On locally trivial G a -actions

If the additive group of complex numbers acts algebraically on a normal affine variety, then the associated ring of invariants need not be finitely generated, but is an ideal transform of some normal affine algebra (Theorem 1). We investigate such normal affine algebras in the case of a locally trivial action on a factorial variety. If the variety is a complex affine space and the ring of invariants is isomorphic to a polynomial ring, then the action is conjugate to a translation (Theorem 3). Equivalently, ifC ⁿ , is the total space for a principalG _a -bundle over some open subset ofC ^n–1 then the bundle is trivial. On the other hand, there is a locally trivialG _a -action on a normal affine variety with nonfinitely generated ring of invariants (Theorem 2).Supported in part by NSA Grant No. MDA904-96-1-0069     

A family of affine Weyl group representations

In this paper we explicitly determine the virtual representations of the finite Weyl subgroups of the affine Weyl group on the cohomology of the space of affine flags containing a family of elementsn _t in an affine Lie algebra. We also compute the Euler characteristic of the space of partial flags containingn _t and give a connection with hyperplane arrangements.This paper forms part of my Ph.D. thesis at M.I.T. I was supported by an NSF Graduate Fellowship and NSF grant DMS 9304580.     

On 321-Avoiding Permutations in Affine Weyl Groups

We introduce the notion of 321-avoiding permutations in the affine Weyl group W of type A _{n – 1} by considering the group as a George group (in the sense of Eriksson and Eriksson). This enables us to generalize a result of Billey, Jockusch and Stanley to show that the 321-avoiding permutations in W coincide with the set of fully commutative elements; in other words, any two reduced expressions for a 321-avoiding element of W (considered as a Coxeter group) may be obtained from each other by repeated applications of short braid relations.Using Shi's characterization of the Kazhdan–Lusztig cells in the group W, we use our main result to show that the fully commutative elements of W form a union of Kazhdan–Lusztig cells. This phenomenon has been studied by the author and J. Losonczy for finite Coxeter groups, and is interesting partly because it allows certain structure constants for the Kazhdan–Lusztig basis of the associated Hecke algebra to be computed combinatorially.We also show how some of our results can be generalized to a larger group of permutations, the extended affine Weyl group associated to GL _n ()     

Upper Bounds in Affine Weyl Groups under the Weak Order

Björner and Wachs proved that under the weak order every quotient of a Coxeter group is a meet semi-lattice, and in the finite case is a lattice. In this paper, we examine the case of an affine Weyl group W with corresponding finite Weyl group W ₀. In particular, we show that the quotient of W by W ₀ is a lattice and that up to isomorphism this is the only quotient of W which is a lattice. We also determine that the question of which pairs of elements of W have upper bounds can be reduced to the analogous question within a particular finite subposet.     

Weylgruppe und Momentabbildung

Summary Let G be a connected, reductive group defined over an algebraically closed field of characteristic zero. We assign to any G-variety X a finite cristallographic reflection group W _X by means of the moment map on the cotangent bundle. This generalizes the little Weyl group of a symmetric space. The Weyl group W _X is related to the equivariant compactification theory of X. We determine the closure of the image of the moment map and the generic isotropy group of the action of G on the cotangent bundle. As a byproduct we determine the ideal of elements of (g) which act trivially on X as a differential operator.

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Teilweise unterstützt durch den Schweizerischen Nationalfonds     

Hyperoskulierende logarithmische Spiralen in affinen CK-Ebenen

At a pointk ₀ aC ⁴-curve k of an affine Cayley-Klein-plane (CK-plane) has a unique hyperosculating logarthmic spiral. We give a construction of the pole p of this spiral, which consists of an affine and a metric part. This metric part is a similar one in the three CK — planes. It is shown that this result is connected with results dealing with the center of the osculating circle given by R.Bereis in [2,p.248].

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Herrn Prof.Dr.K.STRUBECKER zum 85.Geburtstag gewidmet     

ON TWIN-TRIANGULAR ADDITIVE GROUP ACTIONS ON C4

This paper concerns some of the conditions satisfied by additive group actions on complex affine space which can be written locally as a translation of a variable. Assume X is the affine variety C ⁿ , G_a = (C, +), and σ : G_a × X → X is the action defined by a group monomorphism G _a → Aut _C X. If σ is locally trivial, then the action satisfies what is termed a “GICO” condition.

It will be shown that a large class of G_a -actions on C ⁴, that is, fixed-point free, “twin-triangular” actions with finitely-generated rings of invariants, are at least GICO. Remaining questions are discussed.     

Presentations for crystallographic complex reflection groups

We find presentations for the irreducible crystallographic complex reflection groupsW whose linear part is not the complexification of a real reflection group. The presentations are given in the form of graphs resembling Dynkin diagrams and very similar to the presentations for finite complex reflection groups given in [2]. As in the case of affine Weyl groups, they can be obtained by adding a further node to the diagram for the linear part. We then classify the reflections in the groupsW and the minimal number of them needed to generateW, using the diagrams. Finally we show for more than half of the infinite series that a presentation for the fundamental group of the space of regular orbits ofW can be derived from our presentations. The author gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft.     

Nets in Weyl spaces and their metrically equipped subspaces

Let the surfaceW _m⊂W_n be metrically equipped by a normal planeR _n−m. A correspondence between the nets belonging toW _m andW _n is established. Relations between the coefficients of the derivation equations for the fields of directions of the corresponding nets are found. The apparatus for studying the connection between the properties of the corresponding nets is constructed. The influence of the choice that the net belonging toW _m be Chebyshev of the first kind or geodesic or c-net on the corresponding net belonging toW _n is investigated and the results are formulated in theorems 2.6, 2.9, 2.10, 2.11, 2.12 being a generalization of the results received in [2].

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Sunto Supponiamo che la superficieW _m⊂W_n sia dotata di un piano normaleR _n−m. Si stabilisce una corrispondenza tra le reti diW _m e quelle diW _n e si propone un metodo per studiare la relazione tra le proprietà delle reti corrispettive.

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The present investigation is partially supported by the National Science Fund of the Ministry of Science and Education, Republic of Bulgaria under grant MM 528.     

Automorphism groups of some algebras

The automorphism groups of algebras are found in many papers. Using auto-invariance, we find the automorphism groups of the Laurent extension of the polynomial ring and the quantum n-plane (respectively, twisting polynomial ring) in this work. As an application of the results of this work, we can find the automorphism group of a twisting algebra. We define a generalized Weyl algebra and show that the generalized Weyl algebra is simple. We also find the automorphism group of a generalized Weyl algebra. We show that the generalized Weyl algebra A _m,m+n is the universal enveloping algebra of the generalized Witt algebra W(m,m + n). This work was supported by 2007 Research fund of Hanyang University     

On the Chebyshev nets in a hypersurface of a Weyl space

In [1], Zlatanov introduced the Chebyshev vector fields of the first and second kind and the geodesic vector fields for an n-dimensional net in the Weyl spaceW _n . After having defined, in [2], the Chebyshev and geodesic curvatures of the lines of an arbitrary net,the b-nets and the c-nets, Tsareva and Zlatanov studied, among other things, some properties of the Chebyshev nets. In this paper, we consider an n-dimensional net in the hypersurfaceW _n of the Weyl SpaceW _n+1 and study some properties of the Chebyshev vector fields of the first and second kind and the geodesic vector fields of this net. Finally, two theorems concerning the b-nets and c-nets inW _n are obtained.     

On the determination of Kazhdan–Lusztig cells for affine Weyl groups with unequal parameters

Let W be a Coxeter group and L be a weight function on W. Following Lusztig, we have a corresponding decomposition of W into left cells which have important applications in representation theory. We study the case where W is an affine Weyl group of type . Using explicit computation with COXETER and CHEVIE, we show that (1) there are only finitely many possible decompositions into left cells and (2) the number of left cells is finite in each case, thus confirming some of Lusztig's conjectures in this case. A key ingredient of the proof is a general result which shows that the Kazhdan–Lusztig polynomials of affine Weyl group are invariant under (large enough) translations.     

Actions affines sur les arbres réels

Résumé. Nous commencons par une étude générale des actions affines sur les arbres réels. Nous prouvons que, sauf cas dégénérés, l'action par isométries du sous-groupe des commutateurs détermine l'action affine du groupe total. Ensuite, nous exhibons des actions affines libres de groupes qui ne peuvent agir librement par isométries sur les arbres réels.

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We first study affine actions on real trees in general; we show that, except for some degenerate cases, the isometric action of the commutators group determines the affine action of the whole group. We then give examples of free affine actions of groups which cannot act freely and isometrically on real trees.

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Received: 11 November 1998; in final form: 22 February 2000 / Published online: 25 June 2001     

The spherical part of the local and global Springer actions

The affine Weyl group acts on the cohomology (with compact support) of affine Springer fibers (local Springer theory) and of parabolic Hitchin fibers (global Springer theory). In this paper, we show that in both situations, the action of the center of the group algebra of the affine Weyl group (the spherical part) factors through the action of the component group of the relevant centralizers. In the situation of affine Springer fibers, this partially verifies a conjecture of Goresky–Kottwitz–MacPherson and Bezrukavnikov–Varshavsky. We first prove this result for the global Springer action, and then deduce from it the result for the local Springer action. This gives an application of global Springer theory to more classical problems.     

Calcul differentiel exterieur de degre arbitraire

LetX and V be F-modules. For every integer t, it is developed an algebraic formalism which generates, by an analogous procedure of construction of the exterior de rivation in the usual sense, couples formed by a deriva tion of degree t of A_F(X,F), an endomorphism of degree t of the graded additive group A_F(X,V), and satisfying a rule of derivation of degree t. This formalism is the most general ifX is projective of finite type. If t is odd, the curvature, and ifX - V, the torsion, are discussed at some lenght. If t=1, the differen tial calculus associated to Lie modules, linear connexions and fields of endomorphisms, are special cases.

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Ce travail a été financé par le Conselho Nacio nal de Pesquisas (Brésil), contract TC 8233.     

Separation de semi-algebriques

In this paper we consider the problem of separating by a polynomial function two open disjoint semi-algebraic subsets A and B of a real affine variety M under the assumption that the subsets are already polynomially separated up to a proper algebraic subset. First of all some elementary results in small dimensions are given. When M is non-singular, a hypothesis on the behaviour of the boundaries of A and B is sufficient to obtain the separation. The problem is also analysed if M is singular, and some positive results are obtained in the compact case.

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Les auteurs sont associés au groupe G.N.S.A.G.A. du C.N.R. Recherche partiellement financiée par le M.P.I.     

Irreducible Components of Fixed Point Subvarieties of Flag Varieties

Suppose G is a connected reductive algebraic group, P is a parabolic subgroup of G, L is a Levi factor of P, and e is a regular nilpotent element in Lie L. We assume that the characteristic of the underlying field is good for G. Choose a maximal torus, T, and a Borel subgroup, B, of G, so that T?B∩L, B ? P and e ∈ Lie B. Let β be the variety of Borel subgroups of G and let ??_e be the subset of ?? consisting of Borel subgroups whose Lie algebras contain e. Finally, let W be the Weyl group of G with respect to T. For ω ∈ W let ??_ω be the B-orbit in ?? containing ^ωB. We consider the intersections ??_ω ∩ ??_e. The main result is that if dim ??_ω ∩ ??_e = dim ??_e, then ??_ω ∩ ??_e is an affine space. Thus, the irreducible components of ??_e are indexed by Weyl group elements. It is also shown that if G is of type A, then this set of Weyl group elements is a right cell in W.     

Minuscule alcoves for GLn and GSp2n

Let μ be a minuscule coweight for either GL _n or GSp _2n , and regard μ as an element t _μ in the extended affine Weyl group . We say that an element is μ-admissible if there exists μ′ in the Weyl group orbit of μ such that x≤t _μ′ in the Bruhat order on . Our main result is that is μ-admissible if and only if it is μ-permissible, where μ-permissibility is defined using inequalities arising naturally in the study of bad reduction of Shimura varieties. Received: 5 July 1999     

Double centralizing theorem for wreath products of the alternating group

The double centralizing theorem between the action of the symmetric group S_n and the action of the general linear group on the tensor space T_n(W) was obtained by Schur. Here we obtain a double centralizing theorem when S_n is replaced by the wreath product of a finite group G and the alternating group A_n.     

Die metrische Theorie der linearen Komplexbündel von Typ 1 des einfach isotropen RaumesJ 3 (1)

According toK. Strubecker ([18]–[21]) a three dimensional real affine space with the metricds ²=dx ²+dy ² is called a simply isotropic spaceJ ₃ ⁽¹⁾ . InJ ₃ ⁽¹⁾ exist 41 types of bundles of linear line complexes. In this paper we study the metric theory of a bundle of type 1. Especially we investigate the congruence of axis, we give some isotropic and affine results and we study the complementary bundle.

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Herrn em. o. Prof. Dr. W. Wunderlich zum 75. Geburtstag gewidmet

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Im folgenden beziehen wir uns stets auf die Bezeichnungen und Resultate aus [15].