These are my live-TeXed notes for the course *Math 270x: Topics in Automorphic Forms* taught by Jack Thorne at Harvard, Fall 2013.

Any mistakes are the fault of the notetaker. Let me know if you notice any mistakes or have any comments!

Recommended references for this course:

- Diamond, Shurman, Introduction to modular forms;
- Borel, Casselman, Automorphic forms, representations and L-function (Corvallis);
- Bushnell, Henniart, Local Landlands conjecture for GL(2);
- Bernstein, Gelbart, Introduction to the Langlands program.

09/10/2013

##
Modular forms and number fields

The absolute Galois group contains huge arithmetic information. One can ask algebraic number theoretic questions like: for which prime , is a quotient of (i.e., there exists a number field with such a Galois group)? Or how to describe Galois extensions of with prescribed local ramification behavior at a prime ? Miraculously that these kinds of questions can be answered using *modular/automorphic forms* and furthermore automorphic forms can be understood by the *Langlands dual group* .

(Eisenstein series)
For

, we define the

*weight Eisenstein series* It is easy to show that this series converges absolutely and converges uniformly in any compact subset of

. Notice that

acts on

on the right. The stabilizer of

is

. Thus

The weak modularity of

follows from the modular cocycle property of

.

is uniformly convergent in

and thus

So

is bounded as

. We have verified that

is a modular form of weight on

. A standard fact is that

has a nice

-expansion:

where

is the

-th Bernoulli number (a rational number).

The

*Ramanujan modular function* is defined to be

It is a

*cusp* form of weight 12 on

.

is the unique (up to scalar) cusp form on

of minuscule weight (i.e., there is no cusp forms of weight

on

). Moreover, all the coefficients of

-expansion of

are indeed integers.

*Hecke operators* The quotient has representatives , where and . One can then explicitly compute In other words, It follows directly that for .

*Peterson inner product* Fix an integer . For two holomorphic functions on , we define . One computes that In particular, is -invariant when . For , converges and defines a Hermitian positive definite inner product, called the *Peterson inner product*.

An important fact is that is *self-adjoint* with respect to the Peterson inner product. Therefore we can simultaneously diagonalize these commuting Hecke operators to obtain a basis of such that every is a -eigenvector with real eigenvalue for any .

The upshot is that starting from a weight , we obtain real numbers (the -eigenvalues) and modular forms indexed by the prime numbers . We can associate to this basis *Galois representations* (or *motives*) and doing becomes useful for constructing extensions of with prescribed behavior.

Let

and

be the

-th isotypic part of

. Herbrand and Ribet proved that for

, it is true that

if and only if

. In particular, when

, by class field theory, there exists a unramified

-extension of

corresponding to

. How do we construct this extension? Ribet managed to do so using modular forms. For example:

divides

and the explicit formula gives a congruence between

and

modulo

; the desired extension of

is then constructed using the Galois representation attached to

.

09/12/2013

##
Automorphic representations on

It begins to reveal the remarkable nature of modular/automorphic forms when you add a little bit representation theory.

Let

be a number field and

be the ring of adeles of

. The group

is defined to be the restricted direct product

with respect to the open compact subgroups

. Notice as an abstract group

is simply the group of invertible matrices with entries in

. We now consider

and write

for short.

This complicated definition generalizes the classical notion of modular forms (the advantage is that it puts the representation theory in scope as we consider all levels ).

Suppose the open compact subgroup

satisfies

. Then

is a congruence subgroup of

and

To prove the theorem, we need the following

- Pick an arbitrary lift . We would like a matrix such that and . By multiplying on the left and right by , we may assume . Then and the matrix is desired.
- Using and , we know that there is a bijection between free -modules in of rank and free module of of rank given by , . Consequently, it gives a bijection . In particular the desired result follows.
- Let . Use part (b) to write , where , . Taking determinants, we know . The claim follows from changing by an element of if necessary.
¡õ

(Strong approximation for )
is dense in

.

Let

be any open compact subgroup. We will show that

. We may assume that

, where

. By (c) of the previous lemma, we can write

as

, where

and

. By (a) of the previous lemma, we can find

such that

. Then

.

¡õ
Let

be an open compact subgroup such that

. Then

- .

- It follows from the and the strong approximation of .
- It follows from the first part because acts transitively on .
¡õ

(Theorem 1)
The bijection is constructed as

and

, where

such that

, whose existence is assured by the previous corollary. (One needs to check the latter is well-defined and

thus defined is a modular form of level

).

¡õ
The most interesting thing is that the group acts on both of the spaces by (so if ).

An

*automorphic representation* of weight

is an irreducible representation

of

which is isomorphic to a subquotient of

. Similarly, a

*cuspidal automorphic representation* is a subquotient of

.

##
Representations of locally profinite groups

A representation of means more precisely the following.

A

*representation* of a locally profinite group

is a pair

, where

is a

-vector space and

is a homomorphism. A representation

is called

*smooth* if for any

, there exists an open compact subgroup

such that

. It is called

*admissible* if it is smooth and

is finite dimensional for any open compact subgroup

.

09/17/2013

Let

(here

stands for compactly supported and

stands for locally constant). This becomes a smooth representation of

under left and right translations:

For any

, one can write it as

where

is some compact open subgroup depending on

.

The group

and

are unimodular. In general, if

is any reductive group over

, then

is also unimodular.

Now fix an open compact subgroup

and normalize the Haar measure such that

. We define the

*Hecke algebra* to be the space of compactly supported, locally constant functions on

which are

-bi-invariant. In particular,

.

Let

. Then for any

,

if and only if

.

Since , it follows that is a subalgebra of with the unit .

- Let be a smooth representation of . Then for any , . Therefore is a -module.
- Let be an admissible irreducible representation of . Then either or is an irreducible -module. Moreover, if for such that , we have if and only if as -modules.

Now let us consider the case and . Every compact subgroup is contained in a -conjugate of and is a maximal compact subgroup.

We say an admissible irreducible representation

is

*unramified* if

.

Let

and

. Then

. In particular,

is commutative.

If

is an unramified representation of

. Then

is 1-dimensional, determined by the eigenvalues of

and

.

##
Back to automorphic representations on

When , we can write . Therefore the latter admits an action of , so is a -module, where and .

The operator

then induces an operator, still denoted by

, acting on

.

When

, this adelically defined operator agrees with the classical Hecke operators

acting on

. Indeed, decompose

and call these representatives

and

viewed as elements in

. Starting from

, we want to show that

. Notice

Define

as the

viewed as elements in

. Then

has entry 1 at the prime

and

at other primes

, hence is an elements of

. Thus

as desired when you expand the terms.

Similarly, induces a operator on . One can compute that .

Since is a *semisimple* representation of (Corollary 3), we can write where 's are the irreducibles. This implies that After reordering, we can assume (hence is 1-dimensional) if and only if . The isomorphism classes of , are then determined by the isomorphism class of as a -module, or even better, by the eigenvalues of and ().

09/19/2013

A smooth representation of a locally profinite group

is called

*unitary* if there exists a Hermitian positive definite inner product

that is

-invariant. When

is compact (e.g., finite), every irreducible representation is unitary.

If

is unitary and admissible representation of

, then it is

*semisimple*. Namely

where each

is irreducible and admissible.

By Zorn's lemma, it suffices to show that any

-invariant subspace of

has

-invariant complement, namely

. We will show that we can take

. By positive definiteness,

. It remains to show that

. Choose

an open compact subgroup. Recall that (Remark

6)

We then need to show that

. Because

is

*finite dimensional*, we have

. For any

a smooth irreducible of

,

is perpendicular to

(check this according

or

). We see that

, since

. It follows that

.

¡õ
If

, then

is a semisimple representation of

. So we can write

, where

runs over all cuspidal automorphic representations of

.

Want to show that

is a unitary representation. This is not quite true but there exists a character

such that

is unitary. The unitary structure is a generalization of the Peterson inner product which we will talk about next time. This is enough to show that

is semisimple by untwisting the direct sum decomposition of

.

¡õ
- is an irreducible admissible representation of , which only depends on and not on or 's.
- If is any irreducible admissible representation of , there exists 's such that . Moreover, the isomorphism class of the 's are uniquely determined.

The proof is purely algebraic, see Flath in Corvallis I. We do remark that the some holds if

is replaced by any restricted direct product

with respect to

such that for almost all

, the Hecke algebra

is commutative (e.g.,

for any reductive group

).

¡õ
We now can describe the summands of in classical terms.

- Let and a prime , then acts
*semisimply* on and the , commute for a prime.
- If is an eigenform for all , , then the submodule of generated by is
*irreducible*. Conversely, any irreducible submodules of is obtained in this way.

- Let be the irreducible submodules of such that (there are only finitely many by the admissibility of . As modules for , Then acts on the first factor trivially and acts on the second factor (which is 1-dimension) in the usual way. It follows that the action of is semisimple and , commute.
- We Need some nontrivial global information:

- (Multiplicity one) If are irreducible submodules such that , then . Namely, decomposes with multiplicity one.
- (Strong multiplicity one). If and are cuspidal automorphic representations of of weight such that for almost all (e.g., all unramified primes), , then .

Let be the submodule generated by . By the semisimplicity, we can write , where 's are irreducible. Notice that because these are unramified representations and their isomorphism class is determined by eigenvalue of on . Now we apply strong multiplicity to obtain and the multiplicity one implies that .

For the converse, we need some nontrivial local information:

Let

and define

If

is any irreducible admissible representations of

, then there exists

such that

. The smallest

is called the

*conductor* of

.

By this theorem, there exists such that (do it at one prime a time). If is a cuspidal automorphic representation, then we can chose . Then is a -eigenvector for all and generates .
¡õ

##
General automorphic representations on

Reference on this section:

- Gelbart, Automorphic forms on adeles groups
- Deligne, Formes modulaires et representations de GL(2)

Now we are going to also incorporate the representation theory at and enlarge the notion of automorphic forms and representations on . We write , , and . Then is a maximal compact subgroup of and write .

We define

Then

acts on the left on

(or

) by right translation and

acts on

by

This makes

(or

) into a representation of

and hence extends to a representation of

, the universal enveloping algebra of

. By linearity,

also acts on

(or

, where

.

The vector

spans

.
The vectors

and

are the eigenvectors for the adjoint action of

or

on

with non-zero eigenvalue. We have the commutant relations

Let

be the center of the universal enveloping algebra. One can show that

, where

and

is the

*Casmir operator*.

09/24/2013

We use the Iwasawa decomposition

, so any

has a unique decomposition

Using the fact that

, we see that

and the first part follows. For the second part, one compute the differential operator

and find out that

if and only if

.

¡õ
Let

, we associate to

a function

as above. Then it satisfies that (c.f., Definition

2)

- For any , is holomorphic.
- There exists open compact subgroup such that for any and , .
- For any and , .
- For any and , .
- For any , .
- .
- For any , there exists such that for any , where for , .

The above association

gives an isomorphism of

-modules

and the functions

satisfying the above 7 conditions. This restricts to an isomorphism between

and the

's which also satisfy

It is easy to check the isomorphism by construction. To match the cuspidality condition, it suffices to check on

of the form

,

. Using

and the Cauchy integral formula, we see that

where

.

¡õ
Let

be a

-module. We say that

is

*admissible* if for any

continuous irreducible, the isotypic subspace

is finite dimensional. We say

is

*irreducible* if it is algebraically irreducible, i.e. there is no proper

-submodule.

A

*-module* is a

-vector space

endowed with the structure of

-module and a commuting smooth action of

. We say

is

*admissible* if for any open compact subgroup

and for all continuous irreducible representations

, the isotypic subspace

is finite dimensional. Similarly, we say

is

*irreducible* is it is algebraically irreducible.

Suppose

is an irreducible admissible

-module, then there exists

as an irreducible admissible

-module and for all

, an irreducible admissible

-module, unramified for almost all

, such that

See Flath in Corvallis I.
¡õ

- The spaces and are naturally -modules.
- For any , generates an admissible -module.
- is semisimple (but is not).

This is theorem that requires real work. For part b): one needs reduction theory for

and also finiteness results of solutions to certain differential equations (highly nontrivial). For part c), one needs a generalization of Peterson inner product and the "unitary implies semisimple" result.

¡õ
An

*automorphic representation of * is an irreducible

-module which is isomorphic to a subquotient of

. It is

*cuspidal* if it is isomorphic to a subquotient of

.

- Let . We define a -module as follows. As a vector space As a -module, As a -module, and is defined to be the differentiate action of . Then is an irreducible admissible -module.
- If satisfies and , then generates a -submodule of which is isomorphic to .

09/26/2013

##
Reductive groups over algebraically closed fields

*Convention*. We will assume that is a field of characteristic 0. By a variety over we mean a reduced scheme of finite type over (not necessarily connected or irreducible). Any affine scheme is separated so we will not assume separatedness.

An

*algebraic group* is a variety endowed with morphisms

,

and a point

making

a group object in the category of varieties. A

*representation* of

is a pair

where

is a finite dimensional

-vector space and

is a homomorphism of algebraic groups. We say

is a

*linear* algebraic group if it admits a faithful representation.

A linear algebraic group

is

*reductive* if it is geometrically connected and every representation of

is

*semisimple*.

is a reductive group. If

is a number field and

is a reductive group, we will discuss a good theory of automorphic forms on

, generalizing the classical case

and

.

Reductive groups are nice from the perspective of representation theory. Even better, one can classify all reductive groups over an algebraically closed field using certain combinatoric data and classify them over general fields via descent.

*Convention*. In this section we assume that is algebraically closed.

A

*torus* is a linear algebraic group such that there exists an isomorphism

for some

. We define the

*character group* and the

*cocharacter group* of

,

These are free abelian groups of rank

.

- The natural pairing is a perfect paring.
- The assignment is an equivalence of categories between the category of -tori and the category of finitely generated free abelian groups.
- Every irreducible representation of is 1-dimensional and every representation of is semisimple (so is reductive).

Any representation of

is of the form

, where

.

The

*derived group* of a linear algebraic group

is defined to be

, where

runs over all closed normal subgroup such that

is abelian. The

*derived central series* is

. We say

is

*solvable* if this series terminates at the trivial group.

We recall the *Jordan decomposition*: If is a finite dimensional -vector space and , then there exists unique commuting elements such that ,where is diagonalizable and is unipotent. From this one can deduce the following theorem.

Let

be a linear algebraic group and

. Then there unique exists commuting elements

such that

and for

*any* representation

of

,

is semisimple and

is unipotent.

A linear algebraic group

is

*unipotent* of every

is unipotent.

The following proposition provides a intuitive way to think about the various notion of linear algebraic groups.

- Every connected solvable group admits a faithful representation with image in the subgroup of upper triangular matrices.
- Every unipotent group (in characteristic 0 it is automatically connected) admits a faithful representation with image in the subgroup of the unipotent upper triangular matrices.
- Every connected solvable group contains a unique normal unipotent group such that is a torus (we haven't defined quotients, one way to think about it is that there exists a torus such that is exact in -points). We call the
*unipotent part* of .

The

*radical* of a linear algebraic group

is defined to be the maximal connected normal solvable subgroup. The

*unipotent radical* is defined to be the unipotent part of

.

Let

be a connected linear algebraic group. The

is reductive if and only if

is trivial.

Now we fix a reductive group .

We say a torus

(

means a closed subgroup) is

*maximal* if it is not contained in any strictly larger torus.

Fix a maximal torus . Then has a natural action on : given by differentiating the conjugation. Restricting to we obtain a natural representation of . This gives the Cartan decomposition where is the -eigenspace of for .

. Each

is either 0 or 1-dimensional.

The elements

such that

and

are called the

*roots* of

. We write

for the set of roots.

If

is a root, we define

and

(a reductive subgroup of

), then

is a maximal torus again.

The Weyl group

contains a unique nontrivial element

(so

has order 2) and there exits a unique element

such that

for all

. Namely,

acts on

by reflection and this reflection can be chosen using an

*integral* element

.

The element

is called the

*coroot* of the root

. We write

for the set of coroots. We have a canonical bijection between

and

given by

.

The tuple becomes a root datum in the following sense.

10/01/2013

Let

be a reductive group. A

*Borel subgroup* is a maximal connected solvable subgroup (not normal in general). A

*parabolic subgroup* is a subgroup such that

is projective as an algebraic variety.

Let

be a reductive group.

- contains a unique conjugacy class of Borel subgroups.
- A subgroup is parabolic if and only if it contains a Borel subgroup. In particular, Borel subgroups are parabolic.

. Fix a partition

into positive (= nonnegative) integers . Let

be the block upper triangular subgroup with blocks of size

. Then

is the Grassmanian of filtrations

such that

, which is a projective variety. For the partition

, we obtain the subgroup of upper triangular matrices, a Borel subgroup of

.

Fix

a maximal torus. Let

be the root datum of

. A

*root basis* of

is a subset

such that every

can uniquely expressed as

, where

are integers that are all positive or all negative.

There is a canonical bijection between the root bases and the Borel subgroups

that contains

, characterized by

inside the Cartan decomposition of

. The Weyl group acts simply transitively on both sides of this bijection.

. The Borel subgroups

is in bijection with the orderings

of the standard basis

of

, i.e.,

is given by the stabilizer of the flag

. For example,

gives the subgroup of upper triangular matrices

. The corresponding root basis is

and

as desired.

A

*based root datum* is a tuple

where

is a root basis of the root datum

. So a choice of

gives a based root datum.

A reductive group

is

*semisimple* if

is trivial. A semisimple group

is

*almost-simple* if it has no normal subgroup of dimension

;

*simple* if it has no nontrivial normal subgroup.

An

*isogeny* of semisimple groups is a surjective homomorphism

with finite (automatically central) kernel. If

is semisimple, we say

is

*simply-connected* if there is no proper isogeny

;

*adjoint* if there is no proper isogeny

(equivalently,

has trivial center).

If

is a semisimple group, then there is an ordering-preserving bijection between the elements of the isogeny class of

and the subgroups of

. In particular, the simply-connected group corresponds to the full group and the adjoint group corresponds to the trivial group.

If

is almost simple, we associate to it a graph, the

*Dynkin diagram* of

. Fix a root basis

. The Dynkin diagram has the vertex set

. For

, let

. Then

and we join

,

with

edges. If

, we decorate this multiple edge with an arrow from

to

if

.

Let

be an almost simple group. The Dynkin diagram of

is connected. If

is another almost simple group, then

and

are in the same isogeny class if and only if they have the same Dynkin diagram.

Let

be a reductive group. We denote by

the group of automorphisms of

as an algebraic group over

. We have an exact sequence

Notice

.

A

*pinning* of

is a tuple

where

is a maximal torus,

a Borel subgroup,

a root basis of

and

is a basis of

.

Let

be a pinning and

be a based root datum of

. We define

be the automorphisms of

that fixes

and the

*set* . We define

be the automorphisms of

that fixes

and

. There are natural maps

and

.

- The natural maps and are isomorphisms. Consequently, .
- All pinning of are -conjugate (so all the above splitting are conjugate under the action of ).

##
Reductive groups over general fields

In this section we assume that is an arbitrary field of characteristic 0. Let be an algebraic group over . Let be an algebraic closure of .

We say

is reductive, semisimple, unipotent, solvable, torus if

satisfies the corresponding property.

Notice acts on with fixed points .

A torus

is called

*split* if there exists an isomorphism

defined over

(such an isomorphism always exists over

). If

is any torus over

. We define

. This is a free finitely generated abelian group with an action of the Galois group

.

The assignment

defines an (anti-)equivalence of categories between the category of tori over

and the category of free finitely generated abelian groups with a smooth

-action (equivalently, the conjugacy classes of homomorphisms

.)

10/03/2013

We say that a torus

is

*maximal* if

is maximal. It is a fact that

always contains maximal tori.

A

*maximal split torus* is a split torus, maximal with respect this property. We way

is

*split* if

contains a split maximal torus.

We start off with the *split* reductive groups when we classify reductive groups over general fields.

- Let be a root datum. Then there exists a reductive group and a split maximal torus such that is the root datum of .
- If and be split reductive groups over with the same root datum, then they are isomorphic over .

Let

be a linear algebraic group over

. A

*form* of

is a linear algebraic group

over

such that there exists an isomorphism

. By the above theorem, any reductive group over

is a form of a split reductive group over

.

So our remaining task is to classify the forms of a split group.

Let be a split reductive group and be any reductive group with . For any , acts on and obtain another . One knows that . This is an example of a *1-cocycle* in non-abelian Galois cohomology.

Let

be a group with a continuous action of

(

is endowed with the discrete topology). A

*1-cocycle* is a map

satisfying

. We say two

*1-cocycles* are

*1-cohomologous* if there exists

such that

for all

.

We have constructed a *1-cocycle* . Changing to where doesn't change the class of . Therefore we obtain a map between the isomorphism classes of forms of to . Using Galois descent one can show that

This map is bijective.

Suppose is split and fix a pinning defined over (i.e., is split, is defined over and are defined over ). Let be the associated based root datum. Recall (Theorem 13)the pinning splits the exact sequence acts on this exact sequence and one can show that when is defined over , acts trivially on and the splitting is -equivariant. Thus it gives a homomorphism In particular, any form of gives a homomorphism up to conjugacy (which of course does not determine up to isomorphisms because we lose information when passing from to .)

A reductive group over

is

*quasi-split* if it contains a Borel subgroup (i.e.

such that

is a Borel subgroup).

Let

be reductive groups over

. We say

is an

*inner form* of

if there exists an isomorphism

such that for any

,

. Equivalently, the associated class

lies in the image of

.

The following lemma tells us that passing to , we exactly lose the information distinguishing the inner forms of the same quasi-split group.

has a unique quasi-split inner form. If

and

are forms of the same split group

, then they have the same quasi-split inner form if and only

up to conjugacy.

We summarize the strategy of classifying reductive groups over general fields:

- Construct the split group given a root datum.
- Classify the quasi-split groups (corresponding to homomorphisms ).
- Classify the inner forms of each quasi-split group.

The example is in order.

If

is a central simple algebra of rank

, we define a group

over

by the functor of points:

(this is a representable functor).

depends on

only up to isomorphism, so

*Every inner form* of

is isomorphic to

for some

. For example, when

, the group

has two inner forms

and

.

Fix the standard pinning of . The associated based root datum has automorphism group . So there is a bijection between quasi-split (non-split) forms of and quadratic extensions . Define a Hermitian form on by , where . Notice does map to the nontrivial element in , but the choice of is chosen to be compatible with the splitting given by the standard pinning: preserves the standard Borel and the alternating signs are chosen to preserve each basis .

Define the group by the functor of points The group becomes isomorphic to over . Indeed, for any -algebra , we have and the automorphism of induces the automorphism of . It follows that . One can show the upper triangular subgroup of is a Borel subgroup, therefore is quasi-split as desired.

The inner forms of can be constructed as follows: let be a central simple algebra over and be an anti-involution of that restricts to on (e.g., , ). Then each inner form of is of the form

10/08/2013

##
Automorphic representations on reductive groups

Reference on this section: the article of Borel-Jacquet in Corvallis I.

Let

be a number field and

be a reductive group. Choose an embedding

, then for any place

of

, we endow

the subspace topology from

. For

a finite place, we define

an open compact subgroup of

. If we choose another embedding

, then

for almost all

. We define

to be the restricted product

with respect to

. The topology is the subspace topology from

.

Recall that for any Hausdorff locally compact topological group

, there is a Haar measure on

(Definition

6) unique up to

-multiple. The

*modulus character* is defined as

is called

*unimodular* if

is trivial.

If we further assume

is a closed unimodular subgroup. Then there exists a Haar integral

on the quotient space

which is right

-invariant. Again

is unique up to

-multiple.

Denote , a real Lie group. We choose a maximal compact subgroup (which is unique up to conjugacy). Let and .

An

*automorphic form* on

is a function

satisfying:

- For any , , .
- For any , the function is smooth.
- is -finite, i.e., is a finite dimensional vector space.
- is -finite, i.e., is a finite dimensional vector space.
- There exists an open compact subgroup such that for any and , .
- For any , the function is slowly increasing. A function is
*slowly increasing*, if for one (equivalently, all) embeddings of real algebraic groups, there exists such that for any ,

We say is *cuspidal* if it further satisfies

- For all parabolic subgroups defined over , for any , where is the unipotent radical of . Notice by the left- invariance, if this condition is satisfied for a parabolic subgroup , then it is satisfied for all parabolic subgroups conjugate to .

We write for the space of automorphic forms on and the subspace of cusp forms.

(Harish-Chandra)
If

, then it generates an admissible

-submodule of

.

An

*automorphic representation* on

is an indecomposable

-module, isomorphic to a subquotient of

. We say

is

*cuspidal* if it is isomorphic to a subquotient of

(

is semisimple as the case

).

(Hilbert modular forms)
Let

be a totally real field. Let

be the infinite places. Let

. Then

. We choose

. Recall for any

we defined an irreducible, admissible

-module

(Proposition

6). We choose integers

and consider the cuspidal automorphic representations

of

such that

. These correspond to classical

*Hilbert modular forms* of weight

.

(Quaternion algebras over )
Let

be quaternion algebra over

. Let

be the associated inner form of

. Recall that for

or

, there exists two isomorphic classes of quaternion algebras over

, the split one and the non-split one. For

, the isomorphic classes of quaternion algebras correspond to finite set of places

of

of even cardinality, where

is the set of non-split places of

.

Suppose is nonempty and does not contain , then is anisotropic modulo center since is anisotropic for every . Moreover, . One can check that cuspidal automorphic representations of such that can be described in terms of holomorphic functions on a compact quotient of : there are no cusps since is anisotropic.

Now suppose contains . Then is still anisotropic modulo center but is compact modulo center. The automorphic representations such that is trivial are in bijection with the -constituents of the space such that

- For any , .
- There exists an open compact subgroup which fixes on the right.
- For any , , (due to the triviality at ).

In other words, these are the functions on a finite set . It turns out these simple-looking functions are very interesting and the following Jacquet-Langlands correspondence is the first case of functoriality:

10/10/2013

##
Representation theory of -adic groups

Reference for this section:

- Bushnell, Henniart, Local Landlands conjecture for GL(2);
- Cartier, Corvallis I.

To talk about automorphic representations more intelligently, we need to know more representation theory of local fields, both non-archimedean and archimedean. Then Dick Gross will come and tell you about -groups. Afterwards we are going to discuss Galois representations and it relation with automorphic representations, and as promised, the theorem of Chenevier-Clozel on number fields with limited ramification.

###
Unramified representations

Let be a finite extension of and be an reductive group. We abuse notion and write for the locally compact topological group . Today we will focus on the simplest case: the unramified representations of a unramified -adic reductive group.

is called

*unramified* if

is quasi-split and split over an unramified extension of

. In other words,

is determined by a homomorphism

that factors through an unramified extension of

.

The nice thing about unramified groups is that they admit distinguished open compact subgroups.

Let

be unramified and fix

a maximal split torus. Let

, then

is a maximal torus (for general

, it is the "anisotropic kernel" of

). We denote by

the maximal compact subgroup.

Let

and

be the diagonal torus. Then

is also the diagonal torus and

.

The

*split Weyl group* is a constant group scheme over

It acts faithfully on

and

.

(Satake isomorphism)
There exists a canonical isomorphism between the Hecke algebras

, where

is a hyperspeicial maximal compact subgroup of

.

We can assume

after conjugation. We will write down the

*Satake transform* . Let

be a Borel subgroup containing

(because

is quasi-split,

contains a Borel; a maximal torus is the centralizer of the its maximal split subtorus and all maximal split tori are conjugate, we can conjugate such a Borel to contain

). Let

be the modulus character. Given

, we define

where

is the unipotent radical of

. One can check that it has image in

. It turns out the map does not depend on the choice of

(due to the modulus character factor). For details, see Cartier.

¡õ
Let

be an irreducible admissible representation of

. We say

is

*-unramified* if

. We say a character

*unramified* if it is trivial on

.

The previous theorem together with Theorem 3 have the following corollaries.

If

is

-unramified, then

.

The equivalences of the first two and last two are clear. For the equivalence between b) and c), simply notice that

is a group isomorphism.

¡õ
Now we introduce the notion of parabolic induction for general reductive groups and come back to unramified representations in a moment.

Let

be a parabolic subgroup and

be the Levi decomposition. Let

be an admissible representation of

. We define a

-representation

where

acts by right translation.

is admissible.

Consider any open compact subgroup

. We need to show that

is finite dimensional. Since

is projective and hence compact, we can find

a

*finite* set such that

. Then

is determined by the values

for

. One can check that

. The latter space is finite dimensional as

is admissible by assumption.

¡õ
If

is an admissible representation of

. We define the

*normalized* or

*unitary* induction

.

Suppose

is unitary. Then

is also unitary.

We sketch the case

is a character. Then

is unitary means that

. Then

For

, we write

and try to define the unitary structure by integrating

. However,

is not unimodular and there does not exist

-invaraint integrals on the quotient

. Nevertheless, we do have a right

-invariant integral

on the subspace of locally constant functions

such that

. One can check that

using the extra

factor and

. So integrating

does give a unitary structure on

.

¡õ
Now we come back the the situation where is unramified, hyperspeicial, maximal split torus. and a Borel containing . Let be any smooth character.

Define

. This is called a

*principal series representation* of

. If

is unramified, then

is 1-dimensional. As a consequence,

has exactly one

-unramified subquotient, denoted by

. Conversely, if

is

-unramified, then we get (up to

-conjugacy) an unramified character

via the Satake isomorphism and

.

,

is the diagonal torus,

,

is the standard Borel. Then

and

, where

is the standard cocharacter

and

is the

-th symmetric polynomial in

.

Let

be a classical holomorphic modular forms that is an

-eigenvector for all

with eigenvalue

.

generates an automorphic representation

of

. For

,

is

-unramified for

. Factor the Hecke polynomial

Then the character associated to

via the Satake isomorphism is

One can compute directly that

is irreducible and

. Explicitly,

10/15/2013

Let

be a number field and

be a reductive group over

. For all but finitely many finite places

of

. The group

is unramified and the group

is a hyperspeicial maximal compact subgroup.

(Sketch)
There is an

-torsor

, where

is the quasi-split inner form of

. This torsor has a marked connected component

that is a torsor for

. The assertion that

is quasi-split almost everywhere is equivalent to the assertion that

is non-empty for almost every

.

¡õ
If

is an irreducible admissible representation of

, then for all

, there exists an irreducible admissible representation

of

and an isomorphism

. In particular,

for almost all

(hence is 1-dimensional for almost all

).

See Flath, Corvallis I.
¡õ

###
Hierarchy of representations of -adic groups

Let us come back to the local situation. Let be a -adic field and be a reductive group. We are going to define several classes of irreducible admissible representations of .

Let

be an irreducible admissible representation of

. Then there is a

*central character* such that

for any

and

.

See Bushnell-Henniart.
¡õ

We define

, the

*contragredient* of

as the space of smooth vectors in the algebraic dual

(this definition works for arbitrary smooth representations). If

is

*admissible* (not necessarily irreducible), then the natural map

is an isomorphism.

Let

be an irreducible admissible representations of

with central character

. Choose

,

. We define the

*matrix coefficient* , where

is the natural pairing between

and

.

We say

is

*square-integrable* if

is unitary (i.e.

) and for any

,

,

We say

is

*supercuspidal* if for any

,

, the function

is compactly supported modulo center.

- If is irreducible admissible, then is square-integrable (resp., supercuspidal) if and only if there
*exists* nonzero , such that is square-integrable modulo center (resp. compactly supported modulo center) and is unitary (resp. no condition on ).
- If is unitary, then is supercuspidal implies that is square-integrable.
- If is square-integrable, then is unitary (i.e., admits a -invariant positive definite inner product).

(Sketch)
- Fix , let be the space of such that is compactly supported modulo center. Then one can check that is a nonzero -invariant subspace, hence .
- Obvious.
- Fix nonzero. We define for , One can check it defines a unitary structure.
¡õ

Let

be irreducible admissible with

unitary. We say

is

*tempered* if for any

,

lies in

for any

.

So far everything works well for all locally profinite groups. The following theorem needs more from input from the structure theory of reductive groups.

- An irreducible admissible representation of is supercuspidal if and only if there does
*not* exist a proper parabolic and an admissible representation of together with an embedding .
- Any irreducible admissible representation admits an embedding for some and a supercuspidal representation of .

This is a hard theorem: see Casselman unpublished notes on

-adic groups.

¡õ
###
Local Langlands correspondence for over -adic fields

. A standard parabolic

corresponds to a partition

, where

. Its Levi subgroup

. Therefore every irreducible admissible representation

of

embeds as

, where

is a supercuspidal for

. A special case we have seen is that any unramified representation

is a subquotient of

, where

is an unramified character.

There is a nice interpretation in terms the *local Langlands correspondence* for proved by Harris-Taylor and Henniart. The local Langlands correspondence for is a bijection between

- isomorphism classes of irreducible admissible representations of ;
- conjugacy classes of semisimple homomorphisms (i.e. decomposes as direct sums) such that is algebraic, where is the Weil group of .

is characterized abstractly by some identities relating to their -functions.

If

is the unramified subquotient

, then

is trivial on

, unramified on

and sends a uniformizer

to

.

The local Langlands correspondence

restricts to a bijection between

- classes of
*supercuspidal* representations.
- classes of
*irreducible* representations (trivial on the factor).

Suppose

, and

. Let

, it is a supercuspidal representation of

. Let

, then

is indeed a subquotient of

.

To summarize: building irreducible admissible representations of from supercuspidal representations of Levi subgroups mirrors taking direct sums of irreducible Langlands parameters .

Let

,

and

be irreducibles. Then

is a subquotient of

. One can view the

-factor a way of labeling different subquotients.

Let be any reductive group. There is a canonical square-integrable representation of , the *Steinberg representation* that we are going to construct now. Let be a minimal parabolic, define

- The subrepresentations are in bijections with parabolic subgroups : .
- Let be the quotient of by the span of , . Then is irreducible and square-integrable.

If

. Then

is the unique homomorphism trivial on

and restricts to the unique

-dimensional representation of

. Thus

is a subquotient of

. If

. Then we have an exact sequence

This exact sequence does

*not* split and gives an example of unramified principal series representation (as

is unramified) that has a ramified unitary quotient and a trivial subrepresentation. One also knows that

is non unitary since any unitary admissible representation is semisimple (Lemma

3).

Let

be the cuspidal automorphic representation of

associated to an elliptic curve

. Then

has multiplicative reduction at

if and only if

is an unramified twist of

.

Very recently Gross-Reeder constructed a class of

*simple supercuspidal* representations. For example, consider

,

and

. Let

. It is a character and indeed

is a supercuspidal representation.

10/17/2013

##
Representation theory of real reductive groups

Reference for this section: Wallach in Corvallis I.

Let be a reductive group . We use the usual notation: , , a maximal compact subgroup. , .

###
-representations

A representation

of

is a (separable) Hilbert space

and a homomorphism

to the group of bounded invertible linear operators such that

- then map is continuous.
- for any , is unitary (i.e. an isometry).

We say

is

*irreducible* if there is no nontrivial closed

-invariant subspace of

. We say

is

*unitary* if for any

,

is unitary.

Let us recall the representation theory when is compact

When is compact, there is a unique -conjugacy class of maximal tori (so is several copies of ). Fix a maximal torus and be an associated root datum. Then the natural map is a bijection. Let be an irreducible representation of . Then as a sum of weight spaces. The map is an isomorphism, so we can view the sum as over . In particular, is invariant under , hence under .

If

and

be the diagonal maximal torus. So

,

and

is the usual diagonal torus. The theorem then parametrizes the equivalence classes of irreducible representations of

by tuples

,

.

###
-modules

Let be any reductive group over . There is a unique -conjugacy class of maximal compact subgroup (notice this is *not* true over -adics, cf., Example 16).

,

.

,

.

Let

be a representation of

. By the Peter-Weyl theorem,

where

are irreducibles. We say

is

*admissible* if each isomorphism classes of irreducibles representations of

appears only finitely many times. It is easy to see that if

is admissible, then the subspace

of

-finite vectors is simply the usual direct sum

For

, we define

. If this exists, then we say

is

*differentiable*. We say

is

*smooth* if for any

and any

,

exists.

###
Hierarchy of representations of real reductive groups

Let

,

be two admissible representations of

. We say they

*infinitesimally* equivalent if there associated

modules are algebraically equivalent.

Let

is an admissible representation of

. A

*matrix coefficient* of

is a function of the form

, for

. It is a

*-finite matrix coefficient*, if

and

are further

-finite vectors.

Let

be an irreducible representations of

with unitary central character. We say

is

*square-integrable* if its

-finite matrix coefficients are square-integrable modulo

, where

is the center as an algebraic group. It is

*tempered* if its

-finite matrix coefficients are in

for any

. The square-integrable representations are also called

*discrete series* representations since they occur discretely in the unitary dual.

###
The Harish-Chandra isomorphism

The center of the universal enveloping algebra is isomorphic to a polynomial algebra in generators, where is the rank of (= the dimension of a maximal torus). How does one prove it?

We change notion for convenience: let be a reductive group over . Fix a pinning of .

Let

. Then

for any simple root

. We have

.

We define

to be the projection along

. The

*Harish-Chandra homomorphism* is defined to be the composite

.

(Harish-Chandra)
The map

is an algebra homomorphism, independent of

, and defines a canonical isomorphism

, where

acts on

in the natural way.

10/22/2013

Consider

and

. We define the Casmir element

, where

,

,

. One can directly check that

. We claim that

. Since

in

, we can rewrite

. Fix the root basis

, where

. Then

is a basis of the

-root basis. Hence

. Also

, hence

. So

,

. Therefore

The Weyl group

and acts by

and thus

. Hence

. By the theorem, we know that

.

Now we come back to the situation that

is a reductive group over

. Let

be an irreducible admissible

-module. A version of Schur's lemma says that every element of

acts as a scalar on

. Therefore it defines an algebra homomorphism

, the

*infinitesimal character* of

. This does not determine the representation

uniquely but does tell a lot of information: e.g., the infinitesimal character of an automorphic representation knows about the Hodge-Tate weights of the associated Galois representation.

For

, we define

by composing

and

. It is a fact that every homomorphism

arises in this manner and

if and only if

are in the same

-orbit.

###
Square-integrable representations of real reductive groups

Assume is semisimple group over for simplicity in this section (everything said below are true for any reductive group after appropriate modification).

has a compact inner form

, therefore

has discrete series. On the other hand, for

,

has no compact inner form (there are only two division algebra over

!), thus has no discrete series.

We write

, the

*unitary dual*, for the set of unitary equivalence classes of unitary irreducible admissible representations of

. We write

for the set of discrete series representations.

Assume is non-empty. Then by the previous theorem, there exists a maximal torus and a maximal compact such that and is compact.

The

*real Weyl group* is defined to be

.

naturally embeds into the

*complex Weyl group* .

We say

is

*regular* if

for any

. Notice the regular condition is invariant under the action of

.

We have the following nice parametrization of square-integrable representations.

(Harish-Chandra)
- If is a unitary square-integrable representation of , then has infinitesimal character for some regular.
- There exists a bijection between the square-integrable representations of with infinitesimal character and the set of -orbits inside . In particular there are such representations.

Let

be the unitary group of the Hermitian form

, where

. The subgroup

of determinant 1 is a semisimple group over

, which is a form of

and an inner form of the compact group

. A compact maximal torus in

is the diagonal torus. A maximal compact subgroup is

. We have

and

. The packet of square-integrable representations with same infinitesimal character then has

elements.

###
Representations of

Let , . We have a maximal torus such that . We have and . Therefore the discrete series representations of fall into packets of 2 elements, parametrized by the -orbits on the regular elements .

Let be the diagonal split maximal torus, , (the connected center of ), the standard Borel and the unipotent radical. Then we have and the *Langlands decomposition* . The characters of are then indexed by , i.e. . The parabolic induction in this case gives a representation of .

Consider the space

of smooth functions

such that

, for

,

. Recall the

*Iwasawa decomposition* . Therefore

is uniquely determined by its

*restriction on *. We define an inner product on

by

We define

to be the Hilbert space completion of

.

10/24/2013

##
The trace formula for compact quotients

Announcement: Dick Gross will tell us about -groups next Thursday (Oct 31). There will be no class on Nov 7.

The trace formula is a good tool for constructing interesting automorphic representations. We will talk about the trace formula for the compact quotients today and the simple trace formula for non-compact quotients next time.

Let

be a (separable) Hilbert space. We say a linear operator

is

*Hilbert-Schmidt* if for some (equivalently, any) orthonormal basis

the sum

is convergent. We say

is

*of trace class* if there exists Hilbert-Schmidt operators

,

such that

.

Let

be a Hausdorff

*compact* measure space and

is a continuous function, then we define

by the formula

by integrating the kernel function

, then one can easily check that

is Hilbert-Schmidt (by writing it down using an orthonormal basis).

Now let be a unimodular locally compact Hausdorff topological group. Let be a discrete cocompact subgroup.

and

.

and

a discrete cocompact subgroups, e.g., from a quaternion algebra over

or from a compact Riemann surface uniformized by the upper half plane.

Let and . Then is a unitary representation of under the right translation.

Choose

a continuous function with compact support. We can turn this function into an operator on

by defining

by

Then

is given by integrating over

the kernel function

on

:

- For any , is Hilbert-Schmidt.
- Define . Then .
- Define the convolution on by Then .

We can decompose

, where each

is finite and

is the set of unitary equivalence classes of irreducible unitary representations of

.

If

is of trace class, then

Assume that

(in general, we can write

, a linear combination of two self-adjoint functions). We can choose orthonormal eigenvectors

of

forming an orthonormal basis of the subspace

. These functions

are continuous as they are the images of

. We can write

for

, where the sum converges in

. Write

be the truncated sum of

up to

. Since

is of trace class, we know that

. We have

and

since

in

.

¡õ
The *trace formula* relates two different expressions (the geometric side and the spectral side) for whenever this makes sense.

The *geometric side* is given by (using the previous lemma)
Here means the conjugacy classes in . is the centralizer of in , is the centralizer of in . The quotient measure makes sense since is unimodular.

The *spectral side* is much simpler at this stage:

If

is of trace class, then

where

and

.

Now assume is a number field and is a semisimple anisotropic (= contains no nontrivial split torus) group. In this case, the quotient is compact.

Let

be the space of locally constant/smooth and compactly supported functions. By a nice theorem of Dixmier-Malliavin, every function

is a (finite sum of) convolutions (this is trivial on

by convolving an indicate function on the support, but is far from trivial on

). In particular

is of trace class for such a function

.

- Let be a maximal compact subgroup. If is an irreducible unitary representations of . Them the submodule of -finite vectors for (any compact open subgroup) is an algebraically irreducible admissible -module.
- The above assignment defines a bijection between the unitary equivalence classes of irreducible unitary representations of and the isomorphism classes of unitary irreducible admissible -modules.
- The above assignment restricts to a bijection between the unitary equivalence classes of irreducible representations which appear in and the isomorphism classes of automorphic representations of .

See Flath in Corvallis I.
¡õ

10/29/2013

##
The simple trace formula

References for this section:

- Henniart, Mem. Soc. Math. France 1984;
- Arthur, CJM 1986;
- Arthur, JAMS 1988.

In this section we let be a number field and be a semisimple group (it is easy to extend the results below to the case where is reductive).

Recall form last time that if and is anisotropic, then we gave a formula expressing in two ways The spectral side sums over (countably many) automorphic representations of . The geometric side sums over the conjugacy classes of .

Such a formula does not exist if is not anisotropic (e.g., a group as simple as ). The two main problems are

- does not decompose discretely as the Hilbert direct sum . There exists a certain continuous spectrum.
- The kernel function on is no longer integrable. The operator is no longer of trace class.

Here are two possible ways out:

- Describe the continuous spectrum and truncate to get a well-defined expression. This was done by Arthur after incredible amount of work and the formula thus obtained is called the
*Arthur trace formula*.
- Adopt some simplifying assumptions on the allowable test functions . We shall consider this direction in the sequel.

We define the

*cuspidal subspace* as the subspace of the functions

such that for all parabolics

defined over

, the integral

for almost every

. We only require "almost every" since there is no reason the integral always makes sense.

- The subspace is closed, -invariant and decomposes discreetly as a countable Hilbert direct sum of subrepresentations, each appearing with finite multiplicity. For , , the restriction of on , is of trace class.
- The assignment , where is the -finite vectors in , gives a bijection between the set of unitary equivalence classes of irreducible subrepresentations of and the set of cuspidal automorphic representations of .

The proof Theorem 32 is much easier than Theorem 31. We shall sketch a proof of Theorem 32.

(Sketch)
The image of

.

This only uses the assumptions a) on

:

is a matrix coefficient of a supercuspidal representation. We need to use: if

is a supercuspidal matrix coefficient, then for any parabolic

defined over

, we have

for any

. We take

, then

Choose a parabolic

defined over

. To check the cuspidality, we need to compute the integral

Now

is compact and

is compactly supported, so we can reverse the order of integration to obtain

One can choose the measure on

to be a product measure

, the assumption on

shows that the contribution from the

-factor is zero. Hence the above integral itself is zero.

¡õ
This lemma implies that is of trace class and . One can then show that the kernel function is integrable along and this integral equals to .

We need another lemma which is also easy to prove, though may take us too far afield.

The function

has compact support on

.

This lemma uses the assumption b) on

. The proof uses the reduction theory of Arthur. Morally speaking, the elliptic classes does not end up in the cusps. See Gelbart, Lectures on the Arthur-Selberg trace formula, for an example for

.

¡õ
Consequently, one has The same manipulation as in the compact quotients is now valid and lead to Theorem 32.
¡õ

Here is one application of Theorem 32.

Let

be a semisimple group over a number field

. Let

be a place of

,

a supercuspidal representation of

. Then there exists a cuspidal automorphic representation

of

such that

.

(Sketch)
Use Theorem

32 with

, where

is a matrix coefficient of

and other

's are arbitrary. It suffices to show that

If this holds, then there exists at least one cuspidal automorphic representation

such that

. But if

is any irreducible admissible representation of

and

, then

(one can write an explicit map from

to

). To show the above sum is nonzero, we look at the the finitely many terms on the geometric side. If the measures are chosen appropriately, then the orbital integral

factors as a product of local orbital integrals

. One can manage to choose

such that exactly one class

contributes to the geometric side and

.

¡õ
10/31/2013

##
Langlands dual groups

This is a guest lecture by Dick Gross. Notice the notation is different sometimes.

Reference: Casselman, Survey on -groups.

For each reductive group over , we are going to associate a complex Lie group . For a torus, one simply switches the role of the character group and the cocharacter group .

For a general reductive group , recall that one defines the roots using a torus by looking at the action of on ; one then defines the coroots using a -triple associated to the roots. This gives a natural bijection between the roots and coroots and under the natural pairing. The root datum classifies reductive group over algebraically closed field. The simple reflections preserves hugely restricts the possibility of the root datum and the classification boils to the case of rank 2 root system , , and .

The problem of this classification is that the natural map is not injective. For example, the elements acts non-trivially on . Even worse, is not surjective:

Consider

. Then

, where

takes a diagonal element to its

-th diagonal entry;

, taking

to a diagonal element

;

with root basis

. The coroot

. The sublattice

has corank 1. The Weyl group is

and

. This is not all

because

is always an automorphism of

! It turns out to be the case

.

Another extremal case is that

is a torus of dimension

, then

but

!

Now choosing a Borel subgroup (these are permuted by simply-transitively) gives extra structure: the set of positive roots and the root basis consisting of simple roots. The Weyl group no longer acts on this based root datum . The theorem is that

.

One can not lift the action of to an action of in a natural way: these is no canonical way to split the sequence but one can lift elements of ! The group preserving and is which is smaller than but still too large. The idea is to require that it also preserves a pinning (if you imagine that Borels are like the wings of a butterfly, one "pins" it down): a basis for each . Now acts transitively on the set of pinning, and the stabilizer is exactly . The upshot is that there is no inner automorphism preserving the based root datum.

(Chevalley)
and

.

One can do the same thing for groups which are quasi-split but not necessarily split. Suppose is split over . The acts on . Chevalley noticed that switching the role of gives you another based root datum with the same automorphism group .

For

, the

and

(since they have to product to 2). The dual root datum gives that of

. The dual root datum can be even more wired in general, for example

.

Chevalley noticed this dual operation but didn't know what to do with it. Langlands realized what to do with it: he called it the *dual group* , a connected reduction group over (the choice is a bit artificial) up to isomorphism with the based root datum . In particular, the Galois group acts on .

Langlands generalized this to define a group that is not necessarily connected.

The

*Langlands dual group* , where

acts through pinned automorphism of

. We will later see taking this semidriect product is a bad idea, but this is the first thing you should do.

For

,

.

For

,

. What is this group? It is a subgroup of

(when

is even) or

(when

is odd). Take the Siegel parabolic

(when

is even) or

stabilizing the maximal flag of isotropic subspaces, then the normalizer of the Levi in this Siegel parabolic is exactly the

!

A

*Langlands parameter* for a local field

is a homomorphism

, up to

*conjugation* by

(to be modified later: one shall use the Weil group

instead

so that one can send the Frobenius to any semisimple element (not necessarily of finite order); one shall need an extra

-factor to account for the monodromy operator which is unipotent).

Let

be the stabilizer of

. We define

. It is a finite group attached to the Langlands parameter.

is trivial for

, but can be nontrivial for other groups.

The local Langlands conjecture says

There is a bijection between the classes of

, where

is a Langlands parameter and

is a representation of the finite group

, and the isomorphism classes of irreducible admissible representations of

.

Consider

over

,

. The

. Let

be the reflection around two axes. Its normalizer is

and its centralizer is itself. By Kummer theory, there is a unique

-extension of

since

. One obtains a unique (up to the

-conjugation) Langlands parameter

and

. It should parametrize four representations of

. What are they? All these representations are depth zero supercuspidal constructed by Deligne and Lusztig. There are two maximal compact subgroups up to conjugacy of

:

and its conjugate by

. These four representations are exactly the induced representations from the two half discrete series representations of dimension

of

! The case

is already interesting:

is a double covering of

of order 24, the two half discrete series representations are of dimensional one, i.e., the two cubic character of

.

11/05/2013

We are back to Jack's notation.

References:

- Borel, Corvallis II. It includes everything (expected to be true) about -groups;
- Langlands, Problems in the theory of automorphic forms (historical document;
- Gross-Reeder, From Laplace to Langlands.

Let be a field of characteristic 0 and . Let be a reductive group over . Let be the bases root datum of . Then the Galois action on the root datum of induces homomorphism (which only depends on the quasi-split inner form of ).

We observed that by swapping the role of roots and coroots is still a based root datum and . We let be the split reductive group over (sometimes we also use depending on the situation) with the based root datum equipped with a pinning giving rise to .

Define

, where

acts via

on

. If

is a Galois extension such that

is trivial, people also use the definition

.

##
Langlands parameters

###
Local Langlands conjecture

The introduction of the -group allows us to state the Langlands conjecture. Assume now is a -adic field, or .

If

or

, a

*Langlands parameter* is a semisimple continuous homomorphism (i.e., a direct sum of irreducible ones, for any representation of

)

such that

is the usual map. As the

-adic case, we define similarly the notion of equivalence.

Let be the classes of Langlands parameters of and be the set of isomorphism classes of irreducible admissible -module, where and is a maximal compact subgroup.

(Local Langlands conjecture)
There is a natural partition

into finite disjoint sets

. If

is quasi-split, every

is non-empty. The set

is called a

*-packet*.

Of course this conjecture has no content without assuming extra condition characterizing the -packets . The following cases are known:

- When , the conjecture is known and the set can be written down explicitly using the parametrization of as a starting point. This is due to Langlands.
- If is -adic and is a torus, then the correspondence exists and can be essentially constructed from local class field theory. If , this is nothing but local class field theory.
- When is -adic and . In this case, all the sets has size one and gives a bijection between and . This is due to Harris-Taylor, Henniart. Scholze recently gave a new proof. The correspondence is characterized by some compatibility condition arising from the theory of -functions due to Henniart. Henniart showed there is at most one of such bijection. Harris-Taylor proved the existence of such a bijection satisfying this characterization.
- When is -adic and is a quasi-split classical group. The correspondence exists and is characterized by comparison with due to the recent work of Arthur and Mok. This is established using the fact that these groups are twisted endoscopy groups of and the method of twisted trace formula (including the fundamental lemma).
- If is -adic and is unramified, then there is a natural correspondence between the unramified elements in and the unramified -packets (namely these factor through ).

###
Unramified local Langlands correspondence

Let be a -adic field and be an unramified reductive group. Fix a hyperspeicial maximal compact subgroups.

There is a canonical bijection between

such that

and unramified parameters

.

Let

be a maximal split torus. The centralizer

is a maximal torus of

(Definition

52). The group

is a constant group scheme over

and acts faithfully on

. Let

be the minimal extension splitting

. Then

is unramified. Let

be the arithmetic Frobenius and

be its image. The

contains the subset

, which is normalized by

. If

is an unramified Langlands parameter. Then it is determined by

, i.e., semisimple

-conjugacy classes in

.

On the other hand, the unramified represetnations are parametrized by by the Satake isomorphism (Corollary 6).

Since acts on , hence on by functoriality, hence can view . Let be the inverse image of in . A bit work shows that both are in bijection with .
¡õ

When

is split,

,

and

. When

,

is simply the

-conjugacy classes of parameters

. The unramified parameters simply correspond to semisimple conjugacy classes in

, i.e.,

-orbits of diagonal matrices. On the other hand, an unramified representation

is a subquotient of a parabolic induction

, where

are unramified characters, determined up to

-conjugacy. The bijection is simply

.

###
Global Langlands functoriality conjecture

Let be a number field and be a reductive group. Recall that for almost all places of , the group is unramified. If is an irreducible admissible representation of , then for almost all , is an unramified representation. Choose for every place of an algebraic closure and an embedding extending . This induces an inclusion , hence a map .

For any irreducible admissible representation

of

, the unramified local Langlands correspondence gives for almost all places

a

-conjugacy classes

. We call the collection of elements

defined for almost all

the

*Satake parameters* of

.

(Global Langlands conjecture)
Suppose

is an admissible homomorphism and

is an automorphic representation of

. Assume

is

*quasi-split*. Then there exists an automorphic representation

of

such that

for almost all places

.

The are endless interesting examples of this conjecture by taking different and .

When

is a quasi-split inner form of

and

is the identity, the conjecture says that there is an automorphic representation

of

such that

.

When

(Example

13 ) and

. This is the Jacquet-Langlands correspondence. You may not always go back from

to

due to the local obstruction.

When

and

. A homomorphism

gives gives an admissible homomorphism

. The conjecture says that there exists an automorphic representation

of

such that

is the conjugacy class of

. This is known as the

*strong Artin conjecture*.

11/12/2013

##
Global Langlands correspondence

References for this section:

- Clozel and Milne, Ann Arbor volumes (the article by Clozel is especially relevant);
- Buzzard and Gee, On the conjectural relations between automorphic representations and Galois representations (a more modern treatment).

There are Galois representations which do not correspond to automorphic representations (and vice versa). We have to make restriction on both sides in order to make sense of the global Langlands correspondence. This requires the notion of "algebraic" automorphic representations and "algebraic" Galois representations.

###
Algebraic Galois representations

Let be a number field. Fix an algebraic closure . For a place of , fix an algebraic closure of . Choose an embedding extending the embedding . This induces an map , whose image is the decomposition group at .

For a finite set of finite places of , let be the maximal unramified extension of away from . Write . For , the map factors through . and we write for the image of the *geometric* Frobenius, i.e., the inverse the of the arithmetic Frobenius (which acts on the residue field by ).

Let be an reductive group. Let be a prime and be an algebraic closure of . Let be a finite Galois extension which splits . We view as the dual group defined over and , a linear algebraic group over with connected component . We endow with its natural -adic topology induced from some embedding (it is not locally profinite because is too big).

A continuous homomorphism

is

*admissible* if the composite map

is the natural projection.

- We say that a continuous homomorphism is
*algebraic* if
- there exists a finite set such that factors through (i.e., is unramified almost everywhere).
- For any place of , the restriction is
*de Rham* (the -analogue of being "potentially semistable" at places ).

- We say an admissible homomorphism is
*algebraic* if for any algebraic representation , is algebraic. Equivalently, is algebraic for one faithfully representation .

Let

be a smooth geometrically connected projective variety over

. The

-adic etale cohomology groups

are finite dimensional

-vector spaces on which

acts. The associated Galois representations

are algebraic. To prove that it factors through

, one can apply the proper smooth base change theorem after constructing a proper smooth model of

away from

. To prove that it is de Rham, one needs Faltings' comparison theorem.

When

is the elliptic curve,

is dual to

. After choosing a basis, we obtain a 2-dimensional Galois representation

. When

is a place of good reduction,

is unramified at

and

is an integer related the number of points of

mod

.

###
Algebraic automorphic representations

Let be a number field and be a reductive group. Fix a place of , induced by an embedding . Then is a real Lie group. Let be the complexified Lie algebra. Choose a maximal compact subgroup. Choose a maximal torus and write , .

Recall that we have a Harish-Chandra isomorphism (Theorem 24). Also recall that if is an irreducible admissible -module, then there exists such that the infinitesimal character of is equal to , obtained by . This determines up to -conjugacy. (Definition 73).

Notice has a natural integral lattice given by given by .

- We say is
*-algebraic* if lies in .
- We say is
*-algebraic* if , where is the half sum of the positive roots for some root basis. One can check this definition is independent of the choice of the root basis.
- If is an automorphic representation of . We say is
*-algebraic* (resp. *-algebraic*) if for any , is -algebraic (resp. -algebraic).

Consider

. If

is an automorphic representation of

, then for each

we obtain

. If

is a Galois representation, then for each

we obtain a

*Hodge-Tate-Sen weight* lying in

. These weights all vary continuously. But if

is de Rham, then the Hodge-Tate-Sen weights lie in

. The weights at

can be read off from the Hodge structure of the corresponding motive. People guess these are related to the Hodge-Tate-Sen weights exactly when the de Rham condition is satisfied. There do exist Hodge-Tate but non de Rham Galois representations (e.g., coming from

-adic modular forms), but these don't come from geometry, see the end of the paper of Mazur-Wiles for an example.

Fix

and

. Assume given

an eigenform for

,

with eigenvalue

. We associated

an automorphic representation

of

. For

,

, where

are unramified characters such that t

. Let

Let

for any

. These

are also cuspidal automorphic representations, which we can view as being associated to

too. Sometimes it is convenient to normalize

to be unitary, i.e.

.

Notice , where and , . An algebra homomorphism is induced by an element of if and only if there exists such that , , i.e., . On the other hand, the has infinitesimal character and .

Thus is -algebraic if and only if ; -algebraic if and only if . has a twist which is both -algebraic and unitary if and only is odd. In particular, it explains the case in Remark 82.

Let

be a totally real field of degree

. Let

be a cuspidal Hilbert modular form of weight

, where

. One can associate

a cuspidal automorphic representations

which is defined up to a character twist. When does

have an

-algebraic twist? Let

be a real place of

, then

. The local theory at

is the same for

. Suppose

is

-algebraic, then

,

. Since

is cuspidal, there exists

, such that

is unitary (this is true for any cuspidal automorphic representations on general reductive groups). So

is independent of

. In particular,

is independent of

. We conclude that one can lift

to an

-algebraic

only if the parity of

is independent of

. In fact one can show this is also sufficient (see Clozel in Ann Arbor).

Can associate to each an such that has infinitesimal character.

11/14/2013

###
Global Langlands correspondence

Fix a prime . Choose an isomorphism (One expect everything to be defined algebraically, so this choice not essential. If we know everything is defined over , it is enough instead to fix something weaker: two embeddings and ) (see Remark 90). To state the general conjecture, we'd better fix such an isomorphism .

Let be an reductive group over a number field . Suppose is an automorphic representation of .

Suppose

is

-algebraic. Then there exists a finite set

places of

containing infinite places, places

and the places at which

,

are ramified, and an algebraic Galois representation

satisfying:

- is unramified outside ;
- For any , (Definition 89).

In the case , we have the following more precise conjecture.

###

If is a finite place of , we denote the Artin map from local class field by , normalized so that , where is the *geometric Frobenius*. We denote for .

When is a Hecke character -algebraic? We need to look at the infinite components of .

If is a real place, then is is given by , where , . The condition of -algebraicity asks that the differential of agrees with the differential of an algebraic character of . That is to say, (and no condition on ).

If is a complex place, then . It has the form , where , and the symbol is defined formally so that . We are supposed to think of as a *real* Lie group. Let , it is a rank 2 torus over whose functor points is for any -algebra . In particular, . The -algebraicity condition says that the differential of agrees with the differential of an algebraic character of , i.e., for . That is to say, .

These -algebraic Hecke characters are exactly *the character of type * already introduced by Weil.

Here is a compact way of describing an -algebraic character : there exits integers indexed by embeddings such that , where and , where is the place of induced by .

The following theorem verifies Conjecture 5 for .

Fix a prime

and

. Let

be an

-algebraic character. Then there exists a unique representation

satisfying the condition of Conjecture

5. Explicitly, it is given by

Conversely, every algebraic representation

arises in this way from a unique

-algebraic character

.

###

We state partial results (automorphic to Galois) toward Conjecture 5 for over CM fields.

A number field

is called

*CM* if there exists

such that

, for any embedding

. Let

, then there are only two cases:

- is totally real;
- is totally real and is a totally complex quadratic extension.

###
Local-global compatibility

Now suppose is any number field and is a cuspidal -algebraic automorphic representation of . Suppose is known to exist. The local-global compatibility should give at *ramified* places, where is the local Langlands correspondence. When , one can say exactly what should be using Grothendieck's -adic monodromy theorem (see Tate, Number theoretic background in Corvallis II).

Back to the situation in Theorem 38, the local-global compatibility is known to hold due to Taylor-Yoshida for the case a) and Caraiani for the case b).

We now state a special case we shall need for the application of Chenevier-Clozel on number fields with limited ramification.

Fix a finite place

of

. Suppose

is unramified if

and

is supercuspidal. Then

is irreducible. The representation

is unramified at all

, and

.

11/19/2013

References for this section:

- Chenevier, Number fields with given ramification (Compositio)
- Chenevier-Clozel, Corps de nombres peu ramifies (JAMS)

##
Number fields with given ramification

The remaining of this course will be devoted to the application of Chenevier-Clozel on number fields with prescribed ramification, as promised in the first class. To put things in context, we first recall the following classical result.

Let

a prime and

be a finite extension. Then there exists an extension

such that

and

as

-algebras where

is the unique place of

over

.

Choose

be a primitive element (i.e.,

). Let

be its (monic) minimal polynomial. One can choose

such that

for any

and

. By Krasner's lemma, if

is small enough, then

has a root in

and

. We simply set

. Then

.

¡õ
Choose algebraic closures

of

and

of

and an embedding

. Then the natural map

is injective.

We would like to control the ramification at *almost all* primes. Consider a finite set of primes and a prime . Let the maximal unramified subfield of unramified outside .

We write

for the property that the natural map

is injective. Equivalently, if

is a finite extension, then there exists a number field

unramified outside

, a place

of

above

and an embedding

of

-algebras.

Chenevier-Clozel proved that this property is true as long as contains at least two primes.

(Chenevier-Clozel)
holds if

and

are distinct primes.

More generally,

Let

is any number field,

be a finite set of finite places of

and

. Write

similarly for the property that

is injective.

We are going to construct interesting automorphic representations with prescribed level, whose associated Galois representation helps us to attack this algebraic number theoretic problem .

Let

be integers and

be a prime. Then the set of primes

such that the order of

in

is divisible by

is

*infinite*.

Let

be a finite extension of

. Suppose

acts trivially on

(the tame quotient of the inertia subgroup

) by conjugation. Then

.

If

, then by Kummer theory, its action on

is the multiplication by

, where

is the unramified quotient and

is the size of the residue field of

. If this action is trivial, the image of

in

is trivial for all primes

. By the previous lemma applying to

, we know that

, where

is any prime power. Hence

and

.

¡õ
We can rephrase this lemma in the language of Galois theory.

Let

be a finite extension of

. Let

be a (possibly infinite) Galois extension. If

, then

.

Let

and

. They are closed normal subgroups of

. The assumption shows that

. Hence

. In particular,

commutes with

, hence by the previous lemma,

, which is impossible unless

.

¡õ
Let

be a number field and

be a finite set of finite places of

,

. Choose

a prime, an isomorphism

and an embedding

extending the canonical map

. Suppose for

*every* irreducible continuous representation

there exists a continuous representation

such that

Then the property

holds.

By the previous corollary, it is enough to show that

(so the completion of

is

). Fix

a finite Galois extension inside

, we need to show that

. Let

be the regular representation of

. Applying the hypothesis of the lemma to each irreducible representation

(of

, we obtain a representation

such that

. Let

. Then

. Since the regular representation is faithful, by Galois theory, we obtain that

: indeed, we have

and

, by construction

.

¡õ
Let

be totally complex CM field with its maximally totally real subfield

. Let

be the unique nontrivial element. Fix a prime

, a finite place

of

not dividing

and a finite set

of finite places of

such that the places

and

are contained in

. Suppose that for all integers

and for every supercuspidal representation

of

, there exists a cuspidal conjugate self dual regular

-algebraic automorphic representation

such that

- For any of , is unramified;
- there exists an unramified character such that .

Then holds.

We will show that the conditions in the previous lemma hold. Fix

and a continuous irreducible representation

. We would like to realize it globally. Let

be a supercuspidal representation of

(Example

22). Let

be an automorphic representation of

satisfying the assumption. Then

is a cuspidal regular

-algebraic automorphic representation of

and has associated Galois representation (Theorem

38)

satisfying

- is unramified at all places of as is unramified and .
- . This follows from the local-global compatibility and the compatibility of local Langlands correspondence with twisting by characters. Notice is unramified so the twisting disappears when restricting to the inertia subgroup.
¡õ

We won't prove the full . Instead, we will prove an earlier result of Chenevier:

- Let be a totally complex CM field and let be a place of split over . Let and assume . Then holds.
- Let be a prime and be an integer such that is the discriminant of a quadratic imaginary field in which splits. Let be the set of places dividing . Then holds.

11/21/2013

##
Base change from unitary groups

References:

- Mok, Endoscopic classification of representations of quasi-split unitary groups;
- Clozel et. al., On the stabilization of the trace formula (Paris book project)

In view of Corollary 9, we would like to construct automorphic representations with prescribed local component. In some sense the only way of doing this is to use the trace formula. The problem is that, unlike the supercuspidal representations at finite places, the -algebraic representations of are *not* isolated in the unitary dual of and hence is not easy to pick out by choosing suitable test functions. But they *are* isolated in the subset of conjugate self-dual representations of . So one can use the twisted trace formula which picks out only conjugate self-dual ones. This is what Clozel used in proving . We will take a different approach using functoriality (base change from a unitary group).

Let be a number field and be a quadratic extension with the nontrivial element . Let be a unitary group associated to a non-degenerate Hermitian form on . Write , . So is the reductive group over with functor of points . It is an outer form of , split over .

If is a place of above a place of . There are two cases:

- is split in , then . Hence . Projecting to the first factor shows this is isomorphic to . Namely, (though this isomorphism depends on a choice of the place ).
- is inert or ramified in , then is a "true unitary group", which becomes split only after extension of scalars to .

(See Mok for details)
Notice for any

-algebra

,

So

. One can then figure out the

. To get a map

, we write down an admissible homomorphism between the

-groups:

, given by

,

(one can check that it is admissible). To get a map

, starting with the parameter

. Then

is a pair of parameter

. The map is simply

. See Mok for the proof of the injectivity (resp. bijectivity) between the Langlands parameters.

¡õ
It follows that we have an injection . Since the local Langlands correspondence for is bijective, if we also know the local Langlands correspondence for , then we obtain a map . This map is called the *local base change* when it exists. The following cases are known to exist:

- When is split in , we had an isomorphism and this map is simply .
- When is inert in and is unramified, this map is explicitly described by Minguez in the Paris book project.
- When is archimedean.

The functoriality in this special case says the following:

(Functoriality for )
Let

be an automorphism representation of

. Then there exists an automorphic representation

of

satisfying: for

in the previous above cases,

is given by the local base change applied to

.

By the theory of Eisenstein series, if

is not cuspidal, then for any place

of

,

is a subquotient of a parabolic induction from a

*proper* Levi subgroup (see Langlands supplement to Corvallis I). By the assumption,

is supercuspidal, which is not of the form, hence

itself is cuspidal. In particular,

and

are also cuspidal. Now by strong multiplicity one for

*cuspidal* automorphic representations of

, one only needs to check that

and

are isomorphic for almost all places

. This can be checked by the construction of the base change. For example, in the second case (

is inert and

is unramified), then

has a parameter

. The parameter for

is

, so the parameter for

is

and the parameter for

is

(by the compatibility of the local Langlands correspondence for

with the passage to the contragradient). But

, where

is a lift of

and

is of the form

. A calculation shows that these two parameters agrees

*up to conjugation*. (c.f., Remark

101).

¡õ
The existence of the base change is the content of the next theorem.

For the existence of

, see Mok. We already know that

is cuspidal and conjugate self-dual. It remains to check the regularity and

-algebraicity at

of

. The square-integrable representations of

,

is parametrized by their infinitesimal characters which came from regular elements

. If a

representations

is square-integrable, then

is regular and

-algebraic (Theorem

26). One then deduces the properties of the base change of

by calculating the infinitesimal character under base change.

¡õ
To summarize, to prove Theorem 42,we are reduced to, by the previous theorem and Corollary 9, the following theorem (which gives more than we needed as input for Corollary 9).

Let

be a number field and

be a reductive group. Suppose that for any

of

,

has compact center and

has square-integrable representations (this implies that

is totally real). Fix a finite place

of

and a supercuspidal representation

of

and an open compact subgroup

. Then there exists an automorphic representation

of

satisfying

- For any , is square-integrable;
- There exists an unramified character such that .
- .

In the remaining of this course, we will deduce this theorem from Arthur's simple trace formula.

11/26/2013

##
Application of the trace formula for compact quotients

To make life a bit easier, we are going to prove this theorem in the case where , and has *trivial* center to avoid minor technicality. We will first focus on the case where is *compact* and treat the general case next time.

Since is compact, is anisotropic and we can use the trace formula for the compact quotient (Theorem 29).

We will choose of the form where , , . We pick

- the characteristic function of ;
- the matrix coefficient of (it is compactly supported as is supercuspidal and has trivial center).
- This is the most interesting one: is allowed to vary depending on a parameter . Fix a maximal torus, , set of roots of . a root basis. the set of -positive roots. We parametrize the irreducible representations of with the -dominant weights (Theorem 21). Write the corresponding highest weight representation for . We take .

We need to know about these functions in order to proceed. The key input is the Weyl character formula, which we now review.

Let

be a basis of the free

-module

We say a function

is

*polynomial* if it is given by a polynomial in (the dual basis of)

. We say a function

is

*rational* if it is given by a rational function of

.

(Weyl character formula)
- Fix and suppose is regular semisimple (i.e., for any ). Then Here is the half sum of the positive roots and is the sign character. The denominator does not vanish by the assumption that is regular semisimple.
- Fix . Then is a polynomial function.

. Let

be the diagonal torus. Then

, given by

,

. The dominant weights are the non-negative integers. Then

This makes sense when

(i.e,

is regular semisimple) and recovers exactly the Weyl character formula for

. Moreover,

is the

-th symmetric power of the standard representation and

is a polynomial function.

Let

be a sequence of

-dominant weights satisfying: for any

,

as

(we say

* far from the walls*). Then for all

,

, we have

After conjugation, we may assume that

. Then

Since

far from the walls, the coefficients of the monomials of

go to infinity. Because

,

, thus

. Since

is uniformly bounded as

varies, the result follows.

¡õ
We now return to the trace formula. Fix a sequence of -dominant weights such that far from the walls. We choose by

We observe that

- If is an automorphic representation of such that for some , then has the desired property: is simply the equivariant projection , in particular, if and only if . Since is chosen to be a matrix coefficients, if and only if . So to prove the theorem, it is enough to show where is the regular representation of .
- The global orbital integral splits up as a product of local orbital integrals, provided all the measures are chosen carefully; the constants are not important for the proof.

Now we apply the trace formula and obtain Notice is a *compact* subset of , which does not depend on . So by Remark 106, the sum over can be replaced by a sum on a finite subset . Notice . We can always arrange that by choosing suitable matrix coefficient. Then Dividing by , we obtain that The first term is *nonzero* and is independent of . So if we can show that tends to zero, then for all sufficiently large .

Let us choose a measure on so that for any , This is allowable since is compact. So as is conjugation invariant. We conclude the orbital side is nonzero since by Corollary ##Cor:AFfarfromwalls , which finishes the proof of Theorem 44 in the compact case.

12/03/2013

##
Application of Arthur's simple trace formula

References for this section:

- Chenevier-Clozel, JAMS
- Clozel-Delorme, le theoreme de Paley-Wiener invariant pour les groupes reductifs, I, II

Today we we will remove the compact condition imposed on last time. We are going to use Arthur's simple trace formula (Theorem 31). As last time, we choose the test functions of the form and will ensure that for suitable choices of and , vanishes if is not -elliptic. We may assume that and .

Fix an -elliptic maximal torus. The existence of is equivalent to the existence of a compact inner form of (Theorem 25). The idea is to transfer functions (resp. conjugacy classes) on to functions (resp. conjugacy classes) on .

We fix such an and an isomorphism such that is an inner automorphism. Define . The fact (one needs to know a bit more about real groups) is that one can choose such that is defined *over * and the restriction is also defined *over *. Choose such an and let to be the real torus which becomes after extension of scalars. So we have an isomorphism over .

Suppose

,

. Then the diagonal torus

is an

-elliptic torus. Choose

also to be diagonal torus. Then one can find an isomorphism

which restricts to an isomorphism

defined over

.

Recall (Theorem 26) that the square-integrable representations of fall into packets indexed by -orbits of regular elements . Each packet contains exactly representations with the infinitesimal character . Similarly, the irreducible representations of the compact group are in bijection with -orbits of regular elements . This differs from the highest weight parametrization by a -shift (Example 33).

So another way to phrase the parametrization of discrete series of is to put them in packets , where varies over irreducible representations of . The assignment does not depend on the choice of the isomorphism . This is a special case of the (local) Langlands functoriality (the -groups of and are the same and is "more quasi-split" than ). Dual to this transfer we should have a transfer of conjugacy classes. Let be an -elliptic element. Then is -conjugate to an element of the fixed torus . Without loss of generality, we may assume . We define , well-defined up to -conjugacy.

We can further choose all the 's to have support in a fixed compact subset of . Now applying Arthur's trace formula gives Let vary so that all have support in a fixed compact subset . Then the orbital side becomes where is a finite set of -elliptic elements of . But by the previous theorem, we obtain Let far from walls as last time, we obtain that as an application of Weyl character formula. In particular, if is sufficiently far from the wall, then there exists a cuspidal automorphic representation such that . Thus we have constructed the automorphic representation with all desired local properties in Theorem 44.