Complex Algebraic Group Research Articles

A criterion for holomorphic extension of principal bundles

Let G be a complex affine algebraic group, and let EG be a holomorphic principal G-bundle on the complement M$\backslash$S, where S is a closed complex analytic subset, of complex codimension at least two, of a connected complex manifold M. We give a criterion for EG to extend to M as a holomorphic principal G-bundle. Two applications of this criterion are given.

Analytic R-groups of affine Hecke algebras

We define analytic $R$-groups for affine Hecke algebras, and prove the analog of the Knapp-Stein Dimension Theorem. As a corollary we prove that the commutant algebra of a unitary principal series representation is isomorphic to the complex group algebra of the $R$-group, twisted by a certain 2-cocycle $\gamma$. For classical Hecke algebras we prove that $\gamma$ is always trivial.

Extended weight semigroups of affine spherical homogeneous spaces of non-simple semisimple algebraic groups

The extended weight semigroup of a homogeneous space of a connected semisimple algebraic group characterizes the spectra of the representations of on spaces of regular sections of homogeneous line bundles over , including the space of regular functions on . We compute the extended weight semigroups for all strictly irreducible affine spherical homogeneous spaces , where is a simply connected non-simple semisimple complex algebraic group and is a connected closed subgroup of . In all cases we also find the highest-weight functions corresponding to the indecomposable elements of this semigroup. Among other things, our results complete the computation of the weight semigroups for all strictly irreducible simply connected affine spherical homogeneous spaces of semisimple complex algebraic groups.

Symmetric groups are determined by their character degrees

Let G be a finite group. Let X 1 ( G ) be the first column of the ordinary character table of G. In this paper, we will show that if X 1 ( G ) = X 1 ( S n ) , then G ≅ S n . As a consequence, we show that S n is uniquely determined by the structure of the complex group algebra C S n .

MV-polytopes via affine buildings

We give a construction of MV-polytopes of a complex semisimple algebraic group G in terms of the geometry of the Bott-Samelson variety and the affine building. This is done by using the construction of dense subsets of MV-cycles by Gaussent and Littelmann. They used LS-gallery to define subsets in the Bott-Samelson variety that map to subsets of the affine Grassmannian, whose closure are MV-cycles. Since points in the Bott-Samelson variety correspond to galleries in the affine building one can look at the image of a point in such a special subset under all retractions at infinity. We prove that these images can be used to construct the corresponding MV-polytope in an explicit way, by using the GGMS strata. Furthermore we give a combinatorial construction for these images by using the crystal structure of LS-galleries and the action of the ordinary Weyl group on the coweight lattice.

GROUP COVERS, o-MINIMALITY, AND CATEGORICITY

We study the model theory of of groups H definable in an o-minimal structure M. We pose the question of whether any finite central extension G of H is interpretable in M, proving some cases (such as when H is abelian) as well as stating various equivalences. When M is an o-minimal expansion of the reals (so H is a definable Lie group) this is related to Milnor's conjecture [15], and many cases are known. We also prove a strong relative Lω1, ω-categoricity theorem for universal covers of definable Lie groups, and point out some notable differences with the case of covers of complex algebraic groups (studied by Zilber and his students).

On Positivity in T-Equivariant K-Theory of Flag Varieties

We prove some general results on the T-equivariant K-theory K T (G/P) of the flag variety G/P, where G is a complex semisimple algebraic group, P is a parabolic subgroup, and T is a maximal torus contained in P. In particular, we make a conjecture about a positivity phenomenon in K T (G/P) for the product of two basis elements written in terms of the basis of K T (G/P) given by the dual of the structure sheaf (of Schubert varieties) basis. (For the full flag variety G/B, this dual basis is closely related to the basis given by Kostant-Kumar.) This conjecture is parallel to (but different from) the conjecture of Griffeth-Ram for the structure constants of the product in the structure sheaf basis. We give explicit expressions for the product in the T-equivariant K-theory of projective spaces in terms of these bases. In particular, we establish our conjecture and the conjecture of Griffeth-Ram in this case.

Zariski Density and Genericity

In this paper, we combine a number of results (some due to the author, some not) to show that Zariski density is, in a strong sense, a generic property of subgroups of SL(n, ℤ) and Sp(2n, ℤ). We also extend previous results of the author on generic properties of elements of the special linear and symplectic groups to generic elements in arbitrary Zariski-dense (as subgroups of the ambient complex algebraic group) subgroups of these groups. Jointly with Ilya Kapovich, we also show that a generic free group automorphism is hyperbolic.

Quantum cohomology of G/P and homology of affine Grassmannian

Let G be a simple and simply-connected complex algebraic group, P \subset G a parabolic subgroup. We prove an unpublished result of D. Peterson which states that the quantum cohomology QH^*(G/P) of a flag variety is, up to localization, a quotient of the homology H_*(Gr_G) of the affine Grassmannian \Gr_G of G. As a consequence, all three-point genus zero Gromov-Witten invariants of $G/P$ are identified with homology Schubert structure constants of H_*(Gr_G), establishing the equivalence of the quantum and homology affine Schubert calculi. For the case G = B, we use the Mihalcea's equivariant quantum Chevalley formula for QH^*(G/B), together with relationships between the quantum Bruhat graph of Brenti, Fomin and Postnikov and the Bruhat order on the affine Weyl group. As byproducts we obtain formulae for affine Schubert homology classes in terms of quantum Schubert polynomials. We give some applications in quantum cohomology. Our main results extend to the torus-equivariant setting.

Spherical Varieties and Langlands Duality

Let G be a connected reductive complex algebraic group. This paper is devoted to the space Z of meromorphic quasimaps from a curve into an affine spherical G-variety X. The space Z may be thought of as an algebraic model for the loop space of X. In this paper, we associate to X a connected reductive complex algebraic subgroup $\check H$ of the dual group $\check G$. The construction of $\check H$ is via Tannakian formalism: we identify a certain tensor category Q(Z) of perverse sheaves on Z with the category of finite-dimensional representations of $\check H$. Combinatorial shadows of the group $\check H$ govern many aspects of the geometry of X such as its compactifications and invariant differential operators. When X is a symmetric variety, the group $\check H$ coincides with that associated to the corresponding real form of G via the (real) geometric Satake correspondence.

A rigid irregular connection on the projective line

In this paper we construct a connection ∇ on the trivial G-bundle on P 1 for any simple complex algebraic group G, which is regular outside of the points 0 and ∞, has a regular singularity at the point 0, with principal unipotent monodromy, and has an irregular singularity at the point ∞, with slope 1/h, the reciprocal of the Coxeter number of G. The connection ∇, which admits the structure of an oper in the sense of Beilinson and Drinfeld, appears to be the characteristic 0 counterpart of a hypothetical family of l-adic representations, which should parametrize a specific automorphic representation under the global Langlands correspondence. These l- adic representations, and their characteristic 0 counterparts, have been constructed in some cases by Deligne and Katz. Our connection is constructed uniformly for any simple algebraic group, and characterized using the formalism of opers. It provides an example of the geometric Langlands correspondence with wild ramification. We compute the de Rham cohomology of our connection with values in a representation V of G, and describe the differential Galois group of ∇ as a subgroup of G.

Automorphisms of wonderful varieties

Let G be a complex semisimple linear algebraic group, and X a wonderful G-variety. We determine the connected automorphism group Aut0(X) and we calculate Luna’s invariants of X under its action.

Poisson structures on affine spaces and flag varieties. II

The standard Poisson structures on the flag varieties $G/P$ of a complex reductive algebraic group $G$ are investigated. It is shown that the orbits of symplectic leaves in $G/P$ under a fixed maximal torus of $G$ are smooth irreducible locally closed subvarieties of $G/P$, isomorphic to intersections of dual Schubert cells in the full flag variety $G/B$ of $G$, and their Zariski closures are explicitly computed. Two different proofs of the former result are presented. The first is in the framework of Poisson homogeneous spaces, and the second one uses an idea of weak splittings of surjective Poisson submersions, based on the notion of Poisson–Dirac submanifolds. For a parabolic subgroup $P$ with abelian unipotent radical (in which case $G/P$ is a Hermitian symmetric space of compact type), it is shown that all orbits of the standard Levi factor $L$ of $P$ on $G/P$ are complete Poisson subvarieties which are quotients of $L$, equipped with the standard Poisson structure. Moreover, it is proved that the Poisson structure on $G/P$ vanishes at all special base points for the $L$-orbits on $G/P$ constructed by Richardson, Röhrle, and Steinberg.

YANG–MILLS EQUATION FOR STABLE HIGGS SHEAVES

We establish a Kobayashi–Hitchin correspondence for the stable Higgs sheaves on a compact Kähler manifold. Using it, we also obtain a Kobayashi–Hitchin correspondence for the stable Higgs G-sheaves, where G is any complex reductive linear algebraic group.

Multiplication of Polynomials on Hermitian Symmetric spaces and Littlewood–Richardson Coefficients

Abstract. Let K be a complex reductive algebraic group and V a representation of K. Let S denote the ring of polynomials on V. Assume that the action of K on S is multiplicity-free. If ƛ denotes the isomorphism class of an irreducible representation of K, let ρƛ : K → GL(Vƛ) denote the corresponding irreducible representation and Sƛ the ƛ-isotypic component of S. Write Sƛ ・ Sμ for the subspace of S spanned by products of Sƛ and Sμ. If Vν occurs as an irreducible constituent of Vƛ ⊗ Vμ, is it true that Sν ⊆ Sƛ ・ Sμ? In this paper, the authors investigate this question for representations arising in the context of Hermitian symmetric pairs. It is shown that the answer is yes in some cases and, using an earlier result of Ruitenburg, that in the remaining classical cases, the answer is yes provided that a conjecture of Stanley on the multiplication of Jack polynomials is true. It is also shown how the conjecture connects multiplication in the ring S to the usual Littlewood–Richardson rule.

Quantum subgroups of a simple quantum group at roots of one

AbstractLetGbe a connected, simply connected, simple complex algebraic group and let ϵ be a primitiveℓth root of one,ℓodd and 3∤ℓifGis of typeG2. We determine all Hopf algebra quotients of the quantized coordinate algebra 𝒪ϵ(G).

Hecke operators on quasimaps into horospherical varieties

Let G be a connected reductive complex algebraic group. This paper and its companion citeGNcombo06 are devoted to the space Z of meromorphic quasimaps from a curve into an affine spherical G -variety X . The space Z may be thought of as an algebraic model for the loop space of X . The theory we develop associates to X a connected reductive complex algebraic subgroup \check H of the dual group \check G . The construction of \check H is via Tannakian formalism: we identify a certain tensor category Q(Z) of perverse sheaves on Z with the category of finite-dimensional representations of \check H . In this paper, we focus on horospherical varieties, a class of varieties closely related to flag varieties. For an affine horospherical G -variety X_{\mathrm{horo}} , the category Q(Z_{\mathrm{horo}}) is equivalent to a category of vector spaces graded by a lattice. Thus the associated subgroup \check H_{\mathrm{horo}} is a torus. The case of horospherical varieties may be thought of as a simple example, but it also plays a central role in the general theory. To an arbitrary affine spherical G -variety X , one may associate a horospherical variety X_{\mathrm{horo}} . Its associated subgroup \check H_{\mathrm{horo}} turns out to be a maximal torus in the subgroup \check H associated to X .

Descent of Line Bundles to Git Quotients of Flag Varieties by Maximal Torus

Let G be a connected, simply connected semisimple complex algebraic group with a maximal torus T and let P be a parabolic subgroup containing T. Let $ \mathcal{L}_{P} {\left( \lambda \right)} $ be a homogeneous ample line bundle on the ag variety Y = G = P. We give a necessary and sufficient condition for $ \mathcal{L}_{P} {\left( \lambda \right)} $ to descend to a line bundle on the GIT quotient Y(λ)//T. As a consequence of this result, we get the precise list of P-regular weights λ for which the line bundle $ \mathcal{L}_{P} {\left( \lambda \right)} $ descends to the GIT quotient Y(λ)//T.

A mirror symmetric construction of [formula omitted

Let G be a simple simply connected complex algebraic group. We give a Lie-theoretic construction of a conjectural mirror family associated to a general flag variety G / P , and show that it recovers the Peterson variety presentation for the T-equivariant quantum cohomology rings q H T ∗ ( G / P ) ( q ) with quantum parameters inverted. For SL n / B we relate our construction to the mirror family defined by Givental and its T-equivariant analogue due to Joe and Kim.

On Positivity in T-Equivariant K-Theory of Flag Varieties

We prove some general results on the T-equivariant K-theory K_T(G/P) of the flag variety G/P, where G is a semisimple complex algebraic group, P is a parabolic subgroup and T$ is a maximal torus contained in P. In particular, we make a conjecture about a positivity phenomenon in K_T(G/P) for the product of two basis elements written in terms of the basis of K_T(G/P) given by the dual of the structure sheaf (of Schubert varieties) basis. (For the full flag variety G/B, this dual basis is closely related to the basis given by Kostant-Kumar.) This conjecture is parallel to (but different from) the conjecture of Griffeth-Ram for the structure constants of the product in the structure sheaf basis. We give explicit expressions for the product in the T-equivariant K-theory of projective spaces in terms of these bases. In particular, we establish our conjecture and the conjecture of Griffeth-Ram in this case.