Semisimple Algebraic Group Research Articles

On the Brauer group of the product of a torus and a semisimple algebraic group

Let T be a torus (not assumed to be split) over a field F, and denote by nH et 2 (X,{ie375-1}) the subgroup of elements of the exponent dividing n in the cohomological Brauer group of a scheme X over the field F. We provide conditions on X and n for which the pull-back homomorphism nH et 2 (T,{ie375-2}) → n H et 2 (X × F T, {ie375-3}) is an isomorphism. We apply this to compute the Brauer group of some reductive groups and of non-singular affine quadrics. Apart from this, we investigate the p-torsion of the Azumaya algebra defined Brauer group of a regular affine scheme over a field F of characteristic p > 0.

Metric Diophantine approximation on homogeneous varieties

AbstractWe develop the metric theory of Diophantine approximation on homogeneous varieties of semisimple algebraic groups and prove results analogous to the classical Khintchine and Jarník theorems. In full generality our results establish simultaneous Diophantine approximation with respect to several completions, and Diophantine approximation over general number fields using $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$-algebraic integers. In several important examples, the metric results we obtain are optimal. The proof uses quantitative equidistribution properties of suitable averaging operators, which are derived from spectral bounds in automorphic representations.

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.

Linear preservers and representations with a 1-dimensional ring of invariants

We determine the group of linear transformations on a vector space $V$ that preserve a polynomial function $f$ on $V$ for several interesting pairs $(V,f)$, using the theory of semisimple algebraic groups.

Gelfand pairs and strong transitivity for Euclidean buildings

AbstractLet $G$ be a locally compact group acting properly, by type-preserving automorphisms on a locally finite thick Euclidean building $\Delta $, and $K$ be the stabilizer of a special vertex in $\Delta $. It is known that $(G, K)$ is a Gelfand pair as soon as $G$ acts strongly transitively on $\Delta $; in particular, this is the case when $G$ is a semi-simple algebraic group over a local field. We show a converse to this statement, namely that if $(G, K)$ is a Gelfand pair and $G$ acts cocompactly on $\Delta $, then the action is strongly transitive. The proof uses the existence of strongly regular hyperbolic elements in $G$ and their peculiar dynamics on the spherical building at infinity. Other equivalent formulations are also obtained, including the fact that $G$ is strongly transitive on $\Delta $ if and only if it is strongly transitive on the spherical building at infinity.

Construction of minimal non-abelian left group codes

Algorithms to construct minimal left group codes are provided. These are based on results describing a complete set of orthogonal primitive idempotents in each Wedderburn component of a semisimple finite group algebra \({\mathbb F}G\) for a large class of groups \(G\). As an illustration of our methods, alternative constructions to some best linear codes over \({\mathbb F}_2\) and \({\mathbb F}_3\) are given. Furthermore, we give constructions of non-abelian left group codes.

Representatives of elliptic Weyl group elements in algebraic groups

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Automorphism groups of positive entropy on projective threefolds

We prove two results about the natural representation of a group $G$ of automorphisms of a normal projective threefold $X$ on its second cohomology. We show that if $X$ is minimal, then $G$, modulo a normal subgroup of null entropy, is embedded as a Zariski-dense subset in a semi-simple real linear algebraic group of real rank $\le 2$. Next, we show that $X$ is a complex torus if the image of $G$ is an almost abelian group of positive rank and the kernel is infinite, unless $X$ is equivariantly non-trivially fibred.

Representations of semisimple Lie algebras in prime characteristic and the noncommutative Springer resolution

We prove most of Lusztig’s conjectures on the canonical basis in homology of a Springer ber. The conjectures predict that this basis controls numerics of representations of the Lie algebra of a semisimple algebraic group over an algebraically closed eld of positive characteristic. We check this for almost all characteristics. To this end we construct a noncommutative resolution of the nilpotent cone which is derived equivalent to the Springer resolution. On the one hand, this noncommutative resolution is closely related to the positive characteristic derived localization equivalences obtained earlier by the present authors and Rumynin. On the other hand, it is compatible with the t-structure arising from an equivalence with the derived category of perverse sheaves on the ane ag variety of the Langlands dual group. This equivalence established by Arkhipov and the rst author ts the framework of local geometric Langlands duality. The latter compatibility allows one to apply Frobenius purity theorem to deduce the desired properties of the basis. We expect the noncommutative counterpart of the Springer resolution to be of independent interest from the perspectives of algebraic geometry and geometric Langlands duality.

On Orbits in Double Flag Varieties for Symmetric Pairs

Let G be a connected, simply connected semisimple algebraic group over the complex number field, and let K be the fixed point subgroup of an involutive automorphism of G so that (G, K) is a symmetric pair. We take parabolic subgroups P of G and Q of K, respectively, and consider the product of partial flag varieties G/P and K/Q with diagonal K-action, which we call a double flag variety for a symmetric pair. It is said to be of finite type if there are only finitely many K-orbits on it. In this paper, we give a parametrization of K-orbits on G/P × K/Q in terms of quotient spaces of unipotent groups without assuming the finiteness of orbits. If one of P ⊂ G or Q ⊂ K is a Borel subgroup, the finiteness of orbits is closely related to spherical actions. In such cases, we give a complete classification of double flag varieties of finite type, namely, we obtain classifications of K-spherical flag varieties G/P and G-spherical homogeneous spaces G/Q.

Degenerate coordinate rings of flag varieties and Frobenius splitting

Recently E. Feigin introduced the $$\mathbb G _a^N$$ -degenerations of semisimple algebraic groups and their associated degenerate flag varieties. It has been shown by Feigin, Finkelberg, and Littelmann that the degenerate flag varieties in types $$A_n$$ and $$C_n$$ are Frobenius split. In this paper, we construct an associated degeneration of homogeneous coordinate rings of classical flag varieties in all types and show that these rings are Frobenius split in most types. It follows that the degenerate flag varieties of types $$A_n, C_n$$ , and $$G_2$$ are Frobenius split. In particular, we obtain an alternate proof of splitting in types $$A_n$$ and $$C_n$$ ; the case $$G_2$$ was not previously known. We also give a representation-theoretic condition on PBW-graded versions of Weyl modules which is equivalent to the existence of a Frobenius splitting of the classical flag variety that maximally compatibly splits the identity.

Shifted generic cohomology

AbstractThe idea that the cohomology of finite groups might be fruitfully approached via the cohomology of ambient semisimple algebraic groups was first shown to be viable in the papers [E. Cline, B. Parshall, and L. Scott, Cohomology of finite groups of Lie type, I, Publ. Math. Inst. Hautes Études Sci. 45 (1975), 169–191] and [E. Cline, B. Parshall, L. Scott and W. van der Kallen, Rational and generic cohomology, Invent. Math. 39 (1977), 143–163]. The second paper introduced, through a limiting process, the notion of generic cohomology, as an intermediary between finite Chevalley group and algebraic group cohomology. The present paper shows that, for irreducible modules as coefficients, the limits can be eliminated in all but finitely many cases. These exceptional cases depend only on the root system and cohomological degree. In fact, we show that, for sufficiently large $r$, depending only on the root system and $m$, and not on the prime $p$ or the irreducible module $L$, there are isomorphisms ${\mathrm{H} }^{m} (G({p}^{r} ), L)\cong {\mathrm{H} }^{m} (G({p}^{r} ), {L}^{\prime } )\cong { \mathrm{H} }_{\mathrm{gen} }^{m} (G, {L}^{\prime } )\cong {\mathrm{H} }^{m} (G, {L}^{\prime } )$, where the subscript ‘gen’ refers to generic cohomology and ${L}^{\prime } $ is a constructibly determined irreducible ‘shift’ of the (arbitrary) irreducible module $L$ for the finite Chevalley group $G({p}^{r} )$. By a famous theorem of Steinberg, both $L$ and ${L}^{\prime } $ extend to irreducible modules for the ambient algebraic group $G$ with ${p}^{r} $-restricted highest weights. This leads to the notion of a module or weight being ‘shifted $m$-generic’, and thus to the title of this paper. Our approach is based on questions raised by the third author in [D. I. Stewart, The second cohomology of simple ${\mathrm{SL} }_{3} $-modules, Comm. Algebra 40 (2012), 4702–4716], which we answer here in the cohomology cases. We obtain many additional results, often with formulations in the more general context of ${ \mathrm{Ext} }_{G({p}^{r} )}^{m} $ with irreducible coefficients.

Cluster ensembles and Kac–Moody groups

We study the relationship between two sets of coordinates on a double Bruhat cell, the cluster variables introduced by Berenstein, Fomin, and Zelevinsky and the X-coordinates defined by the coweight parametrization of Fock and Goncharov. In these coordinates, we show that the generalized Chamber Ansatz of Fomin and Zelevinsky coincides with the cluster ensemble map, a canonical monomial transformation between the cluster variables and X-coordinates defined by a common exchange matrix. We prove this in the setting of an arbitrary symmetrizable Kac–Moody group, generalizing along the way many previous results on the double Bruhat cells of a semisimple algebraic group. In particular, we construct an upper cluster algebra structure on the coordinate ring of any double Bruhat cell in a symmetrizable Kac–Moody group, proving a conjecture of Berenstein, Fomin, and Zelevinsky.

Functional equations of Weng's zeta functions for ( G,P )/Q

It is shown that Weng's zeta functions associated with arbitrary semisimple algebraic groups defined over the rational number field and their maximal parabolic subgroups satisfy the functional equations.

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.

Normalizers of solvable spherical subgroups

For an arbitrary connected solvable spherical subgroup H of a connected semisimple algebraic group G, we compute the group N G (H), the normalizer of H in G. Therebywe complete a classification of all (not necessarily connected) solvable spherical subgroups in semisimple algebraic groups.

A local-to-global result for topological spherical buildings

Abstract Suppose that Δ;Δʹ are two buildings each arising from a semisimple algebraic group over a field, a topological field in the former case, and that for both the buildings the Coxeter diagram has no isolated nodes. We give conditions under which a partially defined injective chamber map, whose domain is the subcomplex of Δ generated by a nonempty open set of chambers, and whose codomain is Δʹ, is guaranteed to extend to a unique injective chamber map. Related to this result is a local version of the Borel-Tits theorem on abstract homomorphisms of simple algebraic groups.

Hecke modifications, wonderful compactifications and moduli of principal bundles

In this paper we obtain parametrizations of the moduli space of principal bundles over a compact Riemann surface using spaces of Hecke modifications in several cases. We begin with a discussion of Hecke modifications for principal bundles and give constructions of universal Hecke modifications of a fixed bundle of fixed type. This is followed by an overview of the construction of the wonderful, or De Concini-Procesi, compactification of a semi-simple algebraic group of adjoint type. The compactification plays an important role in the deformation theory used in constructing the parametrizations. A general outline to construct parametrizations is given and verifications for specific structure groups are carried out.

Wide Subalgebras of Semisimple Lie Algebras

Let G be a connected semisimple algebraic group over $k$, with Lie algebra $\g$. Let $\h$ be a subalgebra of $\g$. A simple finite-dimensional $\g$-module V is said to be $\h$-indecomposable if it cannot be written as a direct sum of two proper $\h$-submodules. We say that $\h$ is wide, if all simple finite-dimensional $\g$-modules are $\h$-indecomposable. Some very special examples of indecomposable modules and wide subalgebras appear recently in the literature. In this paper, we describe several large classes of wide subalgebras of $\g$ and initiate their systematic study. Our approach is based on the study of idempotents in the associative algebra of $\h$-invariant endomorphisms of V. We also discuss a relationship between wide subalgebras and epimorphic subgroups.

Commuting involutions and degenerations of isotropy representations

Let σ 1 and σ 2 be commuting involutions of a semisimple algebraic group G. This yields a $$ {{\mathbb{Z}}_2}\times {{\mathbb{Z}}_2} $$ -grading of $$ \mathfrak{g} $$ = Lie(G), $$ \mathfrak{g}={\oplus_{i,j=0,1 }}{{\mathfrak{g}}_{ij }} $$ , and we study invariant-theoretic aspects of this decomposition. Let $$ \mathfrak{g}\left\langle {{\sigma_1}} \right\rangle $$ be the $$ {{\mathbb{Z}}_2} $$ -contraction of $$ \mathfrak{g} $$ determined by σ 1. Then both σ 2 and σ 3 := σ 1 σ 2 remain involutions of the non-reductive Lie algebra $$ \mathfrak{g}\left\langle {{\sigma_1}} \right\rangle $$ . The isotropy representations related to $$ \left( {\mathfrak{g}\left\langle {{\sigma_1}} \right\rangle, {\sigma_2}} \right) $$ and $$ \left( {\mathfrak{g}\left\langle {{\sigma_1}} \right\rangle, {\sigma_3}} \right) $$ are degenerations of the isotropy representations related to $$ \left( {\mathfrak{g},{\sigma_2}} \right) $$ and $$ \left( {\mathfrak{g},{\sigma_3}} \right) $$ , respectively. These degenerated isotropy representations retain many good properties. For instance, they always have a generic stabiliser and their algebras of invariants are often polynomial. We also develop some theory on Cartan subspaces for various $$ {{\mathbb{Z}}_2} $$ -gradings associated with the $$ {{\mathbb{Z}}_2}\times {{\mathbb{Z}}_2} $$ -grading of $$ \mathfrak{g} $$ and study the special case in which σ 1 and σ 2 are conjugate.