MATH7720 (Fall 2026): Category O and Soergel theory.

Instructor: Prof. Ivan Loseu (email: ivan.loseu@yale.edu)

Lectures: TTh 1.05-2.20, Location: KT205.

Office hours: For the first few weeks: Thursday Sept 10 8.30-9.30pm on zoom; Thursday Sept 17, 10-11am, Friday, Sept 18, 1.30-2.30pm, Tuesday, Sept 22, 10-11am in KT 715.

See here for a course description, prerequisites and some references.

Homework:

  • Problem 1, due Oct 1.

    Schedule:

  • Sept 3 (on zoom), lecture 1, Introduction 1: I will define category O, explain its basic properties (w/o proofs, those will be explained later) and state three theorems of Soergel, which form the origin of the Soergel theory. "Official" notes and notes from the lecture. References.
  • Sept 8 (on zoom), lecture 2, Introduction 2: We will define Soergel (bi)modules and explain result about them "on their own". "Official" notes and notes from the lecture. References.
  • Sept 10 (on zoom), lecture 3, Introduction 3: We briefly discuss the Koszul duality equivalence for the category O. Then we start discussing the relevant geometry: strongly equivariant modules, and a connection between the principal block of category O and equivariant D-modules on the flag variety. "Official" notes and notes from the lecture. References.
  • Sept 15 (on zoom), lecture 4, Introduction 4: This is the last lecture in the intro part (after which we start to actually prove stuff). I give a crash-course on equivariant D-modules and explain a connection between suitably equivariant D-modules on the flag varieties to Soegel (bi)modules/ a geometric meaning of various things introduced in Lec 2. "Official" notes and notes from the lecture. References.
  • Sept 17 (in person!), lecture 5, Basics on category O 1: We start with sketching the proof of Thm 2 from Sec 1 of Lec 3. Then we start a new topic: basics of category O. We will introduce a deformed version of the category O and establish infinitesimal block decompositions and various finiteness statements. notes and references.
  • Sept 22 (again in person!), lecture 6, Basics on category O 2: We prove that objects in category O over a field have finite length and then discuss projective objects and projective functors. Notes. References.
  • Sept 24, no class, to be rescheduled to some Friday later on in the semester.
  • Sept 29 (in person until the end of the semester): lecture 7, Basics on category O 3: We examine special cases of projective functors: translation to and from a wall and reflections. Using these we produce projectives in the deformed principal block with nice properties.