| Sept 20 |
Stefan Dawydiak |
Refresher
|
Based on examples from e.g. Justin
Campbell’s notes. |
| Sept 27 |
Dylan Butson |
Overview |
I will give an overview talk about the first few topics of the
seminar, which are: D
modules, the Riemann-Hilbert correspondence, and
Beilinson-Bernstein localization. I will try to give a very
explicit, calculation-based introduction to these topics,
focusing on the example of ℙ1 (which will
explain the representation theory of 𝔰𝔩2) without
worrying too much about technical points. In the following weeks
we will try to fill in the details of these arguments, but I
thought it would be good for everyone to be on the same page
about what we’re trying to understand, so that we’re motivated
for the more technical talks. |
| Oct 4 |
Benjamin Gammage |
Introduction to Springer Theory |
We survey some main ideas surrounding springer theory, focusing on
simple examples. References: de
Cataldo-Migliorini Section
4.2, Yun Lecture
1, Chris
Elliott’s talk notes on a related paper, and
Nadler’s general perspective of which a brief summary can be
found at the beginning of this
paper. |
| Oct 11 |
Balazs Elek |
Geometry of the Flag Variety |
We will review some geometry of the flag variety and introduce
Bott-Samelson varieties in order to do some computations
generalizing those we saw on ℙ1 for SL2.
|
| Oct 18 |
Dylan Butson |
Technical Introduction to D
modules/Beilinson-Bernstein Localization for SL3
|
In the first half, I will give a more careful introduction to the
theory of D modules.
In the second half, I will give a detailed statement of the
Beilinson-Berstein equivalence and revisit the derivation of the
BGG resolution for L(0) from the
geometry of the flag variety in the SL2
and SL3
examples, following the calculations Balazs Elek explained last
time.
Attached are some notes I wrote about the
6 functors formalism for D modules. I won’t
exactly follow these notes in my talk, but they carefully
explain the conventions that I will use (which are not
completely standard, despite being canonical) and might be
helpful to look at before/during/after the talk. This should be
the last talk on BB localization and D modules for now, but we
will return to these topics when we discuss the proof of the
Kahzdan-Lusztig conjectures. |
| Oct 25 |
Stefan Dawydiak |
Reminder on perverse sheaves and the equivariant
derived category
|
In the first half, I’ll give a reminder on perverse sheaves,
beginning with recalling the definition of the perverse t-structure on Dcb(X)
before recalling the various exactness results we proved last
year. These will be illustrated by a series of example
computations, which will be either parallel to computations
we’ve hopefully all seen in category 𝒪, or will be continuations
of the exercises done in the first reminder talk on
constructible sheaves. The second half will introduce the
Bernstein-Lunts equivariant derived category. Equivariant
derived categories will feature heavily in the applications
chapters, and we’ll begin by briefly motivating them by
considering their analogues on the level of functions. We will
follow a mixture of Achar ch. 6 and Ch. 2 and 3 of Bernstein,
Lunts, Equivariant |
| Nov 1 |
Dylan Butson |
The Riemann-Hilbert correspondence, toward the
proof of the Kazhdan-Lusztig conjecture |
In the first half, I will outline a proof of the Riemann-Hilbert
correspondence. In the second half, I will discuss equivariant
D modules, the
relation of equivariance to integrability of 𝔤-modules under
Beilinson-Bernstein localization, and explain how this allows us
to reduce the Kazhdan-Lusztig conjecture to a statement about
B equivariant sheaves
on G/B. |
| Nov 8 |
Stefan Dawydiak |
Statement of proof of the Kazhdan-Lusztig
conjecture |
In the first part of the talk, I’ll define the Hecke algebra and
formulate Kazhdan-Lusztig conjectures. The Hecke algebra is
equipped with three natural bases and an involution, and we will
see what this data corresponds to in each of principal block
of category 𝒪,
B-equivariant
D-modules on
G/B,
and B-equivariant
sheaves on G/B.
I’ll talk about the category of B-equivariant
perverse sheaves on G/B and how
to convolve them (it is this operation which requires visiting
the equivariant derived category), which will lead to a
categorification of the Hecke algebra in terms of the additive
category of the so-called semisimple complexes. From here we
will be able to prove the conjecture and comment on
categorifications of Hecke algebras in other contexts. The talk
is based primarily on sections 7.1 through 7.3 of Achar, but see
also Kostya’s notes from the seminar last year for an
alternative proof of the Kazhdan-Lusztig conjectures which does
not require the equivariant derived category, but which also
does not give a categorification of the Hecke algebra. |
| Nov 15 |
Pavel Shlykov |
Preliminaries on Springer correspondence |
I will try to review the Chriss-Ginzburg style construction of the
Springer correspondence and show you by the clumsiness of my
computations, that perverse sheaves (which I won’t use at all)
should be a better tool. |
| Nov 22 |
Suriya Raghavendran |
More on the Springer correspondence |
TBA |
| Nov 29 |
Anne Dranowski |
Geometric Satake correspondence 1 |
We will give a definition of the affine Grassmannian, state the classical Satake correspondence, and set up the geometric Satake correspondence by investigating several examples. Our main references are Achar (for the definition), Gross (for the classical Satake), and Lusztig (for the Geometric Satake). For a comprehensive review, we recommend these notes of Goresky. Joel also recommends Baumann and Riche's notes. |
| Dec 6 |
Roger Bai |
Geometric Satake correspondence 2 |
TBA |
| X |
Y |
Z |
References: Coherent sheaves on ℙn
and problems of linear algebra |