On the commutativity of the algebra of invariant differential operators on certain nilpotent homogeneous spaces

Let be a simply connected connected real nilpotent Lie group with Lie algebra , a connected closed subgroup of with Lie algebra and satisfying . Let be the unitary character of with differential at the origin. Let be the unitary representation of induced from the character of . We consider the algebra of differential operators invariant under the action of on the bundle with basis associated to these data. We consider the question of the equivalence between the commutativity of and the finite multiplicities of . Corwin and Greenleaf proved that if is of finite multiplicities, this algebra is commutative. We show that the converse is true in many cases.

     

On supports and associated primes of modules over the enveloping algebras of nilpotent Lie algebras

Let be a nilpotent Lie algebra, over a field of characteristic zero, and its universal enveloping algebra. In this paper we study: (1) the prime ideal structure of related to finitely generated -modules , and in particular the set of associated primes for such (note that now is equal to the set of annihilator primes for ); (2) the problem of nontriviality for the modules when is a (maximal) prime of , and in particular when is the augmentation ideal of . We define the support of , as a natural generalization of the same notion from commutative theory, and show that it is the object of primary interest when dealing with (2). We also introduce and study the reduced localization and the reduced support, which enables to better understand the set . We prove the following generalization of a stability result given by W. Casselman and M. S. Osborne in the case when , as in the theorem, are abelian. We also present some of its interesting consequences.

Theorem. Let be a finite-dimensional Lie algebra over a field of characteristic zero, and an ideal of ; denote by the universal enveloping algebra of . Let be a -module which is finitely generated as an -module. Then every annihilator prime of , when is regarded as a -module, is -stable for the adjoint action of on .

     

Local derivations on -algebras are derivations

Kadison has shown that local derivations from a von Neumann algebra into any dual bimodule are derivations. In this paper we extend this result to local derivations from any -algebra into any Banach -bimodule . Most of the work is involved with establishing this result when is a commutative -algebra with one self-adjoint generator. A known result of the author about Jordan derivations then completes the argument. We show that these results do not extend to the algebra of continuously differentiable functions on . We also give an automatic continuity result, that is, we show that local derivations on -algebras are continuous even if not assumed a priori to be so.

     

Theta lifting of holomorphic discrete series: The case of

Let be a reductive dual pair in the stable range. We investigate theta lifts to of unitary characters and holomorphic discrete series representations of , in relation to the geometry of nilpotent orbits. We give explicit formulas for their -type decompositions. In particular, for the theta lifts of unitary characters, or holomorphic discrete series with a scalar extreme -type, we show that the structure of the resulting representations of is almost identical to the -module structure of the regular function rings on the closure of the associated nilpotent -orbits in , where is a Cartan decomposition. As a consequence, their associated cycles are multiplicity free.

     

Block representation type of reduced enveloping algebras

Let be an algebraically closed field of characteristic , a connected, reductive -group, , and the reduced enveloping algebra of associated with . Assume that is simply-connected, is good for and has a non-degenerate -invariant bilinear form. All blocks of having finite and tame representation type are determined.

     

Bounded Hochschild cohomology of Banach algebras with a matrix-like structure

Let be a unital Banach algebra. A projection in which is equivalent to the identitity may give rise to a matrix-like structure on any two-sided ideal in . In this set-up we prove a theorem to the effect that the bounded cohomology vanishes for all . The hypotheses of this theorem involve (i) strong H-unitality of , (ii) a growth condition on diagonal matrices in , and (iii) an extension of in by an amenable Banach algebra. As a corollary we show that if is an infinite dimensional Banach space with the bounded approximation property, is an infinite dimensional -space, and is the Banach algebra of approximable operators on , then for all .

     

Length, multiplicity, and multiplier ideals

Let be an -dimensional regular local ring, essentially of finite type over a field of characteristic zero. Given an -primary ideal of , the relationship between the singularities of the scheme defined by and those defined by the multiplier ideals , with varying in , are quantified in this paper by showing that the Samuel multiplicity of satisfies whenever . This formula generalizes an inequality on log canonical thresholds previously obtained by Ein, Mustata and the author of this paper. A refined inequality is also shown to hold for small dimensions, and similar results valid for a generalization of test ideals in positive characteristics are presented.

     

Good ideals in Gorenstein local rings

Let be an -primary ideal in a Gorenstein local ring (, ) with , and assume that contains a parameter ideal in as a reduction. We say that is a good ideal in if is a Gorenstein ring with . The associated graded ring of is a Gorenstein ring with if and only if . Hence good ideals in our sense are good ones next to the parameter ideals in . A basic theory of good ideals is developed in this paper. We have that is a good ideal in if and only if and . First a criterion for finite-dimensional Gorenstein graded algebras over fields to have nonempty sets of good ideals will be given. Second in the case where we will give a correspondence theorem between the set and the set of certain overrings of . A characterization of good ideals in the case where will be given in terms of the goodness in their powers. Thanks to Kato's Riemann-Roch theorem, we are able to classify the good ideals in two-dimensional Gorenstein rational local rings. As a conclusion we will show that the structure of the set of good ideals in heavily depends on . The set may be empty if , while is necessarily infinite if and contains a field. To analyze this phenomenon we shall explore monomial good ideals in the polynomial ring in three variables over a field . Examples are given to illustrate the theorems.

     

Noncrossed products over

Noncrossed product division algebras are constructed over rational function fields over number fields by lifting from arithmetic completions . The existence of noncrossed products over -adic rational function fields is proved as a corollary.

     

A universal continuum of weight

We prove that every continuum of weight  is a continuous image of the Cech-Stone-remainder  of the real line. It follows that under  the remainder of the half line is universal among the continua of weight  -- universal in the `mapping onto' sense.

We complement this result by showing that 1) under  every continuum of weight less than  is a continuous image of , 2) in the Cohen model the long segment of length  is not a continuous image of , and 3)  implies that is not a continuous image of , whenever  is a -saturated ultrafilter.

We also show that a universal continuum can be gotten from a -saturated ultrafilter on , and that it is consistent that there is no universal continuum of weight  .

     

Large rectangular semigroups in Stone-Cech compactifications

We show that large rectangular semigroups can be found in certain Stone-Cech compactifications. In particular, there are copies of the rectangular semigroup in the smallest ideal of , and so, a semigroup consisting of idempotents can be embedded in the smallest ideal of if and only if it is a subsemigroup of the rectangular semigroup. In fact, we show that for any ordinal with cardinality at most , contains a semigroup of idempotents whose rectangular components are all copies of the rectangular semigroup and form a decreasing chain indexed by , with the minimum component contained in the smallest ideal of .

As a fortuitous corollary we obtain the fact that there are -chains of idempotents of length in . We show also that there are copies of the direct product of the rectangular semigroup with the free group on generators contained in the smallest ideal of .

     

A generalization of tight closure and multiplier ideals

We introduce a new variant of tight closure associated to any fixed ideal , which we call -tight closure, and study various properties thereof. In our theory, the annihilator ideal of all -tight closure relations, which is a generalization of the test ideal in the usual tight closure theory, plays a particularly important role. We prove the correspondence of the ideal and the multiplier ideal associated to (or, the adjoint of in Lipman's sense) in normal -Gorenstein rings reduced from characteristic zero to characteristic . Also, in fixed prime characteristic, we establish some properties of similar to those of multiplier ideals (e.g., a Briançon-Skoda-type theorem, subadditivity, etc.) with considerably simple proofs, and study the relationship between the ideal and the F-rationality of Rees algebras.

     

-actions whose fixed data has a section

Given a collection of real vector bundles over a closed manifold , suppose that, for some is of the form , where is the trivial one-dimensional bundle. In this paper we prove that if is the fixed data of a -action, then the same is true for the Whitney sum obtained from by replacing by . This stability property is well-known for involutions. Together with techniques previously developed, this result is used to describe, up to bordism, all possible -actions fixing the disjoint union of an even projective space and a point.     

Serre's generalization of Nagao's theorem: An elementary approach

Let be a smooth projective curve over a field . For each closed point of let be the coordinate ring of the affine curve obtained by removing from . Serre has proved that is isomorphic to the fundamental group, , of a graph of groups , where is a tree with at most one non-terminal vertex. Moreover the subgroups of attached to the terminal vertices of are in one-one correspondence with the elements of , the ideal class group of . This extends an earlier result of Nagao for the simplest case .

Serre's proof is based on applying the theory of groups acting on trees to the quotient graph , where is the associated Bruhat-Tits building. To determine he makes extensive use of the theory of vector bundles (of rank 2) over . In this paper we determine using a more elementary approach which involves substantially less algebraic geometry.

The subgroups attached to the edges of are determined (in part) by a set of positive integers , say. In this paper we prove that is bounded, even when Cl is infinite. This leads, for example, to new free product decomposition results for certain principal congruence subgroups of , involving unipotent and elementary matrices.

     

Boundary relations and their Weyl families

The concepts of boundary relations and the corresponding Weyl families are introduced. Let be a closed symmetric linear operator or, more generally, a closed symmetric relation in a Hilbert space , let be an auxiliary Hilbert space, let

and let be defined analogously. A unitary relation from the Krein space to the Krein space is called a boundary relation for the adjoint if . The corresponding Weyl family is defined as the family of images of the defect subspaces , , under . Here need not be surjective and is even allowed to be multi-valued. While this leads to fruitful connections between certain classes of holomorphic families of linear relations on the complex Hilbert space and the class of unitary relations , it also generalizes the notion of so-called boundary value space and essentially extends the applicability of abstract boundary mappings in the connection of boundary value problems. Moreover, these new notions yield, for instance, the following realization theorem: every -valued maximal dissipative (for ) holomorphic family of linear relations is the Weyl family of a boundary relation, which is unique up to unitary equivalence if certain minimality conditions are satisfied. Further connections between analytic and spectral theoretical properties of Weyl families and geometric properties of boundary relations are investigated, and some applications are given.

     

On the number of -equivalent non-isomorphic models

We prove that if is consistent then is consistent with the following statement: There is for every a model of cardinality which is -equivalent to exactly non-isomorphic models of cardinality . In order to get this result we introduce ladder systems and colourings different from the ``standard' counterparts, and prove the following purely combinatorial result: For each prime number and positive integer it is consistent with that there is a ``good' ladder system having exactly pairwise nonequivalent colourings.

     

Extensions of Hopf Algebras and Lie Bialgebras

Let , be finite-dimensional Lie algebras over a field of characteristic zero. Regard and , the dual Lie coalgebra of , as Lie bialgebras with zero cobracket and zero bracket, respectively. Suppose that a matched pair of Lie bialgebras is given, which has structure maps . Then it induces a matched pair of Hopf algebras, where is the universal envelope of and is the Hopf dual of . We show that the group of cleft Hopf algebra extensions associated with is naturally isomorphic to the group of Lie bialgebra extensions associated with . An exact sequence involving either of these groups is obtained, which is a variation of the exact sequence due to G.I. Kac. If , there follows a bijection between the set of all cleft Hopf algebra extensions of by and the set of all Lie bialgebra extensions of by .

     

The automorphism tower of groups acting on rooted trees

The group of isometries of a rooted -ary tree, and many of its subgroups with branching structure, have groups of automorphisms induced by conjugation in . This fact has stimulated the computation of the group of automorphisms of such well-known examples as the group studied by R. Grigorchuk, and the group studied by N. Gupta and the second author.

In this paper, we pursue the larger theme of towers of automorphisms of groups of tree isometries such as and . We describe this tower for all subgroups of which decompose as infinitely iterated wreath products. Furthermore, we fully describe the towers of and .

More precisely, the tower of is infinite countable, and the terms of the tower are -groups. Quotients of successive terms are infinite elementary abelian -groups.

In contrast, the tower of has length , and its terms are -groups. We show that is an elementary abelian -group of countably infinite rank, while .

     

Embeddings of

We show that there is only one embedding of in at the prime , up to self-maps of . We also describe the effect of the group of self-equivalences of at the prime on this embedding and then show that the Friedlander exceptional isogeny composed with a suitable Adams map is an involution of whose homotopy fixed point set coincide with

     

Quadratic maps and Bockstein closed group extensions

Let be a central extension of the form where and are elementary abelian -groups. Associated to there is a quadratic map , given by the -power map, which uniquely determines the extension. This quadratic map also determines the extension class of the extension in and an ideal in which is generated by the components of . We say that is Bockstein closed if is an ideal closed under the Bockstein operator.

We find a direct condition on the quadratic map that characterizes when the extension is Bockstein closed. Using this characterization, we show for example that quadratic maps induced from the fundamental quadratic map given by yield Bockstein closed extensions.

On the other hand, it is well known that an extension is Bockstein closed if and only if it lifts to an extension for some -lattice . In this situation, one may write for a ``binding matrix' with entries in . We find a direct way to calculate the module structure of in terms of . Using this, we study extensions where the lattice is diagonalizable/triangulable and find interesting equivalent conditions to these properties.