|Geometric Satake Correspondence and basis theory|
|Time：||Mon/Thu 15:20-17:20, 2015-09-17~ 2015-12-14|
|Instructor：||Jiuzu Hong [Yale University]|
|Place：||Mon Lecture Hall, Floor 3; Thu Conference Room 3, Floor 2|
Geometric Satake correspondence is a bridge relating the geometry of affine Grassmannian of reductive groups and the representation theory of the Langlands dual groups. It is very fundamental and the first step in geometric Langlands program. As a consequence of geometric Satake, the bases of representations naturally arise. These bases are closely related to the canonical basis of Lusztig and Kashiwara, more precisely they share the same combinatorics with canonical basis.
This course will consist of two main parts, geometric Satake correspondence and the parametrization of bases arising from it. We will mainly focus on the following topics: affine Grassmannian, perverse sheaves, Tannakian formalism, hyperbolic restriction functors, Mirkovic-Vilonen cycles and polytopes, component of convolution varieties, configuration space of decorated flags, positivity and tropicalization, and so on.
Representation theory of reductive Lie groups and Lie algebras; sheaves theory; algebraic geometry
I.Mirković; K.Vilonen. Geometric Langlands duality and representations of algebraic groups over commutative rings. Ann. of Math. (2) 166 (2007), no. 1, 95–143.
A.Braverman; D.Gaitsgory. Crystals via the affine Grassmannian. Duke Math. J. 107(2001), no. 3, 561–575.
J.Kamnitzer,. Mirković-Vilonen cycles and polytopes. Ann. of Math. (2) 171 (2010), no. 1, 245–294.