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: Tuesday, 10-11am, and Friday, 1.30-2.30pm in KT 715.

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

Homework:

  • Problem 1, due Oct 1.
  • Homework 2 (Problems 2 and 3), due Nov 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. Notes. References.
  • Oct 1: lecture 8, Basics on category O 4: we will study equivalences between infinitesimal blocks coming from translation functors as well as the structure of the blocks as A-linear abelian categories. Notes. References.
  • Oct 6: lecture 9, Soergel theory 1: We proceed to proving Soergel's theorems starting with the 2nd: the Soergel functor is fully faithful on the projectives. Notes. References.
  • Oct 8: lecture 10, Soergel theory 2: We finish the proof of the full faithfulness (Soergel's 2nd theorem) and start moving towards the defomed version of the first theorem, in particular, discuss the center of the deformed category O. Notes, the references are the same as for lecture 9.
  • Oct 13: lecture 11, Soergel theory 3: We continue to investigate the structure of the center of the category O_A^\Xi, the endomorphism algebra of P_A(\lambda^-), and a connection between them. Notes, the references are the same as for lecture 9.
  • Oct 15: lecture 12, Soergel theory 4: We continue proving the main result announced in Sec 1.0 of Lec 11. Notes and references.
  • Oct 20: lecture 13, Soergel theory 5: We proceed to the deformed and generalized version of the third theorem of Soergel. Time permitting we also discuss an application of representation theoretic results to Soergel (bi)modules. Notes and references.
  • Oct 27: lecture 14, What's next 1: we discuss some features of Soergel's approach, formulate the problems we are going to study next, and discuss relatives of the category O. Notes, references.
  • Oct 29: lecture 15, What's next 2: we continue our discussion of the structure of affine categories O including negative, positive and critical levels. We state a relation with the quantum category O. Notes, references.
  • Nov 3, lecture 16: What's next 3: we discuss Bezrukavnikov's equivalence between coherent and constructible realizations of affine Hecke category as well as a connection of the constructible side with the affine category O via a Beilinson-Bernstein type localization theorem. Notes, references.
  • Nov 5, lecture 17: What's next 4: we discuss things around Beilinson-Bernstein localization for negative and positive level categories O. Time permitting, we start our discussion of highest weight categories. Notes, references.
  • Nov 10, lecture 18: Highest weight categories 1: We discuss the structure of highest weight categories. Notes. A reference for highest weight categories over rings is Section 4.1 in Rouquier's paper.
  • Nov 12, lecture 19: Highest weight categories 2: We introduce costandard objects and apply them to the study of the category of standardly filtered objects.