Complex Lie Group Research Articles

Giambelli and Degeneracy Locus Formulas for Classical G/P spaces

Let G be a classical complex Lie group, P any parabolic subgroup of G, and X = G/P the corresponding homogeneous space, which parametrizes (isotropic) partial flags of subspaces of a fixed vector space. In the mid 1990s, Fulton, Pragacz, and Ratajski asked for global formulas which express the cohomology classes of the universal Schubert varieties in flag bundles -- when the space X varies in an algebraic family -- in terms of the Chern classes of the vector bundles involved in their definition. This has applications to the theory of degeneracy loci of vector bundles and is closely related to the Giambelli problem for the torus-equivariant cohomology ring of X. In this article, we explain the answer to these questions which was obtained in arXiv:0908.3628, in terms of combinatorial data coming from the Weyl group.

Embeddings of Altn and its perfect covers for n ≥ 6 in exceptional complex Lie groups

We classify Altn and its perfect covers for n≥6 up to conjugacy as subgroups of the exceptional complex Lie groups (with some exceptions where we give lower bounds and discuss the possibilities of Lie primitive subgroups).

Holomorphic functions and the Itô chaos

This paper is concerned with the characterization of spaces of square integrable holomorphic functions on a complex manifold, $G$, in terms of the derivatives of the function at a fixed point $o\in G$. The reproducing kernel properties of square integrable holomorphic functions are reviewed and a number of examples are given. These examples include square integrable holomorphic functions relative to Gaussian measures on complex Euclidean spaces along with their generalizations to heat kernel measures on complex Lie groups. These results are intimately related to the Itô's chaos expansion in stochastic analysis and to the Fock space description of free quantum fields in physics.

Meromorphic quotients for some holomorphic G-actions

Using mainly tools from [B.13] and [B.15] we give a necessary and sufficient condition in order that a holomorphic action of a connected complex Lie group $G$ on a reduced complex space $X$ admits a strongly quasi-proper meromorphic quotient. We apply this characterization to obtain a result which assert that, when $G = K.B$ with $B$ a closed complex subgroup of $G$ and $K$ a real compact subgroup of $G$, the existence of a strongly quasi-proper meromorphic quotient for the $B-$action implies, assuming moreover that there exists a $G-$invariant Zariski open dense subset in $X$ which is good for the $B-$action, the existence of a strongly quasi-proper meromorphic quotient for the $G-$action on $X$.

Invariant Solutions to the Strominger System on Complex Lie Groups and Their Quotients

Using canonical 1-parameter family of Hermitian connections on the tangent bundle, we provide invariant solutions to the Strominger system on complex Lie groups. Both flat and non-flat cases are discussed in detail.

Corwin–Greenleaf multiplicity functions for complex semisimple symmetric spaces

Let Gℂbe a complex simple Lie group, GUa compact real form, and [Formula: see text] the natural projection between the dual of the Lie algebras. We prove that, for any coadjoint orbit [Formula: see text] of GU, the intersection of [Formula: see text] with a coadjoint orbit [Formula: see text] of Gℂis either an empty set or a single orbit of GUif [Formula: see text] is isomorphic to a complex symmetric space.

Massive And Massless Gauge Fields Formed by Flat Connections

The Yang - Mills type massive and massless gauge theories are interpreted in the geometrical frame of holomorphic principal bundles on a complex 2 - manifold. It is seen in this formalism that, the component (1,1) of the curvature of this connection appears because of flat connections generated by holomorphic structure although connection is flat. Thus it is possible to write a Lagrangian for a Yang - Mills theory including massive and massless gauge fields. However, the mass matrix of a massive gauge field on such a bundle isn’t nilpotent and this field is generated by a noncommutative flat connection on the same bundle, then the structure group of this bundle is non - Abelian complex Lie group. However, if the gauge field is massless, then this is generated by commutative flat connection, and so the structure group of the bundle is Abelian complex Lie group. Also one sees that the second Chern number or topological charge is proportional to the total volume of the base manifold for each massless and massive gauge theories and Abelian (massless) gauge theories are indeed the theories of the Kahler potential on the complex projective space $\mathbb {C}P^{2}$ .

STEPWISE SQUARE INTEGRABLE REPRESENTATIONS FOR LOCALLY NILPOTENT LIE GROUPS

In a recent paper we found conditions for a nilpotent Lie group N to have a filtration by normal subgroups whose successive quotients have square integrable representations, and such that these square integrable representations fit together nicely to give an explicit construction of Plancherel for almost all representations of N. The prototype for this sort of group is the group of upper triangular real matrices with 1’s down the diagonal. More generally, this class of groups contains the nilradicals of minimal parabolic subgroups of all (finite-dimensional) reductive real or complex Lie groups, in other words, all groups N in Iwasawa decompositions of reductive real or complex Lie groups.

Complex-analytic quotients of algebraic $$G$$ G -varieties

It is shown that any compact semistable quotient (in the sense of Heinzner and Snow) of a normal algebraic variety by a complex reductive Lie group $G$ is a good quotient. This reduces the investigation and classification of such complex-analytic quotients to the corresponding questions in the algebraic category. As a consequence of our main result, we show that every compact space in Nemirovski's class $\mathscr{Q}_G$ has a realisation as a good quotient, and that every complete algebraic variety in $\mathscr{Q}_G$ is unirational with finitely generated Cox ring and at worst rational singularities. In particular, every compact space in class $\mathscr{Q}_T$, where $T$ is an algebraic torus, is a toric variety.

Semidirect products and invariant connections

Let $S$ be a complex reductive group acting holomorphically on a complex Lie group $N$ via holomorphic automorphisms. Let $K(S)\subset S$ be a maximal compact subgroup. The semidirect product $G := N\rtimes K(S)$ acts on $N$ via biholomorphisms. We give an explicit description of the isomorphism classes of $G$-equivariant almost holomorphic hermitian principal bundles on $N$. Under the assumption that there is a central subgroup $Z= \text{U}(1)$ of $K(S)$ that acts on $\text{Lie}(N)$ as multiplication through a single nontrivial character, we give an explicit description of the isomorphism classes of $G$-equivariant holomorphic hermitian principal bundles on $N$.

Representing Analytic Cohomology Groups of Complex Manifolds

Consider a holomorphic vector bundle L→X and an open cover 𝔘={U a :a∈A} of X, parametrized by a complex manifold A. We prove that the sheaf cohomology groups H q (X,L) can be computed from the complex C hol • (𝔘,L) of cochains (f a 0 ...a q ) a 0 ,...,a q ∈A that depend holomorphically on the a j , provided S={(a,x)∈A×X:x∈U a } is a Stein open subset of A×X. The result is proved in the setting of Banach manifolds, and is applied to study representations on cohomology groups induced by a holomorphic action of a complex reductive Lie group on L.

Anti-holomorphic involutions of the moduli spaces of Higgs bundles

We study anti-holomorphic involutions of the moduli space of principal $G$-Higgs bundles over a compact Riemann surface $X$, where $G$ is a complex semisimple Lie group. These involutions are defined by fixing anti-holomorphic involutions on both $X$ and $G$. We analyze the fixed point locus in the moduli space and their relation with representations of the orbifold fundamental group of $X$ equipped with the anti-holomorphic involution. We also study the relation with branes. This generalizes work by Biswas--Garc\'{\i}a-Prada--Hurtubise and Baraglia--Schaposnik.

Flat bundles on G/Γ and stability

Let G be a connected complex Lie group and Γ a cocompact lattice in G. Let H be a connected reductive complex affine algebraic group and \({\rho\, : \Gamma\, \longrightarrow H}\) a homomorphism such that \({\rho(\Gamma)}\) is not contained in some proper parabolic subgroup of H. Let \({E^\rho_H}\) be the holomorphic principal H–bundle on G/Γ associated to ρ. We prove that \({E^\rho_H}\) is polystable. If ρ satisfies the further condition that \({\rho(\Gamma)}\) is contained in a compact subgroup of H, then we prove that \({E^\rho_H}\) is stable.

The gap phenomenon in parabolic geometries

Abstract 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 ) ${(G,P)}$ , we use Tanaka theory to derive a universal upper bound on the submaximal symmetry dimension. We use Kostant’s version of the Bott–Borel–Weil Theorem to show that this bound is in fact sharp in almost all complex and split-real cases by exhibiting (abstract) models. We explicitly compute all submaximal symmetry dimensions when G is any complex or split-real simple Lie group.

Clifford–Klein Forms and a-Hyperbolic Rank

The porpose of this article is to introduce and investigate properties of a tool (the a-hyperbolic rank) which enables us to obtain new examples of homogeneous spaces G/H which admit and do not admit almost compact Clifford-Klein forms. We achieve this goal by exploring in greater detail the technique of adjoint orbits developed by Okuda combined with the well-known conditions of Benoist. We find easy-to-check conditions on G and H expressed directly in terms of the Satake diagrams of the corresponding Lie algebras, in cases when G is a real form of a complex Lie group of types A, D or E6. One of the advantages of this approach is the fact that we don't need to know the embedding of H into G. Using the a-hyperbolic rank we also show that the homogeneous space G/H of reductive type of the real form of E6 (of the IV type) admits compact Clifford-Klein forms if and only if H is compact. Inspired by the work of Okuda on symmetric spaces G/H we classify all 3-symmetric spaces admitting almost compact Clifford-Klein forms.

Maximum rank of a Legendrian web

We propose the Legendrian web in a contact three manifold as a second order generalization of the planar web. An Abelian relation for a Legendrian web is analogously defined as an additive equation among the first integrals of its foliations. For a class of Legendrian d-webs defined by simple second order ODE's, we give an algebraic construction ofρd=(d−1)(d−2)(2d+3)6 linearly independent Abelian relations. We then employ the method of local differential analysis and the theory of linear differential systems to show that ρd is the maximum rank of a Legendrian d-web.In the complex analytic category, we give a possible projective geometric interpretation of ρd as an analogue of Castelnuovo bound for degree 2d surfaces in the 3-quadric Q3⊂P4 via the duality between P3 and Q3 associated with the rank two complex simple Lie group Sp(2,C).The Legendrian 3-webs of maximum rank three are analytically characterized, and their explicit local normal forms are found. For an application, we give an alternative characterization of a Darboux super-integrable metric as a two dimensional Riemannian metric g+ which admits a mate metric g− such that a Legendrian 3-web naturally associated with the geodesic foliations of the pair g± has maximum rank.

Characters and dimensions of highest-weight representations of the intermediate Lie group

We study highest-weight representations of the non-semisimple complex Lie group used for separating multiple points of the spectrum in the reduction . In particular, we find formulae for the characters and dimensions of these representations, which turn out to be similar to the well-known Weyl formulae for classical Lie groups.

Rigidity of automorphism groups of invariant domains in homogeneous Stein spaces

For a large class of Stein manifolds which are homogeneous under a complex reductive Lie group, we prove a rigidity property of the automorphism groups of domains invariant with respect to a compact form of this complex group.

Levi problem and semistable quotients

A complex space X is in class 𝒬 G if it is a semistable quotient of the complement to an analytic subset of a Stein manifold by a holomorphic action of a reductive complex Lie group G. It is shown that every pseudoconvex unramified domain over X is also in 𝒬 G .

Curvature spectra of simple Lie groups

The Killing form \beta\ of a real (or complex) semisimple Lie group G is a left-invariant pseudo-Riemannian (or, respectively, holomorphic) Einstein metric. Let \Omega\ denote the multiple of its curvature operator, acting on symmetric 2-tensors, with the factor chosen so that \Omega\beta=2\beta. The result of Meyberg [8], describing the spectrum of \Omega\ in complex simple Lie groups G, easily implies that 1 is not an eigenvalue of \Omega\ in any real or complex simple Lie group G except those locally isomorphic to SU(p,q), or SL(n,R), or SL(n,C) or, for even n only, SL(n/2,H), where p\ge q\ge0 and p+q=n>2. Due to the last conclusion, on simple Lie groups G other the ones just listed, nonzero multiples of the Killing form \beta\ are isolated among left-invariant Einstein metrics. Meyberg's theorem also allows us to understand the kernel of \Lambda, which is another natural operator. This in turn leads to a proof of a known, yet unpublished, fact: namely, that a semisimple real or complex Lie algebra with no simple ideals of dimension 3 is essentially determined by its Cartan three-form.