups closely associated with classical Lie groups.

Spin(8,C)-Higgs pairs over a compact Riemann surface

Abstract Let X X be a compact Riemann surface of genus g ≥ 2 g\ge 2 , G G be a semisimple complex Lie group and ρ : G → GL ( V ) \rho :G\to {\rm{GL}}\left(V) be a complex representation of G G . Given a principal G G -bundle E E over X X , a vector bundle E ( V ) E\left(V) whose typical fiber is a copy of V V is induced. A ( G , ρ ) \left(G,\rho ) -Higgs pair is a pair ( E , φ ) \left(E,\varphi ) , where E E is a principal G G -bundle over X X and φ \varphi is a holomorphic global section of E ( V ) ⊗ L E\left(V)\otimes L , L L being a fixed line bundle over X X . In this work, Higgs pairs of this type are considered for G = Spin ( 8 , C ) G={\rm{Spin}}\left(8,{\mathbb{C}}) and the three irreducible eight-dimensional complex representations which Spin ( 8 , C ) {\rm{Spin}}\left(8,{\mathbb{C}}) admits. In particular, the reduced notions of stability, semistability, and polystability for these specific Higgs pairs are given, and it is proved that the corresponding moduli spaces are isomorphic, and a precise expression for the stable and not simple Higgs pairs associated with one of the three announced representations of Spin ( 8 , C ) {\rm{Spin}}\left(8,{\mathbb{C}}) is described.

Arakelov–Milnor inequalities and maximal variations of Hodge structure

In this paper we study the $\mathbb {C}^*$-fixed points in moduli spaces of Higgs bundles over a compact Riemann surface for a complex semisimple Lie group and its real forms. These fixed points are called Hodge bundles and correspond to complex variations of Hodge structure. We introduce a topological invariant for Hodge bundles that generalizes the Toledo invariant appearing for Hermitian Lie groups. An important result of this paper is a bound on this invariant which generalizes the Milnor–Wood inequality for a Hodge bundle in the Hermitian case, and is analogous to the Arakelov inequalities of classical variations of Hodge structure. When the generalized Toledo invariant is maximal, we establish rigidity results for the associated variations of Hodge structure which generalize known rigidity results for maximal Higgs bundles and their associated maximal representations in the Hermitian case.

Órbitas coadjuntas semi-directas y familias Lagrangianas con respecto a la forma Hermitiana

We use the underlying structure of then coadjoint orbits of a semidirect product of a connected Lie group and a vector space to obtain families of Lagrangian submanifolds in the adjoint orbits of complex semisimple Lie groups with respect to the symplectic hermitian form. This construction is a generalization of a type of semi-direct orbit previously studied by the authors.

FRÖLICHER SPECTRAL SEQUENCE AND HODGE STRUCTURES ON THE COHOMOLOGY OF COMPLEX PARALLELISABLE MANIFOLDS

For complex parallelisable manifolds Γ\\G, with G a solvable or semisimple complex Lie group, the Frölicher spectral sequence degenerates at the second page. In the solvable case, the de Rham cohomology carries a pure Hodge structure. In contrast, in the semisimple case, purity depends on the lattice, but there is always a direct summand of the de Rham cohomology which does carry a pure Hodge structure and is independent of the lattice.

Configuration Poisson Groupoids of Flags

Abstract Let $G$ be a connected complex semi-simple Lie group and ${\mathcal {B}}$ its flag variety. For every positive integer $n$, we introduce a Poisson groupoid over ${{\mathcal {B}}}^n$, called the $n$th total configuration Poisson groupoid of flags of $G$, which contains a family of Poisson sub-groupoids whose total spaces are generalized double Bruhat cells and bases generalized Schubert cells in ${\mathcal {B}}^n$. Certain symplectic leaves of these Poisson sub-groupoids are then shown to be symplectic groupoids over generalized Schubert cells. We also give explicit descriptions of symplectic leaves in three series of Poisson varieties associated to $G$.

K-stability of Gorenstein Fano group compactifications with rank two

We give a classification of Gorenstein Fano bi-equivariant compactifications of semi-simple complex Lie groups with rank two, and determine which of them are equivariant K-stable and admit (singular) Kähler–Einstein metrics. As a consequence, we obtain several explicit examples of K-stable Fano varieties admitting (singular) Kähler–Einstein metrics. We also compute the greatest Ricci lower bounds, equivalently the delta invariants for K-unstable varieties. This gives us three new examples on which each solution of the Kähler–Ricci flow is of type II.

Homological methods in certain Picard group computations

Let G be a connected complex semisimple Lie group, Γ be a cocompact, irreducible and torsionless lattice in G and K be a maximal compact subgroup of G. Assume Γ acts by left multiplication and K acts by right multiplication on G. Let MΓ=Γ﹨G, X=G/K and XΓ=Γ﹨X. In this article we prove that for any n≥0, the composition Hn(XΓ,C)→Hn(MΓ,C)→Hn(MΓ,OMΓ) is an isomorphism. As an application when G is simply connected, we compute the Picard group of MΓ for the cases rank(G) =1,2. More precisely we show that if rank(G) =1, Pic(MΓ)=(Cr/Zr)⊕A and if rank(G) =2, then Pic(MΓ)≅A via the first Chern class map, where A is the torsion subgroup of H2(MΓ,Z) and r is the rank of Γ/[Γ,Γ].

The classification of thick representations of simple Lie groups

We characterize finite-dimensional thick representations over $\mathbf{C}$ of connected complex semi-simple Lie groups by irreducible representations which are weight multiplicity-free and whose weight posets are totally ordered sets. Moreover, using this characterization, we give the classification of thick representations over $\mathbf{C}$ of connected complex simple Lie groups.

TEMPERED HOMOGENEOUS SPACES IV

AbstractLet G be a complex semisimple Lie group and H a complex closed connected subgroup. Let and be their Lie algebras. We prove that the regular representation of G in $L^2(G/H)$ is tempered if and only if the orthogonal of in contains regular elements by showing simultaneously the equivalence to other striking conditions, such as has a solvable limit algebra.

Enumeration of Gelfand–⁠Cetlin Type Reduced Words

The combinatorics of reduced words and their commutation classes plays an important role in geometric representation theory. For a semisimple complex Lie group $G$, a string polytope is a convex polytope associated with each reduced word of the longest element $w_0$ in the Weyl group of $G$ encoding the character of a certain irreducible representation of $G$. In this paper, we deal with the case of type $A$, i.e., $G = \mathrm{SL}_{n+1}(\mathbb{C})$. A Gelfand–⁠Cetlin polytope is one of the most famous examples of string polytopes of type $A$. We provide a recursive formula enumerating reduced words of $w_0$ such that the corresponding string polytopes are combinatorially equivalent to a Gelfand–⁠Cetlin polytope. The recursive formula involves the number of standard Young tableaux of shifted shape. We also show that each commutation class is completely determined by a list of quantities called indices.

Alcove Paths and Gelfand–Tsetlin Patterns

In their study of the equivariant K-theory of the generalized flag varieties G/P, where G is a complex semisimple Lie group, and P is a parabolic subgroup of G, Lenart and Postnikov introduced a combinatorial tool, called the alcove path model. It provides a model for the highest weight crystals with dominant integral highest weights, generalizing the model by semistandard Young tableaux. In this paper, we prove a simple and explicit formula describing the crystal isomorphism between the alcove path model and the Gelfand–Tsetlin pattern model for type A.

Geometry of compact complex manifolds associated to generalized quasi-Fuchsian representations

We study the topology and geometry of compact complex manifolds associated to Anosov representations of surface groups and other hyperbolic groups in a complex semisimple Lie group $G$. These manifolds are obtained as quotients of the domains of discontinuity in generalized flag varieties $G/P$ constructed by Kapovich-Leeb-Porti (arXiv:1306.3837), and in some cases by Guichard-Wienhard (arXiv:1108.0733). For $G$-Fuchsian representations and their Anosov deformations, where $G$ is simple, we compute the homology of the domains of discontinuity and of the quotient manifolds. For $G$-Fuchsian and $G$-quasi-Fuchsian representations in simple $G$ of rank at least two, we show that the quotient manifolds are not K\"{a}hler. We also describe the Picard groups of these quotient manifolds, compute the cohomology of line bundles on them, and show that for $G$ of sufficiently large rank these manifolds admit nonconstant meromorphic functions. In a final section, we apply our topological results to several explicit families of domains and derive closed formulas for topological invariants in some cases. We also show that the quotient manifold for a $G$-Fuchsian representation in $\mathrm{PSL}_3(\mathbb{C})$ is a fiber bundle over a surface, and we conjecture that this holds for all simple $G$.

(G,P)-Opers and global Slodowy slices

In this paper, we introduce a generalization of G-opers for arbitrary parabolic subgroups P<G of a complex semisimple Lie group. For parabolic subgroups associated to “even nilpotents”, we parameterize (G,P)-opers by an object generalizing the base of the Hitchin fibration. In particular, we describe and parameterize families of opers associated to higher Teichmüller spaces.

Quantization of a Poisson structure on products of principal affine spaces

We give the analogue for Hopf algebras of the polyuble Lie bialgebra construction by Fock and Rosli. By applying this construction to the Drinfeld–Jimbo quantum group, we obtain a deformation quantization \mathbb C_\hslash[(N \backslash G)^m] of a Poisson structure \pi^{(m)} on products (N \backslash G)^m of principal affine spaces of a connected and simply connected complex semisimple Lie group G . The Poisson structure \pi^{(m)} descends to a Poisson structure \pi_m on products (B \backslash G)^m of the flag variety of G which was introduced and studied by the Lu and the author. Any ample line bundle on (B \backslash G)^m inherits a natural flat Poisson connection, and the corresponding graded Poisson algebra is quantized to a subalgebra of \mathbb C_\hslash[(N \backslash G)^m] . We define the notion of a strongly coisotropic subalgebra in a Hopf algebra, and explain how strong coisotropicity guarantees that any homogeneous coordinate ring of a homogeneous space of a Poisson Lie group can be quantized in the sense of Ciccoli, Fioresi, and Gavarini.

Bott–Samelson atlases, total positivity, and Poisson structures on some homogeneous spaces

Let G be a connected and simply connected complex semisimple Lie group. For a collection of homogeneous G-spaces G/Q, we construct a finite atlas $${{\mathcal {A}}}_{{\scriptscriptstyle BS}}(G/Q)$$ on G/Q, called the Bott–Samelson atlas, and we prove that all of its coordinate functions are positive with respect to the Lusztig positive structure on G/Q. We also show that the standard Poisson structure $$\pi _{{\scriptscriptstyle G}/{\scriptscriptstyle Q}}$$ on G/Q is presented, in each of the coordinate charts of $${{\mathcal {A}}}_{{\scriptscriptstyle BS}}(G/Q)$$ , as a symmetric Poisson CGL extension (or a certain localization thereof) in the sense of Goodearl–Yakimov, making $$(G/Q, \pi _{{\scriptscriptstyle G}/{\scriptscriptstyle Q}}, {{\mathcal {A}}}_{{\scriptscriptstyle BS}}(G/Q))$$ into a Poisson–Ore variety. In addition, all coordinate functions in the Bott–Samelson atlas are shown to have complete Hamiltonian flows with respect to the Poisson structure $$\pi _{{\scriptscriptstyle G}/{\scriptscriptstyle Q}}$$ . Examples of G/Q include G itself, G/T, G/B, and G/N, where $$T \subset G$$ is a maximal torus, $$B \subset G$$ a Borel subgroup, and N the uniradical of B.

Geodesic flows on real forms of complex semi-simple Lie groups of rigid body type

The geodesic flows are studied on real forms of complex semi-simple Lie groups with respect to a left-invariant (pseudo-)Riemannian metric of rigid body type. The Williamson types of the isolated relative equilibria on generic adjoint orbits are determined.

Hermitian curvature flow on unimodular Lie groups and static invariant metrics

We investigate the Hermitian curvature flow (HCF) of leftinvariant metrics on complex unimodular Lie groups. We show that in this setting the flow is governed by the Ricci-flow type equation ∂ t g t = − R i c 1 , 1 ( g t ) \partial _tg_{t}=-{\mathrm { Ric}}^{1,1}(g_t) . The solution g t g_t always exists for all positive times, and ( 1 + t ) − 1 g t (1 + t)^{-1}g_t converges as t → ∞ t\to \infty in Cheeger–Gromov sense to a nonflat left-invariant soliton ( G ¯ , g ¯ ) (\bar G, \bar g) . Moreover, up to homotheties on each of these groups there exists at most one left-invariant soliton solution, which is a static Hermitian metric if and only if the group is semisimple. In particular, compact quotients of complex semisimple Lie groups yield examples of compact non-Kähler manifolds with static Hermitian metrics. We also investigate the existence of static metrics on nilpotent Lie groups and we generalize a result of Enrietti, Fino, and the third author [J. Symplectic Geom. 10 (2012), no. 2, 203–223] for the pluriclosed flow. In the last part of the paper we study the HCF on Lie groups with abelian complex structures.

Cycle Intersection for SOp,q-Flag Domains

A real form G0 of a complex semisimple Lie group G has only finitely many orbits in any given compact G-homogeneous projective algebraic manifold Z=G/Q. A maximal compact subgroup K0 of G0 has special orbits C which are complex submanifolds in the open orbits of G0. These special orbits C are characterized as the closed orbits in Z of the complexification K of K0. These are referred to as cycles. The cycles intersect Schubert varieties S transversely at finitely many points. Describing these points and their multiplicities was carried out for all real forms of SLn,ℂ by Brecan (Brecan, 2014) and (Brecan, 2017) and for the other real forms by Abu-Shoga (Abu-Shoga, 2017) and Huckleberry (Abu-Shoga and Huckleberry). In the present paper, we deal with the real form SOp,q acting on the SO (2n, C)-manifold of maximal isotropic full flags. We give a precise description of the relevant Schubert varieties in terms of certain subsets of the Weyl group and compute their total number. Furthermore, we give an explicit description of the points of intersection in terms of flags and their number. The results in the case of G/Q for all real forms will be given by Abu-Shoga and Huckleberry.