Speaker: Prof. Anthony Henderson (University of Sydney)
I. Geometric modular representation theory
- 16:00-17:00, January 30, 2015
Representation theory is one of the oldest areas of algebra, but many basic questions in it are still unanswered. This is especially true in the modular case, where one considers vector spaces over a field k of positive characteristic; typically, complications arise for particular small values of the characteristic. For example, from a vector space V one can construct the symmetric square S^2(V), which is one easy example of a representation of the group GL(V). One would like to say that this representation is irreducible, but that statement is not always true: if k has characteristic 2, there is a nontrivial invariant subspace. Even for GL(V), we do not know the dimensions of all irreducible representations in all characteristics.
In this talk, I will introduce some of the main ideas of geometric modular representation theory, a more recent approach which is making progress on some of these old problems. Essentially, the strategy is to re-formulate everything in terms of homology of various topological spaces, where k appears only as the field of coefficients and the spaces themselves are independent of k; thus, the modular anomalies in representation theory arise because homology with modular coefficients is detecting something about the topology that rational coefficients do not.
Juteau-Mautner-Williamson, `Perverse sheaves and modular representation theory', arXiv:0901.3322
II. The Springer correspondence
- 16:00-17:00, February 2, 2015
Here the representations are of a Weyl group (finite crystallographic reflection group), and the geometry is of the nilpotent cone in a simple Lie algebra and its desingularization.
Borho-MacPherson, `Representations de groupes de Weyl et homologie d'intersection pour les varietes nilpotentes', C. R. Acad. Sci. Math. 292 (1981)
Shoji, `Geometry of orbits and Springer correspondence', in Asterisque 168 (1988)
Juteau, `Modular Springer correspondence, decomposition matrices and basic sets', arXiv:1410.1471
Ⅲ. Character sheaves
-16:00-17:00, February 3, 2015
Here the representations are of a matrix group over a finite field, and the geometry is of the corresponding algebraic group and its conjugacy classes.
Lusztig, `Intersection cohomology complexes on a reductive group', Invent. Math. 75 (1984)
Lusztig, `Character sheaves' I-V, Adv. in Math. (1985-86)
Mars-Springer, `Character sheaves', in Asterisque 173--174 (1989)
Ⅳ. The geometric Satake equivalence
- 16:00-18:00, February 4, 2015
Here the representations are of an algebraic group, and the geometry is of the affine Grassmannian of the dual group.
Mirkovic-Vilonen, `Geometric Langlands duality and representations of algebraic groups over commutative rings', Ann. of Math. 166 (2007)
Kamnitzer, `Lectures on geometric constructions of the irreducible representations of GL_n', arXiv:0912.0569
Achar-Henderson-Riche, `Geometric Satake, Springer correspondence, and small representations II', arXiv:1205.5089
Place: Room 106, Math Science bldg. (GAIA seminar room)
Date: January 30, February 2, 3, 4, 2015
Contact: Hyemin Shin(email@example.com, Tel.279-8021)