IV Topics in Number Theory - The Langlands correspondence

4The Langlands correspondence

IV Topics in Number Theory

4.3 The Langlands classification

Recall that the group algebra

[

] is an important tool in the representation

theory of finite groups. This decomposes as a direct sum over all irreducible

representations

C[G] =

dim(π)

The same result is true for compact groups, if we replace

[

] by

(

). We

get a decomposition

(G) =

dim(π)

where

is defined with respect to the Haar measure, and the sum is over all

(finite dimensional) irreducible representations of

. The hat on the direct sum

says it is a Hilbert space direct sum, which is the completion of the vector space

direct sum. This result is known as the Peter–Weyl theorem. For example,

(R/Z) =

n∈Z

C · e

2πiny

However, if G is non-compact, then this is no longer true.

Sometimes, we can salvage this a bit by replacing the discrete direct sum

with a continuous version. For example, the characters of

are those of the

form

x 7→ e

2πixy

which are not L

functions. But we can write any function in L

(R) as

x 7→

ϕ(y)e

2πixy

dy.

So in a sense,

(

) is the “continuous direct sum” of irreducible representations.

In general,

(

) decomposes as a sum of irreducible representations, and

contains both a discrete sum and a continuous part. However, there are irre-

ducible representations that don’t appear in

(

), discretely or continuously.

These are known as the complementary series representations. This happens,

for example, for G = SL

(R) (Bargmann 1947).

We now focus on the case

(

), or any reductive

-group (it doesn’t

hurt to go for generality if we are not proving anything anyway). It turns out

in this case, we can describe the representations that appear in

(

) pretty

explicitly. These are characterized by the matrix coefficients.

G → GL

(

) is a finite-dimensional representation, then the matrix

coefficients π(g)

can be written as

π(g)

= δ

(π(g)e

where

∈ C

is the

basis vector and

∈

(

)

∗

is the

dual basis vector.

More generally, if

G → GL

(

) is a finite-dimensional representation, and

v ∈ V , ` ∈ V

∗

, then we can think of

v,`

(g) = `(π(g)v)

as a matrix element of π(g), and this defines a function π

v,`

: G → C.

In the case of the

we care about, our representations are fancy infinite-

dimensional representations, and we need a fancy version of the dual known as

the contragredient.

Definition

(Contragredient)

Let (

π, V

) be a smooth representation. We define

∗

= Hom

(V, C), and the representation (π

∗

, V

∗

) of G is defined by

∗

(g)` = (v 7→ `(π(g

−1

)v)).

We then define the contragredient (˜π, ˜v) to be the subrepresentation

V = {` ∈ V

∗

with open stabilizer}.

This contragredient is quite pleasant. Recall that

V =

[

We then have

V =

[

∗

)

Using the projection π(e

) : V → V

, we can identify

∗

)

= (V

)

∗

So in particular, if

is admissible, then so is

˜π

, and we have a canonical

isomorphism

V →

Definition

(Matrix coefficient)

Let (

π, V

) be a smooth representation, and

v ∈ V , ` ∈

V . The matrix coefficient π

v,`

is defined by

v,`

(g) = `(π(g)v).

This is a locally constant function G → C.

We can now make the following definition:

Definition

(Square integrable representation)

Let (

π, V

) be an irreducible

smooth representation of G. We say it is square integrable if ω

is unitary and

|π

v,`

| ∈ L

(G/Z)

for all (v, `).

Note that the fact that

is unitary implies

|π

v,`

is indeed a function on

(G/Z). In general, it is unlikely that π

v,`

is in L

(G).

is finite, then

is automatically unitary and we don’t have to worry

about quotienting about the center. Moreover,

is square integrable iff

v,`

∈

(

). In this case, if we pick

` ∈

non-zero, then

v 7→ π

v,`

gives an embedding

of V into L

(G). In general, we have

V ⊆ L

(G, ω

) = {f : G → C : f (zg) = ω

(z)f(z), |f| ∈ L

(G/Z)},

A slight weakening of square integrability is the following weird definition:

Definition

(Tempered representation)

Let (

π, V

) be irreducible,

unitary.

We say it is tempered if for all (v, `) and ε > 0, we have

|π

v,`

| ∈ L

2+ε

(G/Z).

The reason for this definition is that

is tempered iff it occurs in

(

), not

necessarily discretely.

Weakening in another direction gives the following definition:

Definition

(Essentially square integrable)

Let (

π, V

) be irreducible. Then

(π, V ) is essentially square integrable (or essentially tempered) if

π ⊗(χ ◦det)

is square integrable (or tempered) for some character χ : F

→ C.

Note that while these definitions seem very analytic, there are in fact purely

algebraic interpretations of these definitions, using Jacquet modules.

A final category of representations is the following:

Definition

(Supercuspidal representation)

We say

is supercuspidal if for all

(v, `), the support of π

v,`

is compact mod Z.

These are important because they are building blocks of all irreducible

representations of GL

(F ), in a sense we will make precise.

The key notion is that of parabolic induction, which takes a list of represen-

tations σ

of GL

(F ) to a representation of GL

(F ), where N =

We first consider a simpler case, where we have an

-tuple

= (

, . . . , χ

) :

(

)

→ C

of characters. The group

(

) containing the Borel

subgroup

of upper-triangular matrices. Then

is the semi-direct product

T N

, where

∼

(

)

consists of the diagonal matrices and

the unipotent

ones. We can then view χ as a character χ : B → B/N = T → C

We then induce this up to a representation of

. Here the definition of an

induced representation is not the usual one, but has a twist.

Definition

(Induced representation)

Let

B → C

be a character. We define

the induced representation

Ind

(

) to be the space of locally constant functions

f : g → C such that

f(bg) = χ(b)δ

(b)

1/2

f(g)

for all b ∈ B and g ∈ G, where G acts by

π(g)f : x 7→ f(xg).

The function δ

(b)

1/2

is defined by

(b) = |det ad

(b)|.

More explicitly, if the diagonal entries of b ∈ B are x

, . . . , x

, then

(b) =

i=1

n+1−2i

= |x

n−1

n−3

···|x

−n+1

This is a smooth representation since

B\G

is compact. In fact, it is admissible

and of finite length.

When this is irreducible, it is said to be a principle series representation of

Example. Recall that P

(F ) = B\GL

(F ). In this terminology,

∞

(F )) = Ind

(δ

−1/2

This is not irreducible, since it contains the constant functions, but quotienting

by these does give an irreducible representation. This is called the Steinberg

representation.

In general, we start with a parabolic subgroup

P ⊆ GL

(

) =

, i.e. one

conjugate to block upper diagonal matrices with a partition

···

This then decomposes into MN, where

∼

(F ), N =

















··· ∗

















This is an example of a Levi decomposition, and M is a Levi subgroup.

To perform parabolic induction, we let (

σ, U

) be a smooth representation of

, e.g.

⊗ ···⊗σ

, where each

is a representation of

(

). This then

defines a representation of

via

P → P/N

, and we define

Ind

(

) to be

the space of all locally constant functions f : G → U such that

f(pg) = δ

(p)σ(p)f(g)

for all p ∈ P, g ∈ G and δ

is again defined by

(p) = |det ad

(p)|.

This is again a smooth representation.

Proposition.

(i) σ is admissible implies π = Ind

σ is admissible.

(ii) σ is unitary implies π is unitary.

(iii) Ind

(˜σ) = ˜π.

(ii) and (iii) are the reasons for the factor δ

1/2

Example. Observe

∞

(P(F )) = Ind(δ

1/2

) = {f : G → C : f (bg) = δ

(b)f(g)}.

There is a linear form to

given by integrating

over

(

) (we can’t use

F since GL

(F ) is not compact). In fact, this map is G-invariant, and not just

(O)-invariant. This is dual to the constant subspace of C

∞

(F )).

The rough statement of the classification theorem is that every irreducible

admissible representation of

(

), is a subquotient of an induced

representation

Ind

for some supercuspidal representation of a Levi subgroup

P = GL

× ···×GL

(F ). This holds for any reductive G/F .

For

(

), we can be more precise. This is called the Langlands classifica-

tion. We first classify all the essentially square integrable representations:

Theorem.

Let

with

m, r ≥

1. Let

be any supercuspidal representation

of GL

(F ). Let

σ(x) = σ ⊗|det

Write ∆ = (

σ, σ

(1)

, . . . , σ

(

r −

1)), a representation of

(

)

× ···×GL

(

Then

Ind

(∆) has a unique irreducible subquotient

(∆), which is essentially

square integrable.

Moreover,

(∆) is square integrable iff the central character is unitary, iff

(

r−1

) is square-integrable, and every essentially square integrable

is a

(∆)

for a unique ∆.

Example. Take n = 2 = r, σ = | · |

−1/2

. Take

P = B =



∗ ∗

0 ∗



Then

Ind

(| · |

−1/2

, | · |

−1/2

) = C

∞

(B\G) ⊇ C,

where

is the constants. Then

∞

(

B\G

) is not supercuspidal, but the quo-

tient is the Steinberg representation, which is square integrable. Thus, every

two-dimensional essentially square integrable representation which is not super-

cuspidal is a twist of the Steinberg representation by χ ◦ det.

We can next classify tempered representations.

Theorem.

The tempered irreducible admissible representations of

(

) are

precisely the representations

Ind

, where

is irreducible square integrable. In

particular, Ind

σ are always irreducible when σ is square integrable.

Example.

For

, we seek a

which is tempered but not square integrable.

This must be of the form

π = Ind

(χ

, χ

where

|χ

= 1. If we want it to be essentially tempered, then we only

need |χ

| = |χ

Finally, we classify all irreducible (admissible) representations.

Theorem.

Let

···

be a partition, and

tempered representation

(

). Let

∈ R

with

> ··· > t

. Then

Ind

(

)

, . . . , σ

(

)) has a

unique irreducible quotient Langlands quotient, and every

is (uniquely) of this

form.

Example.

For

, the remaining (i.e. not essentially tempered) representations

are the irreducible subquotients of

Ind

(χ

, χ

where

|χ

| = | · |

, t

> t

Note that the one-dimensional representations must occur in this set, because

we haven’t encountered any yet.

For example, if we take χ

= | · |

1/2

and χ

= | · |

−1/2

, then

Ind

(χ

, χ

) =

∞

(B\G),

which has the trivial representation as its irreducible quotient.