Complex Reductive Lie Research Articles

Non-connected Lie groups, twisted equivariant bundles and coverings

Let Gamma be a finite group acting on a Lie group G. We consider a class of group extensions 1 rightarrow G rightarrow hat{G} rightarrow Gamma rightarrow 1 defined by this action and a 2-cocycle of Gamma with values in the centre of G. We establish and study a correspondence between hat{G}-bundles on a manifold and twisted Gamma -equivariant bundles with structure group G on a suitable Galois Gamma -covering of the manifold. We also describe this correspondence in terms of non-abelian cohomology. Our results apply, in particular, to the case of a compact or reductive complex Lie group hat{G}, since such a group is always isomorphic to an extension as above, where G is the connected component of the identity and Gamma is the group of connected components of hat{G}.

Equivariant Oka theory: survey of recent progress

We survey recent work, published since 2015, on equivariant Oka theory. The main results described in the survey are as follows. Homotopy principles for equivariant isomorphisms of Stein manifolds on which a reductive complex Lie group G acts. Applications to the linearisation problem. A parametric Oka principle for sections of a bundle E of homogeneous spaces for a group bundle {{mathscr {G}}}, all over a reduced Stein space X with compatible actions of a reductive complex group on E, {{mathscr {G}}}, and X. Application to the classification of generalised principal bundles with a group action. Finally, an equivariant version of Gromov’s Oka principle based on a notion of a G-manifold being G-Oka.

A Characterization of Linearizability for Holomorphic ℂ*-Actions

AbstractLet $G$ be a reductive complex Lie group acting holomorphically on $X=\mathbb{C}^n$. The (holomorphic) Linearization Problem asks if there is a holomorphic change of coordinates on $\mathbb{C}^n$ such that the $G$-action becomes linear. Equivalently, is there a $G$-equivariant biholomorphism $\Phi \colon X\to V$ where $V$ is a $G$-module? There is an intrinsic stratification of the categorical quotient $X/\!\!/G$, called the Luna stratification, where the strata are labeled by isomorphism classes of representations of reductive subgroups of $G$. Suppose that there is a $\Phi $ as above. Then $\Phi $ induces a biholomorphism ${\varphi }\colon X/\!\!/G\to V/\!\!/G$ that is stratified, that is, the stratum of $X/\!\!/G$ with a given label is sent isomorphically to the stratum of $V/\!\!/G$ with the same label. The counterexamples to the Linearization Problem construct an action of $G$ such that $X/\!\!/G$ is not stratified biholomorphic to any $V/\!\!/G$. Our main theorem shows that, for a reductive group $G$ with $\dim G\leq 1$, the existence of a stratified biholomorphism of $X/\!\!/G$ to some $V/\!\!/G$ is not only necessary but also sufficient for linearization. In fact, we do not have to assume that $X$ is biholomorphic to $\mathbb{C}^n$, only that $X$ is a Stein manifold.

Hitchin and Calabi–Yau Integrable Systems via Variations of Hodge Structures

Abstract Since its discovery by Hitchin in 1987, G-Hitchin systems for a reductive complex Lie group G have extensively been studied. For example, the generic fibers are nowadays well-understood. In this paper, we show that the smooth parts of G-Hitchin systems for a simple adjoint complex Lie group G are isomorphic to non-compact Calabi–Yau integrable systems extending results by Diaconescu–Donagi–Pantev. Moreover, we explain how Langlands duality for Hitchin systems is related to Poincaré–Verdier duality of the corresponding families of quasi-projective Calabi–Yau threefolds. Even though the statement is holomorphic-symplectic, our proof is Hodge-theoretic. It is based on polarizable variations of Hodge structures that admit so-called abstract Seiberg–Witten differentials. These ensure that the associated Jacobian fibration is an algebraic integrable system.

Gromov’s Oka Principle for Equivariant Maps

We take the first step in the development of an equivariant version of modern, Gromov-style Oka theory. We define equivariant versions of the standard Oka property, ellipticity, and homotopy Runge property of complex manifolds, show that they satisfy all the expected basic properties, and present examples. Our main theorem is an equivariant Oka principle saying that if a finite group G acts on a Stein manifold X and another manifold Y in such a way that Y is G-Oka, then every G-equivariant continuous map \(X\rightarrow Y\) can be deformed, through such maps, to a G-equivariant holomorphic map. Approximation on a G-invariant holomorphically convex compact subset of X and jet interpolation along a G-invariant subvariety of X can be built into the theorem. We conjecture that the theorem holds for actions of arbitrary reductive complex Lie groups and prove partial results to this effect.

On the existence of infinite-dimensional generalized Harish-Chandra modules

We prove a general existence result for infinite-dimensional admissible $$(\mathfrak {g},\mathfrak {k})$$ -modules, where $$\mathfrak {g}$$ is a reductive finite-dimensional complex Lie algebra and $$\mathfrak {k}$$ is a reductive in $$\mathfrak {g}$$ algebraic subalgebra.

An Oka principle for Stein G-manifolds

Let $G$ be a reductive complex Lie group acting holomorphically on Stein manifolds $X$ and $Y$. Let $p_X\colon X\to Q_X$ and $p_Y\colon Y\to Q_Y$ be the quotient mappings. Assume that we have a biholomorphism $Q:= Q_X\to Q_Y$ and an open cover $\{U_i\}$ of $Q$ and $G$-biholomorphisms $\Phi_i\colon p_X^{-1}(U_i)\to p_Y^{-1}(U_i)$ inducing the identity on $U_i$. There is a sheaf of groups $\mathcal A$ on $Q$ such that the isomorphism classes of all possible $Y$ is the cohomology set $H^1(Q,\mathcal A)$. The main question we address is to what extent $H^1(Q,\mathcal A)$ contains only topological information. For example, if $G$ acts freely on $X$ and $Y$, then $X$ and $Y$ are principal $G$-bundles over $Q$, and Grauert's Oka Principle says that the set of isomorphism classes of holomorphic principal $G$-bundles over $Q$ is canonically the same as the set of isomorphism classes of topological principal $G$-bundles over $Q$. We investigate to what extent we have an Oka principle for $H^1(Q,\mathcal A)$.

Homotopy principles for equivariant isomorphisms

Let G G be a reductive complex Lie group acting holomorphically on Stein manifolds X X and Y Y . Let p X : X → Q X p_X\colon X\to Q_X and p Y : Y → Q Y p_Y\colon Y\to Q_Y be the quotient mappings. When is there an equivariant biholomorphism of X X and Y Y ? A necessary condition is that the categorical quotients Q X Q_X and Q Y Q_Y are biholomorphic and that the biholomorphism φ \varphi sends the Luna strata of Q X Q_X isomorphically onto the corresponding Luna strata of Q Y Q_Y . Fix φ \varphi . We demonstrate two homotopy principles in this situation. The first result says that if there is a G G -diffeomorphism Φ : X → Y \Phi \colon X\to Y , inducing φ \varphi , which is G G -biholomorphic on the reduced fibres of the quotient mappings, then Φ \Phi is homotopic, through G G -diffeomorphisms satisfying the same conditions, to a G G -equivariant biholomorphism from X X to Y Y . The second result roughly says that if we have a G G -homeomorphism Φ : X → Y \Phi \colon X\to Y which induces a continuous family of G G -equivariant biholomorphisms of the fibres p X − 1 ( q ) p_X{^{-1}}(q) and p Y − 1 ( φ ( q ) ) p_Y{^{-1}}(\varphi (q)) for q ∈ Q X q\in Q_X and if X X satisfies an auxiliary property (which holds for most X X ), then Φ \Phi is homotopic, through G G -homeomorphisms satisfying the same conditions, to a G G -equivariant biholomorphism from X X to Y Y . Our results improve upon those of our earlier paper [J. Reine Angew. Math. 706 (2015), 193–214] and use new ideas and techniques.

SUFFICIENT CONDITIONS FOR HOLOMORPHIC LINEARISATION

Let G be a reductive complex Lie group acting holomorphically on X = ℂn. The (holomorphic) Linearisation Problem asks if there is a holomorphic change of coordinates on ℂn such that the G-action becomes linear. Equivalently, is there a G-equivariant biholomorphism Φ: X → V where V is a G-module? There is an intrinsic stratification of the categorical quotient QX, called the Luna stratification, where the strata are labeled by isomorphism classes of representations of reductive subgroups of G. Suppose that there is a Φ as above. Then Φ induces a biholomorphism φ: QX → QV which is stratified, i.e., the stratum of QX with a given label is sent isomorphically to the stratum of QV with the same label.

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.

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 .

An Oka principle for equivariant isomorphisms

Abstract Let G be a reductive complex Lie group acting holomorphically on normal Stein spaces X and Y, which are locally G-biholomorphic over a common categorical quotient Q. When is there a global G-biholomorphism X → Y? If the actions of G on X and Y are what we, with justification, call generic, we prove that the obstruction to solving this local-to-global problem is topological and provide sufficient conditions for it to vanish. Our main tool is the equivariant version of Grauert's Oka principle due to Heinzner and Kutzschebauch. We prove that X and Y are G-biholomorphic if X is K-contractible, where K is a maximal compact subgroup of G, or if X and Y are smooth and there is a G-diffeomorphism ψ : X → Y over Q, which is holomorphic when restricted to each fibre of the quotient map X → Q. We prove a similar theorem when ψ is only a G-homeomorphism, but with an assumption about its action on G-finite functions. When G is abelian, we obtain stronger theorems. Our results can be interpreted as instances of the Oka principle for sections of the sheaf of G-biholomorphisms from X to Y over Q. This sheaf can be badly singular, even for a low-dimensional representation of SL2(ℂ). Our work is in part motivated by the linearisation problem for actions on ℂ n . It follows from one of our main results that a holomorphic G-action on ℂ n , which is locally G-biholomorphic over a common quotient to a generic linear action, is linearisable.

Homogeneous Kähler and Hamiltonian manifolds

We consider actions of reductive complex Lie groups $G=K^C$ on K\"ahler manifolds $X$ such that the $K$--action is Hamiltonian and prove then that the closures of the $G$--orbits are complex-analytic in $X$. This is used to characterize reductive homogeneous K\"ahler manifolds in terms of their isotropy subgroups. Moreover we show that such manifolds admit $K$--moment maps if and only if their isotropy groups are algebraic.

Parabolic and Levi Subalgebras of Finitary Lie Algebras

Let g be a locally reductive complex Lie algebra which admits a faithful countable-dimensional finitary representation V . Such a Lie algebra is a split extension of an abelian Lie algebra by a direct sum of copies of sl1, so1, sp1, and finite-dimensional simple Lie algebras. A parabolic subalgebra of g is any subalgebra which contains a maximal locally solvable (that is, Borel) subalgebra. Building upon work by Dimitrov and the authors of the present paper, [DP2], [D], we give a general description of parabolic subalgebras of g in terms of joint stabilizers of taut couples of generalized flags. The main differences with the Borel subalgebra case are that the description of general parabolic subalgebras has to use both the natural and conatural modules, and that the parabolic subalgebras are singled out by further “trace conditions” in the suitable joint stabilizer. The technique of taut couples can also be used to prove the existence of a Levi component of an arbitrary subalgebra k of gl1. If k is splittable, we show that the linear nilradical admits a locally reductive complement in k. We conclude the paper with descriptions of Cartan, Borel, and parabolic subalgebras of arbitrary splittable subalgebras of gl1. 2000 MSC: 17B05 and 17B65

Radial Components, Prehomogeneous Vector Spaces, and Rational Cherednik Algebras

Let be a finite-dimensional representation of a connected reductive complex Lie group . Denote by G the derived subgroup of and assume that the categorical quotient is one dimensional, i.e. for a nonconstant polynomial f. In this situation there exists a homomorphism , the radial component map, where is the first Weyl algebra. We show that the image of is isomorphic to the spherical subalgebra of a rational Cherednik algebra whose multiplicity function is defined by the roots of the Bernstein-Sato polynomial of f. In the case where is also multiplicity free we describe the kernel of and prove a Howe duality result between representations of G occurring in and lowest-weight modules over the Lie algebra generated by f and the “dual” differential operator ; this extends results of H. Rubenthaler obtained when is a parabolic prehomogeneous vector space. If satisfies a Capelli-type condition, some applications are given to holonomic and equivariant D-modules on V. These applications are related to results proved by Muro [34, 37, 38] or Nang [40, 42, 44] in special cases of the representation .

Cramped subgroups and generalized Harish-Chandra modules

Let G G be a reductive complex Lie group with Lie algebra g \mathfrak g . We call a subgroup H ⊂ G H\subset G cramped if there is an integer b ( G , H ) b(G,H) such that each finite-dimensional representation of G G has a non-trivial invariant subspace of dimension less than b ( G , H ) b(G,H) . We show that a subgroup is cramped if and only if the moment map T ∗ ( K / L ) → k ∗ T^*(K/L)\to \mathfrak k^* is surjective, where K K and L L are compact forms of G G and H H . We will use this in conjunction with sufficient conditions for crampedness given by Willenbring and Zuckerman (2004) to prove a geometric proposition on the intersections between adjoint orbits and Killing orthogonals to subgroups. We will also discuss applications of the techniques of symplectic geometry to the generalized Harish-Chandra modules introduced by Penkov and Zuckerman (2004), of which our results on crampedness are special cases.

Classification des triples de Manin pour les algèbres de Lie réductives complexes: Avec un appendice de Guillaume Macey

We study real and complex Manin triples for a complex reductive Lie algebra g . First, we generalize results of E. Karolinsky (1996, Math. Phys. Anal. Geom 3 , 545–563; 1999, Preprint math.QA.9901073) on the classification of Lagrangian subalgebras. Then we show that, if g is noncommutative, one can attach to each Manin triple in g another one for a strictly smaller reductive complex Lie subalgebra of g . This gives a powerful tool for induction. Then we classify complex Manin triples in terms of what we call generalized Belavin–Drinfeld data. This generalizes, by other methods, the classification of A. Belavin and V. G. Drinfeld of certain r -matrices, i.e., the solutions of modified triangle equations for constants (cf. A. Belavin and V. G. Drinfeld, “Triangle Equations and Simple Lie Algebras,” Mathematical Physics Reviews, Vol. 4, pp. 93–165, Harwood Academic, Chur, 1984, Theorem 6.1). We get also results for real Manin triples. In passing, we retrieve a result of A. Panov (1999, Preprint math.QA.9904156) which classifies certain Lie bialgebra structures on a real simple Lie algebra.

Actions of Borel Subgroups on Homogeneous Spaces of Reductive Complex Lie Groups and Integrability

Let G be a real reductive Lie group, K its compact subgroup. Let A be the algebra of G-invariant real-analytic functions on T*(G/K) (with respect to the Poisson bracket) and let C be the center of A. Denote by 2ε(G,K) the maximal number of functionally independent functions from A\C. We prove that ε(G,K) is equal to the codimension δ(G,K) of maximal dimension orbits of the Borel subgroup B⊂G. Moreover, if δ(G,K)=1, then all G-invariant Hamiltonian systems on T*(G/K) are integrable in the class of the integrals generated by the symmetry group G. We also discuss related questions in the geometry of the Borel group action.

A note on the complete integrability of the Hitchin system

We consider the Hitchin map \( {\cal H}\!: T^{*}{\cal M}_{G}\longrightarrow {\cal K},\) defined on the cotangent bundle of the moduli space of stable principal G-bundles over a compact Riemann surface, G being any reductive complex Lie group. In [5] Hitchin showed that the \(m=\dim{\cal M}_{G}\) components of \(\cal H\) are analytic functions that commute with respect to the natural Poisson structure on \(T^{*}{\cal M}_{G}\). In order to do this, Hitchin used a description of \(T^{*}{\cal M}_{G}\) as a Marsden-Weinstein quotient \(\mu ^{-1}(0)/{\cal G}, \mu \) being the moment map for the action of the Gauge group \(\cal G\) on the infinite dimensional space of stable holomorphic structures on a fixed principal G-bundle. In this note we obtain the same result as an application of a few elementary properties satisfied by the homogeneous Ad G-invariant polynomials on the Lie algebra of G.

The Cohomology of the Regular Semisimple Variety

We use the equivariant cohomology of hyperplane complements and their toral counterparts to give formulae for the Poincaré polynomials of the varieties of regular semisimple elements of a reductive complex Lie group or Lie algebra. As a result, we obtain vanishing theorems for certain of the Betti numbers. Similar methods, usingl-adic cohomology, may be used to compute numbers of rational points of the varieties over the finite field Fq. In the classical cases, one obtains, both for the Poincaré polynomials and for the numbers of rational points, polynomials which exhibit certain regularity conditions as the dimension increases. This regularity may be interpreted in terms of functional equations satisfied by certain power series, or in terms of the representation theory of the Weyl group.