These are some notes for the comprehensible lectures, by Professor Benedict Gross at Columbia (Fall 2011), on interesting results in representation theory and number theory discovered using the local Langlands correspondence (including a motivated introduction to the Langlands program).

##
The Langlands correspondence

The Langlands program relates two major branches of mathematics: number theory and representation theory. The (conjectural) *Langlands correspondence* is a dictionary between them, which enables people to translate problems in number theory to representation theory and vice versa. In its initial form, the local Langlands correspondence is a bijection between the irreducible representations of a reductive group over a local field , and the conjugacy classes of continuous homomorphism from the Galois group of to the dual group of . These homomorphisms are called *Langlands parameters* (to be redefined for three more times later). Local class field theory says that is isomorphic to the profinite completion of , hence can be formulated as the case under the framework of Langlands program. From this point of view, the Langlands program can be regarded as a vast nonabelian generalization of class field theory. In the global case, the Langlands conjecture predicts a correspondence between automorphic representations and Langlands parameters of the global field.

Note that is a profinite group and we endow the complex Lie group with discrete topology, so a Langlands parameters factors through a finite quotient of . So a Langlands parameter is given by a finite subgroup of and an isomorphism up to conjugacy by . Note that for a local field, has a ramification filtration by the inertia group and wild inertia groups , so there are certain restriction on the structure of the finite subgroup .

When is unramified, is generated by the Frobenius element and thus a Langlands parameter corresponds to a conjugacy class of finite order in . In particular, this conjugacy class is semisimple and is classified by the points of the Steinberg variety . Katz completely classified these semisimple conjugacy classes of finite order using the affine diagram. By the very definition of the dual group, corresponds to a character , and hence a representation of given by the parabolic induction . However, is not always irreducible. In other words, in this case a Langlands parameter corresponds to a finite set of irreducible representations of , instead of a single irreducible representation. This finite set is called an *-packet*.

An invariant related to the -packet associated to a Langlands parameter is the component group of the centralizer . Since is the stabilizer of the orbit of , a representation of can be viewed as a local system on the orbit.

For

, this group is always trivial and the

-packet consists of a single irreducible representation. For

, any conjugacy class of finite order in

is of the form

, where

is an

-th root of unity. When

,

but when

,

. So the component group

is trivial when

but is

when

.

has two irreducible representations and it turns out when

has order 2, the corresponding parabolic induction is reducible and has two irreducible components which together form the

-packet. In general,

is an elementary 2-group for all classical groups and can be

,

,

etc. for exceptional groups.

A Langlands parameter is called *discrete* if the centralizer is finite. These parameters corresponds discrete series representations. Note that no unramified parameter is discrete since .

When is tamely ramified, is generated by the Frobenius element and the generator of the tame ramification satisfying . So a tamely ramified Langlands parameter corresponds to a finite group of generated by two elements of finite order with relation . Suppose is *regular* in , i.e., . Since , it maps to an element in the Weyl group and . If is furthermore *elliptic*, then is finite and is a discrete parameter.

For

, we have

. If we choose

to be of order at least 3 and

to be the nontrivial, then

is a group of order 2 consisting the identity and

. Notice that

,

, we know that

is a dihedral group of order

and the Langlands parameter

should correspond to 2 irreducible representation of

. For example, when

, we have

and

. There is a unique tamely ramified extension

with Galois group

. The parameter

corresponds 2 irreducible representations of

given by compact induction from

and

of the characters

.

Wildly ramified representations are even more interesting and contains huge arithmetic information (e.g. Bhargava's recent work). (The video is incomplete at this point.)

##
Artin L-functions

A Langlands parameter is a homomorphism . Before studying them, we need to know both sides of this homomorphism. We shall discuss a bit about the number theory side in this section and the structure of reductive groups in the next section.

The ideas of connecting number theory to representation theory started with Artin. Let be a number field extension and be a representation of . Artin defined the *Artin -function*

When

is the trivial representation of

,

is equal to the

*Dedekind zeta function* . In particular, when

, the Artin

-function

recovers the

*Riemann zeta function* Also, since all eigenvalues of

are roots of unity, by comparing to

, one knows that

convergences on the half plane

.

When

is a quadratic extension of discriminant

and

is the nontrivial 1-dimensional representation of

, by the quadratic reciprocity we know that

is equal to the

*Dirichlet -function* associated to the quadratic character

. In general, Artin noticed that the 1-dimensional representations of

correspond to the Dirichlet

-functions by Artin reciprocity. Since the analytic properties of Dirichlet

-functions are well understood, Artin conjectured these should be also true for higher dimensional representations of

. This is the content of the famous Artin conjecture.

(Artin)
- has meromorphic continuation to the entire plane. Moreover, the continuation is holomorphic if (i.e., contains no trivial representations).
- satisfies a functional equation.

The meromorphic part was proved by Brauer using his induction theorem, however, the holomorphic part is still open. Solving the Artin conjecture was one of the motivation of Langlands program: The holomorphic continuation of an Artin -functions will follow from the holomorphic continuation of the associated automorphic -function, which can be established comparatively easily.

By adding the factors at infinity places, Hecke was able to show the functional equation for the Dedekind zeta function in his thesis (and Tate reproved it using Fourier analysis on adeles in his famous thesis). Let and be the number of real and complex places of and , then Artin formulated the functional equations of Artin -functions analogously. Define the local factors
where are the eigenspaces of the complex conjugation. Then satisfies the functional equation (proved by Brauer) Here is the *root number*. It is a complex number with absolute value 1 satisfying but is hard to compute in general. The factor , where is the discriminant of and is the *Artin conductor* of . The Artin conductor is is defined locally and has a -component iff the inertia group at acts on nontrivially. More precisely, Let be the ramification filtration at , then . Artin proved that this number is actually an integer, thus is an integral ideal of . In fact, using the full power of local class field theory, Artin found a representation of defined *over * such that . The *Swan conductor* . So only when the *wild inertia* acts nontrivially.

Notice that and for is a Galois extension with . We know that because the regular representation decomposes as . Comparing the factors in the functional equation of and , we obtain the *conductor-discriminat formula* The Taylor expansion of contains much arithmetic information from the *class number formula*. The *Stark Conjecture* predicts the information coming from the Taylor expansion of the Artin -functions, which is still largely open.

More generally, when is not necessarily Galois, we take it Galois closure with Galois group . Suppose is the subgroup fixing , then . Since , we have . In particular, when acts 2-transitively, decomposes as two irreducible components: a copy of trivial representation and a copy of an irreducible representation of dimension . Hence and .

Consider

. Let

(Eisenstein) and

, then

is totally ramfied and hence

. Let

be the Galois closure of

. The discriminant of

is equal to

. Since

contains

and 101 is not a square in

, we know that

contains the degree 2 unramified extension of

and

. It follows that

is a subgroup of

of order at least 8, hence is either

or

. Since

does not have a cyclic quotient of order 6, we know that

, thus

. So either

(

) or

(

). Since

, we know that

by the conductor-discriminant formula. This rules out the case

, where

. Therefore

. From

, we know that

. Let

be the orthogonal representation of

with determinant 1, then

and

since it only differs from

by the sign character. This orthogonal representation gives us a Langlands parameter

with Swan conductor 1. It corresponds to the representation

of

induced from the Iwahori subgroup

of the character

.

##
Artin L-functions for function fields

Let

be a function field over a finite field

with

elements and

be the genus of the curve

. Let

be a field extension with Galois group

. For

a representation of

, we can define the Artin

-function

similarly to the number field case. In the number field case, the exponential factor in the functional equation of

is

, where

is the discriminant and

is the Artin conductor of

. For the function field case, the exponential factor is replaced by

, where

is the conductor

*divisor*. The functional equation is similar but has a lot of cancellation since the residue field is always

. For example,

In general, for a curve of genus

, the zeta function will be of the form

which is a rational function of degree

in

. The general fact that Weil discovered is that the Artin

-function

is a rational function of

of degree

(e.g.,

corresponds to the trivial representation and we have

and

.) Even nicer, if the

*geometric inertia group* (where

is the closure of

in

) has trivial invariants, namely

, then

is actually a

*polynomial* in

. In particular, when

and

, then the degree

, thus

and the equality holds if and only if

. Some interesting things happen even in this seemingly silly case.

Let

with

(corresponding to the projective line

) and

be the quadratic extension of

(corresponding to the curve

). Let

be the non-trivial irreducible representation of

. Assume

, then by the above discussion we know

Then the inertia group is non-trivial for

(tamely ramified as

), so

and

.

In characteristic zero, a finite group acting on

can only be a finite subgroup of

, which can only be a cyclic group, dihedral group,

,

or

. But in positive characteristic, the finite automorphism group

of

over

is huge. Let

be the covering obtained by taking the quotient of

by the group

. To find the Artin conductor, let us find the

-orbits on

.

is a

-orbit with stabilizer the Borel subgroup

of

, which has order

.

is a

-orbit with stabilizer the non-split torus

, which has order

. All other orbits are free since a fixed point under

necessarily satisfies a quadratic equation over

. Choose our third orbit to be a free orbit

(note

).

Let us assume those three orbits are mapping to correspondingly. For the first orbit, there is no residue field extension, so everything is in the inertia group . It turns out that the ramification filtration is , where is the unipotent radical which has order . For the second orbit, the inertia group and there is no wild ramification. For the third orbit, .

By the previous discussion, we know for any nontrivial representation of and . But the Artin conductor Now one can have a lot of fun to verify the equality for various representations of . We have the principal series representations with , Steinberg and twisted Steinberg representations with , and discrete series representations with . For the principal series representations, we have , , . We verify that . All these exotic groups acting on in positive characteristic is precisely because of *wild ramification*. There are enormous numbers of examples (e.g., all the Deligne-Lusztig curves) where the covering of is only ramified at two points and is only wildly ramified at one point.

##
Classification of reductive groups

###
Over algebraically closed fields

Over an algebraically closed field (of any characteristic), it is well known that the reductive groups are classified by their root data. Let be a reductive group over , then all maximal tori of are conjugate. Fix a maximal torus . Let be the *character group*. Then the restriction to of the adjoint representation of on decomposes into pieces , where is the set of roots and each is one-dimensional. Fixing a root , one can find a homomorphism (up to conjugacy by ) such that and the Lie algebra of maps into and that of maps into . So we can associate a *coroot* to and by composing with we get the pairing . Even better, , the image of lies in the normalizer . The quadruple satisfies the *reduced root datum axioms*:

- There is a bijection between and the pairing .
- If , then the only integral multiple is .
- The simple reflection carries into itself and the induced map of carries into itself.

The element acts on as the Weyl group acts on . Actually it acts via the simple reflection and all the simple reflections permute the roots and generate the Weyl group. The last axiom extremely limits the possibility of root data (e.g.s there are only three rank two root systems) and allows the classification of the root data. The isomorphism theorem and the existence theorem ensures a reductive group is determined by its root datum *up to isomorphism* and for every root datum there exists a reductive group giving rise to it. The *dual group* is defined by switching the role of roots and coroots, which is again only defined up to isomorphism. We can not even talk about conjugacy classes in the dual since it may have outer automorphisms.

The isomorphism tells us that , however, is usually *not* equal to . For example, when , we know that but .

When is a torus of dimension , and . On the contrary, when is semisimple, and are both of finite index, and the position of between and determines the isogeny class of . The Cartier dual of is . In particular, when , then , namely is adjoint if and only if . On the contrary, is simply-connected if and only if . More generally, splits if and only if is connected and splits if and only if the derived group of is simply connected.

Given a root datum , the simple reflections generate the Weyl group , which sits inside the automorphism group . We also have a canonical automorphism . Sometimes (e.g. for ), but sometimes (e.g. for ). The notion of *Borel subgroups* come into play in order to understand this problem. Fixing a Borel subgroup gives more structure on the root datum by looking at the action of on , namely it picks out the *positive roots* with respect to and also determines a root basis . The -tuple is called the *based root datum* and the Weyl group acts *simply* transitively on . So and has trivial intersection inside . In fact, it turns out that .

Notice that the exact sequence does not split ( has order two, but may have order 4), hence does not act on itself. The amazing thing is that the other part actually acts on . A *pinning* is a choice of , where is a basis of the simple root space . Any inner automorphism preserving a pinning must be trivial. A theorem of Chevalley states that , where is the group of pinned automorphisms for a fixed pinning, which is in fact equal to .

###
Over arbitrary fields

We now understand the classification of reductive groups over an algebraically closed field and the automorphism group . Serre gave a method using Galois cohomology and descent to classify reductive groups over an arbitrary field .

For tori, a theorem of Grothendieck ensures that if two tori and over become isomorphic over , then they are actually isomorphic over a *finite separable* extension of (i.e., the scheme of tori is smooth). For every isomorphism over , conjugating by gives rise to an automorphism of over satisfying . This assignment is a 1-cocycle as and a different isomorphism over gives rise to which is equal to up to a 1-coboundary. In fact, if then . Even better, as a torus has no outer automorphism, we know and have no difference since both of them are isomorphic to . So the above assignment in fact gives a homomorphism (up to conjugacy), in other words, an integral Galois representation of rank . Conversely, for any such Galois representation on , we obtain a desired torus with -points . Hence the category of tori isomorphic to is equivalent to the category of integral -representations of rank .

For

, we obtain the homomorphism

. When

is trivial,

is split over

. When

is nontrivial,

fixes a separable quadratic extension

with Galois group

and

. Every such torus can be viewed as a subgroup of

by considering its action on

. For

, we get a split torus

and a nonsplit torus

. For

, we obtain a split torus

of order

and a nonsplit torus

of order

.

For

and

, we obtain the conjugacy class of a homomorphism

. Let

be the image of the nontrivial element under

. Then we get the tori

,

,

, and

corresponding to

,

,

and

. For

and

, we obtain the homomorphism

. Its image has order 1,2,3,4,6. For order 2, we obtain the four cases as above. For each of the other orders, we obtain a unique torus.

Now let us consider a general reductive group over . Given a root datum, Chevalley showed that there exists a unique split group (called the *Chevalley group*) over with that root datum (even a group over proved by Grothendieck). Then the same argument as in the case of tori shows that the groups defined over that are isomorphic to over are classified by the pointed set . However, due to the existence of for general , these cohomology classes are no longer homomorphisms as in the case of tori. Anyway, we can map any class to given by the map and corresponds to a homomorphism . Since can be viewed as a subgroup of , this map is actually surjective. Such a form corresponding to is called *quasi-split*. Moreover, the fiber over can be identified as the image of (*inner forms*) in .

Let

be a finite field. By Lang's theorem, for any connected algebraic group

over

, we have

. So there is no inner form and

is determined by

up to conjugacy, in other words, conjugacy classes of finite order in the group

. So the forms over a finite field are classified by pure combinatorical data.

Let

be a local field. Kneser proved that

, where

is a finite commutative group scheme over

of multiplicative type. By Tate duality, we have a perfect pairing

. Hence

is the Pontryagin dual of

, where

is the root system of

. In particular, for

, we obtain

.

Let

be a global field. We have the

*Hasse principle* saying that the map

into the local Galois cohomologies is injective.

##
Langlands dual groups and Langlands parameters

Let be a reductive group over an arbitrary field . Then gives a quasi-split form corresponding to the conjugacy class of a homomorphism . Let be the fixed field of . Then is a finite separable Galois extension of and splits the maximal torus of . is isomorphic to the pinned automorphism of , which is also isomorphic to the pinned automorphism of , so we obtain an action of on .

The

*Langlands dual group* is defined to be

, where the connected component

is the complex algebraic group with the root datum dual to

and

acts on

as pinned automorphisms (fixing a pinning). Note that

*only* depends on the quasi-split form of

.

A

*Langlands parameter* is (initially) a homomorphism

such that we get the standard surjection

when composing

with the projection

. Two Langlands parameters are called

*equivalent* if they are conjugate by some

.

When

is split or an inner form, we have

, so

. For

,

and a Langlands parameter is a

-dimensional complex representation of

. For

,

and a Langlands parameter is a 3-dimensional orthogonal representation of

with determinant 1. More generally, for

,

and a Langlands parameter is a

-dimensional orthogonal representation of

with determinant 1.

For

,

. The action of

is

*not* by the inverse transpose. Write

, where

is an

-dimensional vector space. Let

be an orthogonal (

odd) or symplectic (

even) space of dimension

such that

,

are the maximal isotropic space of dimension

and

. Then

is the Levi subgroup of the parabolic stabilizing

. When

is odd, the normalizer of

in

is equal to

, so a Langlands parameter is an

-dimensional orthogonal representation.

Let

be a torus of dimension 1. For

,

. For

,

, where

acts on

by

. More generally, the Langlands dual of a torus

is given by

, where

is the character group. As

is abelian, we know that a Langlands parameter

is exactly a 1-cocycle of

with values in

and the equivalence relation is exactly the coboundary condition, hence Langlands parameters correspond to the classes in

. From the exact sequence

we obtain an isomorphism

, as

is a complex vector space and all its higher Galois cohomologies are trivial. On the other hand, the Galois pairing

induces the cup product pairing

By local class field theory,

for

a local

-adic field. In this case, Tate proved that the above induces a perfect pairing between

and

(the profinite completion of

). Therefore a Langlands parameter of

is nothing but a homomorphism

of

*finite order*. In order to get

*all* continuous homomorphisms

, Langlands replaced

by the

*Weil group* so that we can take the Frobenius to anywhere we like.

A

*Langlands parameter* is a homomorphism

such that we get the standard surjection

when composing

with the projection

.

Under this new and better definition, the Langlands parameters of are exactly all irreducible complex representation of .

There are two more local fields: the real numbers and complex numbers. Their Galois groups are not very interesting, but their Weil groups are. It turns out

and

, where

is the Hamiltonians. In the latter case, we have a (nonsplit) exact sequence

. Since

has an abelian subgroup of index two, its irreducible representations have dimension either 1 or 2. Since

always holds by local class field theory, the 1-dimensional representations of

are the characters of

(e.g. the trivial or the sign character). Let

, we define

and

be a 2-dimensional representation of

. One can check that

and is irreducible unless

(

). The representation

is self-dual and is symplectic if

and is orthogonal if

.

Here comes a piece of evidence of the Langlands correspondence. The two real forms

and

of

has the same Langlands dual group

. The Langlands parameters

are 2-dimensional symplectic representations and the irreducible ones are exactly

parametrized by the half integers

in the previous example. These Langlands parameters correspond to the discrete series representations of

of weights

and irreducible representations of

of dimensions

. More generally, the Langlands parameters

,

correspond to the irreducible representations of

of highest weight

, where

.

##
Unramified parameters and unramified representations

Another piece of evidence of the local Langlands correspondence was discovered for a family of so called unramified representations for -adic groups.

Consider the multiplicative group

over a

-adic field

. The irreducible representations of

are characters

. We say a character

is

*unramified* if it is trivial on the units

, where

is the ring of integers of

. In other words,

is uniquely determined by its value on a uniformizer

. These characters are prevalent as

*almost all* components of a Hecke character

are unramified. By local class field theory,

and the inertia group

maps to the units

, hence an unramified character

corresponds to a Langlands parameter

which is trivial on the inertia group

.

Motivated by this:

We say a Langlands parameter

is

*unramified* if its trivial on the inertia group

. In particular,

is a unramified extension.

Morally speaking, the unramified parameters should correspond to those "unramified" representations of which are "trivial" on , though is not normal in in general.

Let

be a torus split by an unramified extension

. Let

be a maximal split subtorus. Then the cocharacter group

. It turns out

has a maximal compact subgroup, which we denote by

(similarly for

) and

. Fixing a choice of uniformizer

, we have

. Hence an unramified representation of

corresponds to an element in

. On the other hand, an unramified parameter is determined by the image of the Frobenius

in

up to the conjugation of

, which can be identified as

. So we obtain a bijection between the unramified Langlands parameters and unramified representations of

.

We have the following more general notion of unramified representations. Analogously to the 1-dimensional case, *almost all* components of an automorphic representation are unramified in this sense.

We say

is

*unramified* over

if it is a quasi-split group and split by an unramified extension

. If

is unramified, then there exists a

*hyperspecial compact* subgroup

(maximal compact subgroup in the sense of volume, not unique even up to conjugation). We say a representation

of

is

*unramified* if

. (We will see that any irreducible admissible representation of

satisfies

.)

Let us consider the case when is split. By Grothendieck's theorem, is defined over . The hyperspecial group is simply the -points of . Before proceeding, however, we need to correct our earlier definition of unramified Langlands parameters. In the split case, a Langlands parameter is simply a conjugacy class in the complex group determined by the image of the Frobenius .

We say a Langlands parameter

is

*unramified* if it is trivial on the inertia group

and the image of the Frobenius

is

*semisimple*.

The reason comes from the following idea: complex parameters of are reflection of *families of compatible -adic representation* .

Consider the

-adic Tate module (

) of

given by

. Then

is a 1-dimensional

-adic representation of

and

acts as

for each

, where

is the cardinality of the residue field. This family corresponds to a complex 1-dimensional representation where

acts as

.

Let

be an elliptic curve over

and consider its

-adic Tate module

. Then

is a 2-dimensional

-adic

-representation. When

has good reduction, this representation is unramified and the image of arithmetic Frobenius

has an integral characteristic polynomial

for each

, where

. Weil proved that these images are

*semisimple*, hence are determined by this characteristic polynomial. It is believed that for these

-adic representations coming from algebraic geometry (e.g. abelian varieties), the Frobenius elements should map to semisimple classes. When

has bad reduction, it turns out that the inertia

acts nontrivially (ramified). Since the wild inertia

is a pro-

group and

has a pro-

subgroup of finite index, the image of

in

is finite, hence semisimple. However, the tame inertia

can map to elements of infinite order. Indeed, Tate showed that for the elliptic curve

with multiplicative reduction at

, the image of the tame inertia is a subgroup of finite index in

for each

. Moreover, Grothendieck discovered that there exists a unique nilpotent element

such that the action of the tame inertia is given by the exponential of

.

More generally, we will require any Langlands parameter to send to a semisimple element of (note that the image of is of finite order as is pro-finite, hence is automatically semisimple). To account for the phenomenon of tame inertia we have not seen in the torus case, we introduce an extra factor in our definition of Langlands parameter as follows.

A

*Langlands parameter* is a homomorphism

which is algebraic on

and sends the Frobenius

to a semisimple element, such that we get the standard surjection

when composing

with the projection

. The group

is sometimes called the

*Weil-Deligne group* of

.

Let us return to the split case. If is split, then the unramified parameters correspond to semisimple conjugacy classes in , hence by conjugation, correspond to elements of the *Steinberg variety* . The local Langlands correspondence in this case is achieved by the theory of admissible complex representations of the -adic group . Let be the maximal compact subgroup of .

An

*admissible* representation of

is a complex representation

of G such that the subgroup fixing any vector

is open (

*smooth* representation) and

is finite dimensional for any compact open subgroup

. Equivalently,

, where

's are irreducible representations of

(hence are finite dimensional as

is compact). In particular,

is unramified if the trivial representation of

appears in

.

The *Hecke algebra* is defined to be , where is the linear combination of characteristic functions on the double cosets . It acts on an admissible representation via integration and preserves the space .

The map

gives a (non-functorial) bijection between all irreducible admissible representations

satisfying

and all simple

-modules.

The Hecke algebra

is commutative. (We say such a pair

is a

*Gelfand pairing*.)

It follows from this key nontrivial fact that all the simple have dimension one, hence . So to understand the representations of , we need to know more about the structure of the Hecke algebra. From the Cartan decomposition , one looks at the functions on invariant under the action of the Weyl group.

(Satake)
There is an isomorphism

.

An even better version of the Satake isomorphism states that .

Consider

. In this case

has a basis

, where

are integers. Hence the Hecke algebra is generated by

and

(

). In particular, for

one can recognize the classical Hecke operators for modular forms.

Since , is the coordinate ring of the Steinberg variety (which can be also identified as the *representation ring* of ). Thus the characters of are the points of . We now know that the irreducible unramified representations of correspond exactly to unramified parameters . This is the main reason why the *dual group* plays such a big role in the whole story.

##
Steinberg representations and supercuspidal representations

Now let us discuss certain classes of ramified local representations.

Let be any reductive group over a -adic field. Let be the pinning we fixed in the definition of the Langlands dual group. The *principal regular unipotent element* is the element . It determines a homomorphism by sending to on the Lie algebra level. Since is fixed under the action of , we obtain a homomorphism . This induces the canonical *Steinberg parameter* . The *Steinberg representation* of is the canonical representation associated to this Steinberg parameter.

For

, the principal regular nilpotent element

For

,

and the parameter coming from an elliptic curve with multiplicative reduction is the Steinberg parameter.

To construct the Steinberg representation, let us begin with the finite field case which Steinberg originally discovered. Let be a reductive group over , then is a finite group. Any reductive group over a finite field is quasi-split, hence has a Borel subgroup and its unipotent radical . It turns out that is a -Sylow subgroup of . Steinberg discovered a canonical -representation of dimensional . It restriction to is the regular representation and has the property that is 1-dimensional. can be viewed as a representation of the Hecke algebra . We have the Bruhat decomposition and an isomorphism .

The latter isomorphism can be seen as a *-deformation*: the group algebra is generated by satisfying (and other relations) where the 's are the simple roots. Analogously, the Hecke algebra is generated by satisfying . has two distinguished 1-dimensional representations: the trivial representation and the sign representation . Analogously, also has two distinguished 1-dimensional representations: the trivial representation and the Steinberg representation .

Steinberg's idea extends to the -adic settings. We shall assume is simply-connected for simplicity. We define the *Iwahori subgroup* be the lifting of and the *pro- unipotent radical* be the lifting of . These groups play a similar role as the finite field case: the Steinberg representation has the property . However, the Hecke algebra is *noncommutative* in this case. Nevertheless, we have the Iwahori decomposition , where the *affine Weyl group* (generated by the simple roots and the lowest root), and an isomorphism as a -deformation. Again the Steinberg representation corresponds to , which in turn implies that for any large group (called a *parahoric subgroup*), .

Since appears in with multiplicity 1, by Frobenius reciprocity, contains . However, also contains other representations containing (e.g. many unramified representations) and has infinite length. On the other hand, there exist irreducible representations such that is an irreducible representation (called *depth-zero supercuspidal*) and .

For

we have the trivial representation of dimension 1, the Steinberg representation of dimension

, induced representations

(

a character of the split torus) of dimension

and discrete series (or supercuspidal) representations of dimension

. The discrete series are associated to the characters of the non-split torus and are much harder to construct than others. Drinfeld found a way to produce the discrete series by taking the first

-adic cohomology of the curve

.

Drinfeld's idea was generalized by Deligne-Lusztig for any group. Those supercuspidal representations they constructed are indexed by the characters of anisotropic tori. Anisotropic tori are parametrized by the *elliptic conjugacy classes* in the Weyl group (no invariance on the reflection representation). DeBacker and Reeder constructed the Langlands parameters of these Deligne-Lusztig representations as tamely ramified parameters , where the tame inertia maps a cyclic subgroup of of order prime to such that (regular) and the Frobenius maps to an elliptic class in .

Even the Steinberg parameters and depth-zero parameters are not enough for the global application. We need to construct wild representations, i.e., representations which have even no vectors fixed by . We end this section by constructing a class of wild representations called *simple supercuspidal representations*.

The *Frattini subgroup* is the smallest normal subgroup such that the Frattini quotient group is an elementary -group. Moreover, the generators of lifts to a generating set of . When is large enough, Reeder and the speaker can show that as a representation of , where 's are the affine simple roots generating the affine Weyl group. Motivated by Gelfand-Graev's theory of generic representations over finite fields, we say a character of is *affine generic* if it is non-trivial on each line (and trivial on ). It turns out has finite length and has a unique component with trivial central character. This representation thus constructed is *wild*.

For

, we have

So

and

for

a generic character.

##
Motives of reductive groups

Steinberg found a beautiful formula for the order of a reductive group over finite field. As examples, where the powers can be recognized as the degree of the basic invariant polynomials of the Weyl group. We will generalize Steinberg's formula using motives.

The motive of a reductive group over is a collection of Galois representations constructed as follows. Take a quasi-split inner form of and fix a torus and Borel subgroup of defined over . Then () is a representation of (trivial when is split) and also a representation of the Weyl group , hence a representation of the semi-direct product . Let be -invariants of the symmetric algebra of , so acts on . A theorem of Chevalley showed that is a polynomial algebra generated by homogeneous algebraically independent polynomials. Let be the ideal consisting of elements of constant term 0, then consists of the basic invariant polynomials of and the degrees of the basic invariants are uniquely determined. Each is thus a -representation. Now Steinberg's formula can be rewritten as

For

,

, where

is a trivial Galois representation of dimension 1. Steinberg's formula for

coincides with the easy direct computation. For

,

, where

is a

-representation of dimension 1, trivial when

is even and nontrivial when

is odd.

Using the *Tate twist* (as a family of -adic representation), Steinberg's formula can be also written as . More generally:

Let

be a reductive group over an arbitrary field

. The

*motive* of

is defined to be

. (Using

instead of

is sometimes more convenient.)

So for -adic fields and global fields, one has corresponding -functions for the motive . Using , one can check that is finite and nonzero as long as is anisotropic.

(Serre)
Suppose

is a local field, then there exists a unique invariant measure (i.e., a scalar multiple of the Haar measure)

on

such that for any discrete cocompact torsion free subgroup

of

,

The measure

is called the

*Euler-Poincare measure*.

The problem occurring is the possibility of (e.g. ). Nevertheless, Serre showed that if and only if contains an anisotropic maximal torus over (equivalently, has a discrete series in ). Moreover, when is -adic, is also equivalent to being anisotropic, which miraculously matches the condition that is finite and nonzero.

When is -adic and is simply connected, Bruhat-Tits theory gives maximal compact subgroup containing the Iwahori . Serre found a formula for the Euler-Poincare measure where .

For

,

Since

is a subgroup of

and

of index

, we know that

where

is the unique invariant measure assigning the hyperspecial subgroup

measure 1. So Serre's formula shows that

. More generally, if

is unramified and simply-connected, we have

.

Now consider the global case. Suppose is a simply-connected reductive group over such that is compact (so the Euler-Poincare measure assigns measure 1) and is split for every . Let be the invariant measure such that (called the *Tamagawa measure*). Let be the invariant measure assigning measure 1. A great result analogous to the local case is the comparison between the discrete measure and the compact measure where is the Artin -function of the motive.

The *mass formula* then follows from this comparison: where 's are the finite stabilizers of under the right -action.

For

split, we have

, therefore

. But one orbit has stabilizer

of exact order

, so there is only one orbit and

. (There are no other way yet to prove this!)

More generally, suppose is unramified outside a finite set . Let then

##
The trace formula and automorphic representations with prescribed local behavior

For simplicity, we shall assume is a simply-connected simple group over such that is compact (to avoid analytical difficulty of the trace formula). Then is unramified outside a finite set and for we have the hyperspecial subgroup . Under this assumption, is discrete and *cocompact*. So decomposes as a Hilbert sum of irreducible unitary representations of with finite multiplicity.

Let be a continuous function on with compact support, it acts on via integration . Such a function is a product of local functions , where for almost all places . Each irreducible representation of can be decomposed into local factors , where is an irreducible unitary representation of . Since acts on as the projection onto the -fixed space, its trace on is either 1 (when is unramified) or 0 (otherwise). So is a finite product and the sum is a finite sum.

Next we shall choose a suitable function to pick out specific with prescribed local components. Let be a finite set of primes. Kottwitz considers the test function consisting of at , at and the Euler-Poincare function at . Then is nonzero if and only is trivial, is unramified for and is trivial or Steinberg for (when is unitary, a theorem of Casselman says that unless is trivial or Steinberg). When is compact, the Steinberg is the trivial representation. By strong approximation if is not compact and is trivial, then is trivial itself. Consequently, where the sum runs over all such that is trivial, is Steinberg for and is unramified for .

The *trace formula* computes the trace in terms of orbital integrals: So for , the trivial conjugacy class of contributes to the right-hand-side by

Since our test function is supported in an open compact subgroup of , if the orbital integral of is nonzero, has to be of finite order. However, these integrals of torsion classes are quite complicated. (There can be infinitely many of them over !) People sometimes replace conjugacy classes by *stable conjugacy classes* in the trace formula to avoid this complexity.

Here we use a trick to simply the trace formula by introducing simple supercuspidal representations at another finite set of primes disjoint from . For , we choose an affine generic character of the pro- unipotent radical and pick our test function at to be on . When , is an *irreducible* simple supercuspidal representation. So by Frobenius reciprocity, we know that the operator picks out this simple supercuspidal representation when taking the trace. Suppose consists of at least two distinct primes and , if the orbital integral of a conjugacy class is nonzero, then is conjugate to elements of and , thus must have order and simultaneously, hence must be trivial. Consequently, the orbital side only consists one term In sum, the Artin -function of the motive counts the automorphic representations of prescribed local components: is trivial, is Steinberg for , is the simple supercuspidal associated to for and unramified elsewhere:

More generally, when is simple over (but not necessarily simply-connected and is not necessarily trivial), we have the following formula: where and is the decomposition group of . Note that if is simply-connected, then has trivial center and we recover the previous formula.

For

, the above formula gives the number of automorphic representations with prescribed local components:

This is a huge number even if the primes in

and

are chosen to be small!

For

. Let

be the conductor of an irreducible representation

. Then

is Steinberg or unramified twisted Steinberg if and only if

and

is simple supercuspidal if and only if

. Then the number of such representations with prescribed local components is

These representations correspond to new forms of weight 2 and level

. So we obtain an exact formula for the dimension of

.

Let us end these lectures by giving an example over a function field. Take

of genus 0 and

,

. The trace formula works out similarly, but in the function field case

and

, so

identically. In other words, there is a

*unique* automorphic representation with such prescribed local behavior. A natural question is to construct a global Langlands parameter

of this unique representation. This parameter is unramified outside

, hence factors through

. It is tamely ramified at 0 with monodromy a regular unipotent element and wildly ramified at

. For some groups like

, Deligne wrote down these parameters using Kloosterman sums. Frenkel and the speaker constructed complex analogue of these parameters. Later, Heinloth-Ngo-Yun used methods from geometric Langlands theory to construct these parameters in general.