Quantized symplectic singularities and applications to Lie theory.

This is an approximately 10 hour lecture series to be delivered at MIT on June 13-17. We plan to have 2 hours per day in the morning with some exercise/discussion sessions to follow in the afternoon/evening. Mitya Matvieievskyi has kindly agreed to be a TA for this course.

Description: Singular symplectic varieties were introduced by Beauville in the beginning of 2000's. Many of such varieties appear in the geometric representation theory as well (google "Okounkov on symplectic resolutions"), in particular, their "quantizations" (as well as quantizations of their partial resolutions) are often algebras with interesting representation theory. The most famous example is the universal enveloping algebras of semisimple Lie algebras (or sheaves of differential operators on flag varieties). Recent advances in Birational geometry allow to classify the quantizations and perform certain representation theoretic constructions.

Interesting representations of quantized singular symplectic often arise as "quantizations of largangian subvarieties". This is the case with categories O that have been extensively studied in the last decade. Another important example, also motivated by Lie theory, is Harish-Chandra (bi)modules. For the universal enveloping algebras, Harish-Chandra modules were introduced by Harish-Chandra in the 50's in an attempt to understand the representation theory of real semisimple Lie groups algebraically. The general framework of quantized symplectic singularities an their representations allows to get several new results about Harish-Chandra modules that are not accessible to the traditional Lie theoretic techniques.


  • Lecture 1 (2-190, 9.30-11.45): Orbit method, filtered quantizations, nilpotent orbits (part 1). Exercises. Notes.
  • Lecture 2 (2-190, 9.30-11.45): Covers of nilpotent orbits. Singular symplectic varieties. Classification of filtered quantizations. Exercises. Notes.
  • Lecture 3 (2-190, 9.30-11.45& 2.30-3.30): Q-factorial terminalizations. The case of covers of nilpotent orbits. Exercises. Notes.
  • Lecture 4 (2-190, 9.30-11.45& 2.30-3.30). Exercises (the discussion session will be 3.30-4.30). Notes.
  • Lecture 5 (2-190, 9.30-11.45). Notes.

    Office hours: M, 2.30-3.30, T,W, 4.30-5.30, Th, 1.15-2.15 in 2-167. Lecture notes are to be posted after the lecture. On Wednesday and Thursday, there will be overflow lectures, 2.30-3.30, location 2-190.

    Prerequisites: complex semisimple Lie algebras and Lie groups. Their classification and finite dimensional representations, and a few additional things such as the Harish-Chandra isomorphism for the center of the universal enveloping algebra. Some basic background in Symplectic geometry (see the first section in the references below). Somewhat less prominent: algebraic geometry, including birational projective morphisms, line bundles, coherent sheaf cohomology and such.