Deformation spaces of one-dimensional formal modules and their cohomology

Let Mm be the formal scheme which represents the functor of deformations of a one-dimensional formal module over equipped with a level-m-structure. By work of Boyer (in equal characteristic) and Harris and Taylor, the ?-adic étale cohomology of the generic fibre Mm of Mm realizes simultaneously the local Langlands and Jacquet-Langlands correspondences. The proofs given so far use Drinfeld modular varieties or Shimura varieties to derive this local result. In this paper we show without the use of global moduli spaces that the Jacquet-Langlands correspondence is realized by the Euler-Poincaré characteristic of the cohomology. Under a certain finiteness assumption on the cohomology groups, it is shown that the correspondence is realized in only one degree. One main ingredient of the proof consists in analyzing the boundary of the deformation spaces and in studying larger spaces which can be considered as compactifications of the spaces Mm.     

La somme des faux degrés—un mystère en théorie des invariants

We continue the study of the polynomiality of the semicentre Sz(p) of the enveloping algebra of a parabolic subalgebra p of a semisimple Lie algebra g, motivated by its truth when g is of type A or C [F. Fauquant-Millet, A. Joseph, Semi-centre de l'algèbre enveloppante d'une sous-algèbre parabolique d'une algèbre de Lie semi-simple, Ann. Sci. École Norm. Sup. (4) 38 (2) (2005) 155-191] and when p=b, a Borel subalgebra [A. Joseph, A preparation theorem for the prime spectrum of a semisimple Lie algebra, J. Algebra 48 (1977) 241-289] and p=g (Chevalley).We construct a linear map of Sz(b) into Sz(g) and show it to be an isomorphism just in types A and C. We link this to the difficulty of proving the polynomiality of Sz(p) outside types A and C. It leads to “false degrees” defined by underlying combinatorial structure. These are the true degrees when the bounds in [F. Fauquant-Millet, A. Joseph, Semi-centre de l'algèbre enveloppante d'une sous-algèbre parabolique d'une algèbre de Lie semi-simple, Ann. Sci. École Norm. Sup. (4) 38 (2) (2005) 155-191] coincide and polynomiality ensues. We show that these false degrees always sum to which can fail for the true degrees when they are defined. Finally we prove the Tauvel-Yu conjecture on the index of a parabolic.     

Recurrent random walks on homogeneous spaces of p-adic algebraic groups of polynomial growth

Let G be a p-adic algebraic group of polynomial growth and H be a closed subgroup of G. We prove the growth conjecture for the homogeneous space G/H, that is, G/H supports a recurrent random walk if and only if G/H has polynomial growth of degree atmost two. Received: 23 November 2007     

A twist condition and periodic solutions of Hamiltonian systems

In this paper, we investigate existence of nontrivial periodic solutions to the Hamiltonian system

(HS)     

Relatively compact sets of Heisenberg manifolds

We give a necessary and sufficient condition for a set of left invariant metrics on a compact Heisenberg manifold to be relatively compact in the corresponding moduli space.     

Resolvent and lattice points on symmetric spaces of strictly negative curvature

We study the asymptotics of the lattice point counting function for a Riemannian symmetric space X obtained from a semisimple Lie group of real rank one and a discontinuous group of motions in X, such that has finite volume. We show that as , for each . The constant corresponds to the sum of the positive roots of the Lie group associated to X, and n = dimX. The sum in the main term runs over a system of orthonormal eigenfunctions of the Laplacian, such that the eigenvalues are less than . Received: 4 January 1999     

A pathwise approach to the extinction of branching processes with countably many types

We consider the extinction events of Galton–Watson processes with countably infinitely many types. In particular, we construct truncated and augmented Galton–Watson processes with finite but increasing sets of types. A pathwise approach is then used to show that, under some sufficient conditions, the corresponding sequence of extinction probability vectors converges to the global extinction probability vector of the Galton–Watson process with countably infinitely many types. Besides giving rise to a family of new iterative methods for computing the global extinction probability vector, our approach paves the way to new global extinction criteria for branching processes with countably infinitely many types.     

Generalized Littlewood–Richardson rule and sum of coadjoint orbits of compact Lie groups

Let G be a compact connected semisimple Lie group. We extend to all irreducible finite-dimensional representations of G a result of Heckman which provides a relation between the generalized Littlewood–Richardson rule and the sum of G-coadjoint orbits. As an application of our result, we describe the eigenvalues of a sum of two real skew-symmetric matrices.     

Blowup subalgebras of the Sklyanin algebra

We describe some interesting graded rings which are generated by degree-3 elements inside the Sklyanin algebra S, and prove that they have many good properties. Geometrically, these rings R correspond to blowups of the Sklyanin P² at 7 or fewer points. We show that the rings R are exactly those degree-3-generated subrings of S which are maximal orders in the quotient ring of the 3-Veronese of S.