Geometric Representation Theory

Instructor Information

• Lecturer: Dr Swarnava Mukhopadhyay
• Office: MTH 4416
• Email: swarnava@umd.edu
• Phone: (301) 405-5156

Topics Covered

• Review of representation theory of complex semisimple Lie algebras.
• Homegenous spaces as projective varieties and line bundles on G/P.
• Borel-Weil-Bott Theorem with complete proofs.
• Invariant Theory of GL(n) and invariant theory of Grassmannians.
• Strange duality of SL(n) invariants.
• Conformal Blocks in genus zero and discussion on associated open questions in representation theory.
• Enveloping algebras and PBW theorem.
• Generalized Cartan Matrix and Kac-Moody algebras.
• Structure Theory of Kac-Moody algebras, Weyl groups.
• Affine Lie algebras, Loop algebras, Central extensions.
• Invariant bilinear forms, symmetrizable Kac-Moody algebras
• Affine Weyl groups, Kac-Casimir operator.
• Category O and Representations of Kac-Moody algebras.
• Kac-Weyl character formula.
• Integrable modules and semisimplicity of Integrable modules.
• Ind Varieties, basic examples.
• Affine Kac-Moody groups, Parahoric Subgroups and their Flag varieties.
• Affine Grassmannians and Ind variety structure on Affine Grassmannians.
• Discussion on Uniformization theorems, Moduli of Principal Bundles and their Picard groups.

Resources

• Kac Moody groups, Flag Varieties and their representation Theory, S. Kumar
• Infinite dimensional Lie algebras, Victor. G. Kac
• Complex Semisimple Lie algebras, J. P. Serre
• Invariant Theory of GL(n) and invariant theory of Grassmannians, P. Belkale
• Notes on Borel-Weil-Bott Theorem, S. Kumar
• Generalized Theta functions and conformal blocks, A. Beauville, Y. Laszlo
• B-structures on G-bundles and local triviality, V. Drinfeld, C. Simpsion
• Vanishing and identities of conformal blocks divisors, P. Belkale, A. Gibney, S. Mukhopadhyay
• Representation Theory, W. Fulton, J. Harris