DS10 - Défi de tous les savoirs

Geometrical methods in Lie theory – GéoLie

Submission summary

This project is relevant to Lie theory, a well-established domain involving algebra, analysis and geometry. The objectives specifically
deal with the interplay between the algebraic and geometric aspects of the theory.
The geometric methods pop up with a rich variety: complex algebraic geometry,
Schubert calculus, rational points on varieties, affine Grassmannians, Bruhat-Tits buildings, Berkovich spaces...
This project, divided into 4 axes, aims at a better dissemination of ideas from these various fields.

1- Points of algebraic groups, Bruhat-Tits buildings and Berkovich spaces.

Let K be a complete valued field.
Bruhat-Tits theory associates to any reductive K-group $G$ its building,
a cellular metric space with a highly transitive G(K)-action.
This is a deep fact leading to a good understanding of the rational points G(K).
We plan to extend this to the context of pseudo-reductive groups over imperfect fields.
The expected applications involve (rational points of) G-homogeneous spaces under,
and the Kneser-Tits problem for pseudo-reductive groups.
One important expected tool is Berkovich geometry which, among other things,
provides a useful viewpoint on integral structures of G.

2- Affine Grassmannian and Kac-Moody groups.

The geometric Satake correspondence reconstructs a reductive group from the
geometry of the affine Grassmannian of its Langlands dual.
The possibility of embedding the affine Grassmannian in the Bruhat-Tits
building allows for explicit calculations. Certain subvarieties of the affine Grassmannian,
called MV cycles, play a crucial role in the geometric interpretation of combinatorial
formulas and in the construction of bases with nice positivity properties in the
representations. We plan:
(i) to study bases in tensor products of representations that are compatible
with the isotypical decomposition;
(ii) to extend to the case of Kac-Moody groups the possibility of recovering an
MV cycle from its image by the moment map.


3- (Quantum) cohomology of homogeneous spaces.

We will study (quantum, equivariant) cohomology and K-theory of
homogeneous spaces. Two classical approaches exist: to find a presentation
for these rings or to find a combinatorial model for the structure coefficients.
At the combinatorial level, we will look for an analogue of the Littlewood-Richardson
rule for the Belkale-Kumar product. At the geometric level, we will try to prove rational
connectedness of Gromov-Witten varieties, with applications to quantum K-theory.

4- Geometric problems related to (co)-adjoint action.

One typical problem is the study of singularities of closures of nilpotent orbits.
Their desingularizations lead to a deep
connection between nilpotent orbits and irreducible representations of the Weyl group.
Their deformations, realized by the adjoint quotient map and a transversal
(Slodowy) slicet, play an important role in several places:
invariant theory, (affine) W-algebras, etc.
Followin Einsenbud-Frenkel, a (new) point of view is to consider the jet schemes of these
orbit closures. This aspect has also connections with affine Kac-Moody algebras.
Some other variants relevant to our projects are the study of the orbits of a product of
flag varieties, varieties of commuting (nilpotent) elements, and their interplay with some
infinite-dimensional Lie algebras such as affine Lie algebras and 2-toroidal algebras.

These axes have deep and strong interplays. For example, the multiplicative Horn
problem about orbits of compact Lie groups is related to Geometric Invariant Theory,
quantum Schubert calculus, affine Grassmannian and semistability in infinite dimension.
The conjugacy problem in infinite-dimensional Lie algebras, as studied by
Gille et al., is relevant to both items 1 and 2.
The work of de Jong-He-Starr on rationality questions for homogeneous spaces is of interest for
both quantum Schubert calculus and Galois cohomology.

Project coordination

Nicolas Ressayre (Institut Camille Jordan)

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.

Partner

IECL Institut Elie Cartan
ICJ Institut Camille Jordan
ICJ Institut Camille Jordan

Help of the ANR 351,000 euros
Beginning and duration of the scientific project: September 2015 - 48 Months

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