Complex Semisimple Lie Group Research Articles

Components of the Springer fiber and domino tableaux

Consider a complex classical semisimple Lie group along with the set of its nilpotent coadjoint orbits. When the group is of type A, the set of orbital varieties contained in a given nilpotent orbit is described a set of standard Young tableaux. We parameterize both, the orbital varieties and the irreducible components of unipotent varieties in the other classical groups by sets of standard domino tableaux. The main tools are Spaltenstein's results on signed domino tableaux together with Garfinkle's operations on standard domino tableaux.

Adapted algebras for the Berenstein-Zelevinsky conjecture

Let G be a simply connected semisimple complex Lie group and fix a maximal unipotent subgroup U- of G. Let q be an indeterminate and let B* denote the dual canonical basis (cf. [19]) of the quantized algebra Cq[U-] of regular functions on U-. Following [20], fix a ZN≧0-parametrization of this basis, where N = dim U-. In [2], A. Berenstein and A. Zelevinsky conjecture that two elements of B* q-commute if and only if they are multiplicative, i.e., their product is an element of B* up to a power of q. To any reduced decomposition w0 of the longest element of the Weyl group of g, we associate a subalgebra Aw0, called adapted algebra, of Cq[U-] such that (1) Aw0 is a q-polynomial algebra which equals Cq[U-] up to localization, (2) Aw0 is spanned by a subset of B*, (3) the Berenstein–Zelevinsky conjecture is true on Aw0. Then we test the conjecture when one element belongs to the q-center of Cq[U-].

Invariant almost Hermitian structures on flag manifolds

Let G be a complex semi-simple Lie group and form its maximal flag manifold F=G/P=U/T where P is a minimal parabolic (Borel) subgroup, U a compact real form and T= U∩ P a maximal torus of U. We study U-invariant almost Hermitian structures on F . The (1,2)-symplectic (or quasi-Kähler) structures are naturally related to the affine Weyl groups. A special form for them, involving abelian ideals of a Borel subalgebra, is derived. From the (1,2)-symplectic structures a classification of the whole set of invariant structures is provided showing, in particular, that nearly Kähler invariant structures are Kähler, except in the A 2 case.

Dimensional reduction and quiver bundles

The so-called Hitchin-Kobayashi correspondence, proved by Donaldson, Uhlenbeck and Yau, establishes that an indecomposable holomorphic vector bundle over a compact Kahler manifold admits a Hermitian-Einstein metric if and only if the bundle satisfies the Mumford-Takemoto stability condition. In this paper we consider a variant of this correspondence for G-equivariant vector bundles on the product of a compact Kahler manifold X by a flag manifold G=P, where G is a complex semisimple Lie group and P is a parabolic subgroup. The modification that we consider is determined by a filtration of the vector bundle which is naturally defined by the equivariance of the bundle. The study of invariant solutions to the modified Hermitian-Einstein equation over XG=P leads, via dimensional reduction techniques, to gauge-theoretic equations on X. These are equations for hermitian metrics on a set of holomorphic bundles on X linked by morphisms, defin- ing what we call a quiver bundle for a quiver with relations whose structure is entirely determined by the parabolic subgroup P. Similarly, the corresponding stability condition for the invariant filtration over XG=P gives rise to a stability condition for the quiver bundle on X , and hence to a Hitchin-Kobayashi correspondence. In the simplest case, when the flag manifold is the complex projective line, one recovers the theory of vortices, stable triples and stable chains, as studied by Bradlow, the authors, and others.

Prime and Primitive Ideals in Graded Deformations of Algebraic Quantum Groups at Roots of Unity

Let G be a connected, simply connected complex semisimple Lie group of rank n. The deformations employed by Artin, Schelter and Tate, and Hodges, Levasseur and Toro can be applied to the single parameter quantizations, at roots of unity, of the Hopf algebra of regular functions on G. Each of the resulting complex multiparameter quantum groups F∈,p[G] depends on both a suitable root of unity ∈ and an antisymmetric bicharacter p: Z n ×Z n →C×. These quantizations differ significantly from their single parameter (root-of-unity) counterparts, and, in particular, may have infinite-dimensional irreducible representations. Our approach to F∈,p[G] depends on a natural ℋ×ℋ-action thereon, where ℋ is an n-torus, and our main result offers a classification of the primitive ideals: We use a multiparameter quantum Frobenius map to provide a bijection from (Prim F∈,p[G])/ℋ×ℋ onto G/H×H, where H is a maximal torus of G. In the single parameter case, this bijection is a consequence of work by De Concini and Lyubashenko, and De Concini and Procesi; our results require their analysis. Our methods also exploit earlier work by Moeglin and Rentschler concerning actions of algebraic groups on complex Noetherian algebras. In contrast to generic quantizations of the coordinate ring of G, the primitive spectrum of F∈,p[G] is not finitely stratified by the torus action.

The Hodge star operator on Schubert forms

Let X= G/ P be a homogeneous space of a complex semisimple Lie group G equipped with a hermitian metric. We study the action of the Hodge star operator on the space of harmonic differential forms on X. We obtain explicit combinatorial formulas for this action when X is an irreducible hermitian symmetric space of compact type.

A rank-three condition for invariant (1, 2)-symplectic almost Hermitian structures on flag manifolds

This paper considers invariant (1, 2)-symplectic almost Hermitian structures on the maximal flag manifod associated to a complex semi-simple Lie group G. The concept of cone-free invariant almost complex structure is introduced. It involves the rank-three subgroups of G, and generalizes the cone-free property for tournaments related to 𝕊l (n,ℂ) case. It is proved that the cone-free property is necessary for an invariant almost-complex structure to take part in an invariant (1, 2)-symplectic almost Hermitian structure. It is also sufficient if the Lie group is not Bl , l ≥ 3, G2 or F4. For Bl and F4 a close condition turns out to be sufficient.

Uniform estimates for spherical functions on complex semisimple Lie groups

We prove new estimates for spherical functions and their derivatives on complex semisimple Lie groups, establishing uniform polynomial decay in the spectral parameter. This improves the customary estimate arising from Harish-Chandra's series expansion, which gives only a polynomial growth estimate in the spectral parameter. In particular, for arbitrary positive-definite spherical functions \( \varphi_\lambda \) on higher rank complex simple groups, we establish estimates for \( |d^k\varphi_\lambda(a_t)/dt^k| \) which are of the form \( (1\,+ \parallel \lambda \parallel)^{k-\gamma} \) in the spectral parameter \( \lambda \) and have uniform exponential decay in regular directions in the group variable a t . Here \( \gamma \) is an explicit constant depending on G, and \( \lambda \) may be singular, for instance.¶The uniformity of the estimates is the crucial ingredient needed in order to apply classical spectral methods and Littlewood—Paley—Stein square functions to the analysis of singular integrals in this context. To illustrate their utility, we prove maximal inequalities in L p for singular sphere averages on complex semisimple Lie groups for all p in \( (1\,+ {1 \over 2\gamma}, \infty) \). We use these to establish singular differentiation theorems and pointwise ergodic theorems in L p for the corresponding singular spherical averages on locally symmetric spaces, as well as for more general measure preserving actions.

Totally nonnegative and oscillatory elements in semisimple groups

We generalize the well known characterizations of totally nonnegative and oscillatory matrices, due to F. R. Gantmacher, M. G. Krein, A. Whitney, C. Loewner, M. Gasca, and J. M. Peña to the case of an arbitrary complex semisimple Lie group.

Linear cycle spaces in flag domains

Let Z = G/Q, a complex flag manifold, where G is a complex semisimple Lie group and Q is a parabolic subgroup. Fix a real form \(G_0 \subset G\) and consider the linear cycle spaces \(M_D\), spaces of maximal compact linear subvarieties of open orbits \(D = G_0(z) \subset Z\). In general \(M_D\) is a Stein manifold. Here the exact structure of \(M_D\) is worked out when \(G_0\) is a classical group that corresponds to a bounded symmetric domain B. In that case \(M_D\) is biholomorphic to B if a certain double fibration is holomorphic, is biholomorphic to \(B \times \overline{B}\) otherwise. There are also a number of structural results that do not require \(G_0\) to be classical.

Holomorphic diffeomorphisms of complex semisimple Lie groups

Holomorphic diffeomorphisms of complex semisimple Lie groups

A Pieri-Chevalley formula in the K-theory of a 𝐺/𝐵-bundle

Let G G be a semisimple complex Lie group, B B a Borel subgroup, and T ⊆ B T\subseteq B a maximal torus of G G . The projective variety G / B G/B is a generalization of the classical flag variety. The structure sheaves of the Schubert subvarieties form a basis of the K-theory K ( G / B ) K(G/B) and every character of T T gives rise to a line bundle on G / B G/B . This note gives a formula for the product of a dominant line bundle and a Schubert class in K ( G / B ) K(G/B) . This result generalizes a formula of Chevalley which computes an analogous product in cohomology. The new formula applies to the relative case, the K-theory of a G / B G/B -bundle over a smooth base X X , and is presented in this generality. In this setting the new formula is a generalization of recent G = G L n ( C ) G=GL_n({\mathbb C}) results of Fulton and Lascoux.

Unipotent representations attached to spherical nilpotent orbits

A coadjoint nilpotent orbit [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] of a complex semisimple Lie group G is said to be spherical if it contains an open orbit of a Borel subgroup. We determine when and how to attach unitary representations to such an orbit for the real orthogonal and symplectic groups. Our results actually extend to a larger class of nilpotent coadjoint orbits.

A Note on a Symplectic Structure on the Space of G -Monopoles

Let $G$ be a semisimple complex Lie group with a Borel subgroup $B$. Let $X=G/B$ be the flag manifold of $G$. Let $C=P^1\ni\infty$ be the projective line. Let $\alpha\in H_2(X,{\Bbb Z})$. The moduli space of $G$-monopoles of topological charge $\alpha$ (see e.g. [Jarvis]) is naturally identified with the space $M_b(X,\alpha)$ of based maps from $(C,\infty)$ to $(X,B)$ of degree $\alpha$. The moduli space of $G$-monopoles carries a natural hyperk\"ahler structure, and hence a holomorphic symplectic structure. We propose a simple explicit formula for the symplectic structure on $M_b(X,\alpha)$. It generalizes the well known formula for $G=SL_2$ (see e.g. [Atiyah-Hitchin]). Let $P\supset B$ be a parabolic subgroup. The construction of the Poisson structure on $M_b(X,\alpha)$ generalizes verbatim to the space of based maps $M=M_b(G/P,\beta)$. In most cases the corresponding map $T^*M\to TM$ is not an isomorphism, i.e. $M$ splits into nontrivial symplectic leaves. These leaves are explicilty described.

Quantum Cohomology of Flag Manifolds G/B and Quantum Toda Lattices

Let G be a connected semi-simple complex Lie group, B its Borel subgroup, T a maximal complex torus contained in B, and Lie (T ) its Lie algebra. This setup gives rise to two constructions; the generalized nonperiodic Toda lattice ([28], [29]) and the flag manifold G/B. The Toda lattice for (G,B, T ) is the dynamical system on the cotangent bundle T ∗Lie (T ) endowed with the canonical holomorphic symplectic form and the holomorphic hamiltonian function we consider in this paper,

On Bernstein Polynomials for Compact Lie Groups

We give the Bernstein polynomials for basic matrix entries of irreducible unitary representations of compact Lie groupSU(2). We also give an application to the analytic continuation of certain distributions onSU(2), and finally we briefly describe the Bernstein polynomial forB−×B-semi-invariant functions on a semisimple complex Lie group.

Globalization of Holomorphic Actions on Principal Bundles

Suppose G is a Lie group acting as a group of holomorphic automorphisms on a holomorphic principal bundle P → X. We show that if there is a holomorphic action of the complexification GC of G on. X, this lifts to a holomorphic action of GC on the bundle P → X. Two applications are presented. We prove that given any connected homogeneous complex manifold G/H with more than one end, the complexification GC of G acts holomorphically and transitively on G/H. We also show that the ends of a homogeneous complex manifold G/H with more than two ends essentially come from a space of the form S/Γ, where Γ is a Zariski dense discrete subgroup of a semisimple complex Lie group S with S and Γ being explicitly constructed in terms of G and H.

Algebraic Structure of Multiparameter Quantum Groups

Let G be a connected semi-simple complex Lie group. We define and study the multi-parameter quantum group Cq, p[G ] in the case where q is a complex parameter that is not a root of unity. Using a method of twisting bigraded Hopf algebras by a cocycle, [2], we develop a unified approach to the construction of Cq, p [G ] and of the multi-parameter Drinfeld double Dq, p . Using a general method of deforming bigraded pairs of Hopf algebras, we construct a Hopf pairing between these algebras from which we deduce a PeterWeyl-type theorem for Cq, p[G ]. We then describe the prime and primitive spectra of Cq, p[G ], generalizing a result of Joseph. In the one-parameter case this description was conjectured, and established in the SL(n)-case, by the first and second authors in [15, 16]. It was proved in the general case by Joseph in [18, 19]. In particular the orbits in Prim Cq, p[G] under the natural action of the maximal torus H are indexed, as in the one-parameter case by the elements of the double Weyl group W_W. Unlike the one-parameter case there is not in general a bijection between Symp G and Prim Cq, p[G ]. However in the case when

Quantum Schubert polynomials

where In is the ideal generated by symmetric polynomials in x1,... ,xn without constant term. Another, geometric, description of the cohomology ring of the flag manifold is based on the decomposition of Fln into Schubert cells. These are even-dimensional cells indexed by the elements w of the symmetric group Sn. The corresponding cohomology classes oa, called Schubert classes, form an additive basis in H* (Fln 2) . To relate the two descriptions, one would like to determine which elements of 2[xl, ... , Xn]/In correspond to the Schubert classes under the isomorphism (1.1). This was first done in [2] (see also [8]) for a general case of an arbitrary complex semisimple Lie group. Later, Lascoux and Schiitzenberger [22] came up with a combinatorial version of this theory (for the type A) by introducing remarkable polynomial representatives of the Schubert classes oa called Schubert polynomials and denoted Gw. Recently, motivated by ideas that came from the string theory [31, 30], mathematicians defined, for any Kahler algebraic manifold X, the (small) quantum cohomology ring QH* (X, 2), which is a certain deformation of the classical cohomology ring (see, e.g., [28, 19, 14] and references therein). The additive structure of QH* (X , 2) is essentially the same as that of ordinary cohomology. In particular, QH* (Fln , Z) is canonically isomorphic, as an abelian group, to the tensor product H* (Fln , 2) (0 Z[ql,..., qn-1], where the qi are formal variables (deformation parameters). The multiplicative structure of the quantum cohomology is however

On closed radical orbits in homogeneous complex manifolds

Suppose G is a complex Lie group having a finite number of connected components and H is a closed complex subgroup of G with H° solvable. Let RG denote the radical of G. We show the existence of closed complex subgroups I and J of G containing H such that I/H is a connected solvmanifold with I° ⊃ RG, the space G/J has a Klein form SG/A, where A is an algebraic subgroup of the semisimple complex Lie group SG: = G/RG, and, unless I = J, the space J/I has Klein form , where is a Zariski dense discrete subgroup of some connected positive dimensional semisimple complex Lie group Ŝ.