Fall semester: Geometric representation theory.
Geometric representation theory applies algebraic geometry to the problems of
representation theory. Some of the most famous problems of representation
theory were solved on this way during the last 40 years. We will study
representations of the affine Hecke algebras using the geometry of affine
Grassmannians (Satake isomorphism) and Steinberg varieties of triples
(Deligne-Langlands conjecture). This is a course for master students knowing
the basics of algebraic geometry, sheaf theory, homology and K-theory.
Prerequisites:
Basic algebraic geometry (projective varieties, coherent and constructible
sheaves). Basic representation theory (of compact Lie groups).
Some experience in equivariant cohomology and K-theory is desirable.
Curriculum:
Affine Grassmannians.
Stratifications with finite and cofinite strata.
Example for the general linear group.
Semiinfinite orbits.
Hyperbolic localization.
Equivariant perverse sheaves.
Mirkovi\'c-Vilonen fiber functor.
Exactness of convolution.
Drinfeld fusion.
Commutativity constraint.
Tannakian formalism and the Langlands dual group.
Geometric Satake equivalence.
Affine Hecke algebras.
Steinberg variety of triples.
Polynomial representation.
Kazhdan-Lusztig-Ginzburg isomorphism.
Standard representations
Shapovalov form
Deligne-Langlands-Lusztig classification.
Textbooks:
N.Chriss and V.Ginzburg, Representation theory and complex geometry,
Birkh\"auser, Boston, 1997.