# Math 7890: Topics in Representation Theory

**Description:** Conjectured "Langlands dualities"
typically relate "automorphic" objects on a reductive group G to "Galois" objects on
a "dual" reductive group G'. The Satake isomorphism is a classical theorem
in the representation theory of p-adic groups (eg GL_n(Q_p)) that is in
some sense the simplest example of a "Langlands duality"; in particular,
more than any other result, it accounts for the mysterious presence of the
dual group G'. It plays an absolutely basic role in the arithmetic
Langlands conjectures. The geometric Satake equivalence is a
categorification of the classical isomorphism; it is essentially the
"local unramified geometric Langlands correspondence." Geometric Satake is
needed even to begin formulating a geometric Langlands program, and,
remarkably, it also seems to have a great deal to contribute to the
arithmetic story as well.

This course is an introduction to the geometric Satake equivalence, although we will spend most of our time developing preliminary
material in representation theory, sheaf theory, and algebraic geometry.

**Prerequisites:** Some familiarity with basic algebraic geometry, representation theory, sheaf theory, and homological algebra. We will review much of
the background material as the course progresses.

**Topics:**

review of the structure theory and representation theory of compact Lie
groups and reductive algebraic groups (with a geometric emphasis, eg
Borel-Weil theorem); Tannakian categories;
introduction to the smooth representation theory of locally profinite
groups; unramified representations of p-adic groups; the classical
Satake isomorphism;
derived categories of sheaves and Grothendieck's 6 operations; t-structures on triangulated categories; basic geometric theory of perverse sheaves;
geometry of affine Grassmannians; the geometric Satake equivalence;
**References:**

For background in representation theory and algebraic groups, you might look at Fulton & Harris, *Representation Theory*, and Milne's
notes on algebraic groups. For Tannakian categories, see Deligne & Milne, *Tannakian Categories*
(in LNM 900).
For smooth representation theory of locally profinite/p-adic groups and the classical Satake isomorphism, see Bushnell & Henniart,
*The Local Langlands Correspondence for GL(2)*; Cartier, * Representations of p-adic groups* in Proc. Symp. Pure Math. XXXIII, Part 1; Gross, *On the
Satake isomorphism*; and Casselman's notes.
For homological algebra and sheaf theory, see Gelfand & Manin, *Methods of Homological algebra*, or Kashiwara & Schapira, * Sheaves on Manifolds*.
For perverse sheaves, see Beilinson, Bernstein, Deligne, *Faisceaux Pervers*; or Kiehl & Weissauer, *Weil Conjectures, Perverse Sheaves, and l-adic Fourier
Transform*. Also see the survey article of de Cataldo &
Migliorini.
For affine Grassmannians, geometric Satake, and related, see Zhu's notes; and
these notes from Gaitsgory's seminar; and the papers (see the bibliography in Zhu's notes
for detailed citations) of Beauville & Laszlo, Mirkovic & Vilonen, Gaitsgory, Beilinson & Drinfeld (the book with ``Hecke eigensheaves" in the title),
and Richarz.
**Office Hours:** Friday 4-5pm (or get in touch with me to arrange another time)

**Homework:** The primary expectation is that students will review the lecture notes and work to fill in any background material that they have not previously encountered. To
help encourage this, on a rotating basis students will be responsible for carefully TeXing up lecture notes, to be distributed to the rest of the class. I will (correction:
thought I would) also
maintain a running list of exercises here. You are encouraged to discuss these with one another and to include solutions to relevant problems
in the notes you are writing up, but the problems do not otherwise have to be handed in.

**Course Notes:** I will keep a cumulative file **here** of the course notes to date. The note-takers will to some extent fill in the details of
statements I left in class as exercises, but these should be indicated in the notes: you are encouraged to try these exercises before reading on.
If you have comments, questions, or corrections, please
send them to me and the note-taker of the corresponding section (indicated in the section titles). When you are taking notes, please use the following LaTeX template:

main file and preamble. If you define new commands (please do so sparingly), send me an updated copy of
this preamble along with the body of your notes. Here if anyone is interested are the individual body files:
Compact Groups (Michael Zhao)
Algebraic Groups (Adam Brown)
Flag varieties and Bruhat Decomposition (Anna Romanova)
Borel-Weil and Theorem of the Highest Weight (Allechar Serrano López and Sean McAffee)
Tannakian Categories (Christian Klevdal and Shiang Tang)
Introduction to Smooth Representations of p-adic Groups (Kevin Childers)
Hecke Algebras (Sabine Lang)
Statement and Interpretation of the Satake Isomorphism (Shiang Tang)
Proof of Satake isomorphism. Sheaf-function dictionary. (Christian Klevdal)
Introduction to the affine Grassmannian (Marin Petkovic)
Review of homological algebra (triangulated and derived categories) (Allechar Serrano López)
Derived functors, start of sheaf cohomology (Michael Zhao)
Perverse Sheaves (Christian Klevdal)
Glueing t-structures (Sabine Lang)
Intermediate Extension. Examples. (Allechar Serrano López)