# Seminar

## Representation theory seminar 2020, Semester 2 Basic Notions in Representation Theory

Organisers: Kari Vilonen, Ting Xue

Topics: We will cover basic notions in representation theory going back to the geometric methods introduced around 1980. We will be discussing Beilinson-Bernstein localization, the Kazhdan-Lusztig conjectures, Koszul duality, and the Jantzen conjectures. The goal of the seminar is for people to learn and think about the topics we discuss. An original reference, prior to the introduction of Koszul duality, is Beilinson’s ICM talk in 1983. Here is the original Beilinson-Ginzburg preprint on Koszul duality.

Time: Thursdays 2:15pm-4:15pm.

Location: Online

Dec 17 Anna Romanov (Sydney) Jordan-Holder multiplicities of Verma modules with rational highest weight: continued Notes

Dougal Davis (Edinburgh) Re-cap: category O to monodromic D-modules to mixed Hodge modules to the Hecke algebra Notes

Dec 10 No seminar (AustMS meeting)

Dec 3 Anna Romanov (Sydney) Jordan-Holder multiplicities of Verma modules with rational highest weight Notes

Abstract: Earlier this semester, Gufang explained how we can compute the Jordan—Holder multiplicities of Verma modules in the principal block of category O by realising them in terms of constructible sheaves on the flag variety. In this talk, I will attempt to do the same thing for Verma modules with regular rational highest weight. Instead of sheaves on the flag variety, our topological players will be sheaves on line bundles over the flag variety with some extra structure. I will start by explaining the strategy laid out in Lusztig’s chapter “Computation of local intersection cohomology of certain line bundles over a Schubert variety”, then I will sketch another approach using only D-modules.

November 26 Dougal Davis (Edinburgh) Mixed geometry over finite fields: continued Notes

Kari Vilonen (Melbourne) Beilinson functors Notes

November 19 Kari Vilonen (Melbourne) Punctural purity and the socle filtration Notes

Dougal Davis (Edinburgh) Mixed geometry over finite fields Notes

November 12 Kari Vilonen (Melbourne) The Jantzen conjecture: continued Notes

November 5 Kari Vilonen (Melbourne) The Jantzen conjecture: continued Notes

October 29  Kari Vilonen (Melbourne) Comments on mixed geometry and the Jantzen conjecture Notes

Abstract: I will first make a few comments related to Gufang’s talk last week. I will recall some material from the previous talks and will discuss and formulate the Jantzen conjecture. I will also make some preliminary remarks to prepare for its proof.

October 22 Gufang Zhao (Melbourne) Koszulity of category O via mixed geometry: continued Notes

October 15 No seminar

October 8 Gufang Zhao (Melbourne) Koszulity of category O via mixed geometry Notes (pre-seminar) Notes (in-seminar)

Abstract: In this talk we review some basic facts about mixed Hodge modules. After that, we follow Section 4 of Beilinson, Ginzburg, Soergel’s paper to construct a graded lift of the principal block of category O using mixed Hodge modules, and then prove the Koszul property of the graded lift.

October 1 Kari Vilonen (Melbourne) Interlude Notes

September 24 Ting Xue (Melbourne) BGG category $\mathcal{O}$ and Koszul duality: continued Notes

September 17  Dougal Davis (Edinburgh) Beilinson-Bernstein localisation again: continued Notes

Ting Xue (Melbourne) BGG category $\mathcal{O}$ and Koszul duality Notes

Abstract: The study of BGG category $\mathcal{O}$, introduced in the 1970s by Joseph Bernstein, Israel Gelfand, and Sergei Gelfand,  “offers a rewarding tour of the beautiful terrain that lies just beyond the classical Cartan-Weyl theory of finite dimensional representations of $\mathfrak{g}$” (Humphreys). In the first part of the talk we will review the basic properties of the BGG category $\mathcal{O}$, duality in $\mathcal{O}$, BGG reciprocity, blocks of $\mathcal{O}$, and translation functors etc. In the second part we will discuss the Koszul self-duality of the principal block of category $\mathcal{O}$. We will also try to make connections to the previous lectures when we can.

September 10 Yaping Yang (Melbourne) An introduction to the Riemann–Hilbert correspondence: continued Notes

Dougal Davis (Edinburgh) Beilinson-Bernstein localisation again: continued Notes

September 3 (2pm-4pm) Note early start time!

Yaping Yang (Melbourne) An introduction to the Riemann–Hilbert correspondence: continued Notes

Dougal Davis (Edinburgh) Beilinson-Bernstein localisation again Notes

Abstract: In this talk, I will go back over Beilinson-Bernstein localisation in a bit more detail. This time, I will cover the general definition of twisted differential operators for non-integral weights, and the closely related notion of monodromic D-modules. With the aid of some very explicit examples for $G = SL_2$, I will also explain how to go back and forth between representations of the Lie algebra and monodromic D-modules on the flag variety in practice.

August 27 Gufang Zhao (Melbourne) Proof of Kazhdan-Lusztig conjectures on the Hecke algebra: continued

Yaping Yang (Melbourne) An introduction to the Riemann–Hilbert correspondence Notes

Abstract: I will give a general introduction to the Riemann–Hilbert correspondence. Let $X$ be an algebraic variety over $\mathbb{C}$ and let $X^{an}$ be the corresponding complex analytic variety in classical topology. The Riemann–Hilbert correspondence for regular singular connections was proved by Deligne in 1970. It is an equivalence between the category of flat connections on algebraic vector bundles on $X$ with regular singularities and the category of local systems on $X^{an}$. More generally, the Riemann–Hilbert correspondence for regular holonomic D-modules was proved by Kashiwara and Mebkhout independently in 1984. It says the de Rham functor induces an equivalence from the category of regular holonomic D-modules on $X$ to the category of perverse sheaves on $X^{an}$.

August 20  Peter McNamara (Melbourne) Beilinson-Bernstein localisation: continued Notes

Gufang Zhao (Melbourne) Proof of Kazhdan-Lusztig conjectures on the Hecke algebra Notes

Abstract: In this talk, we recall Kazhdan-Lusztig conjectures on the Hecke algebra.  Using the Beilinson-Bernstein localization theorem from Peter’s talk, and the Riemann-Hilbert correspondence,  we give a description of the irreducible modules and Verma modules of the Lie algebra in terms of constructible sheaves on the flag variety. Then, we describe an assignment of a Hecke algebra element to each complex of constructible sheaf. We use the decomposition theorem to show some properties of this assignment with respect to IC sheaves and constant sheaves of Schubert varieties. Finally we deduce Kazhdan-Lusztig conjecture expressing the coefficients of irreducible modules with respect to Verma modules of a Lie algebra in terms of Hecke algebras, and show the non-negativity of the coefficients.

August 13 Peter McNamara (Melbourne) Beilinson-Bernstein localisation Notes

Abstract: I will talk about the Beilinson Bernstein localisation theorem which gives an equivalence of categories between a category of representations of a Lie algebra, and a category of D-modules on the flag variety (we will incorporate the twist into the story). This theorem has since been a fundamental result in geometric representation theory. Examples will be provided when possible.