## Hervé SABOURIN

Director for Strategic projects of the Réseau Figure® (network of 31 universities)

Former Regional Director of the A.U.F (Agence Universitaire de la Francophonie) for the Middle East

Former Vice-President of the University of Poitiers (France)

Title: **Transverse Poisson Structures to adjoint orbits in a complex semi-simple Lie algebra**

Abstract:

The notion of transverse Poisson structure has been introduced by Arthur Weinstein stating in his famous splitting theorem that any Poisson Manifold M is, in the neighbourhood of each point m, the product of a symplectic manifold, the symplectic leaf S at m, and a submanifold N which can be endowed with a structure of Poisson manifold of rank 0 at m. N is called a transverse slice at M of S. When M is the dual of a complex Lie algebra g equipped with its standard Lie-Poisson structure, we know that the symplectic leaf through x is the coadjoint G. x of the adjoint Lie group G of g. Moreover, there is a natural way to describe the transverse slice to the coadjoint orbit and, using a canonical system of linear coordinates (q1, ….., qk), it follows that the coefficients of the transverse Poisson structure are rational in (q1, ….., qk). Then, one can wonder for which cases that structure is polynomial. Nice answers have been given when g is semi-simple, taking advantage of the explicit machinery of semi-simple Lie algebras. One shows that a general adjoint orbit can be reduced to the case of a nilpotent orbit where the transverse Poisson structure can be expressed in terms of quasihomogeneous polynomials. In particular, in the case of the subregular nilpotent orbit the Poisson structure is given by a determinantal formula and is entirely determined by the singular variety of nilpotent elements of the slice.

References:

- Sabourin , H. , Sur la structure transverse à une orbite nilpotente adjointe, Canad.J.Math, 57, (4), 2005, p 750-770.
- Sabourin, H. , Orbites nilpotentes sphériques et représentations unipotentes associées : Le cas SL(n), Representation Theory, 9, (2005), p 468-506.
- Sabourin, H. , Mémoire d'HDR, Quelques aspects de la méthode des orbites en théorie de Lie, Décembre 2005
- Damianou, P. , Sabourin, H. , Vanhaecke, P. , Transverse Poisson structures to adjoint orbits in semi-simple Lie algebras, Pacific J. Math,232, 111-139, 2007.
- Sabourin, H., Damianou, P. , Vanhaecke, P., Transverse Poisson structures : the subregular and the minimal orbits, Differential Geometry and its Applications Proc. Conf. in Honour of Leonhard Euler, Olomouc, August 2007
- Sabourin, H., Damianou, P. , Vanhaecke, P., Nilpotent orbits in simple Lie algebras and their transverse Poisson structures, American Institute of Physics, Conf. Proc. series, Vol. 1023 (2008), pp 148-152.