Complex Semisimple Algebraic Group Research Articles

Torsors on moduli spaces of principal G-bundles on curves

Let [Formula: see text] be a semisimple complex algebraic group with a simple Lie algebra [Formula: see text], and let [Formula: see text] denote the moduli stack of topologically trivial stable [Formula: see text]-bundles on a smooth projective curve [Formula: see text]. Fix a theta characteristic [Formula: see text] on [Formula: see text] which is even in case [Formula: see text] is odd. We show that there is a nonempty Zariski open substack [Formula: see text] of [Formula: see text] such that [Formula: see text], [Formula: see text], for all [Formula: see text]. It is shown that any such [Formula: see text] has a canonical connection. It is also shown that the tangent bundle [Formula: see text] has a natural splitting, where [Formula: see text] is the restriction of [Formula: see text] to the semi-stable locus. We also produce an isomorphism between two naturally occurring [Formula: see text]-torsors on the moduli space of regularly stable [Formula: see text].

A cluster structure on the coordinate ring of partial flag varieties

The main goal of this paper is to show that the (multi-homogeneous) coordinate ring of a partial flag variety C[G/PK−] contains a cluster algebra if G is any semisimple complex algebraic group. We use derivation properties and a special lifting map to prove that the cluster algebra structure A of the coordinate ring C[NK] of a Schubert cell constructed by Goodearl and Yakimov can be lifted, in an explicit way, to a cluster structure Aˆ living in the coordinate ring of the corresponding partial flag variety. Then we use a minimality condition to prove that the cluster algebra Aˆ is equal to C[G/PK−] after localizing some special minors.

Regular functions on spherical nilpotent orbits in complex symmetric pairs: Classical Hermitian cases

Given a classical semisimple complex algebraic group G and a symmetric pair (G,K) of Hermitian type, we study the closures of the spherical nilpotent K-orbits in the isotropy representation of K. We show that all such orbit closures are normal and describe the K-module structure of their ring of regular functions.

Categories of Bott-Samelson Varieties

We consider all Bott-Samelson varieties BS(s) for a fixed connected semisimple complex algebraic group with maximal torus T as the class of objects of some category. The class of morphisms of this category is an extension of the class of canonical (inserting the neutral element) morphisms BS(s)↪BS(s′), where s is a subsequence of s′. Every morphism of the new category induces a map between the T-fixed points but not necessarily between the whole varieties. We construct a contravariant functor from this new category to the category of graded \(\phantom {\dot {i}\!}H^{\bullet }_{T}(\text {pt})\)-modules coinciding on the objects with the usual functor \(\phantom {\dot {i}\!}H_{T}^{\bullet }\) of taking T-equivariant cohomologies. We also discuss the problem how to define a functor to the category of T-spaces from a smaller subcategory. The exact answer is obtained for groups whose root systems have simply laced irreducible components by explicitly constructing morphisms between Bott-Samelson varieties (different from the canonical ones).

NILPOTENT SUBSPACES AND NILPOTENT ORBITS

Let $G$ be a semisimple complex algebraic group with Lie algebra $\mathfrak{g}$. For a nilpotent $G$-orbit ${\mathcal{O}}\subset \mathfrak{g}$, let $d_{{\mathcal{O}}}$ denote the maximal dimension of a subspace $V\subset \mathfrak{g}$ that is contained in the closure of ${\mathcal{O}}$. In this note, we prove that $d_{{\mathcal{O}}}\leq {\textstyle \frac{1}{2}}\dim {\mathcal{O}}$ and this upper bound is attained if and only if ${\mathcal{O}}$ is a Richardson orbit. Furthermore, if $V$ is $B$-stable and $\dim V={\textstyle \frac{1}{2}}\dim {\mathcal{O}}$, then $V$ is the nilradical of a polarisation of ${\mathcal{O}}$. Every nilpotent orbit closure has a distinguished $B$-stable subspace constructed via an $\mathfrak{sl}_{2}$-triple, which is called the Dynkin ideal. We then characterise the nilpotent orbits ${\mathcal{O}}$ such that the Dynkin ideal (1) has the minimal dimension among all $B$-stable subspaces $\mathfrak{c}$ such that $\mathfrak{c}\cap {\mathcal{O}}$ is dense in $\mathfrak{c}$, or (2) is the only $B$-stable subspace $\mathfrak{c}$ such that $\mathfrak{c}\cap {\mathcal{O}}$ is dense in $\mathfrak{c}$.

Regular functions on spherical nilpotent orbits in complex symmetric pairs: Classical non-Hermitian cases

Given a classical semisimple complex algebraic group G and a symmetric pair (G,K) of non-Hermitian type, we study the closures of the spherical nilpotent K-orbits in the isotropy representation of K. For all such orbit closures, we study the normality, and we describe the K-module structure of the ring of regular functions of the normalizations.

Orbits of Real Semisimple Lie Groups on Real Loci of Complex Symmetric Spaces

Let $G$ be a complex semisimple algebraic group and $X$ be a complex symmetric homogeneous $G$-variety. Assume that both $G$, $X$ as well as the $G$-action on $X$ are defined over real numbers. Then $G(\mathbb{R})$ acts on $X(\mathbb{R})$ with finitely many orbits. We describe these orbits in combinatorial terms using Galois cohomology, thus providing a patch to a result of A.Borel and L.Ji.

A SURVEY OF THE ADDITIVE EIGENVALUE PROBLEM (WITH APPENDIX BY M. KAPOVICH)

The classical Hermitian eigenvalue problem addresses the following question: What are the possible eigenvalues of the sum A + B of two Hermitian matrices A and B, provided we fix the eigenvalues of A and B. A systematic study of this problem was initiated by H. Weyl (1912). By virtue of contributions from a long list of mathematicians, notably Weyl (1912), Horn (1962), Klyachko (1998) and Knutson–Tao (1999), the problem is finally settled. The solution asserts that the eigenvalues of A + B are given in terms of certain system of linear inequalities in the eigenvalues of A and B. These inequalities (called the Hom inequalities) are given explicitly in terms of certain triples of Schubert classes in the singular cohomology of Grassmannians and the standard cup product. Belkale (2001) gave a smaller set of inequalities for the problem in this case (which was shown to be optimal by Knutson–Tao–Woodward). The Hermitian eigenvalue problem has been extended by Berenstein–Sjamaar (2000) and Kapovich–Leeb–Millson (2009) for any semisimple complex algebraic group G. Their solution is again in terms of a system of linear inequalities obtained from certain triples of Schubert classes in the singular cohomology of the partial ag varieties G/P (P being a maximal parabolic subgroup) and the standard cup product. However, their solution is far from being optimal. In a joint work with P. Belkale, we define a deformation of the cup product in the cohomology of G/P and use this new product to generate our system of inequalities which solves the problem for any G optimally (as shown by Ressayre). This article is a survey (with more or less complete proofs) of this additive eigenvalue problem. The eigenvalue problem is equivalent to the saturated tensor product problem. We also give an extension of the saturated tensor product problem to the saturated restriction problem for any pair G ⊂ Ĝ of connected reductive algebraic groups. In the appendix by M. Kapovich, a connection between metric geometry and the representation theory of complex semisimple algebraic groups is explained. The connection runs through the theory of buildings. This connection is exploited to give a uniform (though not optimal) saturation factor for any G.

Wonderful subgroups of reductive groups and spherical systems

Let G be a semisimple complex algebraic group, and H⊆G a wonderful subgroup. We prove several results relating the subgroup H to the properties of a combinatorial invariant S of G/H, called its spherical system. It is also possible to consider a spherical system S as a datum defined by purely combinatorial axioms, and under certain circumstances our results prove the existence of a wonderful subgroup H associated with S. As a byproduct, we reduce for any group G the proof of the classification of wonderful G-varieties, known as the Luna conjecture, to its verification on a small family of cases, called primitive.

The geometric meaning of Zhelobenko operators

Let \( \mathfrak{g} \) be the complex semisimple Lie algebra associated to a complex semisimple algebraic group G, \( \mathfrak{b} \) a Borel subalgebra of \( \mathfrak{g} \), \( \mathfrak{h}\subset \mathfrak{b} \) the Cartan sublagebra, and N ⊂ G the unipotent subgroup corresponding to the nilradical \( \mathfrak{n}\subset \mathfrak{b} \). We show that the explicit formula for the extremal projection operator for \( \mathfrak{g} \) obtained by Asherova, Smirnov, and Tolstoy and similar formulas for Zhelobenko operators are related to the existence of a birational equivalence \( N\times \mathfrak{h}\to \mathfrak{b} \) given by the restriction of the adjoint action. Simple geometric proofs of formulas for the “classical” counterparts of the extremal projection operator and of Zhelobenko operators are also obtained.

Normality and smoothness of simple linear group compactifications

If G is a complex semisimple algebraic group, we characterize the normality and the smoothness of its simple linear compactifications, namely those equivariant GxG-compactifications which possess a unique closed orbit and which arise in a projective space of the shape P(End(V)), where V is finite dimensional rational G-module. Both the characterizations are purely combinatorial and are expressed in terms of the highest weights of V. In particular, we show that Sp(2r) (with r > 0) is the unique non-adjoint simple group which admits a simple smooth compactification.

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.

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.

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.

Some topological properties of quotients modulo semisimple algebraic groups

We will prove a general result in Invariant Theory, viz. for a quotient ℂ^n/\!/G, where G is a connected complex semisimple algebraic group, the local first homology group at any point in the quotient ℂ^n/\!/G is trivial and the local second homology group is finite. Using this we will prove that the completion of the local ring of any point in ℂ^n/\!/G is a unique factorization domain (UFD).

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.

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.

A Deodhar-type stratification on the double flag variety

We describe a stratification on the double flag variety $G/B^+\times G/B^-$ of a complex semisimple algebraic group $G$ analogous to the Deodhar stratification on the flag variety $G/B^+$, which is a refinement of the stratification into orbits both for $B^+\times B^-$ and for the diagonal action of $G$, just as Deodhar's stratification refines the orbits of $B^+$ and $B^-$. We give a coordinate system on each stratum, and show that all strata are coisotropic subvarieties. Also, we discuss possible connections to the positive and cluster geometry of $G/B^+\times G/B^-$, which would generalize results of Fomin and Zelevinsky on double Bruhat cells and Marsh and Rietsch on double Schubert cells.

Eigenvalue problem and a new product in cohomology of flag varieties

Let G be a connected semisimple complex algebraic group and let P be a parabolic subgroup. In this paper we define a new (commutative and associative) product on the cohomology of the homogenous spaces G/P and use this to give a more efficient solution of the eigenvalue problem and also for the problem of determining the existence of G-invariants in the tensor product of irreducible representations of G. On the other hand, we show that this new product is intimately connected with the Lie algebra cohomology of the nil-radical of P via some works of Kostant and Kumar. We also initiate a uniform study of the geometric Horn problem for an arbitrary group $G$ by obtaining two (a priori) different sets of necessary recursive conditions to determine when a cohomology product of Schubert classes in G/P is non-zero. Hitherto, this was studied largely only for the group \SL(n).

LS galleries, the path model, and MV cycles

We give an interpretation of the path model of a representation [18] of a complex semisimple algebraic group G in terms of the geometry of its affine Grassmannian. In this setting, the paths are replaced by LS–galleries in the affine Coxeter complex associated to the Weyl group of G. To explain the connection with geometry, consider a Demazure–Hansen–Bott–Samelson desingularization ˆ Σ(λ) of the closure of an orbit G(C[[t]]).λ in the affine Grassmannian. The homology of ˆ Σ(λ) has a basis given by Bia lynicki–Birula cell’s, which are indexed by the T –fixed points in ˆ Σ(λ). Now the points of ˆ Σ(λ) can be identified with galleries of a fixed type in the affine Tits building associated to G, and the T –fixed points correspond in this language to combinatorial galleries of a fixed type in the affine Coxeter complex. We determine those galleries such that the associated cell has a non-empty intersection with G(C[[t]]).λ (identified with an open subset of ˆ Σ(λ)), and we show that the closures of the strata associated to LS-galleries are exactly the MV–cycles [24], which form a basis of the representation V (λ) for the Langland’s dual group G ∨ .