Pre-Talbot seminar, lecture 1 Scott Carnahan
This talk is an overview. For simplicity, everything is done over thecomplex numbers.
What is Langlands duality? It relates two reductive algebraic groups
G and G∨ through their maximal tori. In particular, the weights of G,
defined as maps T → Gm , are identified with the coweights of G∨ ,defined as maps Gm → T ∨ .
Go∪
T DDt|/ G∨∪∨"* ; Txxx

D!

Gm
There is also an identification of roots with coroots, which I will notdescribe. The typical example is GLn , which happens to be dual toitself. Sp2n is dual to SO2n+1 .
A standard strategy in the Langlands program is the use of geometry
involving one group to study the representation theory of the Langlands
dual group. This is sufficiently vague that I can say it with someconfidence.
Representations of G - the finite dimensional representations form
an abelian category. Since G is reductive, they are completely reducible, i.e., they can be decomposed as direct sums of irreducibles.
The standard strategy for understanding them is looking at the actionof the subgroup T . Since T is abelian, any representation decomposes
as a sum of one-dimensional irreducibles, known as weights. Irreducible
representations of G are parametrized by dominant integral weights.
There is a geometric description of the irreducible representations,provided by Borel-Weil-Bott. It asserts a bijection:
Irreducible
Equivariant line
reps of G
bundles on G/B
The right arrow is given by the associated bundle construction: A
dominant integral weight is a representation of T , and we can extend it
uniquely to a representation of the Borel B (this is a maximal solvable
subgroup of G containing T - for G = GLn , it is conjugate to the groupof upper triangular matrices). The projection G → G/B has fiber B,
and we replace B by this one-dimensional representation to get a line
bundle on G/B.
The left arrow is given by taking global sections.
This is a nice picture, but it does not capture the full category. In
particular, the tensor structure is non-obvious. Also, if G is a torus,
12then G/B is a point, equivariant line bundles are tautologically representations, and we have no new information.
There is an alternative geometric description of representations of thetorus T , namely as sheaves of vector spaces on the weight lattice Λ. The
lattice is a discrete topological space, so a sheaf is just an assignmentof vector spaces to points. The torus acts on any particular vector
space by the corresponding weight. Irreducibles are one-dimensional
spaces supported at a single point, direct sums are obvious, and tensor
products are given by convolution:
M(V ⊗ W )λ =Vµ ⊗ Wλ−µµ∈Λ

We can also bring the Langlands dual into this picture:Λ = Hom(T, Gm ) = Hom(Gm , T ∨ )The object on the right is called the space of “polynomial loops inT .” There is another algebraic notion of loops, known as Laurent
loops: T ∨ ((t)) := HomSpec C (Spec C((t)), T ∨ ). This space is rather
large, but if we quotient out by T ∨ [[t]], the space of “jets in T ∨ ,” wehave a natural isomorphism:∨Hom(Gm , T ∨ ) = (T ∨ ((t))/T ∨ [[t]]) (C)If T is one dimensional we can see this in the following way. An
invertible Laurent series has the form aN tN + aN +1 tN +1 + . . . where N
is some integer, and aN 6= 0. For each such series, there is a uniqueinvertible Taylor series b0 + b1 t + b2 t2 + . . . which yields a monic monomial upon multiplication. Thus, points of the quotient are identified
with integers by taking the exponent (in this case, N ).
T ∨ ((t))/T ∨ [[t]] is called GrT ∨ , the affine Grassmannian of T ∨ . Thereis another way to look at GrT ∨ , using algebraic geometry.
Let X be a complete curve, and let x ∈ X be a point. Make a trivial∨T bundle on the punctured curve X \ {x}. To extend this to a T ∨bundle on X, we glue a trivial bundle over a small disc, along a small
punctured disc. Our disc will be Spec C[[t]], and the punctured discwill be its generic point Spec C((t)). There is a transitive action ofT ∨ ((t)) on the space of gluing data, with stabilizer T ∨ [[t]] (since it justchanges the trivialization of the bundle on the disc). The quotient canthen be written:T ∨ -bundles on X with
GrT ∨ ,x :=trivialization away from x

3Similarly, we can remove more than one point, so we have a family
over X n for any n. In particular, for any pair (x1 , x2 ) ∈ X 2 , we haveT ∨ -bundles on X with
GrT ∨ ,x1 ,x2 :=trivialization away from x1 , x2
When x1 is far away from x2 , the fiber is GrT ∨ ,x1 ×GrT ∨ ,x2 . However,the fiber only depends on the punctured curve X \ {x1 , x2 }, so whenx1 = x2 , we just have GrT∨ . In fact, when x1 and x2 are infinitesimally
nearby, we get an identification of fibers, and this endows the familywith an integrable connection. This property of the Grassmannian,where we can degenerate GrT ∨ × GrT ∨ to GrT ∨ , is called factorization,
and we say that the Grassmannian is a factorization space. Here is a
picture of the fibers near the diagonal:
TTTT jjj
jTjjjj TTT
TTTT jjj
jTjjjj TTT
TTTT jjjjTjjjj TTT•∆

Each point above the diagonal is given by joining infinitely many
points away from it, following the convolution law mentioned above.
I should point out that the Grassmannian is not just a countable collection of points. If you have a flat family in which multiple points
combine into a single point as above, that single point is non-reduced,i.e., there is some nilpotent fuzz (more precisely, maps from Spec R toGrT ∨ are not in natural bijection with the weight lattice when a ringR has nilpotent elements). In our case, this non-reduced behavior isquite extreme, since those multiple points in question are themselves
fuzzy, each being isomorphic to the single point.
Recall that representations of T are sheaves on GrT ∨ . Because ofthis factorization property, we can define the tensor product of modules
geometrically, by mashing two points on the curve X together, fusing
the T -representations over them. This idea is due to Beilinson andDrinfeld.
We would like to make this picture work for any reductive group G.
As before, we can choose a curve X and a point x ∈ X.
G∨ -bundles on X with
GrG∨ ,x :=trivialization away from x
This also gives us a factorization space, but for nonabelian G, it
has highly nontrivial geometry. However, GrG∨ can be described as a

4union of finite-dimensional G∨ [[t]] orbits, where G∨ [[t]] naturally actson the left through its inclusion into G∨ ((t)). These orbits are locallyclosed, and parametrized by dominant integral weights of G (or morecanonically, W -orbits of weights).
We’d like to associate some kind of sheaf data to these orbits, so thatthe sheaves are in bijection with irreducible representations of G. A
first approximation would be taking the constant sheaf Cλ on the orbit
Grλ corresponding to the dominant integral weight λ, and pushing it
somehow into GrG∨ along the inclusion map. We should choose a good
category of sheaves on GrG∨ , so that our pushforward is well-behaved.
The Grassmannian is a stratified space (with strata given by G∨ [[t]]orbits), and perverse sheaves are more or less designed to work on
stratified spaces. In particular, G∨ [[t]]-equivariant perverse sheaves are
locally constant on the strata of GrG∨ . The following theorem (knownas the geometric Satake equivalence) indicates that our guess is moreor less correct:
Theorem (Mirkovič, Vilonen) There is an equivalence of symmetrictensor categories:

Finite dimensional
representations of G∼=G∨ [[t]]-equivariant perverse
sheaves on GrG∨

The irreducible representation with highest weight λ is taken to theperverse sheaf ICλ , which is supported on the closure of Grλ in GrG∨ ,and its restriction to Grλ is constant in cohomological degree −2hρ, λi.
AnyLequivariant perverse sheaf F is taken to its global cohomology
space
H i (GrG∨ , F).
Quantum groups
The category of representations of group G has a tensor structure,induced by the diagonal inclusion ∆ : G → G × G, which is a grouphomomorphism. We’d like to deform it somehow, but groups are too
rigid, so we need to pass to an equivalent category with a floppierdeformation theory. If G is simple and simply connected, we have anequivalence Rep G ∼= Rep g ∼= Rep U (g), where g is the Lie algebra ofG, and U (g) is its universal enveloping algebra.
U (g) has a Hopf algebra structure, which means in particular that
we have an algebra homomorphism ∆ : U (g) → U (g) ⊗ U (g) satisfying
some identities. This gives the representation category a monoidalstructure (the same as for G), but since we are now working with analgebra structure over vector spaces, there is some additional freedom.

5Theorem (Drinfeld) If G is simple, then the tensor category Rep U (g)has a one dimensional space of deformations in braided tensor categories.
Drinfeld proved it for infinitesimal deformations, but it was latershown that these could be integrated to a one-parameter family Uq (g)for q ∈ C× not a root of unity. When q is a root of unity, one can still
construct Hopf algebras, but it is more complicated.
Is there a geometric way to see representations of Uq (g)? One answer comes from Bezrukavnikov, Finkelberg, and Schechtman, whoconstructed a category F S c (G) of factorizable sheaves, and showedthat it is equivalent to representations of the small quantum group for
q = e−2πic . The word “small” here refers to a distinction that only
matters at roots of unity.
We’d like to use GrG∨ here somehow. One idea is to use the RiemannHilbert correspondence, which gives an equivalence between perverse
sheaves and D-modules, to construct a category of G∨ [[t]]-equivariant
twisted D-modules on GrG∨ . Unfortunately, Gaitsgory found thatwhen the twisting corresponds to q not a root of unity, the resulting
category is equivalent to representations of the trivial group.
Lurie’s idea was to use an additional equivalence of categories proved
by Frenkel, Gaitsgory, and Vilonen, between G∨ [[t]]-equivariant sheaves
on GrG∨ and the Whittaker category of N ((t))-equivariant sheaves, andtwist the Whittaker category. Gaitsgory then showed that this works:
Theorem (Gaitsgory) For c ∈/ Q (i.e., q not a root of unity),
W hit(GrG∨ ) ∼= F S c (G).