Volume 2012, Issue 673

We develop a Lefschetz theory in a combinatorial category associated to a root system and derive an upper bound on the exceptional characteristics for Lusztig's formula for the simple rational characters of a reductive algebraic group. Our bound is huge compared to the Coxeter number.

For contact manifolds ( M ,  η ) a complexification M c is constructed to which the contact form η extends such that the exterior derivative of the extended form is Kählerian. In the case of a proper action of an extendable Lie group this construction is realized in an equivariant way. In a simultaneous stratification of M and M c according to the isotropy type, it is shown that the Kählerian reduction of the complexification can be seen as the complexification of the contact reduction. For this, the complexification is constructed as a stratified space.

Whenever the mapping class group of a closed orientable surface of genus g acts by semisimple isometries on a complete CAT(0) space of dimension less than g , it fixes a point.

We introduce systems of objects and operators in linear monoidal categories called Ψ̂-systems. A Ψ̂-system satisfying several additional assumptions gives rise to a topological invariant of triples (a closed oriented 3-manifold M , a principal bundle over M , a link in M ). This construction generalizes the quantum dilogarithmic invariant of links appearing in the original formulation of the volume conjecture. We conjecture that all quantum groups at odd roots of unity give rise to Ψ̂-systems and we verify this conjecture in the case of the Borel subalgebra of quantum 𝔰𝔩 2 .

A manifold M with a foliation ℱ is minimizable if there exists a Riemannian metric g on M such that every leaf of ℱ is a minimal submanifold of ( M ,  g ). For a closed manifold M with a Riemannian foliation ℱ, Álvarez López [1] defined a cohomology class of degree 1 called the Álvarez class whose triviality characterizes the minimizability of ( M , ℱ). In this paper, we show that the family of the Álvarez classes of a smooth family of Riemannian foliations on a closed manifold is continuous with respect to the parameter. The Álvarez class has algebraic rigidity under certain topological conditions on ( M , ℱ) as the author showed in [21]. As a corollary of these two results, we show that under the same topological conditions the minimizability of Riemannian foliations is invariant under deformations.

The Hom closed colocalizing subcategories of the stable module category of a finite group are classified. Along the way, the colocalizing subcategories of the homotopy category of injectives over an exterior algebra, and the derived category of a formal commutative differential graded algebra, are classified. To this end, and with an eye towards future applications, a notion of local homology and cosupport for triangulated categories is developed, building on earlier work of the authors on local cohomology and support.

Let ( X ,  ω ) be a symplectic orbifold which is locally like the quotient of a ℤ 2 action on ℝ n . Let be a deformation quantization of X constructed via the standard Fedosov method with characteristic class being ω . In this paper, we construct a deformation of the algebra parametrized by codimension 2 components of the associated inertia orbifold X̃ . This partially confirms a conjecture of Dolgushev and Etingof in the case of ℤ 2 orbifolds. To do so, we generalize the interpretation of the Moyal star-product as a composition of symbols of pseudodifferential operators in the case where partial derivatives are replaced with Dunkl operators. The star-products we obtain can be seen as globalizations of symplectic reflection algebras.

An even unimodular 72-dimensional lattice Γ having minimum 8 is constructed as a tensor product of the Barnes lattice and the Leech lattice over the ring of integers in the imaginary quadratic number field with discriminant −7. The automorphism group of Γ contains the absolutely irreducible rational matrix group ( PSL 2 (7) ×  SL 2 (25)) : 2.