Last revised 10:53 a.m. July 24, 2015
Langlands’ classification of representations of real tori
Bill Casselman
University of British Columbia
cass@math.ubc.ca
In this essay I’ll attempt to explain Langlands’ classification of characters of algebraic tpori defiednoverR.
This is in terms of homomorphisms from the Weil group into the associated L-group. I’ll follow more or less
the treatment in §§2 and 3 of [Langlands:1974/1989], avoiding most subsequent improvements.
Langlands’ classification for tori is a susbet of his results for more general reductive groups. The basic idea
can be stated very briefly for an important subset of cases. Suppose F to be an arbitrary local field, F a
separable algebraic closure. Let WF be the associated Weil group, which is a dense subgroup of the Galois
group G(F /F ).
Suppose G to be a quasi-split reductive group defined over F . To it is associated a connected complex group
G∨ whose root datum is dual to that of G, and the structure of G determines an algebraic action of G(F /F ) on
G∨ . Define L G to be the semidirect product G∨ ⋊ WF . Langlands has conjectured a partition of a large part of
the set of irreducible representations of G(F ) into finite packets (called L packets, because each constituent in
a packet is associated to the same local L-function). The conjecture is that these are parametrized by certain
continuous homomorphisms from WF to L G compatible with the projection from L G to WF . The bijection
should relate in a natural way to the local factors of L-functions Langlands associates to automorphic forms.
If F is C or R this partitions the entire set of irreducible representations, but if F is p-adic one must introduce
a group slightly larger than WF in order to account for certain ℓ-adic representations of the Galois group.
In this essay I’ll be mainly concerned with tori, but also discuss some applications to certain features of the
general conjecture.
Another, briefer, survey of the same material can be found in Chapter III of [Borel:1979].
Contents
1.
2.
3.
4.
5.
6.
Complex tori
The classification of real tori
Appendix. The decomposition of real tori
Admissible homomorphisms
Characters
References
1. Complex tori
In this section I’ll recall some general facts about algebraic tori, show what happens in particular for complex
tori, and then prove Langlands’ conjecture for them.
Langlands’ parametrization of characters of complex tori is essentially just a matter of definition. The group
WC is just C× , and all tori defined over C are just products of copies of the multiplicative group Gm . But,
before I explain Langlands’ parametrization of characters in this simple case, I’ll recall the relevant definitions.
For the moment, let F be an arbitrary field.
A split algebraic torus defined over F is an algebraic group isomorphic to a product of copies of the
multiplicative group Gm . Its group of points rational over F is isomorphic to a product of copies of F × .
ALGEBRAIC CHARACTER AND COCHARACTER GROUPS .
If T is a split algebraic torus of dimension n defined over F , the lattice X ∗ (T ) is the group of algebraic
characters—i.e. algebraic homomorphisms from T to Gm . Given coordinates on T , this becomes isomorphic
to Zn , with (ni ) corresponding to
Y
(xi ) 7−→
xni i .
Langlands’ classification of representations of real tori
2
Let X∗ (T ) be the free Z-module dual to X ∗ (T ). This may be identified canonically with the cocharacter
group of T , the group of algebraic homomorphisms from Gm to T , according to the rule
∨
λ(µ∨ (x)) = xhλ,µ
i
(x ∈ C× ) .
The torus T is completely specified by X ∗ (T ). Its affine ring AF [T ] is the group ring F [X ∗ (T )], which is
also F [x±1
i ]. If S is another split torus defined over F , then the group of algebraic homomorphisms from
S to T may be identified with the group of lattice homomorphisms from X ∗ (T ) to X ∗ (S), or equivalently
from X∗ (S) to X∗ (T ). Explicitly, f in Hom(X∗ (S), X∗ (T )) corresponds to the homomorphism αf from S to
T characterized by the condition
αf : λ∨ (x) 7−→ [f λ∨ ](x) (λ∨ ∈ X∗ (S)) .
If F is algebraically closed, then any torus defined over F is split.
A complex algebraic torus T is the product of several—say n—copies of C× .
× n
n
It has a maximal compact subgroup S isomorphic to Sn . Since C× = S×R×
>0 , we can factor T as S ×(R>0 ) .
Let z 7→ kzk be the projection onto the second factor, so that z 7→ z/kzk is projection onto S . This notation is
chosen to be compatible with that common in local analysis (for example in in Tate’s thesis), where |z| = zz ,
the modulus of the action of the] action on additive measures. The continuous characters of S are of the form
s
s 7→ sn , and those of R×
>0 of the form x 7→ x for s in C, so the continuous characters of T are of the form
CONTINUOUS CHARACTERS .
z 7−→ kzks (z/kzk)n
for s in C ⊗ X ∗ (T ) and n in X ∗ (T ). We can also express this as
z 7−→ χλ,µ (z) = z λ z µ (λ = (s + n)/2,
µ = (s − n)/2) .
In other words, the continuous characters are parametrized by pairs λ, µ in C ⊗ X ∗ (T ) such that λ − µ lies
in X ∗ (T ). I shall call these coherent pairs. In particular, if z lies in the maximal compact subgroup S of T
then z λ z µ = z λ−µ .
There is another way to interpret this. The complex vector space C ⊗ X∗ (T ) may be effectively identified
with the covering space of T (C). The covering map is the exponential map characterized by the formula
λ(ex ) = ehλ,xi
for λ in X ∗ (T ), x in C ⊗ X∗ (T ). The kernel is 2πiX∗ (T ). In these terms
z λ z µ = ehλ,xi ehµ,xi
(z = ex ) .
In summary:
1.1. Proposition. The map taking (λ, µ) to χλ,µ is a bijection between coherent pairs and continuous characters
of the complex torus T .
We shall also want to know the continuous homomorphisms from C× to T (C). These are of the dual form
z 7−→ z λ z µ
λ, µ ∈ C ⊗ X∗ (T ), λ − µ ∈ X∗ (T ) .
Langlands’ classification is now, as I said before, a matter of definition. Suppose T to be an algebraic torus
defined over C. Its dual T ∨ is the complex torus such that X ∗ (T ∨ ) = X∗ (T ). If a homomorphism
ϕ: WC = C× −→ T ∨ (C)
is specified by the pair λ, µ then the character of T is characterized by the same pair. This makes sense
precisely because X ∗ (T ) = X∗ (T ∨ ).
Langlands’ classification of representations of real tori
3
2. The classification of real tori
Let T = (Gm )n . According to the theory of Galois descent, together with the observation Corollary 4.2,
isomorphism classes of tori of dimension n defined over R are in bijection with conjugacy classes of involutions
in GL(X ∗ (T )). If θ∗ is such an involution, let θ be the corresponding involution of T (C) θ∗ that of X∗ (T ).
The corresponding conjugation of T (C) is
σ = σθ : z 7−→ z θ .
The R-rational points of the real torus defined by it are those z in T (C) such that z = z σ .
The simplest tori over R are the split ones, for which θ∗ = I . Other tori are classified, in effect, by comparison
with this one.
What do one-dimensional real tori look like? There are two involutions in GL1 (Z), hence two real tori of
dimension one. One is the split torus Gm , the other is S, whose group of real points is isomorphic to the
group of z in C with kzk = 1. In the second case, the associated involution is scalar multiplication by −1 in
Z.
Another real torus is that whose group of real points is isomorphic to C× . Its real dimension is two. The
corresponding involution is a swap of factors in Z2 . It is the group RC/R (Gm ) obtained from Gm by restriction
of scalars from C to R. In fact, this gives us a complete classification of two-dimensional tori.
2.1. Proposition. Any involution in GL2 (Z) is conjugate to exactly one of the following:
1
0
0
,
1
−1
0
0
,
1
−1 0
,
0 −1
0
1
1
0
.
2.2. Corollary. Every two-dimensional real torus is a product of one-dimensional tori or isomorphic to the
restriction of Gm from C to R.
Proof. Let L = Z2 , let L± be the intersection of L with the eigenspace on which the involution ι acts by ±1,
and let M = L+ + L− with basis (e± ). It is a direct sum, and since
2v = (v + θ(v)) + (v − θ(v))
we see that 2L ⊆ M . It cannot happen that M = 2L, since the e± are primitive vectors in L. If L is not equal
to M , then according to the principal divisor theorem there exists a basis λ, µ for L for which 2λ, µ is a basis
for M . We then have
2λ = ae− + be+
a b
∈ GL2 (Z)
c d
µ = ce− + de+
.
In particular, a and b, a and c are relatively prime. If a were even, then e+ would lie in 2L, which would
mean that e+ is not a basis element of L+ . Hence a is odd, and a is relatively prime to 2c.
We have
(θ − I)λ = a e−
(θ − I)µ = 2c e− .
Therefore
(θ − I)α = 0 (α = −2cλ + aµ) .
Since a and 2c are relatively prime, α is part of a basis (α, β) of L. Since ι has order two, ι swaps β and θ(β).
If these formed a basis of L, we’d be through. The matrix of ι in the basis (α, β) must be of the form
1
0
m
−1
.
Langlands’ classification of representations of real tori
4
Say m = 2n + ε, with ε = 0 or 1. Then
θ(β) = −β + (2n + ε)α,
θ(β − nα) = −(β − nα) + εα .
If m were even (and ε = 0) this is similar to a diagonal matrix, so ε = 1. If γ = β − nα then γ and ι(γ) = γ + α
form a basis of L and are swapped by ι.
A slightly more complicated argument (which I learned from du Cloux) proves this generalization:
2.3. Proposition. If ι is an involution on Zn , then Zn decomposes into a direct sum of lattices on which ι acts
by one of (a) the identity; (b) multiplication by −1; (c) swaps of two copies of Z.
A proof of this can be found in [Casselman:2008]) I’ll include a slightly expanded version of it in an appenidx.
2.4. Corollary. Every real torus is a direct product of copies of one of the three basic tori Gm , S or RC/R (Gm ).
The factors themselves in this result are not unique. This can be shown already by the torus C× × R× , in
which any image of the map
t 7−→ (tn , t)
may be taken as TR . However, their dimensions are.
2.5. Proposition. Suppose T to be a real torus, T ◦ the connected component of T (R). Then T (R)/T ◦ is
isomorphic to TR (R)/TR◦
Proof. The groups TS and TC× are connected.
Projection from the torsion points of TR onto the component group T (R)/TR◦ is an isomorphism. Hence the
canonical projection T to T /T ◦ splits. The splitting is not canonical.
On X ∗ (TR ), θ∗ = I , so I + θ∗ amounts to multiplication by 2, and the torsion subgroup may be identified
with its kernel. Consequently:
2.6. Proposition. The map from X ∗ (T ) to continuous characters of T (R) induces an isomorphism of
Ker(θ∗ + I)
Im(θ∗ − I)
with the dual of T (R)/T ◦ .
This can be proved directly, without using du Cloux’s factorization. If we let Λ = X∗ (T ) and tensor with the
following exact sequence induced by the exponential map z 7→ ez :
exp
0 −→ 2πiZ −→ C −→ C× −→ 1
with Λ we get
0 −→ 2πiΛ −→ C ⊗ Λ −→ T (C) −→ 1 .
Let G be the Galois group of C/R. The associated long exact cohomology sequence gives us
0 −→ (2πiΛ)G −→ (C ⊗ Λ)G −→ T (R) −→ H 1 (G, 2πiΛ) −→ 0 .
This identifies T (R)/T ◦ with
H 1 (G, 2πiΛ) =
Ker(σ∗ + I | 2πiΛ)
.
Im(σ∗ − I | 2πiΛ)
∨
This may in turn be identified with the dual of Ker(θ∗ + I)/Im(θ∗ − I) via the exponential pairing ehλ,µ i/2 .
This observation was made in [Kottwitz:1984], and indeed a much more general result, valid for all local
fields, can be found in §3.3 of that paper.
Langlands’ classification of representations of real tori
5
3. Appendix. The decomposition of real tori
In this appendix I am going to explain how to find the factorization of real tori into simple components. We
are given an involution τ in GLn (Z), and I shall decompose Zn into pieces of dimension one on which τ acts
as ±I and pieces of dimension two on which τ acts as a swap.
I’ll begin with a preliminary result:
3.1. Lemma. The canonical projection from SLn (Z) to GLn (Z/2) = SLn (Z/2) is surjective.
This is a very simple case of the strong approximation theorem for simply connected semi-simple algebraic
groups defined over Dedekind domains. In the particular case at hand it is extremely easy to prove and to
implement constructively.
Proof. If F is any field, then applying Gauss-Jordan elimination, we may write any matrix in GLn (F ) as
n1 wan2 where the ni are upper triangular, a is diagonal, and w is a permutation matrix. We may multiply
w by a diagonal matrix with entries ±1 if necessary to make it a monomial matrix with entries ±1 and
det(w) = 1. The matrices ni are certainly in the image of the projection, as is w. And in our case, since all
units in Z/2 are 1, a happens to be I .
Let L = Zn , and suppose that we are given an integral matrix τ of order 2.
Let M± be the intersection of the ±1 eigenspaces with L. Each of these is a summand of L. Let M = M+ ⊕M− .
If M = L, there is no problem—we set L± = M± , Lswap = {0}. Otherwise, things are more interesting.
Some motivation for the proof will come from understanding how τ acts on the quotient L/2L. This is a
finite-dimensional vector space over F2 . According to the Jordan decomposition theorem it must break up
into a direct sum of spaces on which τ acts by
λ
0

0


1
λ
0
...
0 0

0 ... 0
1 ... 0

λ ... 0

0 ... λ
but since τ 2 = 1 it must in fact be a sum of one-dimensional spaces on which τ is the identity, plus a direct
sum of two-dimensional space on each of which τ acts as the matrix
1 1
0 1
(over F2 ) .
This matrix is equivalent to a swap (in characteristic 2), as we can deduce from the following observation:
3.2. Lemma. (Swap Lemma) The integral matrix
1
1
0 −1
.
is that of a swap.
Proof. If
τ (e) = e
τ (f ) = −f + e
then
τ (e − f ) = e − (−f + e) = f
so τ swaps e − f and f .
Langlands’ classification of representations of real tori
6
The lattice decomposition we are looking for is closely related to this decomposition of L/2L, except that the
two lattices L± of course both collapse to the single eigenspace modulo 2.
I now start the proof in detail. The first step is to find the Smith normal form of τ + I . This gives us a basis
e1 , . . . , ej , ej+1 , . . . ek , ek+1 , . . . , en of L where the 2ei for i ≤ j and the ei for j < i ≤ k make up a basis of
(τ + I)L. This implies that the ei for i ≤ k are basis of M+ . This implies also that the ei with i ≤ k make up
a basis of M+ . The calculation also gives us equations
(τ + I)eℓ =
X
2ci ei +
i≤j
X
ci e i
j<i≤k
for ℓ > k . But then
X
X
X
X
X
2ci ei +
(τ + I) eℓ −
ci e i −
ci e i =
2ci ei =
ci e i .
i≤j
i≤j
j<i≤k
i≤j
j<i≤k
P
If we replace the eℓ for ℓ > k by eℓ − i≤j ci ei , we see that the lattice L∗ spanned by the ei with i > j is
stable under τ , as is its complement, the lattice L+ spanned by the ei with i ≤ j . We can therefore split off
L+ , on which τ = I , and work only with L∗ . We may as well take L to be L∗ . We may also assume we have
a basis e1 , e2 , . . . , em of L with the ei for i ≤ k a basis of (τ + I)L which is now the same as M+ . With the
new basis the matrix of τ is of the form
I
C
0 −I
.
Our goal now is to show that L may be decomposed into L− ⊕ Lswap .
We next find the Smith normal form of
τ −I =
0
C
0 −2I
which means, in effect, finding that of
C
−2I
.
Because the ei for i ≤ k are a basis of M+ , the reduction modulo 2 of the columns of C are sufficiently linearly
independent. This is already in a particularly good Hermite normal form. Since the pivot entries are all 2,
the calculation of its Smith normal form is relatively simple. At any rate, we get a basis fi of L such that the
fi for 1 ≤ i ≤ j and 2fi for j < i ≤ m − k are a basis of (τ − I)L. Let L− be the lattice spanned by the fi
with i > j . As before, we can find a τ -stable complement L∗ of L− , and split off L− .
As a result of what we have done so far, we have a decomposition Zn = L+ ⊕ L∗ ⊕ L− , where τ = ±I on
L± . Furthermore, in L∗ , which I may as well now take to be L, the intersection M± of the ±1-eigenspace of
τ coincides with (τ ± I)L. We also have (a) a basis (ei ) of L such that the ei with i ≤ k form a basis of M+
and (b) a basis (fi ) (i ≤ m − k ) of M− , expressed in terms of the ei .
3.3. Lemma. Suppose τ to be an involution of the lattice L, M± the intersection of the ±1-eigenspaces of τ
with L. If (τ ± I)L = M± , then τ is a direct sum of swaps.
This is really the crux of the matter. The proof will find an explicit decomposition. In practice, however, this
explicit decomposition is rarely necessary. What one most often needs to know are the dimensions of the
spaces L± and Lswap and that, given the truth of this lemma, we know now.
Proof. I claim first that
• the kernel of the map (τ ± I) from L to M± /2M± is M = M− ⊕ M+ .
Because if (τ + I)e = 2m = (τ + I)m for m in M+ , then (τ + I)(e − m) = 0 so that e − m lies in M− and
e = (e − m) + m lies in M . Therefore τ ± I is an embedding of the F2 -vector space L/M into M± /2M±.
Langlands’ classification of representations of real tori
7
As a consequence of this, we must have m − k = k , m = 2k , and L/M ∼
= (Z/2)k .
The ei with i ≤ k and the fi with i ≤ k make up a basis of M+ ⊕ M− . We next find the Smith normal form of
the matrix whose columns are these ei and the fi , assuming the ei to be the basis of L. This matrix looks like
I F1
0 F2
.
Applying elementary column operations, we may reduce this to
I 0
0 G
.
But since the index of L/M ∼
= (Z/2)k , the Smith normal form of G is the diagonal matrix with all diagonal
entries equal to 2. Therefore G is itself divisible by 2. So the columns gi of G/2, together with the ei for i ≤ k ,
are a basis of L such that these ei together with the 2gi are a basis of M .
Now (τ + I) induces an isomorphism of L/M with M+ /2M+ ⊆ L/2L. The images g i in L/M make up a
basis of that vector space over F2 , as are the images ei of the ei in M+ /2M+ . We therefore have
(τ + I)g i =
X
cji ej
j
where C = (ci ) is an invertible k × k matrix over F2 . Since according to strong approximation the canonical
projection from SLn (Z) to SLn (Z/2) is surjective, we may replace the basis ei for M+ , and assume that
(τ + I)gi = ei modulo 2M+ .
But if (τ + I)gi = ei + 2mi with mi in M+ , then
(τ + I)(gi − mi ) = ei + 2mi − 2mi = ei .
We may replace the basis gi by the gi − mi so that
(τ + I)gi = ei ,
τ (gi ) = −gi + ei
The lattice L is the direct sum of the τ -stable two-dimensional lattices spanned by gi and ei , and on this the
matrix of τ is
1
1
0 −1
.
The Swap Lemma tells us that we are through.
4. Admissible homomorphisms
Suppose T to be an algebraic torus defined over R, corresponding to the involution θ. Its Langlands dual is
the complex torus T ∨ with character group X ∗ (T ∨ ) = X∗ (T ). Its L-group LT in this essay is the semi-direct
product T ∨ (C) ⋊ G , generated by 1 × σ 6= 1 and a copy of T ∨ (C) such that
(1 × σ)2 = 1
(1 × σ)(z × 1) = (z θ × 1)(1 × σ) .
I write z 7→ z θ for the involution of T ∨ (C), but it is important to keep in mind that although T and
its Langlands dual are both complex tori, they are in truth very different. For one thing, there are two
involutions of T (C) in play, conjugation x 7→ x as well as θ, while on T ∨ (C) there is just θ. This is reflected in
another curious difference. I shall define C ⊗ X∗ (T ) as the covering space of T (C) vis the exponential map
x 7→ ex . Its kernel is 2πiX∗ (T ). But as the covering space of T ∨ (C) I take C ⊗ X∗ (T ∨ ) with the exponential
Langlands’ classification of representations of real tori
8
map x 7→ e2πix whose kernel is X∗ (T ∨ ). These are the right choices to make in order to get the proper
duality, as we shall see.
The claim to be proved, roughly speaking, is that continuous characters of T (R) are in bijection with conjugacy
classes of admissible homomorphisms from WR to LT . We have already seen a proof of the analogous claim
for complex tori.
THE WEIL GROUP .
The group WR is an extension of C× by G = G(C/R) = {1, σ}, fitting into an exact
sequence
1 −→ C× −→ WR −→ G −→ 1 .
It is generated by the copy of C× and an element σ mapping onto the non-trivial element of G , with relations
σ 2 = −1
z · σ = σ · z,
Since −1 is not the norm of a complex number, the extension does not split. It is not a coincidence that it is
isomorphic to the normalizer of a copy of C× in the unit group of the Hamilton quaternions H. If E/F is any
finite Galois extension of local fields, then WE/F may be embedded into the normalizer of a copy of E × in
the multiplicative group of the division algebra over F defined by the fundamental class in the Brauer group
H 2 (G(E/F ), E × ).
The norm map NM from WR to R× extends the norm on its subgroup C× , mapping
z 7−→ zz,
σ 7−→ −1 .
It is a surjective homomorphism with kernel S, and identifies the maximal abelian quotient of WR with R× .
Suppose given an admissible homorphism ϕ from WR to LT . Restricted
to C it gives a continuous homomorphism into T ∨ (C), hence is defined by what we have seen earlier a pair
λ, µ such that λ − µ lies in X∗ (T ∨ ). I recall that I call this a coherent pair. The image of σ will be of the form
α × σ , with α in T ∨ (C).
ADMISSIBLE HOMOMORPHISMS .
×
4.1. Lemma. Suppose λ, µ to be an coherent pair, and let ϕ be the map from C× to T ∨ (C) that they define.
Then α in T ∨ (C) defines an extension to WR if and only if
(a)
(b)
λ = µθ
α1+θ = (−1)λ−µ .
Two elements in T ∨ (C) define conjugate admissible homomorphisms if and only if their quotient is of the
form β 1−θ .
I leave the proof as exercise. The possible α may therefore be considered as elements of the group
(4.2 )
Ker(I + θ)
.
Im(I − θ)
The operators here are on T ∨ (C).
Any factorization T = TR× × TC× × TS gives rise to an analogous factorization of T ∨ (C). For the second
two, the quotient in (4.2) is trivial. For the first, it is the subgroup of 2-torsion points. As we have seen, its
size is the same as that of the component group of T (R) or, equivalently, the 2-torsion points in TR× . The
equality of sizes is explained by the fact that these two groups are duals, as we shall now see.
Suppose given α in T ∨ (C) such that ααθ = 1. Choose a in C ⊗ X∗ (C) such that
α = e2πia .
Langlands’ classification of representations of real tori
9
θ
Suppose τ to lie in T (R). Then τ = τ θ . It will be of the form et with t in C ⊗ X∗ (T ) and t − t in 2πiX∗ (T ).
Define
θ
χα (τ ) = eha,t−t i .
(4.3 )
θ
The element τ will lie in T ◦ if and only if t − t = 0, and in this case χα (τ ) = 1. It is therefore well defined
on T (R)/T ◦.
Why is the definition independent of choice of α?
4.4. Lemma. If t − t
θ∗
lies in 2πiX∗ (T ), then
∗
θ
hb − bθ , t − t ∗ i = 0
for any b in C ⊗ X∗ (T ∨ (C).
Proof. This reduces to the equation
(t + t) − (t + t)θ∗ = 0 .
Since this equation is compatible with direct sums, we may simply look at each of the three basic real tori in
du Cloux’s factorization.
If T (R) = R× , then θ = I , and the equation is trivially valid.
In the other cases, T (R) is simply connected. Thus T (R) is the image of the exponential map on its real Lie
algebra. Consequently
θ
t = t ∗,
t = t θ∗ ,
and again the equation is immediate.
4.5. Proposition. The map α 7→ χα induces an isomorphism of
Ker(I + θ | T ∨ (C))
Im(I − θ | T ∨ (C))
with the dual of T /T ◦ .
Remark. There is an asymmetry here that I would again like to call attention to. In the exponential of T there
is no factor 2πi, while in that for T ∨ there is one. (This is what Langlands also does.) It is analogous to the
asymmetric Fourier-Laplace transform.
5. Characters
This has been covered in [Langlands:1974/1989] but, as I have already mentioned, a subsequent observation
of Fokko du Cloux is useful. Another approach can be found in [Adams-Vogan:1992]. Suppose T to be
an algebraic torus defined over R. It is defined by an involution θ∗ of X∗ (T ). If we identify T (C) with a
product of copies of C× , the elements of T (R) are those z in T (C) such that z = z θ . Langlands’ idea is that
continuous characters of T (R) are parametrized by admissible homomorphisms from WR to LT .
We start with an admissible homomorphism ϕ corresponding to (λ, µ) and α in T ∨ (C) for which conditions
(a) and (b) of Lemma 4.1 hold. As in the discussion leading up to Proposition 4.5, choose a in C ⊗ X∗ (T ∨ )
such that
α = e2πa .
Since ααθ = (−1)λ−µ we have
(5.1 )
a + aθ −
λ−µ
∈ X∗ (T ∨ ) .
2
Langlands’ classification of representations of real tori
10
Recall also that λ = µ θ . For τ = et Langlands sets
χϕ (τ ) = eha,t−t
(5.2 )
θ
θ
i+hλ,(t+t )/2i
).
Lemma 4.4 tells us that this is independent of the choice of a. If τ = 1, then t lies in 2πiX∗ (T ) and according
to (5.1)
θ
θ
ha, t − t i + hµ/2, t + t i = ha + aθ + λ/2 − λθ /2, ti
lies in 2πiZ, so the character is well defined.
θ
If τ lies in T ◦ then t = t , so the first term in the exponent vanishes. The second becomes hλ, ti. Therefore
the second term defines in a simple (and natural) form the restriction of the character to T ◦ . If this vanishes,
then Langlands’ formula agrees with Proposition 4.5.
5.3. Theorem. (Langlands) The map taking ϕ to χϕ is a bijection between admissible homomorphisms from
WR to LT and continuous characters of T (R).
What is new in this exposition is that it points out how this formula is derived from the extension
1 −→ T ◦ −→ T −→ T /T ◦ −→ 1 .
The proof I’ll give here will follwo from the following discussion of what happens for the basic tori.
(a) Suppose T = Gm . In this case θ = I and T (R) is just R× . The admissible homomorphisms from WR
therefore correspond exactly to continuous homomorphisms from WR to C. But any such homomorphism
must factor through the maximal abelian quotient of WR , which may be identified through the norm map
NM with R× . We have α2 = 1. The pair λ, α corresponds to the character z 7→ |z|λ , σ 7→ α.
(b) Next suppose θ = −I on Z. Then T (R) is the unit circle in C× . The group LT is the semi-direct product
C× ⋊ G , with σ acting as inverse on C× .
The group T (R) is connected, and this correlates with the fact that every α satisfies α1+θ = 1, and that every
α = β 2 for some β . We may therefore take α = 1, a = 0.
We know that λ − µ = m is an integer. But since α1+θ = (−1)λ−µ , m must be an even integer 2n. But then
λ = n, and the associated character is
τ 7−→ τ n .
(c) Finally, suppose θ to be the coordinate swap on Z2 . Therefore T (R) is the subset of pairs (z, w) with
z = w , which may in turn be identified with C× . Admissible homomorphisms when restricted to C× are of
the form
z 7−→ (z λ z µ , z µ z λ )
with λ − µ in Z. This also also describes characters of T (R) = C× . This is a special case of compatibility
between Langlands’ parametrization and restriction of scalars.
It follows from this list of examples, together with du Cloux’s factorization, that the map from admissible
homomorphisms to continuous characters is a bijection.
It is natural to ask, given this proof: is there a simple way to identify directly the characters of T ◦ ? The
example of S shows that the answer is the set of all λ such that (a) λ = λθ lies in X ∗ (T ) and (b) there exists
θ
some α in T ∨ (C) such that α1+θ = (−1)λ−λ . The extension condition (b) is necessary, even when the choice
of possible α is essentially unique. So there is no characterization that is quite trivial.
Remark. Langlands proved directly that the bijection of this theorem and its analogue for WC are compatible
with restriction of scalars. It is also functorial with respect to homomorphisms from one torus to another.
Remark. How do we know that this bijection between admissible homomorphisms and characters is in any
sense the correct one? It is certainly compatible with du Cloux’s factorization, so we must only decide if it
Langlands’ classification of representations of real tori
11
is what we desire for the basic tori. For these, the assignment is consistent with the assignment of Γ-factors
to representations of the Weil group. Two of these basic examples are directly related to characters of the
multiplicative groups of local fields, for which all is certainly correct, while the third comes from these
examples together with functoriality.
6. References
1. Jeffrey Adams and David Vogan, ‘L-groups, projective representations, and the Langlands classification’,
American Journal of Mathematics 114 (1992), 45–138.
2. Armand Borel, ‘Automorphic L-functions’, pp. 27–61 in part 2 of [Borel and Casselman:1979].
3. —— and William Casselman (editors), Automorphic forms, representations, and L-functions, proceedings of a conference in Corvallis, Proceedings of Symposia in Pure Mathematics 33, A.M.S., 1979.
4. Bill Casselman, ‘Computations in real tori’, pp. 137–151 in Representation theory of real reductive Lie
groups, Contemporary Mathematics 472, American Mathematical Society, 2008.
5. R. E. Kottwitz, ‘Stable trace formula: cuspidal tempered terms’, Duke Mathematicas Journal 51 (1984),
611–650.
6. Robert Langlands, ‘On the classification of irreducible representations of real groups’, pp. 101–179 in Representation theory and harmonic analysis on semisimple Lie groups, Mathematical Surveys Monographs
31, American Mathematical Society, 1989. Originally a preprint from the IAS, dated 1974.
7. ——, ‘Representations of abelian algebraic groups’, Pacific Journal of Mathematics (special issue dedicated
to Olga Taussky Todd) (1997), 231–250. Originally a Yale University preprint.
8. John Tate, ‘Number theoretic background’, pp. 3–26 in part 2 of [Borel-Casselman:1979].