R. van Dobben de Bruyn
The Local Langlands Correspondences
Part III Essay, 3 May 2012
Supervisor: Dr T. Yoshida
Trinity College, University of Cambridge
Preface
The main aim of this essay is to state the local Langlands correspondences for
GLn , and to define all the objects involved in the statement of the theorem.
This work is aimed at graduate students. We will assume knowledge about
local fields, basic (infinite) Galois theory and some understanding of the representation theory of finite groups. We will develop representation theory for
a particular type of infinite topological groups (called locally profinite groups).
Some knowledge about commutative algebra might prove useful, but is not required. The language of category theory (“abstract nonsense”) is used freely
(especially adjunctions), but one should also be able to read the work without
any knowledge about such matters.
2
Contents
Introduction
6
1 Locally profinite groups
10
1.1
Topological groups . . . . . . . . . . . . . . . . . . . . . . . . . .
10
1.2
Smooth representations . . . . . . . . . . . . . . . . . . . . . . .
13
1.3
Induced representations . . . . . . . . . . . . . . . . . . . . . . .
16
1.4
Dual representations . . . . . . . . . . . . . . . . . . . . . . . . .
19
2 Tate’s Thesis
24
2.1
Haar measures . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.2
The additive group of a local field
. . . . . . . . . . . . . . . . .
28
2.3
The multiplicative group of a local field . . . . . . . . . . . . . .
31
2.4
Epsilon factors . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
3 Weil groups
40
3.1
Local Class Field Theory . . . . . . . . . . . . . . . . . . . . . .
40
3.2
Representations of the Weil group . . . . . . . . . . . . . . . . .
44
3.3
L-functions and epsilon factors . . . . . . . . . . . . . . . . . . .
46
3.4
Deligne representations . . . . . . . . . . . . . . . . . . . . . . . .
48
4 Representations of the General Linear Group
52
4.1
Parabolic subgroups . . . . . . . . . . . . . . . . . . . . . . . . .
52
4.2
L-functions and epsilon factors . . . . . . . . . . . . . . . . . . .
56
5 The Local Langlands Correspondences
60
References
62
4
Introduction
Class field theory provides a description of the abelian extensions of a field K
(global or local) in terms of its own arithmetic. The Langlands programme
is a series of conjectures generalising class field theory to include information
about non-abelian extensions as well. Like class field theory, the Langlands
programme has global and local versions.
In this essay, we will only be concerned with the local case. Then an important
object associated to a field is the Weil group, which is defined in Chapter 3.
It is a dense subgroup of the absolute Galois group of the field, and local class
field theory gives an isomorphism between the abelianisation of the Weil group
and the multiplicative group K ˆ .
This isomorphism has important consequences for representation theory of the
two groups. Representations of the absolute Galois group arise for instance
naturally from elliptic curves (as Tate modules) or more generally via étale
cohomology. Such representations give (by restriction) a representation of the
Weil group, and if the representation is 1-dimensional (i.e. a character), then
class field theory asserts that it corresponds to a character of K ˆ .
More generally, the Langlands correspondence asserts that n-dimensional representations of the Weil group correspond to representations of GLn pKq in a
natural way. This naturality is expressed by associating to representations of
the Weil group, as well as representations of GLn pKq, a function called the Lfunction. However, it turns out that this L-function does not contain as much
information for higher-dimensional representations as it does for 1-dimensional
ones. Therefore, one also defines ε-factors on both sides.
The local Langlands correspondence then says that there is a bijection between
the set Gn pKq of isomorphism classes of a particular kind of n-dimensional
representations of the Weil group and the set An pKq of isomorphism classes
of irreducible representations of GLn pKq. This bijection is natural in many
ways; traditionally one especially emphasises that it preserves L-functions and
ε-factors.
The local Langlands correspondence was turned into a theorem in 2001 by Harris
and Taylor [6], and independently in 2002 by Henniart [7]. A much more general
version, where GLn is replaced by a more general algebraic group, is still an open
problem (if only because it not clear what the precise formulation has to be).
The main aim of this essay is to state the local Langlands conjecture in the
way it was proven by Harris-Taylor and Henniart. This involves defining Lfunctions and ε-factors on both the Weil group side and the GLn pKq-side of the
correspondence. For the 1-dimensional case, this is the local theory from Tate’s
thesis [16]. For higher dimensions, we only state the results, and refer the reader
to the literature for the proofs. We note that one of the proofs requires a global
argument.
6
Notation
Throughout this essay, K will denote a non-archimedian local field, that is, a
finite extension of either Qp for some prime p, or of Fq ppT qq for some finite field
Fq . We fix a separable algebraic closure K̄ of K, and write GK for the absolute
Galois group of K.
We write OK for the ring of integers in K, pK for the (unique) prime ideal in
OK and vK for the valuation of K. The valuation is a surjective homomorphism
K ˆ Ñ Z, and the valuation of 0 is defined to be `8.
The residue field OK {pK is denoted k, and its number of elements is denoted
q. The characteristic of k is denoted p. The absolute value } ¨ }K on K is
normalised via the rule }x}K “ q ´vK pxq . When the field K is clear, we will drop
the subscript from the notations } ¨ }K and pK .
A basis for the open neighbourhoods around 0 on K is given by the sets pn for
n P Z, and K is the union of these sets.
All finite extensions L{K will be assumed to lie within K̄. This way, we identify
GL with a subgroup of GK . If L is not normal, such an identification is only
canonical up to an inner automorphism of GK .
The group K ˆ fits in the short exact sequence
v
K
ˆ
1 Ñ OK
Ñ K ˆ ÝÑ
Z Ñ 1,
which splits by projectivity of Z, i.e. by the choice of a uniformiser. A basis for
the open neighbourhoods of the identity is given by the sets
n
UK
“ 1 ` pn ,
ˆ
for n ą 0. We write UK for OK
, and we observe that
1
UK {UK
– kˆ ,
whereas (for i ą 0):
i`1
i
UK
{UK
– k.
The group UK is the unique maximal compact open subgroup of K ˆ . Therefore,
in contrast with the additive case, K ˆ is not the union of its compact open
subgroups.
7
Acknowledgements
Most of this essay owes its presentation to [3]. The most important other works
that were used are [17], [14] and [16]. For the statement of the theorems in
Chapter 4, I turned to [4], where also the proofs can be found. For a complete
list of the used literature, see the References.
Besides the literature, I want to express my gratitude towards all the people who
helped me with this essay. First of all, I would like to thank the essay setter and
supervisor Dr T. Yoshida for the useful explanations he gave, both motivational
(“why do we want to study this?”) and technical. Secondly, I want to thank the
other person taking the essay, for the interesting discussions we had about the
contents, of which I wish there were more. Also, I would like to thank my good
friend and formerly fellow student Johan Commelin for proofreading the essay.
His suggestions ranged from typographical to mathematical ones, and made an
improvement on the entire work. Finally, I want to thank my good friend and
fellow Part III student Manuel Bärenz for the advanced LATEX-related tips and
tricks he taught me.
8
1
Locally profinite groups
In this chapter, we will set up some of the basics we need in further chapters.
We will develop the theory of locally profinite groups, and discuss some aspects
of their representation theory.
This chapter is based on [3], although we explain the basics in more depth here.
On the other hand, we cover only the necessary material.
1.1
Topological groups
We recall the following definition:
Definition 1.1.1. A topological group is a group G endowed with a topology
such that the maps
GˆGÑG
GÑG
x ÞÑ x´1
px, yq ÞÑ xy,
are continuous.
Remark 1.1.2. In terms of abstract nonsense, it is a group object in the category Top.
Lemma 1.1.3. Let G be a topological group. Then G is Hausdorff if and only
if it is T1 .
Proof. Clearly every Hausdorff space is T1 . Conversely, let G be a T1 topological
group. Since both multiplication and inversion are continuous, so is the map
GˆGÑG
px, yq ÞÑ xy ´1 .
Since G is T1 , the singleton t1u is closed, hence so is its inverse image under
the above map. This is the diagonal, hence G is Hausdorff.
Lemma 1.1.4. Let G be a topological group, and let H be a subgroup. Then the
closure H of H is also a subgroup. If furthermore H is normal, then so is H.
Proof. By a standard theorem of topology, the closure H ˆ H of H ˆH Ď GˆG
is the product H ˆ H. The map
f: GˆGÑG
px, yq ÞÑ xy
is continuous, so f pH ˆ Hq Ď f pH ˆ Hq Ď H. Hence, H is closed under multiplication, and a similar argument shows that H is closed under inversion.
10
If H is normal, then gHg ´1 “ H for all g P G. If X Ď G is any closed set
containing H, then also the set gXg ´1 is closed and contains H. Hence,
č
č
H“
X“
gXg ´1 “ gHg ´1 .
XĚH
closed
XĚH
closed
Remark 1.1.5. A quotient of G by a subgroup H is T1 (or Hausdorff) if and
only if H is closed. In particular, the quotient of G by the closure of t1u is T1 .
From now on, we will assume that all topological groups are T1 , unless otherwise specified. For that reason, we will only ever consider quotients by normal
subgroups that are closed.
An important example is given by the following construction.
Definition 1.1.6. If G is a topological group, then the T1 -abelianisation is the
quotient of G by the closure of the commutator subgroup. It is denoted Gab .
Remark 1.1.7. Note that the T1 -abelianisation is T1 and abelian (hence the
name). Hence, it is also Hausdorff, so we could also call it the T2 -abelianisation.
Lemma 1.1.8. Let G be a topological group. Then the T1 -abelianisation Gab
satisfies the following universal property:
For every continuous homomorphism f : G Ñ A into an abelian, T1 topological
group A factors uniquely through
f
G
π
A
f ab
ab
G
where π : G Ñ Gab is the natural projection.
Proof. Because A is abelian, we necessarily have rG, Gs Ď ker f . Because A is
T1 and f is continuous, the kernel of f is closed. Hence, we are done by the
fundamental theorem of homomorphisms.
Corollary 1.1.9. There is a natural isomorphism
HompG, Aq – HompGab , Aq.
That is, the T1 -abelianisation functor TopGp Ñ T1 -Ab is left adjoint to the
inclusion functor. In particular, it is right exact.
Definition 1.1.10. A topological group G is locally profinite if every neighbourhood of the identity contains a compact open subgroup of G.
Lemma 1.1.11. Let G be a locally profinite group. If U Ď G is a neighbourhood
of the identity, then it contains a compact open normal subgroup of G.
11
Proof. By the definition of locally profinite groups, we can w.l.o.g. assume that
U is a compact open subgroup. Then all its conjugates xU x´1 are open as well,
hence so is the intersection
č
xU x´1 .
U1 “
xPG
1
Clearly, U is an open normal subgroup contained in U . It is compact since any
open subgroup of the profinite group U is compact.
An important class of locally profinite groups is given by the following lemma.
Lemma 1.1.12. Every profinite group is locally profinite.
Proof. Assume G “ lim
Gα , where α runs over some directed set A, and all
ÐÝ
groups Gα are finite. Let U be ś
a neighbourhood of the identity, and w.l.o.g.
assume that U is open. Let V Ď αPA Gα be an open set with V X G “ U .
ś
By the definition of the product topologyśon αPA Gα , V contains an open
neighbourhood of the identity
ś of the form αPA Uα , where Uα “ Gα for almost
all α P A. Then set V 1 “ αPA Uα1 , where
"
Gα if Uα “ Gα ,
1
Uα “
t1u if Uα Ĺ Gα .
ś
Clearly, V 1 Ď V , and V 1 is an open (normal) subgroup of αPA Gα . Hence, the
intersection U 1 “ V 1 X G is an open subgroup contained in U . In particular, U 1
is closed, hence compact because G is.
Lemma 1.1.13. Let G be a locally profinite group, and H a closed subgroup.
Then H is locally profinite. If furthermore H is normal, then G{H is locally
profinite.
Proof. This can be checked easily.
Lemma 1.1.14. Let G be a locally profinite group. Then G is locally compact.
Furthermore, G is compact if and only if it is profinite.
Proof. The open neighbourhood G of the identity contains some open compact
subgroup. Hence, G is locally compact.
If G is profinite, then clearly G is compact. Conversely, suppose G is compact.
By Lemma 1.1.11, a basis for the open neighbourhoods of the identity is given
by the open normal subgroups. If we put
G1 “ lim
G{U,
ÐÝ
U ŸG
open
then there is a natural homomorphism f : G Ñ G1 . It is continuous since the
composition with any projection G1 Ñ G{U is. Moreover, since G is compact
and G1 is Hausdorff, f is a closed map.
12
´1
On the other hand, let V Ď G1 be an open set of the form πU
ptxuq for some
open normal subgroup U Ď G and some x P G{U , where πU : G1 Ñ G{U denotes
the projection. Then choosing some y P G with y ” x mod U gives an element
with f pyq P V . Since a basis of the topology on G1 is given by the sets V of this
form, this shows that the image of f is dense.
Finally, f is injective: if x P G satisfies f pxq “ 1, then x P U for all open
normal U Ď G. However, G is Hausdorff, so if x ‰ 1, there exists an open
neighbourhood of 1 which does not contain x. This is impossible, so x “ 1.
Since f is a closed map with dense image, it is surjective. Since f is a bijective
closed (continuous) map, it is a topological isomorphism.
This justifies the definition of locally profinite groups, as we conclude:
Corollary 1.1.15. A topological group G is locally profinite if and only if every
open neighbourhood of the identity contains an open subgroup of G that is a
profinite group.
The following result generalises a well-known result about profinite groups:
Lemma 1.1.16. A topological group is locally profinite if and only if it is locally
compact and totally disconnected.
We will not prove this, because we do not need it for our purposes.
1.2
Smooth representations
From here on, G will denote a locally profinite group. All group representations
(not necessarily finite-dimensional) will be over C. We denote by pρ, V q the
representation ρ : G Ñ AutpV q.
Remark 1.2.1. Recall that a representation pρ, V q is called semisimple if it
satisfies one of the following equivalent conditions:
• ρ is the direct sum of irreducible representations,
• every G-stable subspace W has a G-stable complement W 1 :
V – W ‘ W 1,
• V is the sum of its irreducible G-subspaces.
Definition 1.2.2. A representation pρ, V q of a locally profinite group G is called
smooth if the map
¨: G ˆ V Ñ V
pg, vq ÞÑ gv
is continuous, when V is endowed with the discrete topology.
13
Lemma 1.2.3. Let pρ, V q be a representation of the locally profinite group G.
Then the following are equivalent:
(1)
(2)
(3)
(4)
pρ, V q is smooth;
the stabiliser of every element v P V is open;
V is the union of the sets V U for U an open subgroup;
V is the union of the sets V U for U an open compact normal subgroup.
Proof. Clearly (2) and (3) are equivalent. Since by Lemma 1.1.11 every open
subgroup contains a compact open normal subgroup, (2) and (4) are equivalent.
For every v P V , the inverse image under ¨ : G ˆ V Ñ V of tvu is the set
ď
tpg, wq P G ˆ V : gw “ vu.
tpg, wq P G ˆ V : gw “ vu “
wPGv
If we choose for every w P Gv an element gw P G such that gw w “ v, then
ď
ď
tpg, wq P G ˆ V : gw “ vu “
pgw Stabpvqq ˆ twu.
wPGv
wPGv
Since V has the discrete topology, the latter is open in G ˆ V if and only if
Stabpvq is open in G.
From property (2), it follows that subrepresentations and quotient representations of smooth representations are again smooth.
Remark 1.2.4. The category of abstract (i.e. not necessarily smooth) representations of G is equivalent to the category of functors G Ñ C-Vec, where G
is viewed as a one-object category whose morphisms are the elements of G. It
is an abelian category, and we denote it by Repa pGq.
The full subcategory of smooth representations is also abelian, and we denote
it ReppGq.
If pρ1 , V1 q and pρ2 , V2 q are abstract or smooth representations of G, we write
HomG pρ1 , ρ2 q for the set of G-homomorphisms from pρ1 , V1 q to pρ2 , V2 q.
Lemma 1.2.5. If G is profinite, then any smooth representation is semisimple.
Proof. Let pρ, V q be a smooth representation of G. Any vector v P V is fixed
by some compact open normal subgroup U Ď G. Then the subrepresentation
generated by v is an irreducible representation of the finite group G{U , hence
finite-dimensional. Such a representation is semisimple, hence it is the sum of
its irreducible subrepresentations. We repeat the argument for all v P V , and
the result follows.
Definition 1.2.6. A character χ of G is a continuous homomorphism
χ : G Ñ Cˆ
with respect to the usual topology on Cˆ .
14
Any character χ defines a one-dimensional representation pχ, Cq.
Proposition 1.2.7. Let χ : G Ñ Cˆ be a homomorphism. Then the following
are equivalent:
(1) χ is a character,
(2) the representation pχ, Cq is smooth,
(3) the kernel of χ is open.
Proof. Since the ker χ is the stabiliser of any nonzero vector, the equivalence of
(2) and (3) follows from the lemma above. It is obvious that (3) implies (1).
Now assume that χ is continuous. Let U be an open neighbourhood of the
identity in Cˆ , and choose U such that it contains no non-trivial subgroup of
Cˆ . Since χ is continuous, the set χ´1 pU q is open in G, hence it contains a
compact open subgroup U 1 of G. Then χpU 1 q is a subgroup of Cˆ contained in
U , hence it is trivial. Hence, U 1 Ď ker χ, and the kernel of χ contains an open
neighbourhood of the identity.
Definition 1.2.8. Given an abstract (i.e. not necessarily smooth) representation pρ, V q of G, we can construct a smooth representation pρ8 , V 8 q as follows.
Let
ď
V8 “
V U.
U ŸG
open
Remark 1.2.9. If U is any open normal subgroup, then V U is fixed by G, since
for every g P G:
´1
gV U “ V gU g “ V U .
Therefore, V 8 is a G-invariant subspace of V , and we define
ρ8 : G Ñ AutpV 8 q
ˇ
g ÞÑ ρpgqˇ 8 .
V
It is the maximal smooth subrepresentation of V .
Lemma 1.2.10. The functor pρ, V q ÞÑ pρ8 , V 8 q is right adjoint to the inclusion
functor from ReppGq into Repa pGq.
Proof. Let pρ, V q be a smooth representation of G and pσ, W q an abstract representation. Let f : V Ñ W be a G-linear map. If U Ÿ G is an open normal
subgroup, then f pV U q Ď W U . Hence,
ˆ ď
˙
ď
ď
U
f pV q “ f
V
“
f pV U q Ď
W U “ W 8.
U ŸG
open
U ŸG
open
U ŸG
open
Therefore, there is a natural isomorphism
HomG pρ, σq – HomG pρ, σ 8 q.
15
Corollary 1.2.11. In particular, the functor pρ, V q ÞÑ pρ8 , V 8 q is left exact.
Remark 1.2.12. Abstract representations of G correspond to modules over the
group algebra CrGs. There is not really a nice way to define smoothness directly
in terms of the group algebra, but one can study ‘smooth modules’ over another
algebra, called the Hecke algebra. For details, see [3].
1.3
Induced representations
We write pλG , CrGsq for the regular representation of G. The group homomorphism λG : G Ñ AutpCrGsq maps an element g P G to the automorphism
defined by g 1 ÞÑ gg 1 . Then λG is given by left translation, i.e.
˜
¸
n
n
ÿ
ÿ
λG pgq
ai gi “
ai ggi .
i“1
i“1
Similarly, pρG , CrGsq denotes the representation induced by right translation:
¸
˜
n
n
ÿ
ÿ
ai gi “
ai gi g ´1 .
ρG pgq
i“1
i“1
It corresponds to CrGs as a right module over itself, viewed as a left module by
inverting the action.
Definition 1.3.1. Let G be a locally profinite group, and let H be a closed
subgroup. Let pσ, W q be an abstract representation of H. Then define a representation of G on HomH pλG , σq by
˜
¸
¸
˜
n
n
ÿ
ÿ
pgf q
ai gi :“ f
ai gi g ,
i“1
i“1
for g P G and f P HomH pλG , σq. It is called the abstract induced representation,
and is denoted a-IndG
H pσq.
Just like in the case for finite groups, it satisfies the following property.
Lemma 1.3.2 (Abstract Frobenius Reciprocity). Let pρ, V q and pσ, W q be abstract representations of G and H respectively. There is a natural isomorphism
´
¯
` ˇ
˘
HomG ρ, a-IndG
pσq
– HomH ρˇH , σ .
H
Proof. The unit for the adjunction is given by
` ˇ ˘
ˇ
ηρ : ρ Ñ a-IndG
H ρH
˜˜
¸
˜
¸¸
n
n
ÿ
ÿ
v ÞÑ
ai gi ÞÑ
ai ρpgi qv
,
i“1
i“1
16
for all v P V . Observe that ηρ pvq is indeed an H-linear map λG Ñ ρ|H , and
that ηρ is G-linear.
The counit for the adjunction is given by
ˇ
ˇ
εσ : a-IndG
pσq
ˇ
H
H
Ñσ
f ÞÑ f p1q,
for all f P HomH pλG , σq. This map is H-linear.
One checks easily that the maps
´
¯
˘
` ˇ
ˇ
HomG ρ, a-IndG
H pσq ÐÑ HomH ρ H , σ
ˇ
φ ÞÝÑ εσ ˝ φˇH
a-IndG
H pf q ˝ ηρ ÐÝß f
are each others inverses.
Corollary 1.3.3. We have a composite adjunction
ReppGq Õ Repa pGq Õ Repa pHq.
In particular, if pρ, V q is a smooth representation of G and pσ, W q an abstract
representation of H, then
¯
´
` ˇ
˘
8
pσq
– HomH ρˇH , σ .
HomG ρ, a-IndG
H
Definition 1.3.4. If pσ, W q is a smooth representation of H, then the smoothly
G
8
induced representation IndG
of G.
H pσq is the smooth representation a-IndH pσq
Because ReppHq is a full subcategory of Repa pHq and the restriction to H
of a smooth representation of G is smooth, the above corollary also gives the
following result.
Corollary 1.3.5 (Smooth Frobenius Reciprocity). Let pρ, V q and pσ, W q be
smooth representations of G and H respectively. There is a natural isomorphism
´
¯
` ˇ
˘
ˇ
HomG ρ, IndG
H pσq – HomH ρ H , σ .
Remark 1.3.6. Because CrGs is a free CrHs-module, in particular it is projective, so the functor a-IndG
H is exact. By abstract nonsense, it is also clear that
IndG
is
left
exact
(being
a right adjoint functor). One can in fact show that it
H
is exact, cf. Proposition 2.4 of[3].
A very similar construction is given by the following definition.
17
Definition 1.3.7. Let U Ď G be an open subgroup, and let pσ, W q be an
abstract representation of U . Then define the compactly induced representation
c-IndG
U pσq of σ as the representation of G on CrGs bCrU s W by
gpx b wq “ gx b w.
Here, CrGs is viewed as a right CrU s-module in the obvious way, so that the
tensor product makes sense.
In other words, it is obtained by extension of scalars from CrU s to CrGs. In
particular, we get the following two results for free.
Lemma 1.3.8. The functor c-IndG
U is exact.
Proof. This is because CrGs is a free CrU s-module, hence in particular flat.
Lemma 1.3.9 (Abstract Frobenius Reciprocity for Compact Induction). Let
pρ, V q and pσ, W q be abstract representations of G and U respectively. Then
there is a natural isomorphism
¯
´
ˇ ˘
`
ˇ
HomG c-IndG
U pσq, ρ – HomU σ, ρ U .
Proof. This is immediate from the adjoint property of extension of scalars.
It turns out that compact induction preserves smoothness:
Lemma 1.3.10. Let pσ, W q be a smooth representation of U . Then c-IndG
U pσq
is smooth as well.
Proof. Let a1 , . . . , an P C, g1 , . . . , gn P G and w1 , . . . , wn P W be given. Then
each wi is fixed by some open subgroup Ui of U . Hence, the element
n
ÿ
pai gi b wi q
i“1
is fixed by the intersection of the Ui , which is an open subgroup of U , hence
also of G (since U is open in G).
Corollary 1.3.11 (Smooth Frobenius Reciprocity for Compact Induction). Let
pρ, V q and pσ, W q be smooth representations of G and U respectively. Then there
is a natural isomorphism
´
¯
ˇ ˘
`
HomG c-IndG
pσq,
ρ
– HomU σ, ρˇU .
U
Remark 1.3.12. Note that compact induction is left adjoint to restriction,
while smooth induction is right adjoint to restriction.
18
Remark 1.3.13. We can identify HomU pλG , σq8 with the set of maps f : G Ñ
W satisfying
f phgq “ σphqf pgq,
h P U, g P G,
and we can identify CrGs bCrU s W with the subset consisting of maps with finite
support modulo U . As such, we get a natural G-linear injection
G
c-IndG
U pσq ÝÑ IndU pσq
mapping an element of the form g b w to the U -linear map CrGs Ñ W defined
by g ÞÑ w. It is an isomorphism if and only if rG : U s is finite.
In [3] as well as in [14], this is used as the definition of Ind and c-Ind. Also, the
definition of c-Ind can then be extended to closed subgroups H, by replacing
finite support by compact support in Remark 1.3.13 above. Note that if H is
open, then HzG is discrete, so a subset of HzG is compact if and only if it is
finite.
1.4
Dual representations
Definition 1.4.1. Let pρ, V q be an abstract representation of G. Then the
abstract dual representation pρ˚ , V ˚ q is the representation of G on the dual
space
V ˚ :“ HomC pV, Cq
given by
ˆ
ρ˚ pgqf :“
˙
x ÞÑ f pρpg ´1 qxq .
One easily checks that this is indeed a representation of G.
Definition 1.4.2. Let pρ, V q and pσ, W q be abstract representations of G, and
let φ : ρ Ñ σ be a G-linear map. Then define the map
φ˚ : ρ˚ Ñ σ ˚
f ÞÑ f ˝ φ.
Then φ˚ is a G-linear map. The construction pρ, V q ÞÑ pρ˚ , V ˚ q becomes a
functor D : Repa pGqop Ñ Repa pGq.
Lemma 1.4.3. Let pρ, V q be an abstract representation of G. There is a natural
G-linear map
evρ : ρ ÝÑ ρ˚˚
v ÞÝÑ pf ÞÑ pf pvqqq.
It is injective, and it is an isomorphism if and only if V is finite-dimensional.
Proof. The map is clearly C-linear. If g P G and v P V are given, then for all
f P HomC pV, Cq we have
evρ pρpgqvqpf q “ f pρpgqvq “ pρ˚ pg ´1 qf qpvq “ evρ pvqpρ˚ pg ´1 qf q,
19
so evρ pρpgqvq is given by
f ÞÑ evρ pvqpρ˚ pg ´1 qf q,
which is the definition of ρ˚˚ pgq evρ pvq. Hence, evρ is G-linear.
The final statement is a well-known result from linear algebra.
Lemma 1.4.4. Let pρ, V q and pσ, W q be abstract representations of G. Then
there is a natural isomorphism
HomG pρ, σ ˚ q – HomG pσ, ρ˚ q.
Proof. Define maps
HomG pρ, σ ˚ q ÐÑ HomG pσ, ρ˚ q
φ ÞÝÑ φ˚ ˝ evσ
ψ ˚ ˝ evρ ÐÝß ψ.
We can identify any φ P HomG pρ, σ ˚ q with the C-bilinear map
fφ : V ˆ W Ñ C
pv, wq ÞÑ φpvqpwq.
Similarly, any ψ P HomG pσ, ρ˚ q can be identified with the C-bilinear map
gψ : W ˆ V Ñ C
pw, vq ÞÑ ψpwqpvq.
Under this correspondence, the first map is given by
ˆ
˙
fφ ÞÑ pw, vq ÞÑ fφ pv, wq ,
and the second by
ˆ
gψ ÞÑ
˙
pv, wq ÞÑ gψ pw, vq ,
and it is clear that the maps are each others inverses.
Remark 1.4.5. The lemma shows that the functor D : Repa pGqop Ñ Repa pGq
is right adjoint to the functor Dop : Repa pGq Ñ Repa pGqop .
Definition 1.4.6. Let pρ, V q be a smooth representation of G. Then define the
smooth dual representation pρ̌, V̌ q as ppρ˚ q8 , pV ˚ q8 q.
Corollary 1.4.7. Let pρ, V q and pσ, W q be smooth representations of G. Then
there is a natural isomorphism
HomG pρ, σ̌q – HomG pσ, ρ̌q.
Proof. This follows from the lemma, together with the adjunction property of
the functor pτ, Xq ÞÑ pτ 8 , X 8 q.
20
Example 1.4.8. Let χ be a character of G. Then pχ, Cq is a one-dimensional
representation, and we can identify the dual space C˚ with C via f ÞÑ f p1q.
Then χ˚ is given by the homomorphism
χ˚ : G Ñ C
g ÞÑ χpg ´1 q.
This is obviously a character as well, so χ˚ is already smooth. Hence, χ̌ “ χ˚
is given by g ÞÑ χpg ´1 q.
Lemma 1.4.9. Let pρ, V q be a smooth representation of G, and let U Ď G be a
compact open subgroup. Then the map V̌ U Ñ pV U q˚ given by f Ñ
Þ f |V U is an
isomorphism.
Proof. Note that V̌ U is the set of functions f : V Ñ C that satisfy
f pρph´1 qvq “ f pvq
for all v P V . That is, f is U -linear, when we equip C with trivial U -action, so
V̌ U “ HomU pρ|U , 1U q.
Also, any C-linear map pV U q˚ Ñ C is automatically U -linear, since the action
on both sides is trivial. Therefore,
pV U q˚ “ HomU pV U , 1U q.
The representation pρ|U , V q of U is semisimple since U is profinite (Lemma
1.2.5). Hence, it composes as a direct sum of irreducible U -representations.
Therefore any U -homomorphism f : ρ|U Ñ 1U is trivial on the irreducible components that are not isomorphic to 1U . The components that are isomorphic to
1U are exactly the ones contained in V U , and the result follows.
Remark 1.4.10. From now on, we will identify pV U q˚ with the subset V̌ U of
V̌ .
Corollary 1.4.11. We can identify V̌ with
ď ` ˘˚
VU .
U ŸG
open
compact
Proof. This is clear from the definition of V̌ .
Lemma 1.4.12. Let pρ, V q, pσ, W q and pτ, Xq be smooth representations of G,
and let φ : ρ Ñ σ and ψ : σ Ñ τ be G-linear maps. Then the sequence
φ
ψ
V ÝÑ W ÝÑ X
is exact if and only if for every compact open subgroup U Ď G the sequence
φ
ψ
V U ÝÑ W U Ñ X U
is exact.
21
Proof. If U Ď G is a compact open subgroup, then every smooth representation
of U is the direct product of some family of irreducible representations. Furthermore, V U is the direct sum over the subfamily of trivial representations.
Therefore,
φ´1 pW U q Ď V U
and
φpV q X W U “ φpV U q.
If the sequence V Ñ W Ñ X is exact, therefore so is the sequence
V U Ñ W U Ñ XU .
The other implication is trivial, as W is the union of the sets W U for U Ď G
open compact.
Definition 1.4.13. A smooth representation pρ, V q of G is called admissible if
for every compact open subgroup U Ď G the space V U has finite dimension.
Proposition 1.4.14. Let pρ, V q be a smooth representation of G. Then the
image of the G-linear map
˚
ev8
ρ : ρ Ñ pρ̌q
ˆ
˙
v ÞÑ f ÞÑ f pvq
is inside ρ̌ˇ. Furthermore, the map ev8
ρ is injective, and it is an isomorphism if
and only if ρ is admissible.
Proof. The first assertion follows from the universal property of p´q8 (cf. the
proof of Lemma 1.2.10). By Corollary 1.4.11, we have
ď ` ˘˚˚
V̌ˇ “
VU
.
U ŸG
open
compact
The map
ev8
ρ :
ď
VU Ñ
U ŸG
open
compact
ď
`
VU
˘˚˚
U ŸG
open
compact
is the obvious one, so we are done by Lemma 1.4.12.
Proposition 1.4.15. The functor ReppGqop Ñ ReppGq given by pρ, V q ÞÑ
pρ̌, V̌ q is exact.
Proof. Let 0 Ñ V Ñ W Ñ X Ñ 0 be a short exact sequence of representations
of G. Then for each compact open subgroup U Ď G, the sequence
0 Ñ V U Ñ W U Ñ XU Ñ 0
is exact (Lemma 1.4.12). By linear algebra, also the sequence
`
˘˚
`
˘˚
` ˘˚
0 Ñ XU Ñ W U Ñ V U Ñ 0
is exact. The result then follows from Lemma 1.4.12 and Corollary 1.4.11.
22
Corollary 1.4.16. An admissible representation pρ, V q of G is irreducible if
and only if pρ̌, V̌ q is irreducible.
23
2
Tate’s Thesis
The reader is not assumed to be familiar with measure theory or Fourier analysis.
In fact, the structure of locally profinite groups is so strong that the theory is
almost entirely algebraic, as we will see.
The treatment of this chapter is influenced by sections 3 and 23 of [3], as well
as by Tate’s thesis [16].
2.1
Haar measures
We will focus more on the Haar integral than on the Haar measure, because
we do not want to worry about the intricacies of measure theory. (Even the
definition of a measure is quite involved.)
Definition 2.1.1. Let G be a locally profinite group, and let f : G Ñ C be a
function. Then f is smooth if it is locally constant.
We denote by C 8 pGq the C-vector space of smooth functions f : G Ñ C. The
subspace of functions that have compact support is denoted Cc8 pGq.
Definition 2.1.2. A smooth function f : G Ñ C is called positive if f pgq P Rě0
for all g P G. We write f ľ 0.
Example 2.1.3. Let U Ď G be any compact open subset. Then the indicator
function IU is the function
"
1 if g P U,
IU pgq “
0 else.
It is locally constant since U is both open and closed, and it has compact support
since U is compact.
Lemma 2.1.4. The space Cc8 pGq is spanned by the functions IgU for g P G and
U Ď G a compact open subgroup.
Proof. Let f P C8
c pGq. Since f is locally constant, for each g P G there exists
a compact open subgroup Ug Ď G such that f pgUg q “ f pgq. Let S “ tg P G :
f pgq ‰ 0u be the support of f . The open sets gUg (for g P S) cover the compact
set S, so there is a finite subset g1 , . . . , gn of S such that
S“
n
ď
gi Ugi .
i“1
Set U to be the intersection of the Ugi and Ug for some g R S (if such an element
exists).
Then f pghq “ f pgq for all g P G and all h P U , and f is essentially a map
G{U Ñ C with compact support. Since the topology on G{U is discrete, f in
fact has finite support modulo U , and the result follows.
24
Remark 2.1.5. Similarly, one can show that the space Cc8 pGq is spanned by
the functions IU g for g P G and U Ď G a compact open subgroup.
We have a representation pρG , Cc8 pGqq of G by translation on the right:
pρG pgqf qpxq :“ f pxgq
for g P G, f P Cc8 pGq and x P G. The proof of the lemma shows that this
representation is smooth.
Definition 2.1.6. A right Haar integral on G is a G-linear map I : ρG Ñ 1G
such that Ipf q P Rě0 for all f ľ 0.
Proposition 2.1.7. There exists a right Haar integral I : ρG Ñ 1G . Moreover,
if I 1 is another right Haar integral, then there exists a constant c P Rą0 such
that I “ cI 1 .
Proof. For every compact open subgroup, the functions tIU g : g P U zGu span a
G-subspace VU of Cc8 pGq. It is isomorphic to c-IndG
U p1U q by Remark 1.3.13, so
by Frobenius reciprocity we have
dim HomG pVU , 1G q “ dim HomU p1U , 1U q “ 1.
On the other hand, Remark 2.1.5 asserts that
ď
Cc8 pGq “
VU .
(1)
(2)
U ĎG
open
compact
Now fix some compact open subgroup U0 . Then define I : Cc8 pGq Ñ C as
IU g ÞÑ
rU : U X U0 s
.
rU0 : U X U0 s
This map clearly has the desired properties, and it is unique up to scaling by a
positive constant because of (1) and (2).
We also have a representation pλG , C 8 pGqq of G by translation on the left:
pλG pgqf qpxq :“ f pg ´1 xq
for g P G, f P C 8 pGq and x P G. Once again, Cc8 pGq is a G-subspace of C 8 pGq.
Definition 2.1.8. A left Haar integral on G is a G-linear map I : λG Ñ 1G
such that Ipf q P Rě0 for all f ľ 0.
Definition 2.1.9. Denote the underlying set of the group G by X. Endow the
set X with the group structure Gop as the unique group structure making the
bijection
G Ñ Gop
g ÞÑ g ´1
25
an isomorphism. That is, in Gop , the multiplication map ˚ is given by
pg1 , g2 q ÞÑ g1 ˚ g2 :“ g2 g1 .
Giving Gop the topology of G makes the isomorphism G Ñ Gop a topological
one, since inversion on G is a homeomorphism. In particular, Gop is a topological
group.
Remark 2.1.10. We now have a representation pρGop , Cc8 pXqq given by
pρGop pgqf qpxq :“ f px ˚ gq “ f pgxq.
Under the topological isomorphism G – Gop , this representation corresponds
to the representation pλG , Cc8 pXqq.
We use the set X in the notation here to emphasise that the underlying vector
spaces of ρGop and λG are the same. The notation Cc8 pGop q would have been
ambiguous, since it is not clear whether we identify Cc8 pGq with Cc8 pGop q via
g ÞÑ g or via g ÞÑ g ´1 . We use the former.
Proposition 2.1.11. There exists a left Haar integral I : λG Ñ 1G . Moreover,
if I 1 is another left Haar integral, then there exists a constant c P Rą0 such that
I “ cI 1 .
Proof. A left Haar integral on G corresponds to a right Haar integral on Gop .
Hence, the result follows from Proposition 2.1.7.
Remark 2.1.12. By construction (see the proof of Proposition 2.1.7), there is
some compact open subgroup U1 with IpIU1 q “ 1 for some left Haar integral I.
If U Ď G is any compact open subgroup, then U 1 :“ U X U1 is also a compact
open subgroup, and
ÿ
1 “ IpIU1 q “
IpIhU 1 q “ rU1 : U 1 sIpIU 1 q.
hPU1 {U 1
Hence, IpIU 1 q “
1
rU1 :U 1 s ,
and a similar argument shows that
IpIU q “
rU : U 1 s
.
rU1 : U 1 s
In particular, we conclude that
IpIU q ą 0,
for any left Haar integral I on G.
Definition 2.1.13. Let I be a left Haar integral. Then the induced left Haar
measure is the map
µG : tU Ď G : U compact openu Ñ Rě0
U ÞÑ IpIU q.
26
One can show that this ‘is’ a measure, but we will not even state the definition
of a measure. From now on, we will fix some left Haar measure µG . In specific examples, we might force a condition on µG to hold, defining the measure
completely.
If f P Cc8 pGq is a smooth function of compact support, then we denote Ipf q by
ż
f pgq dµG pgq.
G
Sometimes the variable g is dropped, so the notation becomes
ż
f dµG .
G
If U Ď G is a compact open subset, then we will write
ż
ż
f pgq dµG pgq :“
IU f pgq dµG pgq.
U
G
Now if g P G is given, then the map
Cc8 pGq Ñ C
ż
f ÞÑ
f pxgq dµG pxq
G
is a left Haar integral, hence there exists an element δG pgq P Rą0 such that
ż
ż
δG pgq
f pxgq dµG pxq “
f pxq dµG pxq.
G
G
The last expression can also be given as
ż
f pxgq dµG pxgq,
G
so dµG pxgq “ δG pgq dµG pxq.
Definition 2.1.14. The modulus of G is the map G Ñ Cˆ given by g ÞÑ δG pgq.
Remark 2.1.15. If U Ď G is a compact open subgroup, then the indicator
function IU is invariant under left translation by U . The integral
ż
IU pxq dµG pxq
G
is nonzero by Remark 2.1.12, which shows that δG phq “ 1 for all h P U .
Lemma 2.1.16. The modulus of G is a character that is independent of the
choice of Haar measure.
Proof. It is clearly a homomorphism. If U Ď G is any compact open subgroup,
then U Ď ker δG . Hence, the kernel of δG is open, so δG is a character by
Proposition 1.2.7. The modulus clearly does not depend on µG .
27
Remark 2.1.17. If f P Cc8 pGq is a smooth function of compact support, then
´1
δG
f is smooth and of compact support as well. One can easily check that
ż
´1
f ÞÑ
δG
f dµG
G
is a right Haar integral.
Definition 2.1.18. A locally profinite group G is called unimodular if any left
Haar integral is also a right Haar integral.
Lemma 2.1.19. A locally profinite group G is unimodular if and only if δG “ 1.
Proof. Clear from the previous remark and the definition of the modulus.
Example 2.1.20. If G is compact (i.e. profinite), then δG is trivial on any
compact open subgroup U Ď G. In particular, δG is trivial on G, so G is
unimodular.
Lemma 2.1.21. Let χ be a character of G. Let U be a compact open subgroup
of G. Then
ˇ
"
ż
µG pU q if χˇU “ 1U ,
χ dµG “
0
else.
U
Proof. Note that χ is a smooth function by our assumptions on characters.
Furthermore, U is compact, so the integral exists.
Since χ is a character, there is a normal compact open subgroup U 1 Ď G on
which χ is the trivial character. The quotient G1 “ U {pU X U 1 q is a finite group,
and there is a unique character ρ on G1 such that
ˇ
χˇU “ ρ ˝ π,
where π : U Ñ G1 is the canonical projection. Then the integral is a sum
ÿ
µG pU X U 1 qρpgq,
gPG1
and the result follows from Schur orthogonality for finite groups.
2.2
The additive group of a local field
In this section and the next, we will cover a simple version of what is basically
(the local part of) Tate’s Thesis. The original thesis can be found in [16], but
we will follow the treatment of [3].
Observe that K is a locally profinite group. A basis for the open neighbourhoods
of the zero element is given by the sets pn for n P Z. Furthermore, K is the
union of its compact open subgroups, since these are just the pn .
28
Lemma 2.2.1. Let µK be a left Haar measure on K. Let a P K ˆ . Then
dµK paxq “ }a} dµK pxq.
Proof. Let n P Z be given. Then
ż
ż
ż
´1
n
n
Ip pxq dµK paxq “
Ip pa xq dµK pxq “
Iapn pxq dµK pxq.
K
K
K
The latter equals µK ppn`vK paq q, which is q ´vK paq times the size of µppn q. Since
the Haar integral is translation invariant, this gives
ż
ż
Ib`pn pxq dµK paxq “ }a}
Ib`pn dµK pxq
K
K
for all b P K, n P Z. The functions Ib`pn span Cc8 pKq by Lemma 2.1.4.
Definition 2.2.2. Let K be a local field. Then the character group of K is the
p
group (under multiplication) of characters ψ : K Ñ Cˆ . It is denoted K.
p is a nontrivial character, then the kernel of ψ is open, so it contains
If ψ P K
the set pn for some n P Z.
p be a nontrivial character. The least integer n P Z
Definition 2.2.3. Let ψ P K
n
such that p Ď ker ψ is called the level of ψ.
Definition 2.2.4. If ψ is a character, then so is the map x ÞÑ ψpaxq for any
a P K. It is denoted aψ.
Remark 2.2.5. If ψ is a nontrivial character of level n, and a P K ˆ is a nonzero
element, then the character aψ has level n ´ vK paq.
p be a nontrivial character. Then the map
Proposition 2.2.6. Let ψ P K
p
fψ : K Ñ K
a ÞÑ aψ
is an isomorphism.
Proof. It is clearly a homomorphism. If a P K is such that fψ paq “ 1, then in
particular ψpaxq “ 1 for all x P K. But that forces a “ 0, as otherwise ψ “ 1.
Now let ϕ be any nontrivial character of K, say of level m. Let π be a uniformiser
of K, and let n be the level of ψ. Then the character π n´m ψ is trivial on pm ,
as is ϕ.
The characters on pm´1 that are trivial on pm correspond bijectively to the
1
characters on pm´1 {pm – k. The q ´ 1 characters uπ n´m ψ for u P UK {UK
are
m´1
all distinct, so by a counting argument they exhaust all characters on p
that
are trivial on p. Hence, there is some u1 P UK such that u1 π n´m ψ is identical
to ϕ on pm´1 .
29
Proceeding inductively, we get a sequence u1 , u2 , . . . of elements in UK such that
ui π m´n ψ and ϕ agree on pm´i and such that ui ” uj mod pj for i ą j. It is
clear that the limit u of this sequence satisfies uπ m´n ψ “ ϕ.
We now fix not only a left Haar measure dµ on K (we drop the subscript K
from the notation), but also a nontrivial character ψ of K of level m.
Remark 2.2.7. If Φ P Cc8 pKq, then for any ξ P K the function Φpxqψpξxq is
locally constant since the kernel of ψ is open. Also, the function Φpxqψpξxq has
compact support, since Φ has compact support.
Definition 2.2.8. Given Φ P Cc8 pKq. Then the Fourier transform (with respect to dµ and ψ) of Φ is the function K Ñ C defined by
ż
Φ̂pξq “
Φpxqψpξxq dµpxq.
K
The integral is well defined by Remark 2.2.7.
Lemma 2.2.9. Let Φ P Cc8 pKq, and let a P K. Write Φ1 pxq “ Φpx ´ aq. Then
Φ̂1 pξq “ aψpξqΦ̂pξq.
Proof. We compute
ż
Φ̂1 pξq “
Φpx ´ aqψpξxq dµpxq
żK
Φpxqψpξpx ` aqq dµpxq “ aψpξqΦ̂pξq.
“
K
Lemma 2.2.10. Let Φ P Cc8 pKq, and let a P K. Let Φ1 “ aψ ¨ Φ. Then
Φ1 P Cc8 pKq, and
Φ̂1 pξq “ Φ̂pξ ` aq.
Proof. The first assertion follows from Remark 2.2.7. We compute
ż
Φ̂1 pξq “
aψpxqΦpxqψpξxq dµpxq
K
ż
“
Φpxqψppξ ` aqxq dµpxq “ Φ̂pξ ` aq.
K
Lemma 2.2.11. Let n P Z be given. Then
Îpn “ µppn qIpm´n .
Proof. Let ξ P K be given. We have
ż
ż
Ipn pxqψpξxq dµpxq “
ψpξxq dµpxq.
pn
G
When ξ P pm´n , the character ξψ is trivial on pn , and the result follows.
30
Hence assume ξ R pm´n , and set i “ vK pξq. Then i ă m ´ n, and
ż
ż
ψpξxq dµpxq “
ψpxq dµpxq.
pn
pn`i
This is zero by Lemma 2.1.21, since ψ|pn`i is a nontrivial character on pn`i .
Proposition 2.2.12. For Φ P Cc8 pKq, we have Φ̂ P Cc8 pKq. Furthermore,
there exists a number c P Rą0 (depending on dµ and ψ) such that
ˆ
Φ̂pxq “ cΦp´xq
for all Φ P Cc8 pKq, x P K.
Proof. We take c “ µpOK q2 q ´m . Firstly, let Φ “ Ipn for some n P Z. Then
Lemma 2.2.11 shows that Φ̂ P Cc8 pKq, and we compute
ˆ
Φ̂ “ µppn qÎpm´n “ µppn qµppm´n qIpm .
Because µppn q “ q ´n µpOK q and µppm´n q “ q n´m µpOK q, the result holds for
Φ “ Ipm .
Now consider the function Φ1 “ Ia`pn for a P K, n P Z. By Lemma 2.2.9, we
have Φ̂1 “ aψ ¨ Φ̂, where Φ “ Ipn . Then Lemma 2.2.10 asserts that Φ̂1 P Cc8 pKq,
and that
ˆ
ˆ
Φˆ1 pξq “ Φ̂pξ ` aq.
Therefore, the result also holds for the function Ia`pn . Since these functions
span Cc8 pKq by 2.1.4, we are done.
m
Remark 2.2.13. From now on, we will w.l.o.g. assume that µpOK q “ q 2 . This
determines the Haar measure completely, and this measure is called self-dual.
Theorem 2.2.14 (Fourier inversion formula). Let Φ P Cc8 pKq. Then
ˆ
Φ̂pxq “ Φp´xq,
for all x P K.
2.3
The multiplicative group of a local field
Firstly, note that K ˆ is a locally profinite group in the topology induced from
K. This implies in particular that the set Cc8 pK ˆ q can be identified with the
subset of Cc8 pKq consisting of functions that vanish at the origin.
Definition 2.3.1. Let χ be a character of K ˆ . Then χ is called unramified if
its restriction to UK is trivial. Otherwise, χ is called ramified. This terminology
will be explained in Remark 3.2.3 and Remark 3.2.4.
We will fix a character χ of K ˆ and a Haar measure µˆ on K ˆ .
31
Remark 2.3.2. Let π P p be a uniformiser. Then the set π n UK “ pn zpn`1
does not depend on π. It will be denoted Sn , and it is a compact open subset
of K ˆ . It satisfies
ISn “ Ipn ´ Ipn`1 .
In particular, if Φ P Cc8 pKq is a smooth function of compact support, then ISn Φ
is also smooth and of compact support. Since it vanishes at 0, we have
ISn Φ P Cc8 pK ˆ q.
Since Φ has compact support, there is some m P Z such that Φ vanishes outside
pm . Hence, for all j ă m, it holds that
ISj Φ “ 0.
Definition 2.3.3. If Φ P Cc8 pKq and a P K ˆ are given, then we write aΦ for
the function x ÞÑ Φpa´1 xq.
In this notation, it follows that ISn “ π n IUK . More generally, if U Ď K is any
compact open set, and Φ is the indicator function of U , then aΦ is the indicator
function of aU .
Lemma 2.3.4. The space Cc8 pKq is spanned by Cc8 pK ˆ q and IOK .
Proof. This can be proven using explicit knowledge about Cc8 pKq and Cc8 pK ˆ q,
cf. Lemma 2.1.4. However, a much easier argument goes as follows:
We have identified Cc8 pK ˆ q with the subset of Cc8 pKq of functions that vanish
at 0. That is, we have a short exact sequence of C-vector spaces
φ
0 Ñ Cc8 pK ˆ q Ñ Cc8 pKq ÝÑ C Ñ 0,
where φ is given by Φ ÞÑ Φp0q. Picking any function Φ P Cc8 pKq that does not
vanish at 0 induces a splitting, and IOK is such a function.
Definition 2.3.5. Let Φ P Cc8 pKq. Then we define the formal Laurent series
ÿ
ZpΦ, χ, Xq “
zn pΦ, χq X n P CppXqq,
nPZ
where
ż
Φpxqχpxq dµˆ pxq.
zn pΦ, χq “
Sn
Note that Φ ¨ χ is a smooth function with compact support, so the integral is
defined. Note also that zn pΦ, χq “ 0 for n sufficiently small, so the series is
indeed a Laurent series.
Definition 2.3.6. Let Φ P Cc8 pKq. Then the zeta function of Φ (with respect
to χ) is the function
ζpΦ, χ, sq “ ZpΦ, χ, q ´s q.
For now, the zeta function is only a formal Laurent series in q ´s . However, we
will show that it is actually a rational function in q ´s , so it defines a bona fide
meromorphic function on C in the variable s.
32
Remark 2.3.7. It is clear that the function
Z : Cc8 pKq Ñ CppXqq
Φ ÞÑ ZpΦ, χ, Xq
is C-linear.
Lemma 2.3.8. Let Φ P Cc8 pKq and a P K ˆ . Then
ZpaΦ, χ, Xq “ χpaqX vK paq ZpΦ, χ, Xq.
Proof. Write m “ vK paq. For any n P Z, we compute
ż
zn paΦ, χq “
Φpa´1 xqχpxq dµˆ pxq
Sn
ż
“
Φpyqχpayq dµˆ payq
Sn´m
ż
“ χpaq
Φpyqχpyq dµˆ pyq “ χpaqzn´m pΦ, χq,
Sn´m
and the result follows.
Lemma 2.3.9. The image of Cc8 pK ˆ q under Z is CrX, X ´1 s.
Proof. Let Φ P Cc8 pK ˆ q. Then Φ vanishes at 0, hence also in some open
neighbourhood pn of 0. Therefore, the integral
ż
Φ ¨ χ dµˆ
Sm
vanishes for m ě n, so ZpΦ, χ, Xq is a Laurent polynomial in X.
i
Since χ is a character, it has an open kernel; say that UK
Ď ker χ. Let Φ0 be
i
the indicator function of UK . Note that
" i
UK m “ 0,
i
UK X Sm “
∅
m ‰ 0.
i
Hence, using that χ is trivial on UK
, we get
" ˆ i
ż
ż
µ pUK q m “ 0,
ˆ
ˆ
Φ0 ¨ χ dµ “
χ dµ “
0
m ‰ 0.
Sm
U i XSm
K
It follows that ZpΦ0 , χ, Xq is a nonzero constant Laurent polynomial. We are
done by Lemma 2.3.8.
Lemma 2.3.10. Let Φ be the indicator function of OK . Then
"
1
p1 ´ χpπqXq´1 if χ is unramified,
ZpΦ,
χ,
Xq
“
0
if χ is ramified.
µˆ pUK q
33
Proof. For n ă 0, we have Sn X OK “ ∅, so
ż
zn pΦ, χq “
Φ ¨ χ dµˆ “ 0.
Sn
For n ě 0, we have Sn Ď OK , so
ż
ż
ˆ
zn pΦ, χq “
Φ ¨ χ dµ “
χpxq dµˆ pxq
Sn
Sn
ż
ż
n
ˆ n
n
“
χpπ yq dµ pπ yq “ χpπq
χpyq dµˆ pyq.
S0
UK
Hence,
ˆż
ZpΦ, χ, Xq “
χ dµ
ˆ
χ dµ
ˆ
˙ÿ
8
UK
n“0
˙
ˆż
“
pχpπqXqn
UK
1
.
1 ´ χpπqX
The integral is µˆ pUK q if χ is unramified and 0 if χ is ramified.
Corollary 2.3.11. Let Φ P Cc8 pKq. There exists a constant s0 P R such that
the integral
ż
Φpxqχpxq }x}s dµˆ pxq
Kˆ
converges absolutely and uniformly on compact subsets of the right half plane
tRe s ą s0 u. On this right half plane, the integral is equal to ζpΦ, χ, sq.
Proof. The key observation is that
ż
ż
ÿ
s
ˆ
´ns
Φpxqχpxq }x} dµ pxq “
q
Kˆ
Φpxqχpxq dµˆ pxq,
Sn
nPZ
which is equal to ZpΦ, χ, q ´s q as a formal Laurent series in q ´s . For all functions
Φ P Cc8 pK ˆ q the result follows since the sum is finite.
For the function IOK , we take s0 “
8
ÿ
lnpχpπqq
ln q .
Then the series
pχpπqq ´s qn
n“0
converges absolutely and uniformly on compact subsets of tRe s ą s0 u, and the
result follows from the computations in Lemma 2.3.10.
The vector space Cc8 pKq is spanned by Cc8 pK ˆ q and IOK , so we are done.
Remark 2.3.12. Some authors use the integral above as the definition of the
zeta function. In Chapter 4, we will do something similar for functions on
Mn pKq with respect to a representation of GLn pKq.
34
Proposition 2.3.13. The image of the map Z is equal to Pχ pXq´1 CrX, X ´1 s,
where
"
1 ´ χpπqX if χ is unramified,
Pχ pXq “
1
if χ is ramified.
Proof. If χ is ramified, then IOK is mapped to 0 under Z. We are then done by
Lemma 2.3.4 and Lemma 2.3.9.
If χ is unramified, then the same two lemmata imply that the image of Z is
the linear subspace of CppXqq spanned by CrX, X ´1 s and p1 ´ χpπqXq´1 . The
result follows by linear algebra. Alternatively, one can use Lemma 2.3.8 to
conclude.
Definition 2.3.14. Define the L-function associated to the character χ as
Lpχ, sq “ Pχ pq ´s q´1 .
2.4
Epsilon factors
Throughout this section, we will fix a nontrivial character χ of K ˆ and an
m
Ď ker χ. We will also fix a nontrivial character ψ
integer m P Zą0 such that UK
of K, and we let µ be the self-dual Haar measure of K, cf. Remark 2.2.13. We
will implicitly fix a Haar measure µˆ of K ˆ , just like in the previous section.
Proposition 2.4.1. Denote by Λ the CpXq-vector space of C-linear functions
λ : Cc8 pKq Ñ CpXq satisfying
λpaΦq “ χpaqX vK paq λpΦq,
for all a P K ˆ and all Φ P Cc8 pKq. Then Λ has dimension 1 over CpXq.
Proof. We will show that any λ P Λ is uniquely determined by λpIUKm q. To be
precise, consider the CpXq-linear map
evIU m : Λ Ñ CpXq
K
λ ÞÑ λpIUKm q.
We will show that it is injective. Thus, let λ P Λ be given such that λpIUKm q “ 0.
m
If a P UK
, then χpaq “ 1 and vK paq “ 1. Therefore, the condition on λ forces
λpIaUKn q “ λpIUKn q
whenever n ě m. Hence, for n ě m, we have
ÿ
λpIUKm q “
λpIaUKn q “ q n´m λpIaUKn q.
m {U n
aPUK
K
Therefore, λpIUKn q “ 0 for all n ě m. It follows that λpIaUKn q “ 0 for all a P K ˆ
and all n ě m.
35
All functions of the form IbUKi and all functions of the form IbUK for b P K ˆ
and i P Zą0 are finite linear combination of the functions IaUKn as above. They
span Cc8 pK ˆ q by Lemma 2.1.4, so λ vanishes on Cc8 pK ˆ q.
Hence, λ factors through the map φ : Cc8 pKq Ñ C of the proof of Lemma 2.3.4.
That is, the value of λpΦq only depends on Φp0q. But for any a P K ˆ it holds
that paΦqp0q “ Φp0q, so the condition on λ gives
λpΦq “ λpaΦq “ χpaqX vK paq λpΦq,
for all a P K ˆ . This immediately forces λpΦq “ 0 for all Φ P Cc8 pKq. Hence,
λ “ 0, and evIU m is injective.
K
Theorem 2.4.2. There is a unique cpχ, ψ, Xq P CpXq such that
1
ZpΦ̂, χ̌, qX
q “ cpχ, ψ, XqZpΦ, χ, Xq
for all Φ P Cc8 pKq.
Proof. The vector space Λ of the previous proposition has dimension 1 over
CpXq. The function
Φ ÞÑ ZpΦ, χ, Xq
is an element of Λ by Lemma 2.3.8. It is nonzero by Lemma 2.3.9.
If Φ P Cc8 pKq and a P K ˆ are given, then for all ξ P K it holds that
ż
x
aΦpξq
“
Φpa´1 xqψpξxq dµpxq
K
ż
“
Φpyqψpaξyq dµpayq.
K
By Lemma 2.2.1, the latter is equal to
ż
}a}
Φpyqψpaξyq dµpyq “ }a} Φ̂paξq.
K
This gives
x “ }a} a´1 Φ̂.
aΦ
Hence, for all Φ P Cc8 pKq, a P K ˆ , it holds that
x χ̌, 1 q “ }a} Zpa´1 Φ̂, χ̌, 1 q
ZpaΦ,
qX
qX
´1
“ }a} χ̌pa´1 qpqXq´vK pa
q
1
ZpΦ̂, χ̌, qX
q
1
“ χpaqX vK paq ZpΦ̂, χ̌, qX
q.
This shows that the function
1
Φ ÞÑ ZpΦ̂, χ̌, qX
q
is also an element of Λ. It is again nonzero by Lemma 2.3.9. The result then
follows since Λ has dimension 1 over CpXq.
36
Definition 2.4.3. We put γpχ, s, ψq “ cpχ, ψ, q ´s q.
Corollary 2.4.4. We have the functional equation
ζpΦ̂, χ̌, 1 ´ sq “ γpχ, s, ψqζpΦ, χ, sq,
for all Φ P Cc8 pKq.
Proposition 2.4.5. The function γpχ, s, ψq satisfies the functional equation
γpχ, s, ψqγpχ̌, 1 ´ s, ψq “ χp´1q.
Proof. The Fourier inversion formula gives
ˆ
Φ̂pxq “ Φp´xq
for all Φ P Cc8 pKq, x P K. Hence, for all n P Z, we have
ż
ˆ
zn pΦ̂, χq “
Φp´xqχpxq dµˆ pxq
Sn
ż
“
Φpyqχp´yq dµˆ p´yq
Sn
“ χp´1q} ´ 1} zn pΦ, χq “ χp´1qzn pΦ, χq.
ˆ
Hence, ZpΦ̂, χ, Xq “ χp´1qZpΦ, χ, Xq, so
ˆ
ζpΦ̂, χ, sq “ χp´1qζpΦ, χ, sq.
On the other hand, the previous corollary gives
ˆ
ζpΦ̂, χ, Xq “ γpχ̌, 1 ´ s, ψqζpΦ̂, χ̌, 1 ´ sq
“ γpχ̌, 1 ´ s, ψqγpχ, s, ψqζpΦ, χ, sq,
which gives the result.
Definition 2.4.6. Define the function
ΞpΦ, χ, sq “
ζpΦ, χ, sq
.
Lpχ, sq
It is a rational function in q ´s by Proposition 2.3.13.
Definition 2.4.7. Define the function
εpχ, s, ψq “ γpχ, s, ψq
Lpχ, sq
.
Lpχ̌, 1 ´ sq
Corollary 2.4.8. The function ΞpΦ, χ, sq satisfies the functional equation
ΞpΦ̂, χ̌, 1 ´ sq “ εpχ, s, ψq ΞpΦ, χ, sq.
Proof. This is immediate from Corollary 2.4.4.
37
Corollary 2.4.9. The function εpχ, s, ψq satisfies the functional equation
εpχ, s, ψqεpχ̌, 1 ´ s, ψq “ χp´1q.
Furthermore, εpχ, s, ψq is of the form aq ´ns for some a P Cˆ , n P Z.
Proof. The functional equation follows from Proposition 2.4.5.
Pick some Φ for which ΞpΦ, χ, sq ‰ 0. Then the previous corollary shows that
εpχ, s, ψq is a Laurent polynomial in q ´s . Similarly, εpχ̌, 1 ´ s, ψq is a Laurent
polynomial in q ´s . The functional equation asserts that εpχ, s, ψq is an invertible
element in Crq ´s , q s s, so it is a monomial.
38
3
Weil groups
The aim of this chapter is to define the L-function and ε-factors associated to
certain types of representations of the Weil group of K.
We will start with an axiomatic description of the Weil group. The reader who
is not familiar with local class field theory can take the theorems on faith; we
will only give a brief summary. A good reference on class field theory is for
instance [13] or [1].
After that, we will move on to discuss several aspects of the representation
theory of the Weil group. We will define L-functions and ε-factors associated to
finite-dimensional semisimple representations of the Weil group.
Finally, we introduce the notion of Deligne representations. We will extend the
definitions of L-functions and ε-factors to all semisimple Deligne representations
of the Weil group.
The presentation of this chapter is guided by [3]. In some cases we will use the
slightly more general approach of [17].
3.1
Local Class Field Theory
We denote by GK the absolute Galois group of K. The maximal unramified
extension of K is denoted K nr , and its corresponding subgroup of GK is denoted
IK . It is called the inertia subgroup of GK . We can identify GalpK nr {Kq with
Galpk̄{kq.
There is an isomorphism φ : Ẑ Ñ Galpk̄{kq “ GalpK nr {Kq mapping an element
pan qnPZą0 to the automorphism of k the inverse of which is defined by
´1 ˇ
φ ppan qn q ˇFn “ px ÞÑ xan q.
q
The image of 1 under this isomorphism is called the geometric Frobenius substitution, and its inverse is called the arithmetic Frobenius substitution. The
former one is more important for our purposes, and is denoted ΦK . Any element of GK whose restriction to K nr is ΦK is called a Frobenius element (over
K). Usually, we denote such an element by ΦK as well.
By Galois theory, there is a short exact sequence
1 Ñ IK Ñ GK Ñ Ẑ Ñ 1,
where we identify Ẑ with GalpK nr {Kq as above.
Definition 3.1.1. The Weil group WK of K is the subgroup of GK that is the
inverse image of Z Ď Ẑ in the above short exact sequence. The map WK Ñ Z
is called the valuation on WK , and is denoted vK .
We also write } ¨ }K “ q ´vK p¨q , and drop the subscript from the notation when
it is clear what field we are working over.
40
Remark 3.1.2. Note that WK consists of the elements of GK which act like a
power of ΦK on K nr . There is a short exact sequence
v
K
Z Ñ 1.
1 Ñ IK Ñ WK ÝÑ
(3)
Furthermore, if K Ď L is another algebraic extension (inside K̄), then we have
a commutative diagram
1
IL
WL
1
IK
WK
vL
Z
1
Z
1,
vK
since every element of GL that acts on Lnr as a power of Frobenius also acts on
K nr Ď Lnr as a power of Frobenius.
Remark 3.1.3. Note that the Weil group of K, just like the absolute Galois
group, depends on the chosen separable closure of K. As such, it is only defined
up to an inner automorphism of GK .
Remark 3.1.4. Note also that a choice of a Frobenius element Φ P WK induces
a right splitting of the exact sequence (3), and hence a decomposition of WK as
a semi-direct product
WK “ IK ¸ Z.
As a set, we can identify WK with IK ˆ Z; this depends on our choice of Φ.
Definition 3.1.5. The topology on WK is given by the product topology of
IK ˆ Z as above, where IK has its natural profinite topology, and Z has the
discrete topology.
Proposition 3.1.6. The topology on WK does not depend on the choice of a
Frobenius element. Furthermore, WK is a topological group, and the inclusion
ιK : WK Ñ GK
is continuous and has dense image.
Proof. By the definition of product topology, using that Z has the discrete
topology, we see that the topology on WK is the coarsest topology satisfying
the following properties:
• IK is an open subgroup endowed with its natural topology;
• a set U Ď WK is open if and only if ΦU is open.
It follows that the topology is invariant under translation by elements of WK .
Hence, it is also the coarsest translation-invariant topology in which IK is open
and has its natural topology. This description is independent of Φ.
We will show continuity of multiplication. Inversion can be treated similarly,
and is left as an exercise to the reader. We consider the map
pIK ¸ Zq ˆ pIK ¸ Zq Ñ IK ¸ Z
41
that is given by multiplication. Explicitly, it is given by
ppx, Φn q, py, Φm qq ÞÑ pxΦn yΦ´n , Φn`m q.
By the universal property of the product topology, it suffices to show that the
maps
pIK ¸ Zq ˆ pIK ¸ Zq Ñ IK
px, Φn q, py, Φm q ÞÑ xΦn yΦ´n
and
pIK ¸ Zq ˆ pIK ¸ Zq Ñ Z
px, Φn q, py, Φm q ÞÑ Φn`m
are continuous. The first one is continuous since conjugation in GK is continuous. The second one is continuous since it factors through Z ˆ Z Ñ Z, which is
continuous because the topology on both sides is discrete.
Let U Ď GK be an open subgroup, and write H “ U X WK . Then H{pIK X Hq
is a subgroup of Z, say it is nZ for some n P Zě0 . Then pick h P H with
h ” n mod IK X H. Then clearly
H “ pH X IK qxhy,
and the latter is an open set of WK since it is a union of Φ-translates of open
subsets of IK . This implies that the map ιK : WK Ñ GK is continuous. It is
clear that it has dense image.
Lemma 3.1.7. Let L{K be a finite extension, such that L Ď K̄. Then
WL “ GL X WK .
The topology on WL coincides with the subspace topology in WK , and WL is
open in WK .
Proof. The inclusion WL Ď GL X WK is obvious. For the reverse inclusion
and the comparison of topologies, we will treat the cases where L{K is either
unramified or totally ramified. The general case follows since every extension
L{K admits an intermediate field M such that M {K is unramified and L{M is
totally ramified.
If L{K is unramified, then IK is contained in GL . On the quotients GL {IL – Ẑ
and GK {IK – Ẑ, the map induced by GL Ď GK is multiplication by n, where
n “ rL : Ks. The reverse inclusion follows from the fact that the inverse image
in Ẑ of Z under the multiplication by n map is Z.
If ΦK P WK is a Frobenius element over K, then ΦnK P WL is a Frobenius
element over L. Under the identification WK “ IK ¸ Z, the subgroup WL
corresponds to the subgroup IK ¸nZ. Hence, the topology on WL is the subspace
topology of WK , and WL is open in WK .
42
If L{K is totally ramified, then GL X IK “ IL . On the quotients GL {IL – Ẑ
and GK {IK – Ẑ, the map induced by GL Ď GK is the identity. Hence, the
equality WL “ GL X WK is obvious.
Any Frobenius element ΦK P WK over K is also a Frobenius element over L.
Under the identification WK “ IK ¸ Z, the subgroup WL corresponds to the
subgroup IL ¸ Z. Hence, the topology on WL is the subspace topology of WK ,
and WL is open in WK .
Lemma 3.1.8. Let L{K be a finite extension, such that L Ď K̄. Then the
canonical map WK {WL Ñ GK {GL is a bijection, and WL Ď WK is normal iff
L{K is Galois.
Proof. The canonical map WK {WL Ñ GK {GL is injective by the previous
lemma. Furthermore, the topology on GK {GL is discrete since L{K is finite.
Hence, the map above is surjective since the image is dense (Proposition 3.1.6).
If L{K is Galois, then GL Ď GK is normal, hence WL “ GL X WK is normal in
WK .
Conversely, if WL is normal in WK , let g P GK be given. Let g0 P WK be such
that g0´1 g P GL (it exists by the first part of the lemma). Write x “ g0´1 g, so
that g “ g0 x. Then
gGL g ´1 “ g0 GL g0´1 ,
and the latter is clopen since GL is. It contains WL since WL is normal in WK .
But GL is the closure of WL in GK , since GL is closed in GK and WL is dense
in GL . Hence GL Ď gGL g ´1 , so they are equal and GL is normal in GK .
We are now ready to state the results from class field theory, without proofs.
Theorem 3.1.9 (Local Class Field Theory). There exists a unique continuous
group homomorphism
ArtK : WK Ñ K ˆ
such that the following properties hold:
• The induced map Artab
K is a topological isomorphism
„
ab
ˆ
Artab
K : WK ÝÑ K ;
• The valuations on WK and K ˆ coincide;
• If L{K is a finite extension contained in K̄, then the diagram
WL
ArtL
Lˆ
NL{K
WK
ArtK
commutes;
43
Kˆ
• If L{K is a finite extension contained in K̄, then the diagram
WK
ArtK
Kˆ
verL{K
WL
ArtL
Lˆ
commutes.
3.2
Representations of the Weil group
Lemma 3.2.1. Let pρ, V q be an irreducible smooth representation of WK . Then
V is finite-dimensional.
Proof. If v P V is any nonzero element, then there is an open subgroup U Ď WK
fixing v; assume w.l.o.g. that U Ď IK . Then there exists an open U 1 Ď GK with
U “ U 1 X IK . Let L{K be the (finite) field extension corresponding to U 1 , and
assume w.l.o.g. that L{K is (finite) Galois.
Now the subgroup U Ď WK is the intersection of the normal subgroups IK and
U 1 X WK , hence it is normal (in WK ) as well. Since its conjugates xU x´1 “ U
fix the elements ρpxqv, we see that U in fixes the subspace V 1 spanned by the
elements ρpxqv. But since pρ, V q is irreducible, we have V 1 “ V , so U fixes all
of V . Hence, U Ď ker ρ.
If we choose a Frobenius element Φ P WK , then we get a decomposition
WK “ IK ¸ Z.
Hence, Φ acts on IK by conjugation, hence also on the finite group IK {U .
Therefore, some positive power Φn acts trivially on IK {U . In particular, the
conjugation action of Φn on ρpIK q, is trivial, i.e. ρpΦn q commutes with ρpIK q.
Hence, ρpΦn q actually commutes with all of ρpWK q, so by Schur’s lemma (see
[3, Lemma 2.6]), it acts on V as a scalar.
Hence, for every w P V the space spanned by tρpΦm qw : m P Zu is finitedimensional. Letting w range over a basis of the finite-dimensional vector space
spanned by tρpxqv : x P IK u, we get the result.
Definition 3.2.2. A character χ of WK (or of GK ) is called unramified if it is
trivial on IK . Otherwise, χ is called ramified.
Remark 3.2.3. Any character χ of K ˆ gives a character χ ˝ ArtK of WK .
Moreover, χ is unramified if and only if χ ˝ ArtK is unramified.
Remark 3.2.4. An intermediate field K Ď L Ď K̄ defines a rL : Ks-dimensional
representation of GK over the field K. The extension L{K is unramified if and
only if IK acts trivially on L; hence the terminology.
44
Definition 3.2.5. Let n P Zą0 . Then we write Gnss pKq for the set of isomorphism classes of n-dimensional semisimple smooth representations of WK . We
write G ss pKq for the union of the sets Gnss pKq.
ss
Definition 3.2.6. Consider the free abelian group Z‘G pKq generated by the set
G ss pKq. Let A be the subgroup generated by elements of the form rσs ´ rρs ´ rτ s
for every short exact sequence
0 Ñ ρ Ñ σ Ñ τ Ñ 0.
The quotient Z‘G
denoted RpWK q.
ss
pKq
{A is called the Grothendieck group of WK , and it is
Every semisimple smooth representation of WK is the direct sum of irreducible
subrepresentations. Therefore, the Grothendieck group of WK is isomorphic to
the free abelian group generated by isomorphism classes of irreducible smooth
representations of WK .
Lemma 3.2.7. Let A be an abelian group, and let φ : G ss pKq Ñ A be a function.
Then φ can be extended to a homomorphism φ : RpWK q Ñ A if and only if
φpσq “ φpρq ` φpτ q for every short exact sequence
0 Ñ ρ Ñ σ Ñ τ Ñ 0.
Proof. This is clear from the definition of RpWK q.
Definition 3.2.8. A function φ : G ss pKq Ñ A satisfying one of the equivalent
conditions in the lemma above is called additive.
Example 3.2.9. Let L{K be a finite extension contained in K̄. Then the maps
G ss pKq Ñ RpWL q
” ˇ ı
pρ, V q ÞÑ ρˇ
WL
and
G ss pLq Ñ RpWK q
”
ı
K
pρ, V q ÞÑ IndW
ρ
WL
are both additive, so they define homomorphisms
RpWK q Õ RpWL q.
We denote these homomorphisms by ResL{K and IndL{K respectively.
Remark 3.2.10. In the above example, one needs to check that the induction
of a semisimple representation is again semisimple. The proof is similar to the
case of finite groups, and it is carried out in Lemma 2.7 of [3].
Example 3.2.11. The map dim : G ss pKq Ñ Z is clearly additve, so it defines a
homomorphism
dim : RpWK q Ñ Z.
The kernel of this homomorphism is denoted R0 pWK q.
45
Proposition 3.2.12. The group R0 pWK q is generated by elements of the form
IndL{K prχs ´ rχ1 sq,
where L{K is a finite extension and χ, χ1 are characters of WL .
Proof. This is basically a version of Brauer’s induction theorem for WK instead
of a finite group. We refer to Proposition 2.3.1 of [17] or Lemma 30.1.1 of [3]
for a proof.
Corollary 3.2.13. The group RpWK q is generated by elements of the form
IndL{K rχs, where L{K is a finite extension and χ is a character of WL .
Proof. The element r1WK s is surely of the required form (taking L “ K). The
result follows since RpWK q is spanned by R0 pWK q and r1WK s.
Definition 3.2.14. Let A be an abelian group. A family λL : RpWL q Ñ A of
maps (where L ranges over the finite extensions of K) is called additive over K
if for each L{K the map λL is a homomorphism.
Definition 3.2.15. Let A be an abelian group. A family λL : RpWL q Ñ A of
maps is called inductive in degree 0 if it is additive over K and for each tower
K Ď L Ď M of finite extensions the diagram
R0 pWM q
λM
A
IndM {L
λL
R0 pWL q
commutes.
Remark 3.2.16. By Proposition 3.2.12, a family that is inductive in degree 0
is uniquely determined by its values on rχs P RpWL q for L{K finite and χ a
character of WL .
3.3
L-functions and epsilon factors
The definition of the L-function is not too involved. We will not prove the
existence of ε-factors (Theorem 3.3.7). The proof can be found in [3]. It requires
a global argument, which can be found in Tate’s Thesis [16].
Note that if pρ, V q is a representation of WK , then V IK is fixed by WK , since
IK is normal in WK . Furthermore, any two Frobenius elements in WK differ
by an element of IK , hence their actions on V IK are identical. That is, the
automorphism
ˇ
ρpΦqˇV IK
does not depend on the choice of Frobenius element Φ P WK .
46
Definition 3.3.1. Let pσ, V q be a finite-dimensional, semisimple smooth representation of WK . Let Φ P WK be a Frobenius element over K. Then define
ˇ
`
˘´1
Zpσ, tq “ det 1 ´ σpΦqˇV IK t
.
The L-function associated to pσ, V q is defined as
Lpσ, tq “ Zpσ, q ´s q.
Example 3.3.2. If χ is a character of WK , then we have two cases:
• If χ is unramified, then CIK “ C, so
˘´1
`
.
Lpσ, sq “ 1 ´ χpΦqq ´s
• If χ is ramified, then CIK “ 0, so
Lpσ, sq “ 1.
Example 3.3.3. If pσ, V q is an irreducible representation of WK of (finite)
dimension at least 2, then V IK is a subrepresentation. Hence, either V IK “ 0
or V IK “ V .
Suppose that V IK “ V , then IK Ď ker σ, so V is an irreducible representation
of the quotient WK {IK “ Z. But then any eigenvector of σp1q spans a onedimensional subrepresentation, which contradicts irreducibility of pσ, V q. Hence,
V IK “ 0, and
Lpσ, sq “ 1.
Lemma 3.3.4. If pσ1 , V1 q, pσ2 , V2 q are finite-dimensional, semisimple smooth
representations of WK , then
Lpσ1 ‘ σ2 , sq “ Lpσ1 , sqLpσ2 , sq.
I
Proof. Observe that pV1 ‘ V2 q K “ V1IK ‘ V2IK . Hence, the characteristic polynomial of pσ1 ‘σ2 qpΦq|pV1 ‘V2 qIK is the product of the characteristic polynomials
of σ1 pΦq|V IK and σ2 pΦq|V IK . Hence,
1
2
Zpσ1 ‘ σ2 , tq “ Zpσ1 , tqZpσ2 , tq,
and the result follows.
Remark 3.3.5. We could have defined the L-function of irreducible representations by the two examples above, extending to all semisimple representations
using Lemma 3.3.4, cf. [3]. This is also why we only define the L-function for
semisimple representations: we can reduce most statements to the irreducible
case.
Definition 3.3.6. If L{K is a finite extension within K̄, and ψ is a nontrivial
character of K, then we put ψL “ ψ ˝ TrL{K .
47
Theorem 3.3.7. Let ψ be a nontrivial character of K. Then there exists a
unique family of functions
λL : RpWL q Ñ Crq ´s , q s sˆ
that is inductive in degree 0 and satisfies
λL prχ ˝ ArtL sq “ εpχ, s, ψL q
whenever L{K is finite and χ is a character of Lˆ .
Proof. See Theorem 29.4 of [3].
Remark 3.3.8. Uniqueness is clear, since characters on WL correspond to characters on Lˆ (Theorem 3.1.9), and a family that is inductive in degree 0 is
uniquely determined by its values on characters.
Definition 3.3.9. Let ψ be a nontrivial character of K, let L{K be finite and
let pρ, V q be a semisimple smooth representation of WL . We write
εpρ, s, ψL q :“ λL prρsq,
where λL is the map from the theorem. In particular, if χ is a character of Lˆ ,
we have
εpχ ˝ ArtL , s, ψL q “ εpχ, s, ψL q.
Corollary 3.3.10. Let ψ be a nontrivial character of K, and let L{K be finite.
Then the ε-factor satisfies
εpρ1 ‘ ρ2 , s, ψL q “ εpρ1 , s, ψL qεpρ2 , s, ψL q,
whenever pρ1 , V1 q and pρ2 , V2 q are semisimple smooth representations of WL .
Proposition 3.3.11. Let pρ, V q be a semisimple smooth representation of WK .
Then the ε-factor satisfies the functional equation
εpρ, s, ψqεpρ̌, 1 ´ s, ψq “ det ρp´1q,
where the character det ρ of WK is viewed as a character of K ˆ via ArtK .
Proof (sketch). One checks that the family of functions
µL : RpWL q ÝÑ Crq ´s , q s sˆ
σ ÞÝÑ det σp´1qpεpσ̌, 1 ´ s, ψL qq´1
is also inductive in degree 0. The result then follows from the functional equation
for K ˆ (Corollary 2.4.9).
3.4
Deligne representations
We only give the definitions. For motivation, see for instance section 32 of [3].
48
Definition 3.4.1. A Deligne representation of WK is a triple pρ, V, N q where
pρ, V q is a finite-dimensional smooth representation of WK , and N P EndC pV q
is a nilpotent endomorphism satisfying
ρpgqN ρpgq´1 “ }g}N,
for all g P WK .
Definition 3.4.2. A Deligne representation pρ, V, N q is called semisimple if
pρ, V q is semisimple. We write Gn for the set of isomorphism classes of ndimensional semisimple Deligne representations of WK .
Example 3.4.3. Any smooth representation pρ, V q of WK gives rise to a Deligne
representation by setting N “ 0. Hence, we can identify Gnss pKq with a subset
of Gn pKq.
Example 3.4.4. Let V “ Cn with standard basis e1 , . . . , en . Let ρ be the
representation defined by
ρpgqei “ }g}i´1 ¨ ei .
Let N be the nilpotent element defined by
"
ei`1 i ă n,
N ei “
0
i “ n.
Then for g P WK and i P t1, . . . , n ´ 1u, it holds that
ρpgqN ρpgq´1 ei “ }g}1´i ρpgqN ei
“ }g}1´i ρpgqei`1
“ }g}ei`1 “ }g}N ei ,
so pρ, V, N q is indeed a Weil deligne representation. It is denoted Sppnq, and it
is semisimple since ρ – 1WK ‘ } ¨ } ‘ . . . ‘ } ¨ }n´1 .
Definition 3.4.5. Given a Deligne representation pρ, V, N q, define the dual
representation
pρ,­
V, N q :“ pρ̌, V̌ , ´Ň q.
Definition 3.4.6. Given Deligne representations pρi , Vi , Ni q for i P t1, 2u, define
pρ1 , V1 , N1 q ‘ pρ1 , V1 , N1 q :“ pρ1 ‘ ρ2 , V1 ‘ V2 , N1 ‘ N2 q
and
pρ1 , V1 , N1 q b pρ1 , V1 , N1 q :“ pρ1 b ρ2 , V1 b V2 , N1 b IV2 ` IV1 b N2 q.
Remark 3.4.7. Given a Deligne representation pρ, V, N q of WK , the subspace
VN :“ ker N is a WK -subspace, since for every g P WK , v P V the nilpotent
element N acts on ρpgqv as }g}´1 ρpgqN ρpgq´1 .
Definition 3.4.8. Let pρ, V, N q be a Deligne representation of WK . Then define
Lppρ, V, N q, sq :“ LpρN , sq,
49
and
εppρ, V, N q, s, ψq :“ εpρ, s, ψq
Lpρ̌, 1 ´ sq
LpρN , sq
.
Lpρ, sq Lpρ̌´Ň , 1 ´ sq
Remark 3.4.9. If pρ, V, N q comes from an ordinary smooth representation
pρ, V q, i.e. if N “ 0, then VN “ V , and
Lppρ, V, N q, sq “ Lpρ, sq;
εppρ, V, N q, s, ψq “ εpρ, s, ψq.
In that sense, the L-functions and ε-factors above are really an extension of the
previous definitions.
50
4
Representations of the General Linear Group
In this chapter, we will discuss some aspects of the representation theory of
GLn pKq, where K is a non-archimedian local field. We will write Gn for the
group GLn pKq.
The treatment of this chapter is inspired by [14], and for the proofs we refer to
[4].
4.1
Parabolic subgroups
Firstly, we will put a topology on Gn , making it a locally profinite group.
Definition 4.1.1. The topology on the matrix ring Mn pKq is the product topol2
ogy, where we identify Mn pKq with K n .
2
Clearly, the product topology makes K n a topological group. Hence, addition and additive inversion in Mn pKq are continuous. The more important fact
we need is that multiplication and multiplicative inversion are continuous on
GLn pKq. These can, however, easily be deduced from the fact that multiplication and multiplicative inversion are continuous on K.
It is also not hard to see that Gn is locally profinite. If U Ď Gn is any open
neighbourhood of the identity, then 1`pm Mn pOK q Ď U for m sufficiently large.
The set 1 ` pm Mn pOK q is an open subgroup of Gn , and it is clearly compact.
Remark 4.1.2. In general, in a topological ring R, taking inverses is not necessarily continuous. Therefore, the topology given above generally does not make
GLn pRq into a topological group.
Definition 4.1.3. A composition of the integer m is a sequence pa1 , . . . , ar q of
positive integers such that
r
ÿ
m“
ai .
i“1
Two compositions pa1 , . . . , ar q, pb1 , . . . , bs q of m are called equivalent if r “ s
and there exists a permutation σ P Sr such that
paσp1q , . . . , aσprq q “ pb1 , . . . , bs q.
An equivalence class for this equivalence relation is called a partition of m.
Some authors (e.g. [14]) use the word (ordered) partition instead of composition.
Definition 4.1.4. Let α “ pa1 , . . . , ar q be a composition of n. Then define the
following groups.
52
$¨
’
’ A1
’
&˚ 0
˚
Gα :“ ˚ .
’ ˝ ..
’
’
%
0
0
A2
..
.
...
...
..
.
0
...
$¨
A1
’
’
’
&˚ 0
˚
Pα :“ ˚ .
’
˝ ..
’
’
%
0
$¨
Ia1
’
’
’
&˚ 0
˚
Uα :“ ˚ .
’
˝ ..
’
’
%
0
˚
A2
..
.
...
...
..
.
0
...
˚
Ia2
..
.
...
...
..
.
0
...
ˇ
ˇ A1 P Ga1
ˇ
ˇ A2 P Ga2
‹
ˇ
‹
‹ P Gn ˇ
..
ˇ
‚
.
ˇ
ˇ An P G a
Ar
r
,
/
/
/
.
ˇ
ˇ A1 P Ga1
ˇ
ˇ A2 P Ga2
‹
ˇ
‹
‹ P Gn ˇ
..
ˇ
‚
.
ˇ
ˇ An P Ga
Ar
r
,
˛
˚
/
/
/
.
˚ ‹
‹
P
G
.
.. ‹
n
/
. ‚
/
/
Iar
,
/
/
/
.
0
0
..
.
˚
˚
..
.
˛
˛
/
/
/
-
/
/
/
-
The groups Pα are called the standard parabolic subgroups of Gn .
Remark 4.1.5. Clearly, Pα “ Uα Gα , and Uα is normal in Pα . Hence,
Pα {Uα – Gα ,
Pα – Uα ¸ Gα ,
as abstract groups.
Note also that the three groups defined above are closed subgroups of Gn .
Example 4.1.6. Two easy partitions of n are given by pnq and p1, . . . , 1q. If
α “ pnq, then Gα “ Pα “ Gn , and Uα “ tIn u. If α “ p1, . . . , 1q, then Gα
consists of the invertible diagonal matrices, Pα consists of the invertible upper
triangular matrices, and Uα consists of the upper triangular matrices with only
1 on the diagonal. In the latter case, Pα is the standard Borel subgroup of Gn .
Definition 4.1.7. Let pρ, V q be a smooth representation of Gn , and let α “
pa1 , . . . , ar q be a composition of n. Then define
V pUα q “ spantv ´ ρpuqv : v P V, u P Uα u.
It is clearly a Uα -subspace of V , and in fact it is Pα -stable: if u P Uα , p P Pα
and v P V , then
ρppq pv ´ ρpuqvq “ ρppqv ´ ρppup´1 qρppqv,
and pup´1 P Uα since Uα is normal in Pα .
In particular, the quotient V {V pUα q has a natural representation of Pα , which
is trivial on Uα . Therefore, it defines a representation of Gα on V {V pUα q, which
is denoted pρUα , VUα q.
Let φ : ρ Ñ σ be a Gn -homomorphism between two smooth representations
pρ, V q, pσ, W q of Gn . Then for each v P V and each u P Uα , we have
φpv ´ ρpuqvq “ φpvq ´ φpρpuqvq “ φpvq ´ σpuqφpvq,
53
so φ maps V pUα q into W pUα q. Hence, φ induces a Gα -homomorphism
φUα : ρUα Ñ σUα
on the quotient. It is clearly functorial.
Definition 4.1.8. The functor ReppGn q Ñ ReppGα q given by
pρ, V q ÞÑ pρUα , VUα q
is called the Jacquet functor.
Lemma 4.1.9. Let U Ď Uα be a compact subgroup. Let pρ, V q be a smooth
representation of Uα , and let V 1 be a subspace stable under the action of Uα .
Then
V 1 X V pU q “ V 1 pU q.
Proof. By Lemma 1.1.14, U is a profinite group, since it is compact. Now the
representation V of U is semisimple by Lemma 1.2.5. Hence, V is the direct
sum of irreducible representations. The result follows from
˜
¸
à
à
Vi pU q “
Vi pU q,
i
i
as V 1 is a direct sum over a subset of the irreducible summands of V .
Corollary 4.1.10. Let pρ, V q be a smooth representation of Uα , and let V 1 be
a subspace stable under the action of Uα . Then
V 1 X V pUα q “ V 1 pUα q.
Proof. Note that Uα is the union of the compact subgroups
,
$¨
˛ˇ
Ia1 A1,2 . . . A1,r ˇˇ
/
’
/
’
/
’˚
.
‹ˇ
&
0
I
.
.
.
A
a
2,r
2
‹ˇ
˚
mpj´iq
m
,
A
P
p
M
pa
,
a
,
O
q
Uα :“ ˚ .
ˇ
‹
.
.
i j
K
..
..
.. ‚ ˇ i,j
/
’
˝ ..
.
/
’
/
ˇ
’
%
0
0
. . . Iar ˇ
for m P Z. Hence,
ď
V pUα q “
V pUαm q,
iPZ
and the result follows from the lemma.
Proposition 4.1.11. The Jacquet functor is additive and exact.
Proof. Additivity is clear. Let pρ, V q, pσ, W q and pτ, Xq be smooth representations of Gα , and suppose we have a short exact sequence
f
g
0 ÝÑ ρ ÝÑ σ ÝÑ τ ÝÑ 0.
54
Then we get a sequence
0 ÝÑ V pUα q ÝÑ W pUα q ÝÑ XpUα q ÝÑ 0,
which we will prove to be exact.
We can interpret V as a subrepresentation of W , and then exactness at W pUα q
follows from the corollary: if w P W pUα q satisfies gpwq “ 0, then w is in W pUα q
and in ker g “ V , hence w P V pUα q.
Clearly, the restriction of f to V pUα q is still injective, so exactness at V pUα q is
obvious. If x P X and u P Uα are given, then pick w P W with gpwq “ x. Then
gpw ´ σpuqwq “ x ´ τ puqx,
so all generators of XpUα q are in the image. Hence, the sequence is exact.
The result now follows from the snake lemma, applied to the diagram
0
V pUα q
W pUα q
XpUα q
0
0
V
W
X
0,
using that the vertical maps are injective.
Lemma 4.1.12. Let pρ, V q and pσ, W q be smooth representations of Gn and Gα
respectively. There is a natural isomorphism
¯
´
n
HomGn ρ, IndG
Pα pσq – HomGα pρUα , σq .
Proof. By smooth Frobenius reciprocity, there is a natural isomorphism
¯
¯
´
´ ˇ
n
ˇ
HomGn ρ, IndG
Pα pσq – HomPα ρ Pα , σ ,
where σ is viewed as a representation of Pα by making it act trivially on Uα .
But then any homomorphism f : ρ|Pα Ñ σ factors as
f
V
W
π
V {V pUα q
where π : V Ñ V {V pUα q is the natural projection. That is, there is a natural
isomorphism
´ ˇ
¯
HomPα ρˇP , σ – HomGα pρUα , σq .
α
Definition 4.1.13. A smooth irreducible representation pρ, V q of Gn is called
cuspidal if ρUα “ 0, for all α ‰ pnq.
55
Corollary 4.1.14. A smooth irreducible representation pρ, V q of Gn is cuspidal
n
if and only if it is not a subrepresentation of IndG
Pα pσq for a smooth representation pσ, W q of Gα , for any α ‰ pnq.
Remark 4.1.15. Some authors (e.g. [4]) use the term absolutely cuspidal instead of cuspidal. Also the term supercuspidal is sometimes used.
Given (abstract) representations σi of Gai for i P t1, . . . , ru, we get a representation σ “ σ1 b . . . b σr of Gα . Since Gα is topologically isomorphic to the group
G1 ˆ . . . ˆ Gr , the product of open subgroups of the Gi is an open subgroup of
Gα . Therefore, if σ1 , . . . , σr are smooth, so is σ.
Definition 4.1.16. Let σ1 , . . . , σr be smooth representations of Ga1 , . . . , Gar
respectively. View σ “ σ1 b . . . b σr as a representation of Pα that is trivial on
Uα . Then we obtain a representation of Gn by considering the representation
n
σ1 ˆ . . . ˆ σr :“ IndG
Pα pσq.
It is smooth by definition of Ind, since σ is smooth.
Remark 4.1.17. From Lemma 4.1.12, we get a natural isomorphism
HomGn pρ, σ1 ˆ . . . ˆ σr q – HomGα pρUα , σ1 b . . . b σr q ,
whenever pρ, V q is a smooth representation of Gn .
4.2
L-functions and epsilon factors
This is only an outline, and we do not include any of the proofs. A full account
can be found in [4], sections 3-5 of chapter 1 (Local Theory).
Throughout this section, G will denote the group Gn for some n P Zą0 . We
fix a left Haar measure µˆ on G. All representations of G are assumed to be
admissible.
We will write A for the ring Mn pKq of nˆn-matrices. It is a locally profinite
group, cf. Definition 4.1.1. If ψ is a character of K, we will write ψA for the
character ψ ˝ Tr of Mn pKq.
Definition 4.2.1. Let pπ, V q be an irreducible smooth representation of G.
Then define Cpπq to be the subspace of MappG, Cq spanned by the functions
γv̌bv : G Ñ C
g ÞÑ v̌pπpgqvq.
Lemma 4.2.2. The map V̌ b V Ñ Cpπq given by v̌ b v ÞÑ γv̌bv is C-linear and
surjective.
56
Proof. This is because for each g P G the map
V̌ b V Ñ C
č b c ÞÑ v̌pπpgqvq
is linear. Surjectivity is by definition.
Definition 4.2.3. A function f : G Ñ C of the form γv̌bv is called a (matrix)
coefficient of π.
Lemma 4.2.4. Let f be a coefficient of π. Then the map fˇ given by g ÞÑ f pg ´1 q
is a coefficient of π̌.
ˇ
Proof. Assume that f “ γv̌bv . Since π is admissible, the map ev8
π : V Ñ V̌ is
an isomorphism. Then the map
γev8
:GÑC
π pvqbv̌
g ÞÑ ev8
π pvqpπ̌pgqv̌q
coincides with fˇ, since
´1
ev8
qvq,
π pvqpπ̌pgqv̌q “ pπ̌pgqv̌qpvq “ v̌pπpg
for every g P G, by the definition of π̌.
Definition 4.2.5. Let µ be a left Haar measure on G, and let ψ be a nontrivial
character of K. Then define the Fourier transform Φ̂ of any Φ P Cc8 pAq as
ż
Φ̂pξq “
ΦpxqψA pξxq dµpxq.
A
Proposition 4.2.6. Assume Φ P Cc8 pAq. Then Φ̂ P Cc8 pAq. Furthermore,
there exists a left Haar measure µA on A (depending on ψ) such that
ˆ
Φ̂pxq “ Φp´xq
for all x P A and all Φ P Cc8 pAq.
Proof. Omitted. The proof is analogous to that of Proposition 2.2.12.
Definition 4.2.7. Let pπ, V q be an irreducible smooth representation of G, let
Φ P Cc8 pAq be a smooth function and let f P Cpπq be a coefficient of π. Then
the zeta function of Φ with respect to π and f is
ż
ζpΦ, f, sq :“
Φpxqf pxq} det x}s dµˆ pxq.
G
Theorem 4.2.8. Let pπ, V q be an irreducible smooth representation of G. Then
there exists a constant s0 P R such that the integral
ż
Φpxqf pxq} det x}s dµˆ pxq
G
57
converges absolutely and uniformly on compact subsets of tRe s ą s0 u, for all
functions Φ P Cc8 pAq and all coefficients f of π. Moreover, on that half plane,
the function is a rational function in q ´s .
Proof. See Theorem 3.3(1) of [4].
Remark 4.2.9. Similarly to Remark 2.3.7, we get a C-linear map
Z : Cc8 pAq b Cpπq Ñ Cpq ´s q
Φ ÞÑ ζpΦ, f, sq.
Proposition 4.2.10. There is a unique polynomial Pπ P CrXs with Pπ p0q “ 1
such that the image of the map Z above is equal to
Pπ pq ´s q´1 Crq ´s , q s s.
Proof. See Theorem 3.3(2) of [4].
Definition 4.2.11. Let pπ, V q be an irreducible smooth representation of G.
Then define the L-function associated to π as
Lpπ, sq “ Pπ pq ´s q´1 .
Remark 4.2.12. Any two Haar measures µˆ on G differ by a scalar. Therefore,
the image of the map Z does not depend on the chosen Haar measure. Hence,
the L-series of π does not depend on any choice.
Definition 4.2.13. Define the function
ΞpΦ, f, sq “
ζpΦ, f, s ` 21 pn ´ 1qq
.
Lpπ, sq
It is a rational function in q ´s by Proposition 4.2.10.
Theorem 4.2.14. Let ψ be a nontrivial character of K, let µA be the selfdual Haar measure with respect to ψ, and let pπ, V q be an irreducible smooth
representation of G. Then there exists a unique function γpπ, s, ψq P Cpq ´s q
such that
˘
`
ζpΦ̂, fˇ, n ´ sq “ γ π, s ´ 21 pn ´ 1q, ψ ζpΦ, f, sq.
(4)
Proof. This is essentially Theorem 3.3(4) of [4].
Remark 4.2.15. The theorem in [4] actually proves the functional equation
ΞpΦ̂, fˇ, 1 ´ sq “ εpπ, s, ψq ΞpΦ, f, sq,
(5)
where εpπ, s, ψq is defined as
εpπ, s, ψq “ γpπ, s, ψq
Lpπ, sq
.
Lpπ̌, 1 ´ sq
It is easy to check that equations (4) and (5) are equivalent, analogously to
Corollary 2.4.8.
58
Analogously to Proposition 2.4.5 and Corollary 2.4.9, we deduce:
Proposition 4.2.16. The function γpπ, s, ψq satisfies the functional equation
γpπ, s, ψqγpπ̌, s, ψq “ ωπ p´1q,
where ωπ is the central character of π.
Corollary 4.2.17. The function εpπ, s, ψq satisfies the functional equation
εpπ, s, ψqεpπ̌, s, ψq “ ωπ p´1q.
Furthermore, εpπ, s, ψq is of the form aq ´ns for some a P Cˆ , n P Z.
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5
The Local Langlands Correspondences
With all the machinery in place, we can finally get to the theorem that is the
local Langlands correspondence for GLn pKq.
Write An pKq for the set of isomorphism classes of irreducible admissible smooth
representations of GLn pKq. Recall that we write Gn pKq for the set of isomorphism classes of n-dimensional semisimple Deligne representations of WK .
Theorem 5.0.1 (The Local Langlands Correspondences). There is a unique
family of bijections
π n : Gn pKq Ñ An pKq
such that π 1 is given by class field theory, and
Lpπpρq ˆ πpσq, sq “ Lpρ b σ, sq
εpπpρq ˆ πpσq, s, ψq “ εpρ b σ, s, ψq
for all ρ P Gn pKq, σ P Gm pKq.
Proof. This is hard. The first proof was given by Harris and Taylor [6] in 2001,
and independently by Henniart [7] in 2002 (building on an earlier paper by
Harris [5]).
Remark 5.0.2. A proof for n “ 2 was known earlier ([11], [12]), and there is a
textbook [3] about the case.
Remark 5.0.3. A new approach for the proof of the local Langlands correspondence was given by Scholze in [15].
Remark 5.0.4. The map π 1 is given by class field theory. Explicitly, this means
that
πpχ ˝ ArtK q “ χ
for all characters χ of K ˆ . We will at least check the following.
Lemma 5.0.5. The correspondence π 1 satisfies the conditions
Lpπ 1 pχq ˆ π 1 pχ1 q, sq “ Lpχ b χ1 , sq
εpπ 1 pχq ˆ π 1 pχ1 q, s, ψq “ εpχ b χ1 , s, ψq
for all characters χ, χ1 of WK , as promised by the theorem.
Proof. Note that for characters χ, χ1 of WK , the tensor product χ b χ1 is again
a character. Furthermore, π 1 pχq ˆ π 1 pχ1 q is none other than π 1 pχ b χ1 q. It
therefore suffices to prove
Lpχ ˝ ArtK , sq “ Lpχ, sq
εpχ ˝ ArtK , s, ψq “ εpχ, s, ψq
for all characters χ of K ˆ . The first one follows from the computations in
Proposition 2.3.13 and Example 3.3.2, and the second one is by definition.
60
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