C. De Concini

Wonderful models of subspace arrangements

The motivation stems from our attempt to understand Drinfelds construction (el. [Dr2]) of special solutions of the Khniznik-Zamolodchikov equation (of. [K-Z]) with some prescribed asymptotic behavior and its consequences for some universal constructions associated to braiding: universal unipotent monodromy representations of braid groups, the construction of a universal Vassiliev invariant for knots, braided categories etc. The K-Z connection is a special flat meromorphic connection on C ~ with simple poles on a family of hyperplanes. It turns out that the prescription of the asymptotic behavior for such connections is controlled by the geometry of a suitable modification of C ~ in which the union of the polar hyperplanes is replaced by a divisor with normal crossings. In the process of developing this geometry we realized that our constructions could be developed more generally for subspace arrangements and became aware of the paper of Fulton-MaePherson [F-M] in which a Hironaka model is described for the complement of the big diagonal in the power of a smooth variety X. It became clear to us that our techniques were quite similar to theirs and so we adopted their notation of nested set in the appropriate general form. Although we work in a linear subspaces setting it is clear that the methods are essentially local and one can recover their results from our analysis applied to certain special configurations of subspaces. In fact the theory can be applied whenever we have a subvariety of a smooth variety which locally (in the gtale topology) appears as a union of subspaces.

Homology of the zero-set of a nilpotent vector field on a flag manifold

0.1. Let X be a linear transformation of a finite-dimensional vector space V. The configuration of flags in V which are fixed by X has rather remarkable properties when X is unipotent. Though this case is especially interesting, the proper generality in which to study such configurations is in the theory of reductive algebraic groups, where their definition can be reformulated in the language of Borel subalgebras as follows. Let G be a connected reductive group over C, with Lie algebra g, and let N E g be a nilpotent element. Let q be the variety of all Borel subalgebras of g and let

Some remarkable degenerations of quantum groups

We show that an irreducible representation of a quantized enveloping algebraUε at a ℓth root of 1 has maximal dimension (=ℓN) if the corresponding symplectic leaf has maximal dimension (=2N). The method of the proof consists of a construction of a sequence of degenerations ofUε, the last one being aq-commutative algebraUε(2N). This allows us to reduce many problems concerningUε to that concerningUε(2N).

Compactification of symmetric varieties

The symmetric varieties considered in this paper are the quotientsG/H, whereG is an adjoint semi-simple group over a fieldk of characteristic ≠ 2, andH is the fixed point group of an involutorial automorphism ofG which is defined overk. In the casek=C, De Concini and Procesi (1983) constructed a “wonderful” compactification ofG/H. We prove the existence of such a compactification for arbitraryk. We also prove cohomology vanishing results for line bundles on the compactification.

Box splines and the equivariant index theorem

In this article, we start to recall the inversion formula for the convolution with the Box spline. The equivariant cohomology and the equivariant K-theory with respect to a compact torus G of various spaces associated to a linear action of G in a vector space M can be both described using some vector spaces of distributions, on the dual of the group G or on the dual of its Lie algebra. The morphism from K-theory to cohomology is analyzed and the multiplication by the Todd class is shown to correspond to the operator (deconvolution) inverting the semidiscrete convolution with a box spline. Finally, the multiplicities of the index of a G-transversally elliptic operator on M are determined using the infinitesimal index of the symbol.

Spontaneous breakdown of symmetry and group

Abstract In spontaneously broken symmetry theories, the symmetry group that appears in observations proves to be a group contraction of the dynamical invariance group. Infrared effects play a crucial role in the dynamical rearrangement of symmetry which leads to the group contraction. Many examples are considered. General theorems are given for SU( n ) and SO( n ). Low-energy theorems and ordered-state symmetry patterns are observable manifestations of group contractions. These results seem to support the conjecture that the transition from quantum to classical physics involves a group contraction mechanism.

The infinitesimal index

In this note, we study an invariant associated to the zeros of the moment map generated by an action form, the infinitesimal index. This construction will be used to study the compactly supported equivariant cohomology of the zeros of the moment map and to give formulas for the multiplicity index map of a transversally elliptic operator.

The top-cohomology of Artin groups with coefficients in rank-1 local systems over Z☆

Abstract Let W be a Coxeter group and let G w be the associated Artin group . We consider the local system over k ( G w , 1) with coefficients in R = Z [q, q −1 ] which associates to the standard generators of G w the multiplication by q . For the all list of finite irreducible Coxeter groups we calculate the top-cohomology of this local system. It turns out that the ideal which we compute is a sort of Alexander ideal for a hypersurface. In case of the classical braid group Br n this ideal is the principal ideal generated by the nth cyclotomic polynomial . We use these results to calculate the topological category of k ( G w , 1): we prove that it equals the obvious bound given by obstruction theory (so, in case of braid group Br n , it is exactly n ).

A Characteristic Free Approach to Invariant Theory

Publisher Summary This chapter focuses on a portion of classical invariant theory that goes under the name of the first and second fundamental theorem for the classical groups in a characteristic-free way, that is, where the base ring A is any commutative ring, in particular the integers or an arbitrary field. The ring of polynomial functions over A in the entries of n m-vectors x1,…, xn and n m-covectors 1, …, n left formally invariant under the action of GL(m, –) is generated over A by the scalar products The chapter presents a theorem that states that the ring Rn has a basis over A formed by the double-standard tableaux with every row of length ≤ n.