Table Of ContentGeometrization of the local Langlands correspondence
Laurent Fargues and Peter Scholze
Abstract. Followingthe idea of[Far16], wedevelopthe foundations ofthe geometricLanglands
program on the Fargues–Fontaine curve. In particular, we define a category of (cid:96)-adic sheaves on
thestackBun ofG-bundlesontheFargues–Fontainecurve,proveageometricSatakeequivalence
G
over the Fargues–Fontaine curve, and study the stack of L-parameters. As applications, we prove
finiteness results for the cohomology of local Shimura varieties and general moduli spaces of local
shtukas, and define L-parameters associated with irreducible smooth representations of G(E), a
map from the spectral Bernstein center to the Bernstein center, and the spectral action of the
category of perfect complexes on the stack of L-parameters on the category of (cid:96)-adic sheaves on
Bun .
G
Contents
Chapter I. Introduction 7
I.1. The local Langlands correspondence 7
I.2. The big picture 8
I.3. The Fargues–Fontaine curve 18
I.4. The geometry of Bun 20
G
I.5. (cid:96)-adic sheaves on Bun 23
G
I.6. The geometric Satake equivalence 26
I.7. Cohomology of moduli spaces of shtuka 30
I.8. The stack of L-parameters 32
I.9. Construction of L-parameters 34
I.10. The spectral action 37
I.11. The origin of the ideas 39
I.12. Acknowledgments 43
I.13. Notation 44
Chapter II. The Fargues–Fontaine curve and vector bundles 45
II.1. The Fargues–Fontaine curve 47
II.2. Vector bundles on the Fargues–Fontaine curve 56
II.3. Further results on Banach–Colmez spaces 74
Chapter III. Bun 85
G
III.1. Generalities 86
III.2. The topological space |Bun | 87
G
III.3. Beauville–Laszlo uniformization 95
III.4. The semistable locus 98
III.5. Non-semistable points 101
Chapter IV. Geometry of diamonds 105
IV.1. Artin stacks 105
IV.2. Universally locally acyclic sheaves 112
IV.3. Formal smoothness 128
IV.4. A Jacobian criterion 131
IV.5. Partial compactly supported cohomology 149
IV.6. Hyperbolic localization 152
IV.7. Drinfeld’s lemma 161
Chapter V. D (Bun ) 165
´et G
3
4 CONTENTS
V.1. Classifying stacks 166
V.2. E´tale sheaves on strata 169
V.3. Local charts 171
V.4. Compact generation 175
V.5. Bernstein–Zelevinsky duality 177
V.6. Verdier duality 178
V.7. ULA sheaves 180
Chapter VI. Geometric Satake 183
VI.1. The Beilinson–Drinfeld Grassmannian 186
VI.2. Schubert varieties 192
VI.3. Semi-infinite orbits 197
VI.4. Equivariant sheaves 202
VI.5. Affine flag variety 204
VI.6. ULA sheaves 207
VI.7. Perverse Sheaves 211
VI.8. Convolution 219
VI.9. Fusion 221
VI.10. Tannakian reconstruction 226
VI.11. Identification of the dual group 230
VI.12. Cartan involution 234
Chapter VII. D(cid:4)(X) 237
VII.1. Solid sheaves 239
VII.2. Four functors 246
VII.3. Relative homology 251
VII.4. Relation to D 255
´et
VII.5. Dualizability 258
VII.6. Lisse sheaves 262
VII.7. D (Bun ) 265
lis G
Chapter VIII. L-parameter 271
VIII.1. The stack of L-parameters 273
VIII.2. The singularities of the moduli space 276
VIII.3. The coarse moduli space 280
VIII.4. Excursion operators 285
VIII.5. Modular representation theory 287
Chapter IX. The Hecke action 307
IX.1. Condensed ∞-categories 310
IX.2. Hecke operators 311
IX.3. Cohomology of local Shimura varieties 314
IX.4. L-parameter 317
IX.5. The Bernstein center 317
IX.6. Properties of the correspondence 320
IX.7. Applications to representations of G(E) 324
CONTENTS 5
Chapter X. The spectral action 331
X.1. Rational coefficients 332
X.2. Elliptic parameters 337
X.3. Integral coefficients 339
Bibliography 343
CHAPTER I
Introduction
I.1. The local Langlands correspondence
The local Langlands correspondence aims at a description of the irreducible smooth representa-
tions π of G(E), for a reductive group G over a local field E. Until further notice, we will simplify
our life by assuming that G is split; the main text treats the case of general reductive G, requiring
only minor changes.
The case where E is archimedean, i.e. E = R or E = C, is the subject of Langlands’ classical
work [Lan89]. Based on the work of Harish-Chandra, cf. e.g. [HC66], Langlands associates to
each π an L-parameter, that is a continuous homomorphism
ϕπ : WE → G(cid:98)(C)
where WE is the Weil group of E = R,C (given by WC = C× resp. a nonsplit extension 1 → WC →
WR → Gal(C/R) → 1), and G(cid:98) is the Langlands dual group. This is the split reductive group over
Z whose root datum is dual to the root datum of G. The map π (cid:55)→ ϕ has finite fibres, and a lot of
π
work has been done on making the fibres, the so-called L-packets, explicit. If G = GL , the map
n
π (cid:55)→ ϕ is essentially a bijection.
π
Throughout this paper, we assume that E is nonarchimedean, of residue characteristic p > 0,
with residue field F . Langlands has conjectured that one can still naturally associate an L-
q
parameter
ϕπ : WE → G(cid:98)(C)
to any irreducible smooth representation π of G(E). In the nonarchimedean case, W is the
E
dense subgroup of the absolute Galois group Gal(E|E), given by the preimage of Z ⊂ Gal(F |F )
q q
generated by the Frobenius x (cid:55)→ xq. This begs the question where such a parameter should come
from. In particular,
(1)How does the Weil group W relate to the representation theory of G(E)?
E
(2)How does the Langlands dual group G(cid:98) arise?
The goal of this paper is to give a natural construction of a parameter ϕ (only depending on
π
a choice of isomorphism C ∼= Q ), and in the process answer questions (1) and (2).
(cid:96)
7
8 I.INTRODUCTION
I.2. The big picture
In algebraic geometry, to any ring A corresponds a space SpecA. The starting point of our
investigations is a careful reflection on the space SpecE associated with E.1 Note that the group
G(E) is the automorphism group of the trivial G-torsor over SpecE, while the Weil group of E is
essentially the absolute Galois group of E, that is the (´etale) fundamental group of SpecE. Thus,
G(E) relates to coherent information (especially G-torsors) on SpecE, while W relates to ´etale
E
information on SpecE. Moreover, the perspective of G-torsors is a good one: Namely, for general
groups G there can be nontrivial G-torsors E on SpecE, whose automorphism groups are then
the so-called pure inner forms of Vogan [Vog93]. Vogan realized that from the perspective of the
local Langlands correspondence, and in particular the parametrization of the fibres of π (cid:55)→ ϕ , it
π
is profitable to consider all pure inner forms together; in particular, he was able to formulate a
precise form of the local Langlands conjecture (taking into account the fibres of π (cid:55)→ ϕ ) for pure
π
inner forms of (quasi)split groups. All pure inner forms together arise by looking at the groupoid
of all G-bundles on SpecE: This is given by
(cid:71)
[∗/G](SpecE) = [∗/G (E)],
α
[α]∈H1(SpecE,G)
´et
whereH1(SpecE,G)isthesetofG-torsorsonSpecE uptoisomorphism,andG thecorresponding
´et α
pure inner form of G. Also, we already note that representations of G(E) are equivalent to sheaves
on [∗/G(E)] (this is a tautology if G(E) were a discrete group; in the present context of smooth
representations, it is also true for the correct notion of “sheaf”), and hence sheaves on
(cid:71)
[∗/G](SpecE) = [∗/G (E)],
α
[α]∈H1(SpecE,G)
´et
are equivalent to tuples (π ) of representations of G (E).2
α [α] α
Looking at the´etale side of the correspondence, we observe that the local Langlands correspon-
dence makes the Weil group W of E appear, not its absolute Galois group Gal(E|E). Recall that
E
WE ⊂ Gal(E|E) is the dense subgroup given as the preimage of the inclusion Z ⊂ Gal(Fq|Fq) ∼= Z(cid:98),
where Gal(F |F ) is generated by its Frobenius morphism x (cid:55)→ xq. On the level of geometry, this
q q
change corresponds to replacing a scheme X over F with the (formal) quotient X /Frob.
q F
q
In the function field case E = F ((t)), we are thus led to replace SpecE by SpecE˘/ϕZ where
q
E˘ = F ((t)). We can actually proceed similarly in general, taking E˘ to be the completion of the
q
maximal unramified extension of E. For a natural definition of π , one then has π (Spec(E˘)/ϕZ) =
1 1
W — for example, SpecE˘ → Spec(E˘)/ϕZ is a W -torsor, where E˘ is a separable closure.
E E
Let us analyze what this replacement entails on the other side of the correspondence: Looking
at the coherent theory of SpecE˘/ϕZ, one is led to study E˘-vector spaces V equipped with ϕ-linear
automorphisms σ. This is known as the category of isocrystals Isoc . The category of isocrystals
E
is much richer than the category of E-vector spaces, which it contains fully faithfully. Namely, by
1Needless to say, the following presentation bears no relation to the historical developments of the ideas, which
as usual followed a far more circuitous route. We will discuss some of our original motivation in Section I.11 below.
2The point of replacing [∗/G(E)] by [∗/G](SpecE) was also stressed by Bernstein.
I.2. THE BIG PICTURE 9
the Dieudonn´e–Manin classification, the category Isoc is semisimple, with one simple object V
E λ
for each rational number λ ∈ Q. The endomorphism algebra of V is given by the central simple
λ
E-algebra D of Brauer invariant λ ∈ Q/Z. Thus, there is an equivalence of categories
λ
∼ (cid:77)
Isoc = Vect ⊗V .
E Dλ λ
λ∈Q
Here, if one writes λ = s with coprime integers s,r, r > 0, then V ∼= E˘r is of rank r with σ given
r λ
by the matrix
0 1 0 ...
0 1 0
... ... ...
0 0 1
πs 0 0
where π ∈ E is a uniformizer.
Above,wewereconsideringG-torsorsonSpecE,thusweshouldnowlookatG-torsorsinIsoc .
E
TheseareknownasG-isocrystalsandhavebeenextensivelystudiedbyKottwitz[Kot85],[Kot97].
Theirstudyhasoriginallybeenmotivatedbytherelationofisocrystalstop-divisiblegroupsandac-
cordingly a relation of G-isocrystals to the special fibre of Shimura varieties (parametrizing abelian
varieties with G-structure, and thus p-divisible groups with G-structure). Traditionally, the set
of G-isocrystals is denoted B(E,G), and for b ∈ B(E,G) we write E for the corresponding G-
b
isocrystal. In particular, Kottwitz has isolated the class of basic G-isocrystals; for G = GL , a
n
G-isocrystal is just a rank n isocrystal, and it is basic precisely when it has only one slope λ. There
is an injection H1(SpecE,G) (cid:44)→ B(E,G) as any G-torsor on SpecE “pulls back” to a G-torsor in
´et
Isoc ; the image lands in B(E,G) . For any b ∈ B(E,G) , the automorphism group of E is
E basic basic b
an inner form G of G; the set of such inner forms of G is known as the extended pure inner forms
b
of G. Note that for G = GL , there are no nontrivial pure inner forms of G, but all inner forms
n
of G are extended pure inner forms, precisely by the occurence of all central simple E-algebras
as H for some slope λ. More generally, if the center of G is connected, then all inner forms of
λ
G can be realized as extended pure inner forms. Kaletha, [Kal14], has extended Vogan’s results
on pure inner forms to extended pure inner forms, giving a precise form of the local Langlands
correspondence (describing the fibres of π (cid:55)→ ϕ ) for all extended pure inner forms and thereby
π
showing that G-isocrystals are profitable from a purely representation-theoretic point of view. We
will actually argue below that it is best to include G for all b ∈ B(E,G), not only the basic b; the
b
resulting automorphism groups G are then inner forms of Levi subgroups of G. Thus, we are led
b
to consider the groupoid of G-torsors in Isoc ,
E
∼ (cid:71)
G-Isoc = [∗/G (E)].
b
[b]∈B(E,G)
Sheaves on this are then tuples of representations (π ) of G (E). The local Langlands
b [b]∈B(E,G) b
conjecture, including its expected functorial behaviour with respect to passage to inner forms and
Levi subgroups, then still predicts that for any irreducible sheaf F — necessarily given by an
irreducible representation π of G (E) for some b ∈ B(E,G) — one can associate an L-parameter
b b
ϕF : WE → G(cid:98)(C).
10 I.INTRODUCTION
To go further, we need to bring geometry into the picture: Indeed, it will be via geometry that
(sheaves on the groupoid of) G-torsors on SpecE˘/ϕZ will be related to the fundamental group W
E
of SpecE˘/ϕZ. The key idea is to study a moduli stack of G-torsors on SpecE˘/ϕZ.
There are several ways to try to define such a moduli stack. The most naive may be the
following. The category Isoc is an E-linear category. We may thus, for any E-algebra A, consider
E
G-torsors in Isoc ⊗ A. The resulting moduli stack will then actually be
E E
(cid:71)
[∗/G ],
b
b∈B(E,G)
an Artin stack over E, given by a disjoint union of classifying stacks for the algebraic groups
G . This perspective is actually instrumental in defining the G as algebraic groups. However, it
b b
is not helpful for the goal of further geometrizing the situation. Namely, sheaves on [∗/G ] are
b
representations of the algebraic group G , while we are interested in representations of the locally
b
profinite group G (E).
b
A better perspective is to treat the choice of F as auxiliary, and replace it by a general F -
q q
algebra R. In the equal characteristic case, we can then replace E˘ = F ((t)) with R((t)). This carries
q
Z
a Frobenius ϕ = ϕ acting on R. To pass to the quotient SpecR((t))/ϕ , we need to assume that
R
the Frobenius of R is an automorphism, i.e. that R is perfect. (The restriction to perfect R will
become even more critical in the mixed characteristic case. For the purpose of considering (cid:96)-adic
sheaves, the passage to perfect rings is inconsequential, as ´etale sheaves on a scheme X and on its
perfection are naturally equivalent.) We are thus led to the moduli stack on perfect F -algebras
q
G-Isoc : {perfect F -algebras} → {groupoids} : R (cid:55)→ {G-torsors on SpecR((t))/ϕZ}.
q
These are also known as families of G-isocrystals over the perfect scheme SpecR. (Note the curly
I in G-Isoc, to distinguish it from the groupoid G-Isoc.)
This definition can be extended to the case of mixed characteristic. Indeed, if R is a perfect
Fq-algebra,theanalogueofR[[t]]istheuniqueπ-adicallycompleteflatOE-algebraR(cid:101) withR(cid:101)/π = R;
explicitly, R(cid:101) = WOE(R) = W(R)⊗W(Fq) OE in terms of the p-typical Witt vectors W(R) or the
ramified Witt vectors W (R). Thus, if E is of mixed characteristic, we define
OE
G-Isoc : {perfect F -algebras} → {groupoids} : R (cid:55)→ {G-torsors on Spec(W (R)[1])/ϕZ}.
q OE π
WewillnotusethestackG-Isocinthispaper. However,ithasbeenhighlightedrecentlyamong
others implicitly by Genestier–V. Lafforgue, [GL17], and explicitly by Gaitsgory, [Gai16, Section
4.2], andZhu, [Zhu20], andonecanhopethattheresultsofthispaperhaveaparallelexpressionin
terms of G-Isoc, so let us analyze it further in this introduction. It is often defined in the following
slightly different form. Namely, v-locally on R, any G-torsor over R((t)) resp. W (R)[1] is trivial
OE π
by a recent result of Anschu¨tz [Ans18]. Choosing such a trivialization, a family of G-isocrystals is
given by some element of LG(R), where we define the loop group
LG(R) = G(R((t))) (resp. LG(R) = G(W (R)[1])).
OE π
Changing the trivialization of the G-torsor amounts to σ-conjugation on LG, so as v-stacks
G-Isoc = LG/ LG
Ad,σ
is the quotient of LG under σ-conjugation by LG.
