Maximal Parabolic Subgroups in Classical Groups are of Modality Zero

The principal aim of this paper is to show that every maximal parabolic subgroup P of a classical reductive algebraic group G operates with a finite number of orbits on its unipotent radical. This is a consequence of the fact that each parabolic subgroup of a group of type A _n whose unipotent radical is of nilpotent class at most two has this finiteness property.     

Tensor product stabilization in Kac-Moody algebras

We consider a large class of series of symmetrizable Kac-Moody algebras (generically denoted X_n). This includes the classical series A_n as well as others like E_n whose members are of Indefinite type. The focus is to analyze the behavior of representations in the limit n→∞. Motivated by the classical theory of A_n=sl_n+1C, we consider tensor product decompositions of irreducible highest weight representations of X_n and study how these vary with n. The notion of “double-headed” dominant weights is introduced. For such weights, we show that tensor product decompositions in X_n do stabilize, generalizing the classical results for A_n. The main tool used is Littelmann's celebrated path model. One can also use the stable multiplicities as structure constants to define a multiplication operation on a suitable space. We define this so-called stable representation ring and show that the multiplication operation is associative.     

Homogeneous Vector Bundles and Koszul Algebras

Let G be a reductive algebraic group defined over an algebraically closed field of characteristic zero and let P be a parabolic subgroup of G. We consider the category of homogeneous vector bundles over the flag variety G/P. This category is equivalent to a category of representations of a certain infinite quiver with relations by a generalisation of a result in [BK]. We prove that both categories are Koszul precisely when the unipotent radical P_u of P is abelian.     

Maximal Orders of Abelian Subgroups in Finite Chevalley Groups

In the present paper, for any finite group G of Lie type (except for ² F ₄(q)), the order a(G) of its large Abelian subgroup is either found or estimated from above and from below (the latter is done for the groups F ₄ (q), E ₆ (q), E ₇ (q), E ₈ (q), and ² E ₆(q ²)). In the groups for which the number a(G) has been found exactly, any large Abelian subgroup coincides with a large unipotent or a large semisimple Abelian subgroup. For the groups F ₄ (q), E ₆ (q), E ₇ (q), E ₈ (q), and ² E ₆(q ²)), it is shown that if an Abelian subgroup contains a noncentral semisimple element, then its order is less than the order of an Abelian unipotent group. Hence in these groups the large Abelian subgroups are unipotent, and in order to find the value of a(G) for them, it is necessary to find the orders of the large unipotent Abelian subgroups. Thus it is proved that in a finite group of Lie type (except for ² F ₄(q))) any large Abelian subgroup is either a large unipotent or a large semisimple Abelian subgroup.     

Poles of Intertwining Operators via Endoscopy; the Connection with Prehomogeneous Vector Spaces With an Appendix, `Basic Endoscopic Data', by Diana Shelstad To the Memory of Magdy Assem

In this paper, we determine the residues at poles of standard intertwining operators for parabolically induced representations of an arbitrary connected reductive quansisplit algebraic group over a p-acid field whenever the unipotent radical of the parabolic subgroup is Abelian. We then interpret these residues by means of the theory of endoscopy.     

On Noetherian Modules over Minimax Abelian Groups

We consider modules over minimax Abelian groups. We prove that if A is an Abelian minimax subgroup of the multiplicative group of a field k and if the subring K of the field k generated by the subgroup A is Noetherian, then the subgroup A is the direct product of a periodic group and a finitely generated group.     

Index Reduction Formulas for Twisted Flag Varieties, II

This is a continuation of the determination begun in K-Theory 10 (1996), 517–596, of explicit index reduction formulas for function fields of twisted flag varieties of adjoint semisimple algebraic groups. We give index reduction formulas for the varieties associated to the classical simple groups of outer type A _n-1 and D _n, and the exceptional simple groups of type E ₆ and E ₇. We also give formulas for the varieties associated to transfers and direct products of algebraic groups. This allows one to compute recursively the index reduction formulas for the twisted flag varieties of any semi-simple algebraic group.     

Bruhat Order for Two Flags and a Line

The classical Ehresmann-Bruhat order describes the possible degenerations of a pair of flags in a linear space V under linear transformations of V; or equivalently, it describes the closure of an orbit of GL(V acting diagonally on the product of two flag varieties.We consider the degenerations of a triple consisting of two flags and a line, or equivalently the closure of an orbit of GL(V) acting diagonally on the product of two flag varieties and a projective space. We give a simple rank criterion to decide whether one triple can degenerate to another. We also classify the minimal degenerations, which involve not only reflections (i.e., transpositions) in the Weyl group S_VSn = dim(V, but also cycles of arbitrary length. Our proofs use only elementary linear algebra and combinatorics.     

An integrality theorem of Grosshans over arbitrary base ring

We revisit a theorem of Grosshans and show that it holds over arbitrary commutative base ring k. One considers a split reductive group scheme G acting on a k-algebra A and leaving invariant a subalgebra R. Let U be the unipotent radical of a split Borel subgroup scheme. If R ^U = A ^U then the conclusion is that A is integral over R.     

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.     

On the equivalence of Lie symmetries and group representations

We consider families of linear, parabolic PDEs in n dimensions which possess Lie symmetry groups of dimension at least four. We identify the Lie symmetry groups of these equations with the (2n+1)-dimensional Heisenberg group and SL(2,R). We then show that for PDEs of this type, the Lie symmetries may be regarded as global projective representations of the symmetry group. We construct explicit intertwining operators between the symmetries and certain classical projective representations of the symmetry groups. Banach algebras of symmetries are introduced.     

Regular unipotent elements from subsystem subgroups of rank 2 in modular representations of classical groups

Images of regular unipotent elements from subsystem subgroups of type A₂ and B₂ in irreducible modular representations of classical groups are studied. For images of such elements and representations with locally small highest weights, all sizes of Jordan blocks of one and the same parity are found. Bibliography: 17 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 356, 2008, pp. 159–178.     

Subgroups of Lie type groups containing a unipotent radical

In this paper we show, that if G is a Lie type group and R a proper subgroup of G containing some unipotent radical, then R has a nilpotent normal subgroup generated by long root subgroups and R is contained in a proper parabolic subgroup of G. We also obtain some consequences of this result.     

Crystals and total positivity on orientable surfaces

We develop a combinatorial model of networks on orientable surfaces, and study weight and homology generating functions of paths and cycles in these networks. Network transformations preserving these generating functions are investigated. We describe in terms of our model the crystal structure and R-matrix of the affine geometric crystal of products of symmetric and dual symmetric powers of type A. Local realizations of the R-matrix and crystal actions are used to construct a double affine geometric crystal on a torus, generalizing the commutation result of Kajiwara et al. (Lett Math Phys, 60(3):211–219, 2002) and an observation of Berenstein and Kazhdan (MSJ Mem, 17:1–9, 2007). We show that our model on a cylinder gives a decomposition and parametrization of the totally non-negative part of the rational unipotent loop group of GL _n .     

Yet another variation on the theme of decomposition of transvections

The unipotent decomposition method consists in representing elementary matrices as products of factors belonging to proper parabolic subgroups whose images under endomorphisms (e.g., conjugations) remain in proper parabolic subgroup. For the complete linear group, this method was suggested in 1987 by Stepanov, who applied it to simplify the proof of Souslin’s normality theorem. Soon after this, Vavilov and Plotkin transferred the method to other classical groups and the Chevalley groups. Since then, many results in the same spirit have been obtained. The paper suggests yet another variation on this theme. Namely, let R be a commutative ring with identity, and let g ∈ GL(n, R), where n ≥ 4. Then, the elementary group E(n, R) is generated by transvections e + uv, where u ∈ R ⁿ , v ∈ ⁿ R, and vu = 0, such that v, gu, and vg ^?1 have at least one zero component each. This result is related to a simplified proof of theorems of Waterhouse, Golubchik, Mikhalev, Zel’manov, and Petechuk about the automorphisms of the complete linear group being standard, which uses unipotent elements.     

Modular intersection cohomology complexes on flag varieties

We present a combinatorial procedure (based on the W-graph of the Coxeter group) which shows that the characters of many intersection cohomology complexes on low rank complex flag varieties with coefficients in an arbitrary field are given by Kazhdan–Lusztig basis elements. Our procedure exploits the existence and uniqueness of parity sheaves. In particular we are able to show that the characters of all intersection cohomology complexes with coefficients in a field on the flag variety of type A _n for n < 7 are given by Kazhdan–Lusztig basis elements. By results of Soergel, this implies a part of Lusztig’s conjecture for SL(n) with n ≤ 7. We also give examples where our techniques fail. In the appendix by Tom Braden examples are given of intersection cohomology complexes on the flag varities for SL(8) and SO(8) which have torsion in their stalks or costalks.     

Weights of multipartitions and representations of Ariki-Koike algebras

We consider representations of the Ariki-Koike algebra, a q-deformation of the group algebra of the complex reflection group C_r?S_n. The representations of this algebra are naturally indexed by multipartitions of n, and for each multipartition λ we define a non-negative integer called the weight of λ. We prove some basic properties of this weight function, and examine blocks of small weight.     

Small quadratic elements in representations of the special linear group with large highest weights

For almost all p-restricted irreducible representations of the group A_n(K) in characteristic p > 0 with highest weights large enough with respect to p, the Jordan block structure of the images of small quadratic unipotent elements in these representations is determined. It is proved that if φ is an irreducible p-restricted representation of A_n(K) with highest weight

Torsion Abelian afi-Groups

This paper is devoted to the study of Abelian afi-groups. A subgroup A of an Abelian group G is called its absolute ideal if A is an ideal of any ring on G. We will call an Abelian group an afi-group if all of its absolute ideals are fully invariant subgroups. In this paper, we will describe afi-groups in the class of fully transitive torsion groups (in particular, separable torsion groups) and divisible torsion groups.