Support Varieties of Non-Restricted Modules over Lie Algebras of Reductive Groups

Let G be a connected semisimple group over an algebraicallyclosed field K of characteristic p>0, and g=Lie (G). Fixa linear function g* and let Z_g() denote the stabilizer of in g. Set N_p(g)={xg|x^[p]=0}. Let C(g) denote the category offinite-dimensional g-modules with p-character . In [7], Friedlanderand Parshall attached to each MOb(C(g)) a Zariski closed, conicalsubset V_g(M)N_p(g) called the support variety of M. Suppose thatG is simply connected and p is not special for G, that is, p2if G has a component of type B_n, C_n or F₄, and p3 if G has acomponent of type G₂. It is proved in this paper that, for anynonzero MOb(C(g)), the support variety V_g(M) is contained inN_p(g)Z_g(). This allows one to simplify the proof of the Kac–Weisfeilerconjecture given in [18].     

Herman Rings and Arnold Disks

For (,a) C* x C, let f_,a be the rational map defined by f_,a(z)= z² (az+1)/(z+a). If R/Z is a Brjuno number, we let D bethe set of parameters (,a) such that f_,a has a fixed Hermanring with rotation number (we consider that (e²ⁱ,0) D). Resultsobtained by McMullen and Sullivan imply that, for any g D, theconnected component of D(C* x (C/{0,1})) that contains g isisomorphic to a punctured disk. We show that there is a holomorphic injection F:DD such thatF(0) = (e²ⁱ ,0) and , where r is the conformal radius at 0 of the Siegel disk of the quadraticpolynomial z e²ⁱ z(1+z). As a consequence, we show that for a (0,1/3), if f_l,a has afixed Herman ring with rotation number and if m_a is the modulusof the Herman ring, then, as a0, we have e m_a=(r/a) + O(a). We finally explain how to adapt the results to the complex standardfamily z e^(a/2)(z-1/z).     

Weighted Bloch Spaces, Hp, and BMOA

It is determined which Bloch-type conditions on a function f _0<p<H^p ensure that f BMOA.     

Extremal Subgroups in Chevalley Groups

Throughout this paper G(k) denotes a Chevalley group of rankn defined over the field k, where n3. Let be the root systemassociated with G(k) and let ={₁, ₂, ..., _n} be a set of fundamentalroots of , with ⁺ being the set of positive roots of with respectto . For and ⁺, let n() be the coefficient of in the expressionof as a sum of fundamental roots; so =n(). Also we recall thatht(), the height of , is given by ht()=n(). The highest rootin ⁺ will be denoted by . We additionally assume that the Dynkindiagram of G(k) is connected.     

Faithful Representations of Free Products

In 1940 Nisnevi published the following theorem [3]. Let (G) be a family of groups indexed by some set and (F) a family of fields of the same characteristic p0. Iffor each the group G has a faithful representation of degreen over F then the free product* G has a faithful representationof degree n+1 over some field of characteristic p. In [6] Wehrfritzextended this idea. If (G) GL(n, F) is a family of subgroupsfor which there exists ZGL(n, F) such that for all the intersectionGF.1_n=Z, then the free product of the groups *_ZG with Z amalgamatedvia the identity map is isomorphic to a linear group of degreen over some purely transcendental extension of F. Initially, the purpose of this paper was to generalize theseresults from the linear to the skew-linear case, that is, togroups isomorphic to subgroups of GL(n, D) where the D are divisionrings. In fact, many of the results can be generalized to ringswhich, although not necessarily commutative, contain no zero-divisors.We have the following.     

The Isolated Points of G{rho} and the w*-Strongly Exposed Points of P{rho}(G)0

Let G be a separable locally compact group and let be its dualspace with Fell's topology. It is well known that the set P(G)of continuous positive-definite functions on G can be identifiedwith the set of positive linear functionals on the group C^*-algebraC^*(G). We show that if is discrete in , then there exists anonzero positive-definite function associated with such that is a w^*-strongly exposed point of P(G)₀, where P(G)₀={f P(G):f(e)1. Conversely, if some nonzero positive-definite function associatedwith is a w^*-strongly exposed point of P(G)₀, then is isolatedin . Consequently, G is compact if and only if, for every ,there exists a nonzero positive-definite function associatedwith that is a w^*-strongly exposed point of P(G)₀. If, in addition,G is unimodular and , then is isolated in if and only if somenonzero positive-definite function associated with is a w^*-stronglyexposed point of P(G)₀, where is the left regular representationof G and is the reduced dual space of G. We prove that if B(G)has the Radon–Nikodym property, then the set of isolatedpoints of (so square-integrable if G is unimodular) is densein . It is also proved that if G is a separable SIN-group, thenG is amenable if and only if there exists a closed point in. In particular, for a countable discrete non-amenable groupG (for example the free group F₂ on two generators), there isno closed point in its reduced dual space .     

Asymptotics for Fractional Nonlinear Heat Equations

The Cauchy problem is studied for the nonlinear equations withfractional power of the negative Laplacian where (0,2), with critical = /n and sub-critical (0,/n)powers of the nonlinearity. Let u₀ L^1,a L C, u₀(x) 0 in Rⁿ, = . The case of not small initial data is of interest. It is proved that the Cauchy problemhas a unique global solution u C([0,); L L^1,a C) and the largetime asymptotics are obtained.     

Immanant Inequalities, Induced Characters, and Rank Two Partitions

If = {₁, ₂, ..., _s}, where _{1 2} ... _s > 0, is a partitionof n then denotes the associated irreducible character of S_n,the symmetric group on {1, 2, ..., n}, and, if cCS_n, the groupalgebra generated by C and S_n, then d_c(·) denotes thegeneralized matrix function associated with c. If c₁, c₂ CS_nthen we write c₁ c₂ in case (A) (A) for each n x n positivesemi-definite Hermitian matrix A. If cCS_n and c(e) 0, wheree denotes the identity in S_n, then or denotes (c(e))^–1 c. The main result, an estimate for the norms of tensors of a certainanti-symmetry type, implies that if = {₁, ₂, ..., _s, 1^t} isa partition of n such that s > 1 and _s = 2, and ' denotes{₁, ₂, ..., _s-1, 1^t} then (, _{2}) where denotes characterinduction from S_n–2 x S₂ to S_n. This in turn implies thatif = {₁, ₂, ..., _s, 1^t} with s > 1, _s = 2, and ßdenotes {₁ + 2, ₂, ..., _s-1, 1^t} then _ß which,in conjunction with other known results, provides many new inequalitiesamong immanants. In particular it implies that the permanentfunction dominates all normalized immanants whose associatedpartitions are of rank 2, a result which has proved elusivefor some years. We also consider the non-relationship problem for immanants– that is the problem of identifying pairs, (,ß)such that _ß and _ß are both false.     

Antisupercyclic Operators and Orbits of the Volterra Operator

We say that a bounded linear operator T acting on a Banach spaceB is antisupercyclic if for any x B either Tⁿx = 0 for somepositive integer n or the sequence {Tⁿx/||Tⁿx||} weakly convergesto zero in B. Antisupercyclicity of T means that the angle criterionof supercyclicity is not satisfied for T in the strongest possibleway. Normal antisupercyclic operators and antisupercyclic bilateralweighted shifts are characterized. As for the Volterra operator V, it is proved that if 1 p and any f L_p [0,1] then the limit lim_n (n!||Vⁿf||_p)^1/n doesexist and equals 1 – inf supp (f). Upon using this asymptoticformula it is proved that the operator V acting on the Banachspace L_p[0,1] is antisupercyclic for any p (1,). The same statementfor p = 1 or p = is false. The analogous results are provedfor operators when the real part of z C is positive.     

Reducing Subspaces for a Class of Multiplication Operators

Let D be the open unit disk in the complex plane C. The Bergmanspace is the Hilbert space of analytic functions f in D such that where dA is the normalized area measure on D. If are two functions in , then the inner product of f and g is given by We study multiplication operators on induced by analytic functions. Thus for H (D), the space ofbounded analytic functions in D, we define by It is easy to check that M is a bounded linear operator on with ||M||=||||=sup{|(z)|:zD}.     

Constructing {kappa}-like Models of Arithmetic

A model (M, <, ...) is -like if M has cardinality but, forall M, the cardinality of {x M : x < a} is strictly lessthan . In this paper we shall give constructions of -like modelsof arithmetic satisfying an arbitrarily large finite part ofPA but not PA itself, for various singular cardinals . The mainresults are: (1) for each countable nonstandard M ₂–Th(PA)with arbitrarily large initial segments satisfying PA and eachuncountable of cofinality there is a cofinal extension K ofM which is -like; also hierarchical variants of this resultfor _n–Th(PA); and (2) for every n 1, every singular and every M B_n+exp+¬ I_n there is a -like model K elementarilyequivalent to M.     

Identity Theorems for Functions of Bounded Characteristic

Suppose that f(z) is a meromorphic function of bounded characteristicin the unit disk :|z|<1. Then we shall say that f(z)N. Itfollows (for example from [3, Lemma 6.7, p. 174 and the following])that where h₁(z), h₂(z) are holomorphic in and have positive realpart there, while ₁(z), ₂(z) are Blaschke products, that is, where p is a positive integer or zero, 0<|a_j|<1, c isa constant and (1–|a_j|)<. We note in particular that, if c0, so that f(z)0, (1.1) so that f(z)=0 only at the points a_j. Suppose now that z_j isa sequence of distinct points in such that |z_j|1 as j and (1–|z_j|)=. (1.2) If f(z_j)=0 for each j and fN, then f(z)0. N. Danikas [1] has shown that the same conclusion obtains iff(z_j)0 sufficiently rapidly as j. Let _j, _j be sequences of positivenumbers such that _j< and _j as j. Danikas then defines and proves Theorem A.     

The Hausdorff Dimension of Julia Sets of Meromorphic Functions II

A family of transcendental meromorphic functions, f_p(z), p N is considered. It is shown that, if p 6, then the Hausdorffdimension of the Julia set of f_p satisfies dim J(f_p) 1/p, for0 < < 1/6^p, and dim J(f_p) 1–(30 ln ln p/ln p),for p^4p–1/10⁵ ln p < < p^4p–1/10⁴ ln p. Theseresults are used elsewhere to show that, for each d (0, 1),there exists a transcendental meromorphic function for whichdim J(f) = d.     

Real Interpolation and Two Variants of Gehring's Lemma

Let be a fixed open cube in Rⁿ. For r[1, ) and [0, ) we define where Q is a cube in Rⁿ (with sides parallel to the coordinateaxes) and _Q stands for the characteristic function of the cubeQ. A well-known result of Gehring [5] states that if (1.1) for some p(1, ) and c(0, ), then there exist q(p, ) and C=C(p,q, n, c)(0, ) such that for all cubes Q, where |Q| denotes the n-dimensional Lebesguemeasure of Q. In particular, a function fL¹() satisfying (1.1)belongs to L^q(). In [9] it was shown that Gehring's result is a particular caseof a more general principle from the real method of interpolation.Roughly speaking, this principle states that if a certain reversedinequality between K-functionals holds at one point of an interpolationscale, then it holds at other nearby points of this scale. Usingan extension of Holmstedt's reiteration formulae of [4] andresults of [8] on weighted inequalities for monotone functions,we prove here two variants of this principle involving extrapolationspaces of an ordered pair of (quasi-) Banach spaces. As an applicationwe prove the following Gehring-type lemmas.     

Generators and defining relations for ring of invariants of commuting locally nilpotent derivations or automorphisms

Let A be an algebra over a field K of characteristic zero andlet ₁, ..., _sDer _K(A) be commuting locally nilpotent K-derivationssuch that _i(x_j) equals _ij, the Kronecker delta, for some elementsx₁, ..., x_sA. A set of generators for the algebra is found explicitly and a set of defining relationsfor the algebra A is described. Similarly, let ₁, ..., _s Aut_K(A)be commuting K-automorphisms of the algebra A is given suchthat the maps _i – id_A are locally nilpotent and _i (x_j)= x_j + _ij, for some elements x₁, ..., x_s A. A set of generatorsfor the algebra A: = {a A | ₁(a) = ... = _s(a) = a} is foundexplicitly and a set of defining relations for the algebra Ais described. In general, even for a finitely generated non-commutativealgebra A the algebras of invariants A and A are not finitelygenerated, not (left or right) Noetherian and a minimal numberof defining relations is infinite. However, for a finitely generatedcommutative algebra A the opposite is always true. The derivations(or automorphisms) just described appear often in many differentsituations (possibly) after localization of the algebra A.     

Periodic Solutions of Lienard Equations with Asymmetric Nonlinearities at Resonance

The existence of 2-periodic solutions of the second-order differentialequation where a, b satisfy and p(t)=p(t+2),t R, is examined. Assume that limits lim_x±F(x)=F(±)(F(x)=) and lim_x±g(x)=g(±)exist and are finite. It is proved that the equation has atleast one 2-periodic solution provided that the zeros of thefunction ₁ are simple and the zeros of the functions ₁, ₂ aredifferent and the signs of ₂ at the zeros of ₁ in [0,2/n) donot change or change more than two times, where ₁ and ₂ aredefined as follows: Moreover, it is also proved that the given equation has at leastone 2-periodic solution provided that the following conditionshold: with 1 p < q 2.     

Betti Numbers of Semialgebraic and Sub-Pfaffian Sets

Let X be a subset in [–1,1]^n₀R^n₀ defined by the formula X={x₀|Q₁x₁Q₂x₂...Qx ((x₀,x₁,...x)X)}, where Q_i{ }, Q_i Q_i+1, x_i [–1, 1]^n_i, and X may be eitheran open or a closed set in [–1,1]^n₀+...+n, being the differencebetween a finite CW-complex and its subcomplex. An upper boundon each Betti number of X is expressed via a sum of Betti numbersof some sets defined by quantifier-free formulae involving X. In important particular cases of semialgebraic and semi-Pfaffiansets defined by quantifier-free formulae with polynomials andPfaffian functions respectively, upper bounds on Betti numbersof X are well known. The results allow to extend the boundsto sets defined with quantifiers, in particular to sub-Pfaffiansets.     

Regularity Conditions and Bernoulli Properties of Equilibrium States and g-Measures

When T : X X is a one-sided topologically mixing subshift offinite type and : X R is a continuous function, one can definethe Ruelle operator L : C(X) C(X) on the space C(X) of real-valuedcontinuous functions on X. The dual operator always has a probability measure as an eigenvectorcorresponding to a positive eigenvalue ( = with > 0). Necessary and sufficient conditionson such an eigenmeasure are obtained for to belong to twoimportant spaces of functions, W(X, T) and Bow (X, T). For example, Bow(X, T) if and only if is a measure with a certain approximateproduct structure. This is used to apply results of Bradleyto show that the natural extension of the unique equilibriumstate µ of Bow(X, T) has the weak Bernoulli propertyand hence is measure-theoretically isomorphic to a Bernoullishift. It is also shown that the unique equilibrium state ofa two-sided Bowen function has the weak Bernoulli property.The characterizations mentioned above are used in the case ofg-measures to obtain results on the ‘reverse’ ofa g-measure.     

On Artin's Braid Group And Polyconvexity In The Calculus Of Variations

Let R² be a bounded Lipschitz domain and let be a Carathèodory integrand such that F(x,·) is polyconvex for L²-a.e. x . Moreover assume thatF is bounded from below and satisfies the condition as det for L²-a.e. x . The paper describes the effect of domain topologyon the existence and multiplicity of strong local minimizersof the functional wherethe map u lies in the Sobolev space W_id^1,p (, R²) with p 2and satisfies the pointwise condition u(x) >0 for L²-a.e.x . The question is settled by establishing that F[·]admits a set of strong local minimizers on that can be indexed by the group P_n Zⁿ, the directsum of Artin's pure braid group on n strings and n copies ofthe infinite cyclic group. The dependence on the domain topologyis through the number of holes n in and the different mechanismsthat give rise to such local minimizers are fully exploitedby this particular representation.     

Geometry of Critical Loci

Let :(Z,z)(U,0) be the germ of a finite (that is, proper with finite fibres)complex analytic morphism from a complex analytic normal surfaceonto an open neighbourhood U of the origin 0 in the complexplane C². Let u and v be coordinates of C² defined on U. Weshall call the triple (, u, v) the initial data. Let stand for the discriminant locus of the germ , that is,the image by of the critical locus of . Let ()_A be the branches of the discriminant locus at O whichare not the coordinate axes. For each A, we define a rational number d by where I(–, –) denotes the intersection number at0 of complex analytic curves in C². The set of rational numbersd, for A, is a finite subset D of the set of rational numbersQ. We shall call D the set of discriminantal ratios of the initialdata (, u, v). The interesting situation is when one of thetwo coordinates (u, v) is tangent to some branch of , otherwiseD = {1}. The definition of D depends not only on the choiceof the two coordinates, but also on their ordering. In this paper we prove that the set D is a topological invariantof the initial data (, u, v) (in a sense that we shall definebelow) and we give several ways to compute it. These resultsare first steps in the understanding of the geometry of thediscriminant locus. We shall also see the relation with thegeometry of the critical locus.