Torus quotients of homogeneous spaces — minimal dimensional Schubert varieties admitting semi-stable points

In this paper, for any simple, simply connected algebraic group G of type B,C or D and for any maximal parabolic subgroup P of G, we describe all minimal dimensional Schubert varieties in G/P admitting semistable points for the action of a maximal torus T with respect to an ample line bundle on G/P. We also describe, for any semi-simple simply connected algebraic group G and for any Borel subgroup B of G, all Coxeter elements τ for which the Schubert variety X(τ) admits a semistable point for the action of the torus T with respect to a non-trivial line bundle on G/B.     

On the Chow Ring of a Flag

Let G be a reductive linear algebraic group over an algebraically closed field K, let P? be a parabolic subgroup scheme of G containing a Borel subgroup B, and let P = P?_red ? P? be its reduced part. Then P is reduced, a variety, one of the well known classical parabolic subgroups. For char(K) = p > 3, a classification of the P?'s has been given in [W1]. The Chow ring of G/P only depends on the root system of G. Corresponding to the natural projection from G/P to G/P? there is a map of Chow rings from A(G/P?) to A(G/P). This map will be explicitly described here. Let P = B, and let p > 3. A formula for the multiplication of elements in A(G/P?) will be derived. We will prove that A(G/P?) ? A(G/P) (abstractly as rings) if and only if G/P ? G/P? as varieties, i. e., the Chow ring is sensitive to the thickening. Furthermore, in certain cases A(G/P?) is not any more generated by the elements corresponding to codimension one Schubert cells.     

On primary ideals in group algebras

LetG be a -compact locally compact group such thatG/Z(G), whereZ(G) denotes the center ofG, has a relatively compact commutator subgroup. It is shown that primary ideals inL ¹(G) are maximal.     

Maximality of compression semigroups

LetG be a connected semisimple Lie group andr an involution onG. Further letL be an open subgroup of the groupG ^r ofr-fixed points andP⊂-G a parabolic subgroup. The semigroupS(L,P)∶={g∈G∶gLP⊂-LP} is called the compression semigroup of theL-orbit of the base point in the flag manifoldG/P. We show that compression semigroups for open orbits and regular symmetric pairs are maximal semigroups. Supported by a DFG Heisenberg-grant.     

Semigroup actions on homogeneous spaces

LetG be a connected semi-simple Lie group with finite center andS⊄G a subsemigroup with interior points. LetG/L be a homogeneous space. There is a natural action ofS onG/L. The relationx≤y ify ∈Sx, x, y ∈G/L, is transitive but not reflexive nor symmetric. Roughly, a control set is a subsetD ⊄G/L, inside of which reflexivity and symmetry for ≤ hold. Control sets are studied inG/L whenL is the minimal parabolic subgroup. They are characterized by means of the Weyl chambers inG meeting intS. Thus, for eachw ∈W, the Weyl group ofG, there is a control setD _w .D ₁ is the only invariant control set, and the subsetW(S)={w:D _w =D ₁} turns out to be a subgroup. The control sets are determined byW(S)/W. The following consequences are derived: i)S=G ifS is transitive onG/H, i.e.Sx=G/H for allx ∈G/H. HereH is a non discrete closed subgroup different fromG andG is simple. ii)S is neither left nor right reversible ifS #G iii)S is maximal only if it is the semigroup of compressions of a subset of some minimal flag manifold. Research partially supported by CNPq grant n^o 50.13.73/91-8     

A note on the tangent bundle of <Emphasis Type="Italic">G/P</Emphasis>

Let P be a parabolic subgroup of a complex simple linear algebraic groupG. We prove that the tangent bundle T (G/P) is stable.     

Limit distributions of expanding translates of certain orbits on homogeneous spaces

LetL be a Lie group and λ a lattice inL. SupposeG is a non-compact simple Lie group realized as a Lie subgroup ofL and . LetaεG be such that Ada is semisimple and not contained in a compact subgroup of Aut(Lie(G)). Consider the expanding horospherical subgroup ofG associated toa defined as U⁺ ={g&#x03B5;G:a ⁻ⁿ gaⁿ} →e as n → ∞. Let Ω be a non-empty open subset ofU ⁺ andn _i → ∞ be any sequence. It is showed that . A stronger measure theoretic formulation of this result is also obtained. Among other applications of the above result, we describeG-equivariant topological factors of L/gl × G/P, where the real rank ofG is greater than 1,P is a parabolic subgroup ofG andG acts diagonally. We also describe equivariant topological factors of unipotent flows on finite volume homogeneous spaces of Lie groups.     

Weighted bounds for the carleson maximal operator inR n

The Carleson maximal operator is shown to be bounded inL ^p (w) for certain values ofp and certain radial weightsw when acting on products of radial functions and homogeneous harmonic polynomials. Partially supported by DGICYT (MEC Spain). PB 92/187.     

Unbeschränkte Randwerte bei parabolischen Differentialgleichungen

In this paper we will prove the existence of classical solutions u for a quasilinear parabolic differential equation of type $$(1)u_t = \sum\limits_{i = 1}^n {\tfrac{d}{{dx_i }}a_i (x,t,u,u_x )} + a(x,t,u,u_x )$$ in a cylindrical domain G, whereby unbounded boundary values g may be given on the parabolic boundary R_p of G. In general, u is unbounded and u_x?L₂(G). Under certain conditions (see Satz 3) we nevertheless get a reasonable boundary-behavior of u, that is: \(u(Q) \to g(\bar Q)\) when g is continuous in \(\bar Q \in {\text{R(}}Q \to \bar Q{\text{)}}\) , and u(Γ)→g in the L₂-sense for Γ→R_p where Γ means a parallel surface to R_p.     

Explicit Hilbert spaces for certain unipotent representations

Summary LetG be the universal cover of the group of automorphisms of a symmetric tube domain and letP=LN be its Shilov boundary parabolic subgroup. This paper attaches an irreducible unitary representation ofG to each of the (finitely many)L-orbits onn ^*.The Hilbert space of the representation consists of functions on the orbit which are square-integrable with respect to a certainL-equivariant measure. The representation remains irreducible when restricted toP, and descends to a quotient ofG which is, at worst, thedouble cover of a linear group.If theL-orbit isnot open (inn ^*), the construction gives a unipotent representation ofG.Oblatum 28-II-1992This work was supported by an NSF grant at Princeton University, and was carried out in part during a visit to the Mehta Research Institute, Allahabad, India.     

Generic singularities of certain Schubert varieties

Let G be a connected semisimple algebraic group, B a Borel subgroup, T a maximal torus in B with Weyl group W, and Q a subgroup containing B. For , let denote the Schubert variety . For such that , one knows that ByQ / Q admits a T-stable transversal in , which we denote by . We prove that, under certain hypotheses, is isomorphic to the orbit closure of a highest weight vector in a certain Weyl module. We also obtain a generalisation of this result under slightly weaker hypotheses. Further, we prove that our hypotheses are satisfied when Q is a maximal parabolic subgroup corresponding to a minuscule or cominuscule fundamental weight, and is an irreducible component of the boundary of (that is, the complement of the open orbit of the stabiliser in G of ). As a consequence, we describe the singularity of along ByQ / Q and obtain that the boundary of equals its singular locus. Received October 9, 1997; in final form February 19, 1998     

A new formula for the volume of lattice polyhedra

LetL _n be the lattice consisting of all pointsx inR ^N such thatnx belongs to the fundamental latticeL ₁ of points with integer coordinates. When the vertices of a polyhedronP inR ^N are restricted to lie inL ₁ there is a formula which relates the volume ofP to the numbers of points ofL ₁,...,L _N in the interior and on the boundary ofP. The aim of this note is to show that the volume ofP can be determined only by means of the numbers of points ofL ₁,...,L _N lying in the interior ofP and cannot be expressed by the numbers of points ofL ₁,...,L _N lying on the boundary ofP. The latter numbers in turn can be used to compute to comopute the Euler characteristic of the boundary ofP.     

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.     

On J. H. C. Whitehead's aspherical question I

A connected, finite two-dimensional CW-complex with fundamental group isomorphic toG is called a [G, 2] _f -complex. LetL⊲G be a normal subgroup ofG. L has weightk if and only ifk is the smallest integer such that there exists {l ₁,…,l _k}⊆L such thatL is the normal closure inG of {l ₁,…,l _k}. We prove that a [G, 2] _f -complexX may be embedded as a subcomplex of an aspherical complexY=X∪{e ₁ ² ,…,e _k ² } if and only ifG has a normal subgroupL of weightk such thatH=G/L is at most two-dimensional and defG=defH+k. Also, ifX is anon-aspherical [G, 2] _f -subcomplex of an aspherical 2-complex, then there exists a non-trivial superperfect normal subgroupP such thatG/P has cohomological dimension ≤2. In this case, any torsion inG must be inP.     

Parabolic subgroups of positive modality

In this paper we prove a criterion for a parabolic subgroup P of a reductive algebraic group G to be of positive modality, i.e. to act with an infinite number of orbits on the unipotent radical. This condition is simply a lower bound on the length of the descending central series of the radical. We also show that in this situation we can always find a proper P-invariant linear subspace in the Lie algebra of the radical of P which admits an infinite number of P-orbits, unless P is maximal.This research was supported by ARC Grant # A69030627 (chief investigator: Prof. G. Lehrer).     

Compactness and subsets of ordered sets that meet all maximal chains

LetP be a chain complete ordered set. By considering subsets which meet all maximal chains, we describe conditions which imply that the space of maximal chains ofP is compact. The symbolsP ₁ andP ₂ refer to two particular ordered sets considered below. It is shown that the space of maximal chains (P) is compact ifP satisfies any of the following conditions: (i)P contains no copy ofP ₁ or its dual and all antichains inP are finite. (ii)P contains no properN and every element ofP belongs to a finite maximal antichain ofP. (iii)P contains no copy ofP ₁ orP ₂ and for everyx inP there is a finite subset ofP which is coinitial abovex. We also describe an example of an ordered set which is complete and densely ordered and in which no antichain meets every maximal chain.     

Isometric multipliers ofL p (G, X)

Let G be a locally compact group with a fixed right Haar measure andX a separable Banach space. LetL ^p (G, X) be the space of X-valued measurable functions whose norm-functions are in the usualL ^p . A left multiplier ofL ^p (G, X) is a bounded linear operator onB ^p (G, X) which commutes with all left translations. We use the characterization of isometries ofL ^p (G, X) onto itself to characterize the isometric, invertible, left multipliers ofL ^p (G, X) for 1 ≤p ∞,p ≠ 2, under the assumption thatX is not thel ^p -direct sum of two non-zero subspaces. In fact we prove that ifT is an isometric left multiplier ofL ^p (G, X) onto itself then there existsa y ∃ G and an isometryU ofX onto itself such thatTf(x) = U (R _y f)(x). As an application, we determine the isometric left multipliers of L¹ ∩L ^p (G, X) and L¹ ∩C ₀ (G, X) whereG is non-compact andX is not the l^p-direct sum of two non-zero subspaces. If G is a locally compact abelian group andH is a separable Hubert space, we define where г is the dual group of G. We characterize the isometric, invertible, left multipliers ofA ^p (G, H), provided G is non-compact. Finally, we use the characterization of isometries ofC(G, X) for G compact to determine the isometric left multipliers ofC(G, X) providedX ^* is strictly convex.     

Invariant ordering on the simply connected covering of the Shilov boundary of a symmetric domain

The Shilov boundary of a symmetric domain D = G/K of tube type has the form G/P, where P is a maximal parabolic subgroup of the group G. We prove that the simply connected covering of the Shilov boundary possesses a unique (up to inversion) invariant ordering, which is induced by the continuous invariant ordering on the simply connected covering of G and can readily be described in terms of the corresponding Jordan algebra.     

Permutability of minimal subgroups and<Emphasis Type="Italic">p</Emphasis>-nilpotency of finite groups

In this paper it is proved that ifp is a prime dividing the order of a groupG with (|G|,p − 1) = 1 andP a Sylowp-subgroup ofG, thenG isp-nilpotent if every subgroup ofP ∩G ^N of orderp is permutable inN _G (P) and whenp = 2 either every cyclic subgroup ofP ∩G ^N of order 4 is permutable inN _G (P) orP is quaternion-free. Some applications of this result are given. The research of the first author is supported by a grant of Shanxi University and a research grant of Shanxi Province, PR China. The research of the second author is partially supported by a UGC(HK) grant #2160126 (1999/2000).