Cohomological detection and regular elements in group cohomology

Suppose that is a compact Lie group or a discrete group of finite virtual cohomological dimension and that is a field of characteristic . Suppose that is a set of elementary abelian -subgroups such that the cohomology is detected on the centralizers of the elements of . Assume also that is closed under conjugation and that is in whenever some subgroup of is in . Then there exists a regular element in the cohomology ring such that the restriction of to an elementary abelian -subgroup is not nilpotent if and only if is in . The converse of the result is a theorem of Lannes, Schwartz and the second author. The results have several implications for the depth and associated primes of the cohomology rings.

     

Vanishing of the leading term in Harish-Chandra's local character expansion

Harish-Chandra's formula for the character of an irreducible smooth representation of a reductive -adic group expresses near as a linear combination of the Fourier transforms of nilpotent -orbits in the Lie algebra of . In this note, we prove that if is tempered but not in the discrete series, then the coefficient attached to the zero nilpotent orbit vanishes.

     

The equivariant Brauer groups of commuting free and proper actions are isomorphic

If is a locally compact space which admits commuting free and proper actions of locally compact groups and , then the Brauer groups and are naturally isomorphic.

     

Cartan invariants of group algebras of finite groups

We give a result on Cartan invariants of the group algebra of a finite group over an algebraically closed field , which implies that if the Loewy length (socle length) of the projective indecomposable -module corresponding to the trivial -module is four, then has characteristic 2. The proof is independent of the classification of finite simple groups.

     

A reciprocity law for certain Frobenius extensions

Let be a finite Galois extension of algebraic number fields with Galois group . Assume that is a Frobenius group and is a Frobenius complement of . Let be the maximal normal nilpotent subgroup of . If is nilpotent, then every Artin L-function attached to an irreducible representation of arises from an automorphic representation over , i.e., the Langlands' reciprocity conjecture is true for such Galois extensions.

     

On a pattern of reflexive operator spaces

A linear subspace is a separating subspace for an operator space if the only member of annihilating is 0. It is proved in this paper that if has a strictly separating vector and a separating subspace satisfying , then is reflexive. Applying this to finite dimensional leads to more results on reflexivity. For example, if dim , and every nonzero operator in has rank , then is reflexive.

     

Almost disjoint permutation groups

A permutation group on a set of (infinite) cardinality is almost disjoint if no element of except the identity has fixed points, i.e., if is an almost disjoint family of subsets of . We show how almost disjoint permutation groups can be constructed from almost disjoint families of sets.

     

Tate cohomology lowers chromatic Bousfield classes

Let be a finite group. We use recent results of J. P. C. Greenlees and H. Sadofsky to show that the Tate homology of local spectra with respect to produces local spectra. We also show that the Bousfield class of the Tate homology of (for finite) is the same as that of . To be precise, recall that Tate homology is a functor from -spectra to -spectra. To produce a functor from spectra to spectra, we look at a spectrum as a naive -spectrum on which acts trivially, apply Tate homology, and take -fixed points. This composite is the functor we shall actually study, and we'll prove that when is finite. When , the symmetric group on letters, this is related to a conjecture of Hopkins and Mahowald (usually framed in terms of Mahowald's functor ).

     

On Voiculescu's double commutant theorem

For a separable infinite-dimensional Hilbert space , we consider the full algebra of bounded linear transformations and the unique non-trivial norm-closed two-sided ideal of compact operators . We also consider the quotient -algebra with quotient map

For any -subalgebra of , the relative commutant is given by for all in . It was shown by D. Voiculescu that, for any separable unital -subalgebra of ,

In this note, we exhibit a non-separable unital -subalgebra of for which (VDCT) fails.

     

On free group algebras in division rings with uncountable center

Let be a division algebra with uncountable center . If contains a noncommutative free -algebra, then also contains the -group algebra of the free group of rank 2.

     

Representations at fixed points of smooth actions of compact connected Lie groups

Let be a compact connected Lie group acting smoothly on a connected closed manifold with nonempty fixed point set . In this paper, we study the relation between the cohomology of or and the equivalent representations of at fixed points.

     

The second iterate of a map with dense orbit

Suppose that is a Hausdorff topological space having no isolated points and that is continuous. We show that if the orbit of a point under is dense in while the orbit of under is not, then the space decomposes into three sets relative to which the dynamics of are easy to describe. This decomposition has the following consequence: suppose that has dense orbit under and that the closure of the set of points of having odd period under has nonempty interior; then has dense orbit under .

     

Endomorphism rings of simple modules over group rings

If is a finitely generated nilpotent group which is not abelian-by-finite, a field, and a finite dimensional separable division algebra over , then there exists a simple module for the group ring with endomorphism ring . An example is given to show that this cannot be extended to polycyclic groups.

     

Parametrizing maximal compact subvarieties

For the Lie group , let be the open orbit of Lagrangian planes of signature in the generalized flag variety of Lagrangian planes in . For a suitably chosen maximal compact subgroup of and a base point we have that the orbit of is a maximal compact subvariety of . We show that for the connected component containing in the space of translates of which lie in is biholomorphic to , where denotes with the opposite complex structure.

     

A theorem of Briançon-Skoda type for regular local rings containing a field

Let be a regular local ring containing a field. We give a refinement of the Briançon-Skoda theorem showing that if is a minimal reduction of where is -primary, then where and is the largest ideal such that . The proof uses tight closure in characteristic and reduction to characteristic for rings containing the rationals.

     

Non-normal, standard subgroups of the Bianchi groups

Let be a subgroup of , where is a Dedekind ring, and let be the -ideal generated by , where . The subgroup is called standard iff contains the normal subgroup of generated by the -elementary matrices. It is known that, when , is standard iff is normal in . It is also known that every standard subgroup of is normal in when is an arithmetic Dedekind domain with infinitely many units. The ring of integers of an imaginary quadratic number field, , is one example (of three) of such an arithmetic domain with finitely many units. In this paper it is proved that every Bianchi group has uncountably many non-normal, standard subgroups. This result is already known for related groups like .

     

Basic differential forms for actions of Lie groups

A section of a Riemannian -manifold is a closed submanifold which meets each orbit orthogonally. It is shown that the algebra of -invariant differential forms on which are horizontal in the sense that they kill every vector which is tangent to some orbit, is isomorphic to the algebra of those differential forms on which are invariant with respect to the generalized Weyl group of , under some condition.

     

On a class of implicit differential inclusions

The existence of solutions is established for implicit differential inclusions in involving the sum of a maximal monotone mapping and an upper semicontinuous mapping with compact, closed values. A question of Wenzel is answered in the affirmative.

     

Counterexample to a problem of Geoghegan-West

Let be a Banach space and its general linear group. Let denote the operator norm and ``' the pointwise convergence topology on . Is the identity map a homotopy equivalence? The answer is negative. One of the possible counterexamples is a well-known James space ---the ``space of counterexamples in Banach spaces theory'.