ON THE EXISTENCE OF ADMISSIBLE SUPERSINGULAR
REPRESENTATIONS OF p-ADIC REDUCTIVE GROUPSFLORIAN HERZIG, KAROL KOZIOL, AND MARIE-FRANCE VIGNÉRAS
Abstract. Suppose that G is a connected reductive group over a finite extension
F/Qp , and that C is a field of characteristic p. We prove that the group G(F ) admits
an irreducible admissible supercuspidal, or equivalently supersingular, representation
over C.

Contents
1. Introduction
1.1. Acknowledgements
1.2. Notation
2. Iwahori–Hecke algebras
2.1. Definitions
2.2. Dominant monoids
2.3. Bernstein elements
2.4. Supersingular modules
3. On supercuspidal representations
3.1. Supercuspidality criterion
3.2. Change of coefficient field
3.3. Reduction to an absolutely simple adjoint group
4. Proof of the main theorem for most simple groups
4.1. Discrete Iwahori–Hecke modules
4.2. Characters
4.3. Discrete simple modules with supersingular reduction
4.4. Admissible integral structure via discrete cocompact subgroups
4.5. Reduction to rank 1 and PGLn (D)5. Supersingular representations of rank 1 groups
5.1. Preliminaries
5.2. Parahoric subgroups
5.3. Pro-p Iwahori–Hecke algebras
5.4. Diagrams
5.5. Supersingular representations via homology
6. Supersingular representations of PGLn (D)6.1. Notation and conventions
6.2. On the Jacquet–Langlands correspondence

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The first-named author was partially supported by a Simons Fellowship and an NSERC grant.
The second-named author was supported by the National Science Foundation under Award No. DMS1400779, along with Manish Patnaik’s Subbarao Professorship and NSERC grant RGPIN 4622.
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FLORIAN HERZIG, KAROL KOZIOL, AND MARIE-FRANCE VIGNÉRAS

6.3. On lifting non-supersingular Hecke modules
6.4. A reducibility lemma
6.5. On base change and descent for compact unitary groups
6.6. Supersingular representations of GLn (D)Appendix A. Base change (by Sug Woo Shin)
References

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1. Introduction
Suppose that F is a non-archimedean field of residue characteristic p and that G
is a connected reductive algebraic group over F . There has been a growing interestin understanding the smooth representation theory of the p-adic group G := G(F )over a field C of characteristic p, going back to the work of Barthel–Livné [BL94] andBreuil [Bre03] in the case of G = GL2 . The latter work in particular demonstrated therelevance of the mod p representation theory of p-adic reductive groups to the p-adic
Langlands program.
The results of [AHHV17] (when C is algebraically closed) and [HV19] (for a general C of characteristic p) give a classification of irreducible admissible representationsin terms of supercuspidal C-representations of Levi subgroups of G. Here, an irreducible admissible smooth representation π is said to be supercuspidal if it does notoccur as subquotient of any parabolic induction IndG
P σ, where P is a proper parabolic
subgroup of G and σ an irreducible admissible representation of the Levi quotient of
P . Unfortunately, so far, the supercuspidal representations themselves remain mostly
mysterious, outside anisotropic groups, GL2 (Qp ) ([BL94], [Bre03]), and some relatedcases ([Abd14], [Che13], [Koz16], [KX15]). If F/Qp is a non-trivial unramified extension, then irreducible supercuspidal representations of GL2 (F ) were first constructedby Paškūnas [Paš04]; however, it seems hopelessly complicated to classify them [BP12],[Hu10]. One additional challenge in constructing supercuspidal representations is that
irreducible smooth representations need not be admissible in general (unlike what happens over C), as was shown recently by Daniel Le [Le].
There are two ways to characterize supercuspidality in terms of Hecke actions. The
first description assumes C is algebraically closed and uses weights and Hecke eigenvalues for any fixed choice K of special parahoric subgroup (a weight is then an irreducible
representation of K). It was shown to be equivalent to supercuspidality in [AHHV17].
The second description uses the center of the pro-p Iwahori–Hecke algebra. The equivalence between the second Hecke description and supercuspidality was shown in [OV18](when C is algebraically closed) and [HV19] (for a general C of characteristic p). In
either description, supercuspidality is characterized by the vanishing of certain Hecke
operators, which is why supercuspidal representations are also called supersingular.
Our main theorem is the following:
Theorem A. Suppose F is of characteristic 0, G is any connected reductive algebraic
group over F , and C any field of characteristic p. Then G admits an irreducible
admissible supersingular, or equivalently supercuspidal, representation over C.
This theorem is new outside the low rank cases mentioned above. It provides anaffirmative answer to Question 3 of [AHHV] when char F = 0, and carries out the

EXISTENCE OF SUPERSINGULAR REPRESENTATIONS

3

announcement contained in [AHHV17, § III.26]. Note also that the analogous theorem
for supercuspidal representations over C was proved by Beuzart-Plessis [BP16].
We now briefly explain our argument, which uses several completely different ideas.
First, in Section 3 we reduce to the cases where C is finite and G is absolutely simple
adjoint. If G is moreover anisotropic, then G is compact and any irreducible smooth
representation of G is finite-dimensional (hence admissible) and supercuspidal. If G isisotropic, we distinguish three cases.
For most groups G we show in Section 4 that there exists a discrete series representation π of G over C that admits invariants under an Iwahori subgroup B, and that has
moreover the following property: the module π B of the Iwahori–Hecke algebra H(G, B)
admits a Z[q 1/2 ]-integral structure whose reduction modulo the maximal ideal of Z[q 1/2 ]with residue field Fp is supersingular. The Hecke modules π B are constructed either
from characters (using [Bor76]) or reflection modules (using [Lus83] and [GS05]; thelatter is needed to handle unramified non-split forms of PSO8 ).
Suppose from now on that F is of characteristic zero, i.e. that F/Qp is a finite
extension. The p-adic version of the de George–Wallach limit multiplicity formula([DKV84, App. 3] plus [Kaz86, Thm. K]) implies that the representation π above
embeds in C ∞ (Γ\G, C) for some discrete cocompact subgroup Γ of G (as char F = 0).
By construction we deduce that the Hecke module C ∞ (Γ\G/B, Fp ) = C ∞ (Γ\G, Fp )Bof B-invariants admits a supersingular submodule. Crucially, by cocompactness of Γ
we know that C ∞ (Γ\G, Fp ) is an admissible representation of G. Picking any non-zerosupersingular vector v ∈ C ∞ (Γ\G/B, Fp ), the G-subrepresentation of C ∞ (Γ\G, Fp )generated by v admits an irreducible quotient, which is admissible (as char F = 0) andsupersingular.
Unfortunately, this argument does not work for all groups G. We have the following
exceptional cases:(i) PGLn (D), where n ≥ 2 and D a central division algebra over F ;(ii) PU(h), where h is a split hermitian form in 3 variables over a ramified quadratic
extension of F or a non-split hermitian form in 4 variables over the unramified
quadratic extension of F .
Note that for the group PGLn (D) with n ≥ 2 the only discrete series representations
π having B-invariant vectors are the unramified twists of the Steinberg representation(by Proposition 4.1.5(i) and the classification of Bernstein–Zelevinsky and Tadić), butthen π B is one-dimensional with non-supersingular reduction.
In the second exceptional case, where G ∼= PU(h) for certain hermitian forms h,
we use the theory of coefficient systems and diagrams, building on ideas of Paškūnas[Paš04]. See Section 5. Note that G is of relative rank 1, so the adjoint Bruhat–Titsbuilding of G is a tree, and our method works for all such groups. In order to carry itout, we may apply the reductions in Section 3 and assume that G is absolutely simple
and simply connected. Given a supersingular module Ξ for the pro-p Iwahori–Hecke
algebra of G, we naturally construct a G-equivariant coefficient system (or cosheaf)
DΞ on the Bruhat–Tits tree of G. The homology of DΞ admits a smooth G-action,
and any irreducible admissible quotient will be supersingular (by Proposition 3.1.3).
To construct such a quotient, we define an auxiliary coefficient system D0 , which isbuilt out of injective envelopes of representations of certain parahoric subgroups, alongwith a morphism DΞ → D0 . The image of the induced map on homology is admissible,

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FLORIAN HERZIG, KAROL KOZIOL, AND MARIE-FRANCE VIGNÉRAS

and admits an irreducible quotient π 0 which is itself admissible (as char F = 0) andsupersingular.
In the first exceptional case, where G ∼= PGLn (D), we use a global method (seeSection 6). We find a totally real number field Fe+ and a compact unitary group G
over Fe+ such that G(Fev+ ) is isomorphic to GLn (D) for a suitable place v|p of Fe+ .
Then, fixing a level away from v and taking the limit over all levels at v, the space S
of algebraic automorphic forms of G(A∞) over Fp affords an admissible smooth actionFe++eof G(Fv ). Using automorphic induction and descent we construct an automorphic
representation π of G(AFe+ ) whose associated Galois representation rπ has the property
that its reduction modulo p is irreducible locally at v. From π we get a maximal ideal
m in the Hecke algebra (at good places outside p), and we claim that any irreducible
subrepresentation of the localization Sm is supercuspidal.
To prove the claim, we use the pro-p Iwahori–Hecke criterion for supercuspidality
and argue by contradiction. If one of the relevant Hecke operators has a non-zero
eigenvalue, we lift to characteristic zero by a Deligne–Serre argument and construct anautomorphic representation π 0 with Galois representation rπ0 having the same reduction
as rπ modulo p. Using local-global compatibility at p for rπ0 and some basic p-adic
Hodge theory we show that the non-zero Hecke eigenvalue in characteristic p implies
that rπ0 is reducible locally at v, obtaining the desired contradiction.
For our automorphic base change and descent argument we require results goingslightly beyond [Lab11], since our group G is typically not quasi-split at all finite
places. In the appendix, Sug Woo Shin explains the necessary modifications.
Finally, we remark that we would expect Theorem A to be true even when char F = p.
So far this only seems to be known for the groups GL2 (F ) [Paš04], outside trivial cases.
We crucially use that char F = 0 in (at least) the following ways:(i) the existence of discrete cocompact subgroups, which fails for most groups if
char F = p [BH78, §3.4], [Mar91, Cor. IX.4.8(iv)],(ii) admissibility is preserved under passing to a quotient representation,(iii) the automorphic method in case of the group PGLn (D).
1.1. Acknowledgements. The third-named author thanks Boris Pioline, Gordan
Savin and the organizers of the conference on Automorphic Forms, Mock Modular
Forms and String Theory in Banff (10/2017) for emails, discussions and their invitation,
which led to closer examination of unramified minimal representations corresponding to
the reflection modules of the affine Hecke algebras. She also thanks Volker Heiermann
for a discussion on discrete modules, Günter Harder for emails on discrete cocompact
subgroups, Jean-Loup Waldspurger for recollections on the antipode, and Guy Henniart
for providing a proof below.
The first-named author thanks the University of Paris-Sud and the Mathematics
Institute of Jussieu–Paris Rive Gauche, where some of this work was carried out.
We thank Noriyuki Abe for some helpful comments.
1.2. Notation. Fix a prime number p, and let F be a non-archimedean local field ofresidue characteristic p (we will later assume that char F = 0, i.e., that F is a finite
extension of Qp ). The field F comes equipped with ring of integers OF and residue
field kF of cardinality q, a power of p. We fix a uniformizer $, and let valF and | · |Fdenote the normalized valuation and normalized absolute value of F , respectively.

EXISTENCE OF SUPERSINGULAR REPRESENTATIONS

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If H is an algebraic F -group, we denote by H its group of F -points H(F ).
Let G be a connected reductive F -group, T a maximal F -split subtorus of G, B
a minimal F -parabolic subgroup of G containing T, and x0 a special point of theapartment of the adjoint Bruhat–Tits building defined by T. We associate to x0 andthe triple (G, T, B) the following data:◦ the center Z(G) of G,◦ the root system Φ ⊂ X ∗ (T ),◦ the set of simple roots ∆ ⊂ Φ,
◦ the centralizer Z of T,◦ the normalizer N of T,◦ the unipotent radical U of B (hence B = ZU), and the opposite unipotent
radical Uop ,◦ the triples (Gsc , Tsc , Bsc ) and (Gad , Tad , Bad ), corresponding to the simplyconnected covering of the derived subgroup and the adjoint group of G,◦ the apartment A := X∗ (T )/X∗ (Z(G)◦ ) ⊗Z R associated to T in the adjoint
Bruhat–Tits building,◦ the alcove C of A with vertex x0 lying in the dominant Weyl chamber withvertex x0 ,◦ the Iwahori subgroups B and Bsc of G and Gsc , respectively, fixing C pointwise,◦ the pro-p-Sylow subgroup U of B.
Given a field L, we denote by L a fixed choice of algebraic closure. We fix a field
C of characteristic c ∈ {0, 2, 3, 5, 7, . . .}, which will serve as the field of coefficients forthe modules and representations appearing below. In our main result we will assume
c = p.
Suppose K is a compact open subgroup of G, and R is a commutative ring. We
define the Hecke algebra associated to this data to be the R-algebra
HR (G, K) := EndG R[K\G].
If R = Z, we simply write H(G, K). In our applications below, we will often assume
that K = B or K = U.
Given a module or algebra X over some ring R and a ring map R → R0 , we let
XR0 := X ⊗R R0 denote the base change.
Other notation will be introduced as necessary in subsequent sections.
2. Iwahori–Hecke algebras
In this section we review some basic facts concerning Iwahori–Hecke algebras andtheir (supersingular) modules. We will use these algebras extensively in our construction of supercuspidal G-representations. See [Vig16], [Vig14], and [Vig17] for references.
2.1. Definitions.

Recall that we have defined the Iwahori–Hecke ring as
H(G, B) = EndG Z[B\G].

We have an analogous ring H(Gsc , Bsc ) for the simply-connected group. The naturalring homomorphism H(Gsc , Bsc ) → H(G, B) (induced by the covering Gsc → G of thederived subgroup) is injective, so we identify H(Gsc , Bsc ) with a subring of H(G, B).
We first discuss presentations for these rings.

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FLORIAN HERZIG, KAROL KOZIOL, AND MARIE-FRANCE VIGNÉRAS

There is a canonical isomorphism∼

j sc : H(Gsc , Bsc ) −→ H(W, S, qs ),where H := H(W, S, qs ) is the Hecke ring of an affine Coxeter system (W, S) withparameters {qs := q ds }s∈S . The ds are positive integers, which we will abusively also
refer to as the parameters of G. Thus, H(W, S, qs ) is a free Z-module with basis{Tw }w∈W , satisfying the braid and quadratic relations:
Tw Tw0 = Tww0
(Ts − qs )(Ts + 1) = 0

for w, w0 ∈ W, `(w) + `(w0 ) = `(ww0 ),for s ∈ S.

Here ` : W → Z≥0 denotes the length function with respect to S. We identify
H(Gsc , Bsc ) with H via j sc .
In order to describe H(G, B), we require a larger affine Weyl group. We define theextended affine Weyl group to be
f := N /Z0 ,W
f acts on the apartment
where Z0 is the unique parahoric subgroup of Z. The group Wf
A , and permutes the alcoves of A transitively. We let Ω denote the subgroup of Wfstabilizing C. The affine Weyl group W is isomorphic to a normal subgroup of W ,and permutes the alcoves simply transitively. We therefore have a semidirect productdecomposition
f = W o Ω.
W
f by setting `(uw) = `(wu) = `(w) for u ∈ Ω, w ∈ W . In
The function ` extends to Wf.
particular, we see that Ω is the group of length-zero elements of W
Let Σ denote the reduced root system whose extended Dynkin diagram Dyn is equal
to the Dynkin diagram of (W, S), and let Dyn0 denote the Dynkin diagram Dyn decorated with the parameters {ds }s∈S . The quotient of Ω by the pointwise stabilizer of C
in Ω is isomorphic to a subgroup Ψ of Aut(W, S, ds ), the group of automorphisms of
Dyn0 . Thus, Ω acts on Dyn0 and consequently on H(W, S, qs ), and the isomorphism j sc
extends to an isomorphism(2.1.1)∼e H(W, S, qs ),j : H(G, B) −→ Z[Ω] ⊗e :=e denotes the twisted tensor product. The generalized affine Hecke ring H
where ⊗eZ[Ω] ⊗ H(W, S, qs ) as above is the free Z-module with basis {Tw }w∈W
f , satisfying the
braid and quadratic relations:(2.1.2)(2.1.3)

Tw Tw0 = Tww0
(Ts − qs )(Ts + 1) = 0

f , `(w) + `(w0 ) = `(ww0 ),for w, w0 ∈ W
for s ∈ S.

Thus, we see that the Iwahori–Hecke ring H(G, B) is determined by the type of Σ, theparameters {ds }s∈S , and the action of Ω on Dyn0 .
f forms a system of representatives for the space of double cosets
The group W
e for w ∈ W
f corresponds
B\G/B. Under the isomorphism j, the element Tw ∈ H
to the endomorphism sending the characteristic function of B to the characteristic
function of BnB, where n ∈ N lifts w.

EXISTENCE OF SUPERSINGULAR REPRESENTATIONS

7

Finally, let P = MN denote a standard parabolic F -subgroup of G (meaning B ⊂P), and suppose that M contains T. Then the group M ∩ B is an Iwahori subgroupof M , and we may form the algebraH(M, M ∩ B) = EndM Z[(M ∩ B)\M ].
It is not a subalgebra of H(G, B) in general. The basis of H(M, M ∩B) will be denoted
TwM , where w is an element of the extended affine Weyl group associated to M .
2.2. Dominant monoids.

The subgroupΛ := Z/Z0

f = N /Z0 is commutative and finitely generated, and its torsion subgroup is equalof W
e0 denotes the maximal compact subgroup of Z. (When the groupe0 /Z0 , where Z
to Z
G is F -split or semisimple and simply connected, we have Z0 = Ze0 .) The short exact
sequence
1 → Λ → N /Z0 → N /Z → 1
splits, identifying the (finite) Weyl group W0 := N /Z of Σ with StabW (x0 ). We
therefore obtain semidirect product decompositionsfΛ o W0 = W
andΛsc o W0 = W,
where Λsc := Λ ∩ W .
Given a subgroup J of Z, we defineΛJ := JZ0 /Z0 ⊂ Λ.
We analyze ΛJ for various groups J presently.
Let T0 denote the maximal compact subgroup of T , and note that T0 = Z0 ∩ T . This∼implies that the inclusion T ,→ T Z0 induces an isomorphism T /T0 −→ T Z0 /Z0 = ΛT ,and therefore the map(2.2.1)∼X∗ (T ) −→ ΛTµ 7−→ λµ := µ($)Z0 /Z0

is a W0 -equivariant isomorphism.
Recall that we have a unique homomorphismν :Z →A,
determined by the condition
hα, ν(t)i = −valF (α(t))
for t ∈ T and α ∈ Φ. We claim that the kernel of ν is the saturation of Z(G)Ze0 in
Z, i.e., the set of all elements z ∈ Z such that z n ∈ Z(G)Ze0 for some n ≥ 1. Indeed,e0 , and the group Z/Z(G)Z
e0 is commutative andthe kernel of ν contains Z(G) and Z
finitely generated. This gives an induced map(2.2.2)
ν : Z/Z(G)Ze0 → A .
We note the following three facts: (1) the image of T ad in Z/Z(G)Ze0 is of finite index(cf. comments following (16) in [Vig16]); (2) the Z-span of the coroots Φ∨ is of finite

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FLORIAN HERZIG, KAROL KOZIOL, AND MARIE-FRANCE VIGNÉRAS

index in X∗ (T ad ); (3) ν(α∨ ($−1 )) = α∨ for any coroot α∨ ∈ X∗ (T ). Combining these,
we see that the image of (2.2.2) has the same rank as Z/Z(G)Ze0 , which is equal to
the rank of X∗ (T ad ). Therefore, the kernel of (2.2.2) is exactly the torsion subgroup of
e0 . This gives the claim.
Z/Z(G)Z
Since Z0 is contained in the kernel of ν, the group Λ acts by translation on A via ν.
Therefore, Λker ν is the pointwise stabilizer of C in Λ. Similarly, one easily checks that
Λker ν is the pointwise stabilizer of C in Ω. (In fact, we have Λ ∩ Ω = Λker ν , cf. [Vig16,
Cor. 5.11].) Hence, we obtain(2.2.3)∼Ω/Λker ν −→ Ψ,f induceand the embeddings of Λ and Ω into W(2.2.4)∼ f∼
Λ/(Λker ν × Λsc ) −→ W/(Λker ν × W ) ←− Ω/Λker ν .

An element λ ∈ Λ is called dominant (and λ−1 is called anti-dominant), ifz(U ∩ B)z −1 ⊂ U ∩ B
for any z ∈ Z which lifts λ. We let Λ+ denote the monoid consisting of dominantelements of Λ, and similarly for any subgroup Λ0 ≤ Λ we define Λ0+ := Λ0 ∩ Λ+ . Using
the isomorphism (2.2.1), we say a cocharacter µ ∈ X∗ (T ) is dominant if λµ is, and letX∗ (T )+ denote the monoid consisting of dominant elements of X∗ (T ). The group of
invertible elements in the dominant monoid Λ+ is exactly the subgroup Λker ν , and theinvertible elements of Λsc,+ are trivial.
Lemma 2.2.5. The subgroup ΛZ(G) × Λsc (resp. ΛT ) of Λ is finitely generated of finite+
index. The submonoid ΛZ(G) × Λsc,+ (resp. Λ+T ) of the dominant monoid Λ is finitely
generated of finite index.
Here,
S we say that a submonoid N of a commutative monoid M has finite index if
M = ni=1 (N + xi ) for some xi ∈ M . If M is finitely generated, then dM is of finite
index in M for all d ≥ 1.
e0 and Z
e0 /Z0 are finite, and equations (2.2.3) and (2.2.4)
Proof. The groups ker ν/Z(G)Zsc
imply that Λ/(Λker ν × Λ ) is isomorphic to the finite group Ψ. Thus, we see that thecommutative group ΛZ(G) × Λsc is a finitely generated, finite index subgroup of Λ.
Similarly, ΛT is finite free and it is well known that it is of finite index in Λ. Gordan’slemma implies the second assertion (as in the proof of [HV15, 7.2 Lem.]).

f , and let w = us1 · · · sn be a reduced expres2.3. Bernstein elements. Let w ∈ W
sion, with u ∈ Ω, si ∈ S. We set qw := qs1 · · · qsn , and defineTs∗ := Ts − qs + 1

andTw∗ := Tu Ts∗1 · · · Ts∗n .

Then Tw Tw∗ −1 = qw , and the linear map defined by Tw 7→ (−1)`(w) Tw∗ is an automore
phism of H.
For µ ∈ X∗ (T ), we let Oµ ⊂ Λ denote the W0 -orbit of λµ . We then define
X
Eλ ,
zµ :=
λ∈Oµ

EXISTENCE OF SUPERSINGULAR REPRESENTATIONS

9e corresponding to the spherical oriwhere Eλ are the integral Bernstein elements of H
entation induced by ∆ ([Vig16, Cor. 5.28, Ex. 5.30]). Precisely, they are characterized
by the relations(
Tλ if λ is anti-dominant,(2.3.1)Eλ =Tλ∗ if λ is dominant,(2.3.2)Eλ1 Eλ2 = (qλ1 qλ2 qλ−1)1/2 Eλ1 λ2
1 λ2

if λ1 , λ2 ∈ Λ,

where we take the positive square root. (If λ1 , λ2 are both dominant (or anti-dominant),e and when µ ∈ X∗ (T sc ), zµthen Eλ1 Eλ2 = Eλ1 λ2 .) The elements zµ are central in H,
lies in H.
e
We let A denote the commutative subring of the generalized affine Hecke ring H∼ewith Z-basis {Eλ }λ∈Λ . When G = Z, we have H = A = Z[Λ], but A is not isomorphic
e and the center of H
e are finitely generated modules
to Z[Λ] in general. The rings A, H,
over the central subring with basis {zµ }µ∈X∗ (T ) , which is itself a finitely-generated ring.
2.4. Supersingular modules. We now discuss supersingular Hecke modules.
Recall that C is our coefficient field of characteristic c. We define HC := H ⊗ C andee ⊗ C, which are isomorphic to the Iwahori–Hecke algebras HC (Gsc , Bsc ) andHC := H
HC (G, B), respectively.
e C -module. A nonDefinition 2.4.1 (cf. [OV18, §5.1(3)]). Let M be a non-zero right Hnzero element v ∈ M is called supersingular if v · zµ = 0 for all µ ∈ X∗ (T )+ such thate C -module M is called supersingular−µ 6∈ X∗ (T )+ , and all sufficiently large n. The H
if all its non-zero elements are supersingular. We make a similar definition for modules
over HC , using the monoid X∗ (T sc )+ .
We remark that the definition of a supersingular module differs slightly from thatof [Vig17, Def. 6.10]. There it was required that c = p, and that M · zµn = 0 for allµ ∈ X∗ (T )+ such that −µ 6∈ X∗ (T )+ and n sufficiently large.
Lemma 2.4.2.
e C -module is finite dimensional, and is semisimple as an HC (i) Any simple H
module.
e C does not admit any simple supersingular modules.
(ii) If c - p|W0 |, then H
e C -module is supersingular if and only if its restriction to(iii) If c = p, a simple H
HC is supersingular.
Proof. (i) The first statement follows from [Vig07, §5.3]. For the second part, note that
there exists a finite index subgroup Ω0 of Ω which acts trivially on H (for example, we
may take Ω0 = Λker ν ). Set HC0 := C[Ω0 ] ⊗C HC . Any simple HC -module N extends
e C to HC is a finite direct
trivially to an HC0 -module N 0 , and the restriction of N 0 ⊗HC0 H
Luu
sum u∈Ω/Ω0 N of (simple) conjugates N of N by elements u ∈ Ω. If M is a simple
e C -module and N is contained in M |H , then M is a quotient of N 0 ⊗H 0 H
e C (andH
C

C

thus the restriction of M is semisimple).
(ii) It suffices to assume C is algebraically closed. Let M denote a simple supere is commutative and M is finite dimensional,
singular module. Since the center of H

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FLORIAN HERZIG, KAROL KOZIOL, AND MARIE-FRANCE VIGNÉRAS

there exists an eigenvector v ∈ M with eigenvalues χ for the action of the center.
Supersingularity then implies(2.4.3)0 = v · zµ0 = χ(zµ0 )v

for any µ0 ∈ X∗ (T )+ such that −µ0 6∈ X∗ (T )+ .
Choose µ ∈ X∗ (T )+ lying in the interior of the dominant Weyl chamber, so inparticular −µ 6∈ X∗ (T )+ , and let w◦ ∈ W0 denote the longest element. We claim thatX
(2.4.4)zµ z−w◦ (µ) = qλµ |W0 |z0 +aµ0 zµ0µ0 ∈X∗ (T )+`(λµ0 )>0

for some aµ0 ∈ Z. To see this, note that the product of the orbits Oµ ·O−w◦ (µ) consists ofelements of the form λw(µ) λ−w0 w◦ (µ) , where w, w0 ∈ W0 . If the length of λw(µ) λ−w0 w◦ (µ)
is 0, then [Vig16, Cor. 5.11] implies w(µ) − w0 w◦ (µ) is orthogonal to every simple root.
Since this element is also a sum of coroots, we conclude that w(µ)−w0 w◦ (µ) = 0, which
implies w = w0 w◦ , as the W0 -stabilizer of µ is trivial. The product formula (2.3.2) thengives equation (2.4.4).
Now, for µ0 ∈ X∗ (T )+ , the condition −µ0 6∈ X∗ (T )+ is equivalent to `(λµ0 ) > 0.
Applying χ to both sides of (2.4.4) and using (2.4.3) (for varying µ0 ) gives qλµ |W0 | = 0,
a contradiction.
(iii) This follows from [Vig17, Cor. 6.13] and part (i).
Remark 2.4.5. The conditions in part (ii) of the above lemma are necessary: when c 6= p
divides |W0 |, there exist non-zero supersingular modules. For an example, supposee C admits a unique character χ, which
G = SL2 , q is odd, and c = 2. Then HC = H
sends Ts to 1 for each s ∈ S. If we let µ := (1, −1) ∈ X∗ (T )+ , thenzµ = Ts1 Ts2 + Ts2 Ts1 ,where S = {s1 , s2 }. Thus, we have χ(zµ ) = 0. By induction, and using the assumptionc = 2, we see that the element zkµ lies in the ideal of the center generated by zµ , forevery k ≥ 1. From this, we conclude that χ is supersingular.
3. On supercuspidal representations
The aim of this section is to collect various results concerning supercuspidal representations. We first state Proposition 3.1.3, which gives a convenient criterion for checking that a given irreducible admissible representation is supercuspidal when char C = p.
Propositions 3.2.1 and 3.3.2 allow us to make further reductions: in order to prove that
G(F ) admits an irreducible admissible supercuspidal C-representation when char F = 0
and char C = p, it suffices to assume that C is finite and G is absolutely simple, adjoint,
and isotropic.
3.1. Supercuspidality criterion.

We begin with a definition.

Definition 3.1.1. Let R be a subfield of C. We say that a C-representation π of G
descends to R if there exists an R-representation τ of G and a G-equivariant C-linear
isomorphism∼
ϕ : C ⊗R τ −→ π.
We call ϕ (and more often τ ) an R-structure of π, or a descent of π to R.

EXISTENCE OF SUPERSINGULAR REPRESENTATIONS

11

We now describe the scalar extension of an irreducible admissible C-representation
π of G [HV19]. Given such a π, the commutant D := EndC (π) is a division algebra offinite dimension over C. Let E denote the center of D, Es /C the maximal separable
extension contained in E/C and δ the reduced degree of D/E. Let L be an algebraicallyclosed field containing E and πL the scalar extension of π from C to L.
Proposition 3.1.2 ([HV19, Thms. I.1, III.4]). The length of πL is δ[E : C] and
M
πi⊕δ
πL ∼
=
i∈HomC (Es ,L)where each πi is indecomposable with commutant L ⊗i,Es E, descends to a finite extension of C, has length [E : Es ], and its irreducible subquotients are pairwise isomorphic,
say to ρi . The ρi are admissible, with commutant L, AutC (L)-conjugate, pairwise
non-isomorphic, and descend to a finite extension of C. Any descent of ρi to a finite
extension C 0 /C, viewed as C-representation of G, is π-isotypic of finite length.
Proof. By [HV19, Thms. I.1, III.4], it suffices to prove that if ρi descends to a C 0 representation ρ0i with C 0 /C finite, then ρ0i is π-isotypic of finite length. Then (ρ0i )L
injects into πL , and so ρ0i injects into πC 0 by [HV19, Rk. II.2], which implies the claim.
In particular, any irreducible admissible C-representation π with commutant C isabsolutely irreducible in the sense that its base change πL is irreducible for any field
extension L/C. For example, this holds when C is algebraically closed.
Given an irreducible admissible C-representation π, the space π U of U-invariantscomes equipped with a right action of the pro-p Iwahori–Hecke algebra HC (G, U).
This algebra has a similar structure to that of HC (G, B). In particular, we haveanalogous definitions of the Bernstein elements Eλ (λ ∈ Λ) and the central elementszµ (µ ∈ X∗ (T )), as well as an analogous notion of supersingularity for right HC (G, U)modules (cf. Definition 2.4.1). We say an irreducible admissible C-representation π issupersingular if the right HC (G, U)-module π U is supersingular.
Finally, recall that an irreducible admissible C-representation π of G is said to besupercuspidal if it is not a subquotient of IndG
P τ for any parabolic subgroup P = M N (G and any irreducible admissible representation τ of the Levi subgroup M .
Proposition 3.1.3 (Supercuspidality criterion). Assume c = p. Suppose that π is anirreducible admissible C-representation of G. The following are equivalent:(i) π is supercuspidal;(ii) π is supersingular;(iii) π U contains a non-zero supersingular element;(iv) every subquotient of π U is supersingular;(v) some subquotient of π U is supersingular.
Proof. We have (i)⇔(ii)⇔(iii) by [HV19, Thms. I.13, III.17]. Since (ii)⇒(iv)⇒(v), it
suffices to show that (v)⇒(ii). Let C denote an algebraic closure of C. Say π U has
supersingular subquotient M . Then (π U )C ∼= (πC )U has subquotient MC , and MC
is clearly supersingular. By Proposition 3.1.2 there exists an irreducible admissible
constituent ρ of πC such that the HC (G, U)-module ρU shares an irreducible constituent

12

FLORIAN HERZIG, KAROL KOZIOL, AND MARIE-FRANCE VIGNÉRAS

with MC . In particular, ρU has a supersingular subquotient, and [OV18, Thm. 3] implies
ρ is supersingular. Then [HV19, Lem. III.16 2)] implies that π is supersingular.
Remark 3.1.4. When π B 6= 0, the above criterion holds with π U replaced by π B in
items (iii), (iv), and (v). This follows from the fact that π B is a direct summand of π U
as an HC (G, U)-module, and the action of HC (G, U) on π B factors through HC (G, B).
We now discuss how supercuspidality behaves under extension of scalars. We requirea preliminary lemma.
Lemma 3.1.5. Suppose that C 0 /C is a finite extension and that π 0 is an irreducible
admissible C 0 -representation of G. Then π 0 |C[G] ∼= π ⊕n for some irreducible admissible
C-representation π of G and some n ≥ 1.
Proof. Let C be an algebraic closure of C. Then the finite-dimensional C-algebra
A := C 0 ⊗C C is of finite length over itself. The simple A-modules are given by C with
C 0 acting via the various C-embeddings C 0 → C. It follows that π 0 |C[G] ⊗C C ∼= π 0 ⊗C 0 A
is of finite length as C-representation by Proposition
3.1.2. So π 0 |C[G] is of finite length.
P
If π denotes an irreducible submodule, then i λi π = π 0 |C[G] , where {λi }mi=1 is a basis
00⊕n∼of C /C. It follows that π |C[G] = π
for some n ≤ m. Moreover π is admissible, as
π 0 |C[G] is.
Proposition 3.1.6. Let L denote an algebraically closed field containing C. If c 6= p,
we assume that L = C is an algebraic closure of C.
A C-representation π is supercuspidal if and only if some irreducible subquotient ρ ofπL is supercuspidal, if and only if every irreducible subquotient ρ of πL is supercuspidal.
Proof. If c = p, we note that π is supercuspidal if and only if π is supersingular by
Proposition 3.1.3. This is equivalent to some/every subquotient of πL being supersingular [HV19, Lem. III.16 2)], or equivalently supercuspidal (again by [HV19, Thm.
I.13]).
Now suppose that c 6= p and L = C. Recall that parabolic induction IndG
P is exact,and commutes with scalar extensions and restrictions [HV19, Prop. III.12(i)]. If π is notsupercuspidal, then π is a subquotient of IndG
P τ for some proper parabolic P = M N
and irreducible admissible C-representation τ of M . Then πC is a subquotient of
G
0
∼
(IndG
P τ )C = IndP (τC ). In particular, each irreducible (admissible) subquotient π of00
πC is a subquotient of IndG
P τ for some irreducible (admissible) subquotient τ of τC .
0
Hence none of the π are supercuspidal.
For the converse, suppose by contradiction that πC has an irreducible subquotientρ that is not supercuspidal, i.e. ρ is a subquotient of IndG
P τ for some proper parabolic
P = M N and irreducible admissible C-representation τ of M . By [Vig96, II.4.7](as c 6= p), respectively by Proposition 3.1.2, we can choose a finite extension C 0 /C
with C 0 ⊂ C such that τ , respectively all irreducible constituents of IndG
P τ and πC ,can be defined over C 0 . Write τ ∼= (τ 0 )C for some C 0 -representation τ 0 . Say the00irreducible subquotients of IndG
P τ are σ1 , . . . , σn . So by our choice of C , we know
G 0∼that ρ = (σi )C for some i. As σi is a subquotient of IndP τ , we see that σi |C[G] is a00subquotient of IndG
P (τ |C[M ] ). But σi |C[G] is π-isotypic by Proposition 3.1.2 and τ |C[M ]G 00
has finite length by Lemma 3.1.5, so π is a subquotient of IndP τ for some irreducible(admissible) subquotient τ 00 of τ 0 |C[M ] .
EXISTENCE OF SUPERSINGULAR REPRESENTATIONS

3.2. Change of coefficient field.
result.

13

This section contains the proof of the following

Proposition 3.2.1 (Change of coefficient field).
(i) If G admits an irreducible admissible supercuspidal representation over somefinite field of characteristic p, then G admits an irreducible admissible supercuspidal representation over any field of characteristic p.
(ii) If G admits an irreducible admissible supercuspidal representation over somefield of characteristic c 6= p, then G admits an irreducible admissible supercuspidal representation over any algebraic extension of the prime field of characteristic c.
Proof. Let Fc be the prime field of characteristic c (so that F0 = Q and Fc = Fc if
c 6= 0).
Step 1: We show that, if c 6= p and G admits an irreducible admissible supercuspidal
C-representation π, then G admits one over a finite extension of Fc .
Indeed, by Proposition 3.1.6 we can suppose C is algebraically closed. We claim that
we may twist π by a C-character χ of G, so that the central character of π ⊗ χ takes
values in Fc . To see this, we first note that there exists a subgroup ◦ G of G such that
(1) G/◦ G ∼= Zr for some r ≥ 0; (2) the restriction to Z(G) of the map u : G  Zr
has image of finite index; (3) ker(u|Z(G) ) = Z(G) ∩ ◦ G is compact. (For all of this, see[Ber84, §1.12, 2.3].) Let L := im(u|Z(G) ) ⊂ Zr denote the image of u|Z(G) . Since C isalgebraically closed, the restriction map
res

Hom(Zr , C × ) −−→ Hom(L, C × )is surjective. Let ωπ denote the central character of the irreducible admissible Crepresentation π, and note that ωπ |Z(G)∩◦ G takes values in Fc (since π is smooth andZ(G) ∩ ◦ G is compact). Choose a splitting v of the surjection u : Z(G)  L, and letχ00 ∈ Hom(L, C × ) denote the character ωπ−1 ◦ v. We then let χ0 ∈ Hom(Zr , C × ) denoteany preimage of χ00 under res, and let χ : G → C × be the inflation of χ0 to G via u.
Using that ωπ⊗χ = ωπ χ and ωπ⊗χ |Z(G)∩◦ G = ωπ |Z(G)∩◦ G , the construction of χ impliesωπ⊗χ (z) ∈ Fc for all z ∈ Z(G).
We may therefore assume that the central character of π takes values in Fc . As
c 6= p, by [Vig96, II.4.9] the representation π descends to a finite extension Fc0 /Fc .
Since descent preserves irreducibility, admissibility and supercuspidality, we obtain an
irreducible admissible supercuspidal Fc0 -representation of G.
Step 2: We show that if G admits an irreducible admissible supercuspidal representation over a finite extension of Fc then G admits such a representation over Fc .
Suppose C/Fc is a finite field extension and π an irreducible admissible C-representationof G. By Lemma 3.1.5, π|Fc [G] contains an irreducible admissible Fc -representation π 0 .
0 of π 0 from F to C.
By adjunction, π is a quotient of the scalar extension πCc
We now show that if π is supercuspidal, then π 0 is also supercuspidal. Assume that0π 0 is not supercuspidal, so that it is a subquotient of IndG
P τ , where P is a proper0parabolic subgroup of G and τ is an irreducible admissible Fc -representation of theLevi subgroup M of P . Since parabolic induction is compatible with scalar extension
0 is a subquotient of IndG τ 0 , and therefore thefrom Fc to C, the representation πC
P C
same is true of π. The C-representation τC0 of M has finite length and its irreducible

14

FLORIAN HERZIG, KAROL KOZIOL, AND MARIE-FRANCE VIGNÉRAS

subquotients are admissible by [HV19, Thm. III.4]. Hence, π is a subquotient of IndGP ρ
0for some irreducible admissible subquotient ρ of τC , and we conclude that π is notsupercuspidal.
Step 3: We show that if G admits an irreducible admissible supercuspidal Fp representation (resp., Fc -representation, where c 6= p), then G does so over any fieldof characteristic p (resp., any algebraic extension of Fc ). More generally we show that
if L/C is any field extension, assumed to be algebraic if c 6= p, and G admits anirreducible admissible supercuspidal C-representation then the same is true over L.
Let L/C be a field extension as above, and choose compatible algebraic closures
L/C. Suppose π is an irreducible admissible supercuspidal C-representation of G, andlet τ be an irreducible subquotient of the scalar extension πL of π from C to L. By[HV19, Lem. III.1(ii)], τ is admissible. The scalar extension τL of τ from L to L is asubquotient of the scalar extension πL of πL from L to L (the latter being equal to
the scalar extension of π from C to L). By Propositions 3.1.2 and 3.1.6, πL has finite
length and its irreducible subquotients are admissible and supercuspidal. Therefore,the same is true of τL . By Proposition 3.1.6 this implies that τ is supercuspidal.
We now use extension of scalars to prove the following lemma, which will be used in
the proof of Prop. 3.3.9.
Lemma 3.2.2. Let π be an irreducible admissible C-representation of G and H a finite
commutative quotient of G. Then the representation π ⊗C C[H] of G, with the naturalaction of G on C[H], has finite length and its irreducible subquotients are admissible.
Proof. The scalar extension of the C-representation π (resp. C[H]) to C has finite
length with irreducible admissible quotients πi (resp. χj , of dimension 1). Therefore(π ⊗C C[H])C ∼= πC ⊗C C[H] has finite length with irreducible admissible subquotients(namely, the πi ⊗C χj ), implying the same for π ⊗C C[H].
3.3. Reduction to an absolutely simple adjoint group. As is well known, theadjoint group Gad of G is F -isomorphic to a finite direct product of connected reductive
F -groups
Y(3.3.1)Gad ∼ResFi0 /F (G0i ),=H×iwhere H is anisotropic, the Fi0 /F are finite separable extensions, and ResFi0 /F (G0i )are scalar restrictions from Fi0 to F of isotropic, absolutely simple, connected adjoint
Fi0 -groups G0i .
Proposition 3.3.2. Assume that the field C is algebraically closed or finite, and thatchar F = 0. If, for each i, the group G0i (Fi0 ) admits an irreducible admissible supercuspidal C-representation, then G admits an irreducible admissible supercuspidal
C-representation.
The proposition is the combination of Propositions 3.3.3, 3.3.6, 3.3.8, and 3.3.11 below, corresponding to the operations of finite product, central extension, and scalar restriction (all when C algebraically closed or finite). We also note that if G is anisotropic,
then G is compact and any irreducible smooth representation of G is finite-dimensional(hence admissible) and supercuspidal.

EXISTENCE OF SUPERSINGULAR REPRESENTATIONS

15

3.3.1. Finite product. Let G1 and G2 be two connected reductive F -groups, and σ
and τ irreducible admissible C-representations of G1 and G2 , respectively.
Proposition 3.3.3. Assume that C is algebraically closed.
(i) The tensor product σ ⊗C τ is an irreducible admissible C-representation of
G1 × G2 .
(ii) Every irreducible admissible C-representation of G1 × G2 is of this form.
(iii) The C-representation σ ⊗C τ determines σ and τ (up to isomorphism).
(iv) The C-representation σ ⊗C τ is supercuspidal if and only if σ and τ are supercuspidal.
Proof. Note first that σ ⊗C τ is admissible: for compact open subgroups K1 of G1 andK2 of G2 , we have a natural isomorphism ([Bou12, §12.2 Lem. 1])∼

HomK1 (1K1 , σ) ⊗C HomK2 (1K2 , τ ) −→ HomK1 ×K2 (1K1 ⊗C 1K2 , σ ⊗C τ ),where 1Ki denotes the trivial representation of Ki . Thus, the admissibility of σ and τ
implies the admissibility of σ ⊗C τ .
Suppose now C algebraically closed.
(i) Proposition 3.1.2 implies that the commutant of σ is C. Irreducibility of σ ⊗C τ
then follows from [Bou12, §12.2 Cor. 1].
(ii) Let π be an irreducible admissible C-representation of G1 × G2 , and let K1 , K2
be any compact open subgroups of G1 , G2 , respectively, such that π K1 ×K2 6= 0.
0
If c = p, the C-representation of G1 given by π 1×K2 is admissible (since π K1 ×K2
is finite dimensional for any K10 ). By [HV12, Lemma 7.10], it contains an irreducible
admissible C-subrepresentation σ. Set τ := HomG1 (σ, π) 6= 0, with the natural actionof G2 . The representation σ ⊗C τ embeds naturally in π. As π is irreducible, it isisomorphic to σ ⊗C τ , and τ is irreducible. As π is admissible, τ is admissible as well.
(This proof is due to Henniart.)
If c 6= p, the space π K1 ×K2 is a simple right HC (G1 × G2 , K1 × K2 )-module ([Vig96,
I.4.4, I.6.3]), and we have
HC (G1 × G2 , K1 × K2 ) ∼= HC (G1 , K1 ) ⊗C HC (G2 , K2 ).
By [Bou12, §12.1 Thm. 1], the finite-dimensional simple HC (G1 , K1 ) ⊗C HC (G2 , K2 )modules factor, meaning π K1 ×K2 ∼= σ K1 ⊗C τ K2 for irreducible admissible C-representationsσ, τ of G1 , G2 , respectively (this uses [Vig96, I.4.4, I.6.3] again). Thus, we obtainπ∼= σ ⊗C τ .
(iii) As a C-representation of G1 , σ⊗C τ is σ-isotypic. Similarly, as a C-representationof G2 , σ ⊗C τ is τ -isotypic. The result follows.
(iv) The parabolic subgroups of G1 × G2 are products of parabolic subgroups ofG1 and of G2 . Let P, Q be parabolic subgroups of G1 , G2 , respectively, with Levi
subgroups M, L, respectively, and let π 0 be an irreducible admissible C-representationof the product M × L. By part (ii), the C-representation π 0 factors, say π 0 = σ 0 ⊗C τ 0
for irreducible admissible C-representations σ 0 of M and τ 0 of L. We then obtain anatural isomorphism∼

G2 0
G1 ×G2 00
1IndG
P σ ⊗C IndQ τ −→ IndP ×Q π .

16

FLORIAN HERZIG, KAROL KOZIOL, AND MARIE-FRANCE VIGNÉRAS

Since the inductions on the left-hand side have finite length, we conclude that the irre0
1 ×G2
ducible subquotients of IndG
P ×Q π are tensor products of the irreducible subquotients
G2 0
01
of IndGP σ and of IndQ τ , which gives the result.
We assume from now until the end of §3.3.1 that C is a finite field.
Proposition 3.3.4. Assume that C is finite. Let π be an irreducible admissible Crepresentation of G. The commutant of π is a finite field extension D of C and thescalar extension πD of π from C to D is isomorphic to
M
πi ,
πD ∼
=
i∈Gal(D/C)

where the πi are irreducible admissible D-representations of G. Moreover, the πi eachhave commutant D, are pairwise non-isomorphic, form a single Gal(D/C)-orbit, and,viewed as C-representations, are isomorphic to π.
Proof. The commutant D of π is a division algebra of finite dimension over C. Since
the Brauer group of a finite field is trivial, D is a finite Galois extension of C. The
result now follows from [HV19, Thms. I.1, III.4] by taking R0 = D. (Note also that asa C-representation, πD is π-isotypic of length [D : C].)
Recall that we have fixed irreducible admissible C-representations σ and τ of G1 andG2 , respectively. Their respective commutants Dσ and Dτ are finite extensions of C ofdimensions dσ and dτ , respectively. We embed them into C, and consider:◦ the field D generated by Dσ and Dτ , which has C-dimension lcm(dσ , dτ ),◦ the field D0 := Dσ ∩ Dτ , which has C-dimension gcd(dσ , dτ ).
The fields Dσ , Dτ are linearly disjoint over D0 , we have Dσ ⊗D0 Dτ ∼= D and[D0 :C](3.3.5)

Dσ ⊗C Dτ ∼
=

Y

D.

k=1

Proposition 3.3.6. Assume that C is finite. The C-representation σ ⊗C τ of G1 × G2
is isomorphic to
gcd(dσ ,dτ )σ ⊗C τ ∼
=

M

πk ,

k=1

where the πk are irreducible admissible C-representations with commutant D, which
are pairwise non-isomorphic. The C-representations σ and τ are supercuspidal if andonly if all the πk are supercuspidal, if and only if some πk is supercuspidal.
Proof. By Proposition 3.3.4, we have
M
σD ∼
σi ,
=
i∈Gal(Dσ /C)

τD ∼
=

M

τj ,

j∈Gal(Dτ /C)

where the σi (resp. τj ) are irreducible admissible D-representations of G1 (resp. G2 )with commutant D, which are pairwise non-isomorphic, form a single Gal(D/C)-orbit,

EXISTENCE OF SUPERSINGULAR REPRESENTATIONS

17

descend to Dσ (resp. Dτ ) and their descents, viewed as C-representations, are isomorphic to σ (resp. τ ). The C-representation σ ⊗C τ of G1 × G2 is admissible, and its
scalar extension from C to D is equal toM(3.3.7)(σ ⊗C τ )D ∼σi ⊗D τj .
= σD ⊗D τD ∼=(i,j)∈Gal(Dσ /C)×Gal(Dτ /C)

The D-representation σi ⊗D τj of G1 ×G2 is admissible and has commutant D⊗D D = D([Bou12, §12.2 Lem. 1]). Hence, σi ⊗D τj is absolutely irreducible and equation (3.3.7)
implies (σ ⊗C τ )D is semisimple. By [Bou12, §12.7 Prop. 8], this implies that σ ⊗C τ issemisimple; its commutant is isomorphic to Dσ ⊗C Dτ by [Bou12, §12.2 Lem. 1]. Fromequation (3.3.5) we see that σ ⊗C τ has length [D0 : C] = gcd(dσ , dτ ), its irreducible
constituents πk are admissible and pairwise non-isomorphic with commutant D.
Applying Proposition 3.3.3 over C and Proposition 3.1.6 (several times), we seethat σ and τ are supercuspidal if and only if some/every σi and some/every τj aresupercuspidal, if and only if some/every σi ⊗D τj is supercuspidal. From Proposition
3.1.6 again, this is also equivalent to πk being supercuspidal for some/every k.

3.3.2. Central extension. Recall that we have a short exact sequence of F -groupsi

1 → Z(G) → G →− Gad → 1,
which induces an exact sequence between F -pointsi

1 → Z(G) → G →− Gad → H 1 (F, Z(G)).
The image i(G) of G is a closed cocompact normal subgroup of Gad and H 1 (F, Z(G))
is commutative.
Until the end of §3.3.2, we assume that char F = 0. The group H 1 (F, Z(G)) is thenfinite ([PR94, Thm. 6.14]), implying that i(G) is an open normal subgroup of Gad andthe quotient Gad /i(G) is finite and commutative. Our next task will be to prove thefollowing:
Proposition 3.3.8. Suppose that char F = 0. Then Gad admits an irreducible admissible supercuspidal C-representation if and only if G admits such a representation suchthat moreover Z(G) acts trivially.
Inflation from i(G) to G identifies representations of i(G) with representations of G
having trivial Z(G)-action; this inflation functor respects irreducibility and admissibility. The composite functor(inflation from i(G) to G) ◦ (restriction from Gad to i(G))
from C-representations of Gad to representations of G trivial on Z(G) will be denotedby − ◦ i.
Suppose ρe is an irreducible admissible C-representation of G with trivial action ofZ(G). Then ρe is the inflation of a representation ρ of the open, normal, finite-index
subgroup i(G) of Gad . The C-representation ρ of i(G) is irreducible and admissible,ad
ad is admissible of finite length.
and therefore the induced representation IndG
i(G) ρ of GadG
Any irreducible quotient π of Indi(G)ρ is admissible (if c = p, this uses the assumption

18

FLORIAN HERZIG, KAROL KOZIOL, AND MARIE-FRANCE VIGNÉRAS

char F = 0; see [Hen09, §4, Thm. 1]). By adjunction, π|i(G) contains a subrepresentation isomorphic to ρ and, by inflation from i(G) to G, ρe is isomorphic to a subquotientof π ◦ i.
Conversely, suppose π is an irreducible admissible C-representation of Gad . The restriction π|i(G) of π to i(G) is semisimple of finite length, and its irreducible constituents
ρ are Gad -conjugate and admissible (see [Vig96, I.6.12]; note that the condition thatthe index is invertible in C is not necessary and not used in the proof). Hence, theC-representation π ◦i of G is semisimple of finite length, and its irreducible constituentsare the inflations ρe of the irreducible constituents ρ of π|i(G) .
Proposition 3.3.8 now follows from:
Proposition 3.3.9. Suppose that char F = 0 and let π, ρ and ρe be as above. Then π issupercuspidal if and only if some ρe is supercuspidal, if and only if all ρe are supercuspidal.
Proof. We first check first the compatibility of parabolic induction with − ◦ i. The
parabolic F -subgroups of G and of Gad are in bijection via the map i ([Bor91, 22.6
Thm.]). If the parabolic F -subgroup P of G corresponds to the parabolic F -subgroup
Q of Gad , then i restricts to an isomorphism between their unipotent radicals, andsends a Levi subgroup M of P onto a Levi subgroup L of Q. Further, we have an exact
sequence between F -points:i

1 → Z(G) → M →− L → H 1 (F, Z(G)).
We have Gad = Qi(G) and Q ∩ i(G) = i(P ) = i(M )U, where i(M ) is an open normal
subgroup of L having finite commutative quotient, and U is the unipotent radical of
Q. Thus, if σ is a smooth C-representation of L, the Mackey decomposition implies
ad
i(G)
∼
(IndG
Q σ)|i(G) = Indi(P ) (σ|i(M ) ) and, by inflation from i(G) to G, we obtain:(3.3.10)

ad
G
∼
(IndG
Q σ) ◦ i = IndP (σ ◦ i).

We may now proceed with the proof. It suffices to prove:(i) if π is non-supercuspidal, then all ρe are non-supercuspidal,(ii) if some ρe is non-supercuspidal, then π is non-supercuspidal.
To prove (i), let π be an irreducible admissible non-supercuspidal C-representation ofadad and σ an irreducible
Gad , which is isomorphic to a subquotient of IndG
Q σ for Q ( Gadmissible C-representation of L. Therefore, π ◦ i is isomorphic to a subquotient ofad(IndG
e is isomorphic to a subquotient of IndGeQ σ) ◦ i, and by equation (3.3.10), each ρP τ
for some irreducible subquotient τe of σ ◦ i (depending on ρ). Since τe is admissible and
P ( G, all the ρe are non-supercuspidal.
To prove (ii), let π be an irreducible admissible C-representation of Gad such thatsome irreducible constituent ρe of π ◦ i is non-supercuspidal. Suppose ρe is isomorphic to00
a subquotient of IndG
P τ for P ( G and τ an irreducible admissible C-representationof M . The central subgroup Z(G) acts trivially on ρe, and hence also on τ 0 . Therefore
τ 0 = τe for some irreducible subquotient τ of σ|i(G) , where σ is an irreducible admissible C-representation of L. The representation ρe is isomorphic to a subquotient ofGad
IndG
P (σ ◦ i). By equation (3.3.10) and exactness of parabolic induction, Indi(G) (ρ), and
ad

ad

G
hence its quotient π, is isomorphic to a subquotient of IndG
i(G) ((IndQ σ)|i(G) ). This

EXISTENCE OF SUPERSINGULAR REPRESENTATIONS

19

representation is isomorphic to

ad
Gad
Gad
∼
∼
IndL
IndG
i(M ) (σ|i(M ) ) = IndQ (σ ⊗C C[i(M )\L]).
i(M )U (σ|i(M ) ) = IndQ
By Lemma 3.2.2, the C-representation σ ⊗C C[i(M )\L] of L has finite length and its
irreducible subquotients ν are admissible. Therefore π is isomorphic to a subquotientadadof IndGQ ν for some ν and some Q ( G , and therefore π is non-supercuspidal.
3.3.3. Scalar restriction. Now let F 0 /F be a finite separable extension, G0 a connected
reductive F 0 -group and G := ResF 0 /F (G0 ) the scalar restriction of G0 from F 0 to F .
As topological groups, G0 := G0 (F 0 ) is equal to G := G(F ). By [BT65, 6.19. Cor.], G0
and G have the same parabolic subgroups. Hence:
Proposition 3.3.11. G0 admits an irreducible admissible supercuspidal C-representation
if and only if G does.
4. Proof of the main theorem for most simple groups
4.1. Discrete Iwahori–Hecke modules. Let RepC (G, B) denote the category of
C-representations of G generated by their B-invariant vectors, and let Mod(HC (G, B))
denote the category of right HC (G, B)-modules. The functor of B-invariants
RepC (G, B) → Mod(HC (G, B))
π 7→ π B
admits a left adjoint
T : Mod(HC (G, B)) → RepC (G, B)
M 7→ M ⊗HC (G,B) C[B\G].
Proposition 4.1.1. When c 6= p, the functor π 7→ π B induces a bijection between
the isomorphism classes of irreducible C-representations π of G with π B 6= 0 andisomorphism classes of simple right HC (G, B)-modules ([Vig96, I.4.4, I.6.3]). When
C = C, the functors are inverse equivalences of categories (cf. [Ber84, Cor. 3.9(ii)]; seealso [Mor99, Thms. 4.8, 4.4(iii)]).
Remark 4.1.2. The above functors are not as well-behaved when c = p. In this case,the functor of B-invariants may not preserve irreducibility. Similarly, the left adjoint
T may not preserve irreducibility.
θe C is the linear map
When C = C, the Bernstein ring embedding HC (Z, Z0 ) →
− H
−1/2
Z
defined by sending Tλ to θλ := qλ Eλ for λ ∈ Λ. Its image is equal to AC . Note that
if λ ∈ Λ is anti-dominant and z ∈ Z lifts λ, we have qλ = δB (z), where δB denotes themodulus character of B.
We now recall some properties of the category RepC (G, B), including Casselman’s
criterion of square integrability modulo center, before giving the definition of a discrete
simple right HC (G, B)-module. Recall that πU denotes the space of U -coinvariants(i.e., the unnormalized Jacquet module) of a representation π.
Lemma 4.1.3. Suppose that π is an admissible C-representation of G. Then the∼natural map π  πU induces an isomorphism ϕ : π B −→ πUZ0 . Moreover, we have
−1/2
ϕ(v · θλ−1 ) = δB (t)(t · ϕ(v)) for λ ∈ ΛT , t ∈ T lifting λ, and v ∈ π B .

20

FLORIAN HERZIG, KAROL KOZIOL, AND MARIE-FRANCE VIGNÉRAS

Proof. Recall that B has an Iwahori decomposition with respect to Z, U , U op . Then∼
[Cas, Prop. 4.1.4] implies that the map π  πU induces an isomorphism π B · Tλ−1 −→πUZ0 for λ ∈ ΛT with maxα∈∆ |α(λ)|F sufficiently small. By [Vig16, Prop. 4.13(1)] theoperator Tλ−1 is invertible in HC (G, B), so π B · Tλ−1 = π B .
To show the last statement, we may assume that λ ∈ Λ+
T . Then, in our terminology,[Cas, Lemma 4.1.1] says that ϕ(|BtB/B|−1 [BtB]·v) = t·ϕ(v), where [BtB] denotes the
1/2
usual double coset operator on π B . Now [BtB]·v = v ·Tt−1 and Tt−1 = Et−1 = qt−1 θt−1 .
Moreover, |BtB/B| = qt = qt−1 = δB (t−1 ). Putting this all together, we obtain theclaim.

Remark 4.1.4. The lemma and its proof hold when B is replaced by U and Z0 is replacedby Z0 ∩ U.
Proposition 4.1.5. Let π be an irreducible C-representation of G with π B 6= 0.
(i) π is isomorphic to a subrepresentation of IndG
B σ, where σ is a C-character ofZ trivial on Z0 .
(ii) Casselman’s criterion: π is square integrable modulo center (as defined in[Cas, §2.5]) if and only if its central character is unitary and|χ(µ($))|C < 1+
for all µ ∈ X∗ (T ) such that −µ 6∈ X∗ (T )+ , and all characters χ of T
−1/2
tained in δB πU .

con-

Proof. (i) Since π is irreducible and smooth, it is admissible by [Vig96, II.2.8], and [Cas,3.3.1] implies πU is admissible as well. By Lemma 4.1.3 and the assumption π B 6= 0,
we see that πU 6= 0. The claim now follows by choosing an irreducible quotient πU → σ
for which σ Z0 6= 0 and applying Frobenius reciprocity.
(ii) This follows from [Cas, Thm. 6.5.1].
Definition 4.1.6. We say a simple right HC (G, B)-module is discrete if it is isomorphic
to π B for an irreducible admissible square-integrable modulo center C-representation πof G. We say a semisimple right HC (G, B)-module is discrete if its simple subquotientsare discrete.
Proposition 4.1.7. A simple right HC (G, B)-module M is discrete if and only if any
C-character χ of AC contained in M satisfies the following condition: the restrictionof χ to ΛZ(G) is a unitary character, and(4.1.8)|χ(θλ−1)|C < 1µfor any µ ∈ X∗ (T )+ such that −µ 6∈ X∗ (T )+ .
Proof. Note that M = π B for an irreducible (admissible) C-representation π of G. Thenπ has unitary central character if and only if ΛZ(G) acts by a unitary character on M .
As any irreducible AC -module is a character, by Casselman’s criterion (Proposition
4.1.5) and Lemma 4.1.3, M is discrete if and only condition (4.1.8) holds.
Remark 4.1.9. Some authors view π B as a left HC (G, B)-module. One may pass between left and right modules by using the anti-automorphism Tw 7→ Tw−1 ; that is, we
may define
Tw · v = v · Tw−1

EXISTENCE OF SUPERSINGULAR REPRESENTATIONS

21

f , v ∈ π B . The space π B , viewed as either a left or right HC (G, B)-module,
for w ∈ W
is then called discrete if π is square integrable modulo center. For left modules, theproposition above holds with “dominant” replaced by “anti-dominant,” and “θ” replaced by “θe+ ” (for the definition of θe+ , see the paragraph preceding Proposition 8 in
[Vig05]).
Lemma 4.1.10. For a character χ : AC → C such that χ|ΛZ(G) is unitary, the following
conditions are equivalent:(i) |χ(θλ−1)|C < 1 for any µ ∈ X∗ (T )+ such that −µ 6∈ X∗ (T )+ ,µ(ii) |χ(θλ−1 )|C < 1 for any λ ∈ Λsc,+ such that λ−1 6∈ Λsc,+ ,(iii) |χ(θλ−1 )|C < 1 for any λ ∈ Λ+ such that λ−1 6∈ Λ+ .
Proof. We first recall that the invertible elements in Λ+ consist of Λker ν , so |χ(θλ )|C = 1
for all invertible elements of Λ+ .
As ΛT ∼= X∗ (T ), we see that (iii) implies (i) and (ii). To prove that (ii) implies (iii),
we need to show that |χ(θλ−1 )|C = 1 for λ ∈ Λ+ implies λ−1 ∈ Λ+ . By Lemma 2.2.5
pick n ≥ 1 such that λn ∈ ΛZ(G) × Λsc,+ . Then λn λ0 ∈ Λsc,+ for some λ0 ∈ ΛZ(G) . As
|χ(θλ−n λ−1 )|C = 1 we deduce from (ii) that λn λ0 ∈ Λsc,+ ∩ (Λsc,+ )−1 , which is contained0in Λ+ ∩ (Λ+ )−1 . Therefore λn ∈ Λ+ ∩ (Λ+ )−1 . From the definition of dominance it
follows that λ ∈ Λ+ ∩ (Λ+ )−1 .
The proof that (i) implies (iii) is similar but easier.

Proposition 4.1.11. A simple right HC (G, B)-module M is discrete if and only if
ΛZ(G) acts on M by a unitary character and if its restriction to HC (Gsc , Bsc ) is discrete.
Proof. This follows from Proposition 4.1.7 and Lemma 4.1.10.



4.2. Characters. In this section we continue to assume C is a field of characteristicc, and suppose further that G is absolutely simple and isotropic. We determine the
e = H(G, B). This is an exercise,characters H = H(Gsc , Bsc ) → C which extend to H
which is already in the literature when C = C (cf. [Bor76]).
For distinct reflections s, t ∈ S, the order ns,t of st is finite, except if the type of Σ
is A1 . In the finite case, the braid relations (2.1.2) imply(4.2.1)(4.2.2)(Ts Tt )r = (Tt Ts )rr

if ns,t = 2r,

r(Ts Tt ) Ts = (Tt Ts ) Tt

if ns,t = 2r + 1.

The Ts for s ∈ S and the relations (2.1.3), (4.2.1) and (4.2.2) give a presentation of H.
e is given by the Tu , Ts for u ∈ Ω, s ∈ S and the relations (2.1.3),
A presentation of H
(4.2.1), (4.2.2) and(4.2.3)Tu Tu0 = Tuu0

if u, u0 ∈ Ω,

(4.2.4)Tu Ts = Tu(s) Tu

if u ∈ Ω, s ∈ S,

where u(s) denotes the action of Ω on S.
We have a disjoint decomposition
S=

m
G
i=1

Si ,

22

FLORIAN HERZIG, KAROL KOZIOL, AND MARIE-FRANCE VIGNÉRAS

where Si is the intersection of S with a conjugacy class of W . The Si are precisely theconnected components of Dyn when all multiple edges are removed (see [Bou02, VI.4.3
Th. 4] and [Bor76, 3.3]). Thus, we have




1
A` (` ≥ 2), D` (` ≥ 4), E6 , E7 , or E8 ;m= 2when the type of Σ = A1 , B` (` ≥ 3), F4 , or G2 ;




3
C` (` ≥ 2).
When m > 1, we fix a labeling of the Si such that |S1 | ≥ |S2 |, and when the type ofΣ is C` (` ≥ 2), we let S2 = {s2 } and S3 = {s3 } denote the endpoints of Dyn. (Notethat there are two possible labelings in types A1 and C` (` ≥ 2).) The parameters ds
are equal on each component Si ; we denote this common value by di .
Definition 4.2.5. The unique character χ : H → C with χ(Ts ) = qs (resp., χ(Ts ) =−1) for all s ∈ S is called the trivial (resp., special ) C-character.
Lemma 4.2.6. Suppose {Ts }s∈S → C is an arbitrary map.
(i) When c 6= p, the above map extends to a character of H if and only if it isconstant on each Si , and takes the value −1 or q di on each Ts , s ∈ Si . Thereare 2m characters if q di + 1 6= 0 in C for each i.
(ii) When c = p, the above map extends to a character of H if and only if its values
are −1 or 0 on each Ts , s ∈ S. There are 2|S| characters. Such a character issupersingular if and only if it is not special or trivial.
Proof. (i) This follows from the presentation of H and the fact that the Tw are invertible(so that the map must be constant on conjugacy classes).
(ii) This follows from [Vig17, Prop. 2.2]. The claim about supersingularity follows
from [Vig17, Thm. 6.15].
e Since the elements Tu
We wish to determine which characters of H extend to H.
e
for u ∈ Ω are invertible in H, the relations (4.2.4) imply that a character χ : H → C
e if and only if χ(Ts ) = χ(Tu(s) ) for all s ∈ S and u ∈ Ω.
extends to a character of H
For example, if the image Ψ of Ω in Aut(W, S, di ) is trivial, then any character of H
e The extensions are not unique in general. By their very definition, theextends to H.
trivial and special characters always extend, and we also refer to their extensions astrivial and special characters.
Let χ : H → C denote a character, and suppose c 6= p. By Lemma 4.2.6(i), the valueof χ on Ts for s ∈ Si is constant for each 1 ≤ i ≤ m. We define χi := χ(Ts ) ∈ C fors ∈ Si , and identify the character χ with the m-tuple (χi )1≤i≤m .
Lemma 4.2.7. Assume c 6= p. Let χ : H → C denote a character of H, associatede except in the following
to the m-tuple (χi )1≤i≤m . Then χ extends to a character of H
cases:◦ type A1 , equal parameters d1 = d2 , Ψ 6= 1, and χ1 6= χ2 ;◦ type C` (` ≥ 2), equal parameters d2 = d3 , Ψ 6= 1, and χ2 6= χ3 .
Proof. When m = 1, then χ(Ts ) = χ(Tu(s) ) for all u ∈ Ω and s ∈ S, so that χ extends
e We may therefore assume m > 1. We proceed type-by-type:to H.

EXISTENCE OF SUPERSINGULAR REPRESENTATIONS

23◦ Type A1 with equal parameters d1 = d2 . The group Aut(W, S, ds ) ∼= Z/2Zepermutes s1 and s2 . If Ψ = 1 or χ1 = χ2 , then χ extends to H, while if Ψ 6= 1
and χ1 6= χ2 , the character χ cannot extend.
◦ Type B` (` ≥ 3). In this case, Aut(W, S, ds ) ∼= Z/2Z stabilizes the sets S1 ande
S2 , so that χ(Ts ) = χ(Tu(s) ) for all u ∈ Ω and s ∈ S. Thus χ extends to H.
◦ Type C` (` ≥ 2) with equal parameters d2 = d3 . The group Aut(W, S, ds ) ∼=e whileZ/2Z permutes s2 and s3 . If Ψ = 1 or if χ2 = χ3 , then χ extends to H,
if Ψ 6= 1 and χ2 6= χ3 , the character χ cannot extend.
◦ Type A1 with unequal parameters d1 6= d2 ; Type F4 ; Type G2 ; Type C` (` ≥ 2)with unequal parameters d2 6= d3 . In these cases, Aut(W, S, ds ) (and consee
quently Ψ) is trivial, and thus χ extends to H.

Before stating the next result, we require a definition.
e C -module M isDefinition 4.2.8. Let R ⊂ C be a subring of C. We say a right H◦
e R -submodule M ⊂ M such that the natural map
R-integral if there exists an H
C ⊗R M ◦ → M
e C -modules. We call M ◦ an R-integral structure of M . If p is a
is an isomorphism of He R/p -module R/p ⊗R M ◦ is called reduction of M ◦ modulo p.
maximal ideal of R, the H
We make similar definitions for the algebra HC .
The following proposition combines the above results.
Proposition 4.2.9.
(i) HC admits 2m C-characters. They are all Z-integral, and their reductions
modulo p are supersingular except for the special and trivial characters.
(ii) Suppose χ : HC → C is a character, associated to the m-tuple (χi )1≤i≤m , andsuppose we are in one of the following two cases:◦ type A1 , equal parameters d1 = d2 , Ψ 6= 1, and χ1 6= χ2 ;◦ type C` (` ≥ 2), equal parameters d2 = d3 , Ψ 6= 1, and χ2 6= χ3 .
Then the HC -module χ⊕χ extends to a two-dimensional, Z-integral simple (lefte C -module with supersingular reduction modulo p, where χ = (χ2 , χ1 )or right) H
in the A1 case and χ = (χ1 , χ3 , χ2 ) in the C` case.
(iii) Suppose χ : HC → C is a character which does not fall into either of the twocases of the previous point. Then χ extends to a Z-integral complex charactere C , and its reduction modulo p is supersingular if χ is not special or trivial.
of H
Proof. The claims regarding integrality in (i) and (iii) are immediate.
(i) This follows from Lemma 4.2.6.
(ii) and (iii): Let χ0 : H → Z denote the underlying Z-integral structure of χ. If
we are not in one of the two exceptional cases, the result follows from Lemmas 4.2.6,
4.2.7 and 2.4.2(iii). Otherwise, the character χ0 of H extends to a character χ00 of
e is a rightH 0 := Z[Λker ν ] ⊗ H that is trivial on Λker ν . The tensor product χ00 ⊗H 0 H
e
H-modulethat is free of rank 2 (since the subgroup Λker ν of Ω has index |Ψ| = 2, bye C is simple(2.2.3)). If χ0 : H 0 → C denotes the base change of χ00 to C, then χ0 ⊗HC0 H
and its restriction to HC is equal to χ ⊕ χ. Note that the characters χ and χ in (ii) are

24

FLORIAN HERZIG, KAROL KOZIOL, AND MARIE-FRANCE VIGNÉRAS

neither special nor trivial, since the χi are unequal by assumption and therefore havesupersingular reduction modulo p. We conclude by Lemma 2.4.2(iii).

4.3. Discrete simple modules with supersingular reduction.
We continue
to assume G is absolutely simple and isotropic. Let p denote the maximal ideal ofe C -modules
Z[q 1/2 ] ⊂ C with residue field Fp . We now discuss discrete, Z[q 1/2 ]-integral H
with supersingular reduction modulo p.
The following is the key proposition of this section.
Proposition 4.3.1. Suppose the type of Σ is not equal to A` with equal parameters.
e C -module MC such that:
Then there exists a right H
e C -module;◦ MC is simple and discrete as an H
1/2
◦ MC has a Z[q ]-integral structure M which is furthermore free over Z[q 1/2 ];◦ M has supersingular reduction modulo p.
The proposition will follow from Propositions 4.3.2, 4.3.3, and 4.3.4 below. We sketch
the main ideas of the proof.
Consider first the special character χ : HC → C. It is Z[q 1/2 ]-integral, its reduction
modulo p is non-supersingular, and T(χ) is equal to the Steinberg representation of
Gsc over C, so that χ is discrete. Any discrete, non-special character of HC is Z[q 1/2 ]integral (in fact, Z-integral) and Lemma 4.2.6 implies that its reduction modulo p issupersingular (since the trivial character of HC is not discrete). Thus, we first attempt
to find a discrete non-special character of HC ; these have been classified by Borel in[Bor76, §5.8]. (Note that in [Bor76], the Iwahori subgroup is the pointwise stabilizer
e0 B of an alcove; recall again that if G is F -split or semisimple and simply connected weZhave Ze0 = Z0 .) We describe these characters in Proposition 4.3.2, and use PropositioneC.
4.2.9 to determine which of these characters extend to H
When m = 1, there do not exist any discrete non-special characters of HC , and wee 1/2 (see Proposition 4.3.3). It is free of rank
use instead a reflection module of H
Z[q ]
1/2
|S| over Z[q ] and has supersingular reduction modulo p. When the type is A` , thismodule is non-discrete, which is why we must omit this type. (We also use reflection
modules in Proposition 4.3.4 to handle certain groups of type B3 for which Proposition
4.3.2 does not apply.)
We now proceed with the required propositions.
Proposition 4.3.2. Suppose the type of Σ is B` (` ≥ 4), C` (` ≥ 2), F4 , G2 , A1
with parameters d1 6= d2 , or B3 with parameters (d1 , d2 ) 6= (1, 2). Then the algebrae C admits a discrete non-special simple right module MC , induced from or extending aHcharacter of HC , which is Z[q 1/2 ]-integral. Moreover, the dimension of MC is 1, unlessΨ 6= 1 and the type is◦ C2 with parameters (1, 1, 1), (2, 1, 1), or (3, 2, 2);◦ C3 with parameters (1, 1, 1), (1, 2, 2), or (2, 3, 3);◦ C4 with parameters (1, 2, 2), or (2, 3, 3);◦ C5 with parameters (1, 2, 2).
In these cases, MC extends the HC -module (−1, −1, q d ) ⊕ (−1, q d , −1) where d := d2 =d3 , and thus the dimension of MC is 2.

EXISTENCE OF SUPERSINGULAR REPRESENTATIONS

25

Proof. When m = 1, the only discrete character of HC is the special one ([Bor76, §5.7]).
Suppose m > 1. For each choice of irreducible root system Σ, we list in Tables 1 and
2 the possible parameters (d1 , d2 ) or (d1 , d2 , d3 ) for G (from the tables in [Tit79, §4]),and describe if HC has a discrete non-special character (using [Bor76, §5.8]).
We start with m = 2 in Table 1. For every entry marked “Y,” the given discretee C using the condition of Lemma 4.2.7.
non-special character extends to a character of H
Table 1. m = 2∃ discrete non-special
character of HC ?
A1(d, d) (d ≥ 1)
N
(1, 3)
Y(2, 3)
Y(1, 2)
Y(1, 4)
Y(3, 4)
YB` (` ≥ 3)
(1, 1)
Y(1, 2)Y (if ` ≥ 4), N (if ` = 3)
(2, 1)
Y(2, 3)
YF4(1, 1)
Y(1, 2)
Y(2, 1)
YG2(1, 1)
Y(1, 3)
Y(3, 1)YΣParameters

We now consider m = 3 (that is, type C` ) in Table 2. In this case, the tables in[Bor76, §5.8] show that HC always admits a discrete, non-special character. Note alsothat Borel omitted the parameters (3, 2, 2) for type C2 . In order to obtain this missing
case, we use the criterion of [Bor76, Eqn. 5.6(2)] to see that the only discrete non-special
characters of HC are (−1, −1, 1) and (−1, 1, −1) (in the notation of [Bor76]). Note thateC,
the characters corresponding to parameters with d2 6= d3 automatically extend to H
by Lemma 4.2.7.
(We have one more remark about the tables in [Bor76, §5.8]: the character (−1, −1, 1)
for parameters (2, 1, 4) only works for ` ≥ 3.)
Finally, we remark that in all cases, Propositions 4.1.11 and 4.2.9 imply that thee C -module MC constructed above (either as the extension of a character of HC , or as
He C ) is discrete and Z[q 1/2 ]-integral.
the induction of a character from HC to H

We consider now the types D` (` ≥ 4), E6 , E7 , and E8 . The tables in [Tit79, §4]
imply that G is F -split, so that ds = 1 for all s ∈ S, and for distinct s, t ∈ S, the orderns,t of st is 2 or 3.

26

FLORIAN HERZIG, KAROL KOZIOL, AND MARIE-FRANCE VIGNÉRAS

Table 2. m = 3
ΣParametersC` (` ≥ 2)

(1, 1, 1)(2, 1, 1)(2, 3, 3)(2, 1, 3)(1, 1, 2)(2, 2, 3)(2, 1, 2)(1, 2, 2)(2, 1, 4)(2, 3, 4)(3, 2, 2)

C2

Condition that some discreteeC
non-special character of HC extends to H
` ≥ 4, or Ψ = 1` ≥ 3, or Ψ = 1` = 2, ` ≥ 5, or Ψ = 1
none
none
none
none
` = 2, ` ≥ 6, or Ψ = 1
none
none
Ψ=1

Proposition 4.3.3. Assume that the type of Σ is D` (` ≥ 4), E6 , E7 , or E8 . Let M
e 1/2 -module obtained as the twist of the (left) reflection H
e 1/2 denote the right H
Z[q ]
Z[q ]
module by the anti-automorphism Tw 7→ (−1)`(w) Tw∗ −1 . Then M is free of rank |S|over Z[q 1/2 ], has supersingular reduction modulo p, and MC is a discrete simple righte C -module.
H
e 1/2 -module is the free Z[q 1/2 ]-module with basis {et }t∈S ,Proof. The left reflection HZ[q ]e 1/2 -module structure given bywith HZ[q ]
if s = t,−etTs · et = qet
if s 6= t, ns,t = 2,
1/2
qet + q es if s 6= t, ns,t = 3,
Tu · et = eu(t) ,where s, t ∈ S, u ∈ Ω. Twisting this module by the automorphism Tw 7→ (−1)`(w) Tw∗
e 1/2 -module M 0 , satisfying
gives a left H
Z[q ]


if s = t,qet
Ts · et = −et
if s 6= t, ns,t = 2,
1/2
−et − q es if s 6= t, ns,t = 3,
Tu · et = eu(t) .
e 1/2 -module obtained from M 0 by applying the
Finally, we define M to be the right HZ[q ]eanti-automorphism Tw 7→ Tw−1 . The HC -module MC is simple (even as an HC -module,
cf. [Lus83, §3.13]).
By applying Lemma 2.4.2(iii) and Proposition 4.1.11 twice, we may assume that G
is adjoint in order to prove the required properties of M . The reduction modulo p ofe F -module
M is the Fp -vector space with basis {et }t∈S , with the structure of a right H
p

EXISTENCE OF SUPERSINGULAR REPRESENTATIONS

27

given by(0
et · Ts =−et

if s = t,
if s 6= t,

et · Tu = eu−1 (t) .
e F -module is the direct sum of the characters {χs }s∈S ,The restriction to HFp of this H
p
where
(
0
if s = t,χs (Tt ) =−1 if s 6= t.
By Lemmas 2.4.2(iii) and 4.2.6, we deduce that MFp is supersingular. Further, one
checks that the right action of θλ−1(µ ∈ X∗ (T )+ ) on MC is equal to the left action ofµ(−1)`(λµ ) T̂λ−1−1 on (the base change to C of) the reflection module, where T̂λ−1 is definedµµin [Lus83, §4.3] (note that with respect to our normalizations, the elements ωi of op.
cit. are anti-dominant). The discreteness of MC now follows from Proposition 4.1.7
and [Lus83, §3.2, Thm. 4.7]. (See also [Lus83, §4.23].)Finally, we consider one of the omitted cases from Proposition 4.3.2, namely type
B3 with parameters (1, 2).
Proposition 4.3.4. Assume that the type of Σ is B3 with parameters (1, 2). Thene 1/2 admits a right module M , such that M is free of rank 3 over Z[q 1/2 ], hasHZ[q ]e C -module.
supersingular reduction modulo p, and MC is a discrete simple right H
Proof. In this case, the group Gsc is an unramified non-split form of Spin8 , by thetables in [Tit79]. We will use the reflection module as defined in [GS05, §7].
e long the subset of simple affine roots ∆e which are long. We define an
Denote by ∆1/2
action of HZ[q1/2 ] on the free Z[q ]-module of rank 3 with basis {eβ }β∈∆e long as follows.
e long , we set
If α ∈ ∆
if α = β,−eβTsα · eβ = qeβ
if α 6= β, nsα ,sβ = 2,qeβ + q 1/2 eα if α 6= β, nsα ,sβ = 3,e we set
and if α is the unique short root in ∆,Tsα · eβ = q 2 eβ .
Twisting this reflection module by the automorphism Tw 7→ (−1)`(w) Tw∗ gives a newleft HZ[q1/2 ] -module M 0 , with action given by
if α = β,qeβTsα · eβ = −eβif α 6= β, nsα ,sβ = 2,1/2−eβ − q eα if α 6= β, nsα ,sβ = 3,
e long , andif α ∈ ∆Tsα · eβ = −eβ28

FLORIAN HERZIG, KAROL KOZIOL, AND MARIE-FRANCE VIGNÉRAS

e is short. We extend the action of H 1/2 on M 0 to H
e 1/2 by declaring that
if α ∈ ∆
Z[q ]
Z[q ]
Tu · eα = eu(α) .
e 1/2 is generated by H 1/2 and the elements Tu , u ∈ Ω, subject toAs the algebra H
Z[q ]
Z[q ]
e we see thatfor u, v ∈ Ω and α ∈ ∆,the relations Tuv = Tu Tv and Tu Tsα T −1 = Ts
u
u(α)
0
e
e 1/2 M is a well-defined module of HZ[q1/2 ] . Finally, we define M to be the right HZ[q ]0module obtained from M by applying the anti-automorphism Tw 7→ Tw−1 . One checks

directly that MC is simple (even as an HC -module).
By Lemma 2.4.2(iii) and Proposition 4.1.11 we are now reduced to the case where
G is simply connected. The reduction modulo p of M is the Fp -vector space with basis{eβ }β∈∆e long , with the structure of a right HFp -module given by(0
if α = β,eβ · Tsα =−eβ if α 6= β,e Therefore MF is equal to the direct sum of the characters {χβ } e ,for α ∈ ∆.
pβ∈∆long
where(0
if α = β,χβ (Tsα ) =−1 if α 6= β.
e Lemma 4.2.6 therefore implies that MF is supersingular.
for α ∈ ∆.
p
Once again, we see that the right action of θλ−1(µ ∈ X∗ (T )+ ) on MC is equal toµ
1/2

the left action of (−1)`(λµ ) qλµ Tλ−1
−1 on (the base change to C of) the reflection module.
µSection 8.5 of [GS05] gives an explicit description of Hecke operators associated to thefundamental anti-dominant coweights in terms of Tu and the Tsα . Using this descriptione C -module MC is discrete. (See also
along with Proposition 4.1.7, we see that the H[GS05, Prop. 7.11].)
4.4. Admissible integral structure via discrete cocompact subgroups. Let E
be a number field with ring of integers OE , p a maximal ideal of OE with residue field
k := OE /p, and C/E a field extension.
For any extension of fields, the scalar extension functor commutes with the Binvariant functor and its left adjoint T ([HV19, Lem. III.1(ii)]). Therefore, if τ isan E-structure of a C-representation π, then τ B is an E-structure of π B . Conversely,e C -module N , then T(M ) is an E-structure of T(N ).
if M is an E-structure of an H
Definition 4.4.1. We say that an admissible C-representation π of G is OE -integral
if π contains a G-stable OE -submodule τ ◦ such that, for any compact open subgroup
K of G, the OE -module (τ ◦ )K is finitely generated, and the natural map
ϕ : C ⊗OE τ ◦ → π
is an isomorphism. We call ϕ (and more often τ ◦ ) an OE -integral structure of π. The
G-equivariant map τ ◦ → k ⊗OE τ ◦ (and more often the k-representation k ⊗OE τ ◦ ofG) is called the reduction of τ ◦ modulo p. We say that τ ◦ is admissible if k ⊗OE τ ◦ isadmissible for all p.

EXISTENCE OF SUPERSINGULAR REPRESENTATIONS

29

For any commutative ring R and any discrete cocompact subgroup Γ of G, we define
for all γ ∈ Γ, g ∈ G,∞
C (Γ\G, R) := f : G → R f (γgk) = f (g),and k ∈ Kf
where Kf is some compact open subgroup of G depending on f . Letting G act on
this space by right translation, we obtain a smooth R-representation ρΓR . The complex
representation ρΓC of G has an admissible OE -integral structure given by ρΓOE : thereduction of ρΓOE modulo p is the admissible representation ρΓk .
Proposition 4.4.2. Assume char F = 0 and G semisimple. If π is a square-integrable
C-representation of G, then there exists a discrete cocompact subgroup Γ of G such that
HomG (π, ρΓC ) 6= 0.
Proof. Since char F = 0, there exists a decreasing sequence (Γn )n∈N of discrete cocompact subgroups of G with trivial intersection, such that each is normal and of finite
index in Γ0 . (See [BH78, Thm. A]. The construction there is global, and we obtain therequired decreasing sequence by passing to congruence subgroups.) For any discrete
cocompact subgroup Γ, the normalized multiplicity of π in ρΓC is
mΓ,dg (π) := volΓ · dimC HomG (π, ρΓC ) ,where volΓ is the volume of Γ\G for a G-invariant measure induced by a fixed Haar
measure on G. By the square-integrability assumption on π and the limit multiplicity
formula, the sequence (mΓn ,dg (π))n∈N converges to a nonzero real number (see [DKV84,
App. 3, Prop.] and [Kaz86, Thm. K]).

Proposition 4.4.3. Assume char F = 0. Let π be an irreducible C-representation of
G and Γ a discrete cocompact subgroup of G.
∼(i) If ϕ : C ⊗E τ −→ π is an E-structure of π, then the natural map
C ⊗E HomE[G] (τ, ρΓE ) → HomC[G] (π, ρΓC )is an isomorphism.
(ii) Any irreducible subrepresentation τ of ρΓE admits an admissible OE -integral
structure τ ∩ ρΓOE , whose reduction modulo p is contained in ρΓk .
Proof. We recall a general result in algebra from [Bou12, §12.2 Lem. 1]: let C 0 /C be afield extension and A a C-algebra. For A-modules M, N , the natural map(4.4.4)C 0 ⊗C HomA (M, N ) → HomC 0 ⊗C A (C 0 ⊗C M, C 0 ⊗C N )is injective, and bijective if C 0 /C is finite or the A-module M is finitely generated.
(i) Take C 0 /C = C/E, A = E[G], (M, N ) = (τ, ρΓE ). Then (4.4.4) is an isomorphism
because τ is an irreducible E-representation of G.
(ii) For any compact open subgroup K of G, the OE -module (ρΓOE )K is finite freeand ρΓOE contains τ ◦ := τ ∩ ρΓOE as OE -representations of G. Since the ring OE isnoetherian, these facts imply the OE -submodule (τ ◦ )K of (ρΓOE )K is finitely generated.
The natural linear G-equivariant isomorphism∼

E ⊗OE ρΓOE −→ ρΓE

30

FLORIAN HERZIG, KAROL KOZIOL, AND MARIE-FRANCE VIGNÉRAS

restricts to a linear G-equivariant isomorphism∼

E ⊗OE τ ◦ −→ τ,and therefore τ ◦ is an OE -integral structure of τ . It remains to verify that the injectionτ ◦ ,→ ρΓOE stays injective after reduction modulo p. (As ρΓk is admissible, this will also
imply that k ⊗OE τ ◦ is admissible.) More generally, suppose that 0 → M 0 → M →M 00 → 0 is any exact sequence of OE -modules with M 00 torsion-free. Then M 00 is
the direct limit of its finitely generated submodules, and finitely generated torsion-free00
Emodules are projective, as OE is Dedekind. Hence TorO
1 (M , k) = 0, as Tor functors
commute with direct limits, so the sequence stays exact after reduction modulo p. 
The above result will be used in our construction of irreducible, admissible, supersingular C-representations. It also has the following consequence, which may be of
independent interest.
Corollary 4.4.5. Assume char F = 0 and G semisimple. Then any irreducible supercuspidal C-representation admits an admissible OE -integral structure whose reduction
modulo p is contained in ρΓk , for some discrete cocompact subgroup Γ of G.
Proof. When G is semisimple, any irreducible admissible supercuspidal C-representation
π of G descends to a number field (see [Vig96, II.4.9]). Since π is in particular squareintegrable, Proposition 4.4.2 implies that π embeds into ρΓC for some discrete cocompact
subgroup Γ of G. The claim the follows from points (i) and (ii) of Proposition 4.4.3. 
4.5. Reduction to rank 1 and PGLn (D). We now prove that most p-adic reductive
groups admit irreducible admissible supersingular (equivalently, supercuspidal) representations.
Theorem 4.5.1. Assume that c = p and char F = 0. Suppose G is an isotropic,
absolutely simple, connected adjoint F -group, not isomorphic to any of the following
groups:(i) PGLn (D), where n ≥ 2 and D a central division algebra over F ;(ii) PU(h), where h is a split hermitian form in 3 variables over a ramified quadratic extension of F or a non-split hermitian form in 4 variables over theunramified quadratic extension of F .
Then G admits an irreducible admissible supercuspidal C-representation.
Proof. We first note by the tables in [Tit79] that the above exceptional groups are
precisely the ones where Σ is of type A` with equal parameters. (In that reference our
exceptional groups have names Am−1 , d Amd−1 for m ≥ 2, d ≥ 2 in case (i) and C-BC1 ,2 A00 in case (ii).)3
e 1/2 -module M which is free over Z[q 1/2 ],By Proposition 4.3.1 there exists a right HZ[q ]e C -module, and whose reduction MF iswhose base change MC is a discrete simple H
psupersingular. Set E := Q(q 1/2 ), so that Z[q 1/2 ] ⊂ OE . Let π := T(MC ) denotethe irreducible square-integrable C-representation of G corresponding to MC ; then
τ := T(ME ) is a E-structure of π. We know by Proposition 4.4.2 that π injects intoρΓC for some discrete cocompact subgroup Γ of G, and therefore τ injects into ρΓE by
Proposition 4.4.3(i). We identify π and τ with their images in ρΓC and ρΓE , respectively.

EXISTENCE OF SUPERSINGULAR REPRESENTATIONS

31

Proposition 4.4.3(ii) then ensures that τ ◦ := τ ∩ ρΓOE is an admissible OE -integral∼
structure of τ . In particular, we have a G-equivariant map C ⊗OE τ ◦ −→ π.
Define M 0 := (τ ◦ )B ; since E is a localization of OE , the isomorphism above implies∼C ⊗OE M 0 −→ π B ∼= MC , so that M 0 is an OE -integral structure of MC . Let p ⊂ OE
denote the prime ideal lying over p, and let Op ⊂ C denote the localization of OE at p.
0 = O ⊗
0
Then MO
p OE M is a finitely generated, torsion-free module over the discretepvaluation ring Op , which implies it is free.
0 are He Op -modules which are free over Op , and they are isomorphic
Both MOp and MOp0over C. Thus, we see that the reductions MOp /p and MO
agree up to semisimplifip /p00
cation by the Brauer–Nesbitt theorem. In particular, MOp /p = MOis supersingular
E /p0(since the same is true of MOp /p = MFp ) and, by construction, MOE /p is a submoduleof (ρΓOE /p )B (this uses the final claim of Proposition 4.4.3 (ii)). Therefore we can pick a
non-zero supersingular element v of (ρΓOE /p )B . The G-representation ρΓOE /p is admissible, as Γ is cocompact, and hence so is its subrepresentation hG·vi generated by v. Any
irreducible quotient of hG · vi (which exists by Zorn’s lemma) is admissible by [Hen09,§4, Thm. 1], as F is of characteristic zero, and supersingular by Proposition 3.1.3, as
it contains (the nonzero image of) v. The theorem now follows from Proposition 3.2.1.
The two exceptional cases will be dealt with in Sections 5, 6 below. Assuming this,we can now prove our main result.
Proof of Theorem A. Suppose that G is a connected reductive group over F . We want
to show that G = G(F ) admits an irreducible admissible supercuspidal representation
over any field C of characteristic p. By Proposition 3.2.1 we may assume that C is finite
and as large as we like. Then by Proposition 3.3.2 we may assume that G is isotropic,
absolutely simple, and connected adjoint. The result then follows from Theorem 4.5.1,
Corollary 5.5.2, and Corollary 6.6.2.

5. Supersingular representations of rank 1 groups
In this section we verify Theorem A when G is a connected reductive F -group of
relative semisimple rank 1. In particular, this deals with the second exceptional casein Theorem 4.5.1.
5.1. Preliminaries. We suppose in this section that C is a finite extension of Fp
which contains the |G|p0 -th roots of unity, where |G|p0 denotes the prime-to-p part ofthe pro-order of G.
Suppose that char F = 0. We will show that G admits irreducible, admissible,
supercuspidal C-representations. By Proposition 3.3.2, it suffices to assume G is anabsolutely simple and adjoint group of relative rank 1. We make one further reduction.
Let Gsc denote the simply-connected cover of G:
1 → Z(Gsc ) → Gsc → G → 1.
By Proposition 3.3.8, we see that Gsc admits an irreducible, admissible, supercuspidal
representation on which Z(Gsc ) acts trivially if and only if G does. Therefore, we may
assume that our group G is absolutely simple, simply connected, and has relative rank

32

FLORIAN HERZIG, KAROL KOZIOL, AND MARIE-FRANCE VIGNÉRAS

equal to 1. We will then construct irreducible, admissible, supercuspidal representationsof G on which its (finite) center acts trivially.
5.2. Parahoric subgroups. Let B denote the adjoint Bruhat–Tits building of G.
By our assumptions on G, B is a one-dimensional contractible simplicial complex, i.e.,a tree. Recall that C denotes the chamber of B corresponding to the Iwahori subgroupB, and let x0 and x1 denote the two vertices in the closure of C. We let K0 and K1
denote the pointwise stabilizers of x0 and x1 , respectively. We then have B = K0 ∩ K1 .
The vertices x0 and x1 are representatives of the two orbits of G on the set of verticesof B, and the edge C is a representative of the unique orbit of G on the edges of B. By[Ser03, §4, Thm. 6], we may therefore write the group G as an amalgamated product:G∼= K0 ∗B K1 .
Since the group G is semisimple and simply connected, the stabilizers of vertices
and edges in B are parahoric subgroups (see, e.g., [Vig16, §3.7]). For i ∈ {0, 1}, we let
Ki+ denote the pro-p radical of Ki , that is, the largest open, normal, pro-p subgroupof Ki . The quotient Gi := Ki /Ki+ is isomorphic the group of kF -points of a connected
reductive group over kF (see [HV15, §3.7]). Likewise, the pro-p-Sylow U is the largest
open, normal, pro-p subgroup of B, and Z := B/U is isomorphic to the group ofkF -points of a torus over kF . The image of B in Gi is equal to a minimal parabolic
subgroup Bi , with Levi decomposition Bi = Zi Ui . Thus, we identify the quotient Z
with Zi .
5.3. Pro-p Iwahori–Hecke algebras. We work in slightly greater generality thanin §2. Let
HC := HC (G, U) = EndG C[U\G]denote the pro-p Iwahori–Hecke algebra of G with respect to U. We view HC as the
convolution algebra of C-valued, compactly supported, U-bi-invariant functions on G(see [Vig16, §4] for more details). For g ∈ G, we let Tg denote the characteristic functionof UgU. The algebra HC is generated by two operators Ts̃0 , Ts̃1 , where s̃0 and s̃1 are
lifts to the pro-p Iwahori–Weyl group N /(Z ∩ U) of affine reflections s0 , s1 fixing x0 , x1 ,respectively, along with operators Tz for z ∈ Z. (Note that this labeling is different
than the labeling in §4.2.) For i ∈ {0, 1}, we let HC,i denote the subalgebra of HC
generated by Ts̃i and Tz for z ∈ Z; this is exactly the subalgebra of functions in HC
with support in Ki , i.e.,
HC,i = HC (Ki , U) = EndKi C[U\Ki ].
The algebra HC,i is canonically isomorphic to the finite Hecke algebra HC (Gi , Ui ) (see[CE04, §6.1]).
Since Ki+ is an open normal pro-p subgroup of Ki , the irreducible smooth Crepresentations of Ki and Gi are in bijection. Further, the finite group Gi possesses astrongly split BN pair of characteristic p ([Vig16, Prop. 3.25]). Therefore, by [CE04,
Thm. 6.12], the functor ρ 7→ ρU induces a bijection between isomorphism classes ofirreducible smooth C-representations of Ki and isomorphism classes of simple rightHC,i -modules, all of which are one-dimensional.
We briefly recall some facts about supersingular HC -modules (compare Lemma
4.2.6). We refer to [Vig17, Def. 6.10] for the precise definition (which is analogous

EXISTENCE OF SUPERSINGULAR REPRESENTATIONS

33

to Definition 2.4.1) and give instead the classification of simple supersingular HC modules. Since G is simply connected, every supersingular HC -module is a character.
The characters Ξ of HC are parametrized by pairs (χ, J), where χ : Z → C × is acharacter of the finite torus and J is a subset ofSχ := s ∈ {s0 , s1 } : χ(cs̃ ) 6= 0
(here cs̃ is a certain element of C[Z] which appears in the quadratic relation for Ts̃ ;note also that the definition of Sχ is independent of the choice of lift s̃ ∈ N /(Z ∩ U) ofs). The correspondence is given as follows (cf. [Vig17, Thm. 1.6]): for z ∈ Z, we haveΞ(Tz ) = χ(z), and for s ∈ {s0 , s1 }, we have(0
if s ∈ J,
Ξ(Ts̃ ) =χ(cs̃ ) if s 6∈ J.
Since G is simple, [Vig17, Thm. 1.6] implies that Ξ is supersingular if and only if(Sχ , J) 6= ({s0 , s1 }, ∅), ({s0 , s1 }, {s0 , s1 }).
5.4. Diagrams. Since the group G is an amalgamated product of two parahoric subgroups, the formalism of diagrams used in [KX15] applies to the group G. We recall thata diagram D is a quintuple (ρ0 , ρ1 , σ, ι0 , ι1 ) which consists of smooth C-representations
ρi of Ki (i ∈ {0, 1}), a smooth C-representation σ of B, and B-equivariant morphisms
ιi : σ → ρi |B . We depict diagrams as
ρ0
ι0

σ
ι1

ρ1
Morphisms of diagrams are defined in the obvious way (i.e., so that the relevant squares
commute).
Let Ξ denote a supersingular character of HC , associated to a pair (χ, J). We definea diagram DΞ as follows:◦ set σ := χ−1 , which we view as a character of B by inflation;◦ we let ρΞ,i denote an irreducible smooth C-representation of Ki such that
ρUΞ,i ∼
= Ξ|HC,i as HC,i -modules (by the discussion above, ρΞ,i is unique up toisomorphism);∼◦ let ιi denote the B-equivariant map given by σ = χ−1 −→ ρUΞ,i ,→ ρΞ,i |B .
Pictorially, we write


DΞ =





 −1
 χ




ρΞ,0
ι0

ι1

ρΞ,1












34

FLORIAN HERZIG, KAROL KOZIOL, AND MARIE-FRANCE VIGNÉRAS

We now wish to construct an auxiliary diagram D0 into which DΞ injects. This will
be done with the use of injective envelopes. Recall that if G is a profinite group and τ
is a smooth C-representation of G, an injective envelope consists of a smooth injective
C-representation injG τ of G along with a G-equivariant injection j : τ ,→ injG τ which
satisfies the following property: for any nonzero C-subrepresentation τ 0 ⊂ injG τ , wehave j(τ ) ∩ τ 0 6= 0. This data exists and is unique up to (non-unique) isomorphism.
Lemma 5.4.1 ([Paš04, Lem. 6.13]). Let τ denote a smooth C-representation of G, andlet j : τ ,→ injG τ denote an injective envelope. Let I denote an injective representationof G, and suppose we have an injection φ : τ ,→ I. Then φ extends to an injection
φe : injG τ ,→ I such that φ = φe ◦ j.
Lemma 5.4.2. Suppose G has an open, normal subgroup G + . Let τ denote a smooth
C-representation of G such that G + acts trivially, and let j : τ ,→ injG τ denote an+
injective envelope of τ in the category of C-representations of G. Then τ ,→ (injG τ )Gis an injective envelope of τ in the category of C-representations of G/G + .
Proof. This is [Paš04, Lem. 6.14]; its proof does not require that τ be irreducible orthat G + be pro-p, as we assume that G + acts trivially.

We now begin constructing D0 .
Lemma 5.4.3. Let i ∈ {0, 1}. We then haveM
injB ξ ⊕|Bi \Gi | ,(injKi C[Gi ])|B ∼=ξ

where ξ runs over all C-characters of B (or, equivalently, of Zi ), and we have fixed
choices of injective envelopes.
Proof. Consider the B-representation (injKi C[Gi ])U . The action of B factors throughthe quotient B/U ∼= Z, which is commutative of order coprime to p. Therefore, we
obtain a B-equivariant isomorphism
M(5.4.4)(injKi C[Gi ])U ∼
ξ ⊕mξ
=
ξ

for non-negative integers mξ satisfyingmξ = dimC HomB (ξ, injKi C[Gi ])+= dimC HomB ξ, (injKi C[Gi ])Ki= dimC HomBi (ξ, injGi C[Gi ])= dimC HomZi ξ, (injGi C[Gi ])Ui .
(The third equality follows from Lemma 5.4.2.) Since C[Gi ] is injective as a representation of Gi , we have isomorphisms of Zi -representationsM(injGi C[Gi ])Ui ∼ξ ⊕|Bi \Gi | ,= C[Ui \Gi ] ∼=ξ

so that mξ = |Bi \Gi |.

EXISTENCE OF SUPERSINGULAR REPRESENTATIONS

35

The isomorphism (5.4.4) implies we have a B-equivariant injectionMξ ⊕|Bi \Gi | ,→ (injKi C[Gi ])|B .
ξ

As B is open, [Vig96, §I.5.9 d)] implies that the representation on the right-hand side is
injective. Lemma 5.4.1 then says that the above morphism extends to a split injection
between injective B-representations
M
injB ξ ⊕|Bi \Gi | ,→ (injKi C[Gi ])|B .
ξSince the U-invariants of both representations agree, the above injection must be an
isomorphism.
Lemma 5.4.5. Set a := lcm(|B0 \G0 |, |B1 \G1 |). There exists a diagram D0 of the
form



D0

injK0 C[G0 ]⊕a·|B0 \G0 |






 L
⊕a

ξ injB ξ






=

κ0

κ1

injK1 C[G1 ]⊕a·|B1 \G1 |

−1













−1

where κ0 and κ1 are isomorphisms, and a morphism of diagrams
ψK0

ρΞ,0
ι0

ψ:

−1

injK1 C[G1 ]⊕a·|B1 \G1 |

−1

κ0
ψB

χ−1

injK0 C[G0 ]⊕a·|B0 \G0 |Lξ injB ξ

⊕a

κ1

ι1

ρΞ,1

ψK1

in which all arrows are injections.
Proof. We fix the following injections, which are equivariant for the relevant groups:◦ injective envelopes jξ : ξ ,→ injB ξ for each C-character ξ of B;−1−1◦ injective envelopes ji : C[Gi ]⊕a·|Bi \Gi | ,→ injKi C[Gi ]⊕a·|Bi \Gi | for i ∈ {0, 1};L◦ an inclusion c : χ−1 ,→ ξ ξ ⊕a ;−1

◦ an inclusion ci : ρΞ,i ,→ C[Gi ]⊕a·|Bi \Gi | for i ∈ {0, 1}.
Let i ∈ {0, 1}. We first construct the κi . We have a B-equivariant sequence of maps
ιici

χ−1 ,−→ ρΞ,i ,−→ C[Gi ]⊕a·|Bi \Gi |

−1

ji

,−
→ injKi C[Gi ]⊕a·|Bi \Gi |and thus we obtainji ◦ci ◦ιiχ−1 ,−−−−→ injKi C[Gi ]⊕a·|Bi \Gi |

−1

U

.

−1

36

FLORIAN HERZIG, KAROL KOZIOL, AND MARIE-FRANCE VIGNÉRAS

L ⊕a ∼−1
By Lemmas 5.4.3 and 5.4.2, we have
= (injKi C[Gi ]⊕a·|Bi \Gi | )U . We fix an
ξξ
L
−1
∼
isomorphism αi : ξ ξ ⊕a −→ (injKi C[Gi ]⊕a·|Bi \Gi | )U such thatαi ◦ c = ji ◦ ci ◦ ιi .

(5.4.6)

Now consider the maps of C-representations of B:
M
αi
−1 U
−1 
ξ ⊕a ,−→ injKi C[Gi ]⊕a·|Bi \Gi |,→ injKi C[Gi ]⊕a·|Bi \Gi | |B .
ξ

By Lemma 5.4.1, the above map extends to an B-equivariant split injection
M
−1 
κi :
injB ξ ⊕a ,→ injKi C[Gi ]⊕a·|Bi \Gi | |Bξsuch that(5.4.7)κi ◦Mjξ⊕a = αi .

ξL⊕a and (inj C[G ]⊕a·|Bi \Gi |−1 )|Since bothi
B are injective C-representationsKi
ξ injB ξof B and κi induces an isomorphism between their U-invariants (cf. Lemma 5.4.3), we
see that κi must in fact be an isomorphism.
L
We now construct the morphism of diagrams. Set ψKi := ji ◦ci and ψB := ( ξ jξ⊕a )◦c. We have
ψKi ◦ ιi(5.4.6)(5.4.7)= αi ◦ c = κi ◦ ψB ,and therefore we obtain the desired morphism of diagrams.



5.5. Supersingular representations via homology. Recall that a G-equivariant
coefficient system D consists of C-vector spaces DF for every facet F ⊂ B, alongwith restriction maps for every inclusion of facets. This data is required to have acompatible G-action such that each DF is a smooth C-representation of the G-stabilizerof F. The functor sending D to the quintuple (Dx0 , Dx1 , DC , ι0 , ι1 ), where the ιi are
the natural restriction maps, is an equivalence of categories between G-equivariant
coefficient systems and diagrams (cf. [KX15, §6.3]).
We let DΞ and D0 denote the G-equivariant coefficient systems on B associated toDΞ and D0 , respectively. The homology of G-equivariant coefficient systems gives rise
to smooth C-representations of G, and we defineψ∗
π := im H0 (B, DΞ ) −→ H0 (B, D0 ) ,where ψ∗ denotes the map on homology induced by ψ.
Theorem 5.5.1. Suppose char F = 0. Then the C-representation π of G admits anirreducible, admissible, supercuspidal quotient.
Proof. We use language and notation from [Paš04] and [KX15].
Step 1: The representation π is nonzero.
Fix a basis v for χ−1 . Let ω0,ι0 (v) denote the 0-chain with support x0 satisfying ω0,ι0 (v) (x0 ) = ι0 (v) and let ω̄0,ι0 (v) denote its image in H0 (B, DΞ ). Set ω̄ :=EXISTENCE OF SUPERSINGULAR REPRESENTATIONS

37ψ∗ (ω̄0,ι0 (v) ) = ω̄0,ψK0 ◦ι0 (v) ∈ π ⊂ H0 (B, D0 ). This is the image in H0 (B, D0 ) of a Dx0 0 valued 0-chain supported on x0 , and since the maps κ0 , κ1 are isomorphisms and ψ isinjective, we have ω̄ 6= 0 [Paš04, Lem. 5.7]. We also note that therefore ω̄0,ι0 (v) 6= 0.
Step 2: The representation π is admissible.
Since κ0 , κ1 are isomorphisms, [Paš04, Prop. 5.10] givesM
π|B ⊂ H0 (B, D0 )|B ∼injB ξ ⊕a ,= DC0 ∼=
ξ

which by Lemma 5.4.2 implies π U ,→ ξ ξ ⊕a , so that π is admissible.
Step 3: The HC -module π U contains Ξ.
The element ω̄0,ι0 (v) ∈ H0 (B, DΞ ) is U-invariant and stable by the action of HC , andthe vector space it spans is isomorphic to Ξ as an HC -module (for all of this, see theproof of [KX15, Prop. 7.3]). Since ψ∗ is G-equivariant, the same is true for ω̄ ∈ π.
Step 4: The vector ω̄ generates π.
Since ω̄0,ι0 (v) generates H0 (B, DΞ ) as a G-representation and ψ∗ is G-equivariant, ω̄generates π as a G-representation.
Step 5: We construct the quotient π 0 and list its properties.
By the previous step, the representation π is generated by ω̄. Proceeding as in theend of the proof of Theorem 4.5.1, we see that any irreducible quotient of π = hG · ω̄iis admissible (since char F = 0, and such quotients exist by Zorn’s lemma). Let π 0 be
any such quotient.
Step 6: We prove π 0 is supercuspidal.
Since ω̄ generates π, its image in π 0 is nonzero. Thus, we obtain an injection of HC modules Ξ ∼= C ω̄ ,→ (π 0 )U , and supercuspidality follows from Proposition 3.1.3.
L

Corollary 5.5.2. Suppose char F = 0 and G is a connected reductive F -group of
relative semisimple rank 1. Then G admits an irreducible admissible supercuspidal
C-representation.
Proof. By the reductions in §5.1, it suffices to assume G is absolutely simple and simply
connected, and to construct a supercuspidal C-representation on which Z(G) acts trivially. Since the center of G is finite, it is contained in B∩Z = Z0 . Hence, taking Ξ to be
associated to (1Z , J), where 1Z is the trivial character of Z and J 6= ∅, {s0 , s1 } (notingthat S1Z = {s0 , s1 }), Theorem 5.5.1 produces an irreducible admissible supercuspidal
C-representation π 0 with trivial action of the center. This gives the claim.

Remark 5.5.3. The construction of π 0 above shares some similarities with the construction in §4.5. Therein, supercuspidal representations are constructed as subquotientsof C ∞ (Γ\G, C) ∼= IndGΓ 1Γ , where Γ is a discrete, cocompact subgroup of G and 1Γ
denotes the trivial character of Γ. Taking Γ to be torsion-free, we use the Mackey
formula to obtain
M
0(IndG 1Γ )|K ∼IndKi 1 ∼== inj C[Gi ]⊕ai
Γ

{1}

i

Ki

Γ\G/Kiiwhere a0i = |Γ\G/Ki |. (The last isomorphism follows from the fact that IndK
{1} 1 is+injective, by [Vig96, §I.5.9 b)], and (IndKi 1)Ki ∼= C[Gi ]; we may then proceed as in{1}

the proof of Lemma 5.4.3.) The construction of Theorem 5.5.1 produces supercuspidal

38

FLORIAN HERZIG, KAROL KOZIOL, AND MARIE-FRANCE VIGNÉRAS

representations as subquotients of H0 (B, D0 ), for which we have
H0 (B, D0 )|K ∼= inj C[Gi ]⊕ai ,i

Ki

where ai = a · |Bi \Gi |−1 (cf. [Paš04, Prop. 5.10]).
6. Supersingular representations of PGLn (D)In this section we verify Theorem A when G = PGLn (D), where n ≥ 2 and D acentral division algebra over F . In particular, this deals with the first exceptional casein Theorem 4.5.1.
6.1. Notation and conventions. Throughout Section 6, we let Qp denote a fixed
algebraic closure of Qp , with ring of integers Zp and residue field Fp . We normalize thevaluation val of Qp such that val(p) = 1.
Let D denote a central division algebra over F of dimension d2 . Let B = ZU denotethe upper-triangular Borel subgroup of GLn (D) with diagonal minimal Levi subgroup
Z∼= (D× )n and unipotent radical U . Let T ∼= (F × )n denote the diagonal maximal split
optorus, N its normalizer in GLn (D), and U the lower-triangular unipotent matrices.
Let OD denote the ring of integers of D, mD the maximal ideal of OD , and kD theresidue field, so [kD : kF ] = d. Let D(1) := 1 + mD , so D(1) C D× . Let valD : D×  Z
denote the normalized valuation of D. Let I(1) denote the pro-p Iwahori subgroup
I(1) := {g ∈ GLn (OD ) : g ∈ GLn (kD ) is upper-triangular unipotent}.
For any field K let ΓK denote the absolute Galois group for a choice of separable
closure. If K 0 /K is a finite separable extension, then ΓK 0 is a subgroup of ΓK , up to
conjugacy, hence the restriction of a ΓK -representation to ΓK 0 is well defined up toisomorphism.
If K/Qp is finite we let IK denote the inertia subgroup of ΓK and kK the residue
field of K. If ρ : ΓK → GLn (Qp ) is de Rham and τ : K → Qp is continuous, we letHTτ (ρ) denote multi-set of τ -Hodge–Tate weights. We normalize Hodge–Tate weights
so that the cyclotomic character ε has τ -Hodge–Tate weight −1 for any τ . We letWD(ρ) denote the associated Weil–Deligne representation of WK over Qp (defined by
Fontaine, cf. Appendix B.1 of [CDT99]).
We normalize local class field theory so that uniformizers correspond to geometric
Frobenius elements under the local Artin map. Let recF denote the local Langlands correspondence from isomorphism classes of irreducible smooth representations of GLn (F )over C to isomorphism classes of n-dimensional Frobenius semisimple Weil–Deligne representations of WF over C. (See [HT01].)
If L is a global field, we let | · |L denote the normalized absolute value of AL .
6.2. On the Jacquet–Langlands correspondence. We recall some basic facts
about the representation theory of GLn (D) and the local Jacquet–Langlands correspondence. All representations in this section will be smooth and over C.
For a finite-dimensional central simple algebra A let Nrd : A× → Z(A)× (or NrdA
for clarity) denote the reduced norm. Let ν denote the smooth character | Nrd |F ofGLm (D) for any m. If πi are smooth representations of GLni (D), let π1 × · · · × πr
denote the normalized parabolic induction of π1 ⊗ · · · ⊗ πr to GLP ni (D). In particularthese notions also apply to general linear groups over F (by setting D = F ).

EXISTENCE OF SUPERSINGULAR REPRESENTATIONS

39

We will say that a representation is essentially unitarizable if some twist of it isunitarizable.
The Jacquet–Langlands correspondence [DKV84] is a canonical bijection JL between
irreducible essentially square-integrable representations of GLn (D) and irreducible essentially square-integrable representations of GLnd (F ) that is compatible with character twists and preserves central characters. (For short, we say “square-integrable”
instead of “square-integrable modulo center”.)
On the other hand, Badulescu [Bad08] defined a map | LJGLn (D) | in the other direction, from irreducible essentially unitarizable representations of GLnd (F ) to irreducible
essentially unitarizable representations of GLn (D) or zero, which in general is neither
injective nor surjective. (More precisely, [Bad08] only considers unitarizable representations, but we can extend it by twisting.) In the split case | LJGLn (F ) | is the identity.
It follows from Thm. 2.2 and Thm. 2.7(a) in [Bad08] that | LJGLn (D) |(JL(π)) ∼= π forany essentially square-integrable representation π of GLn (D).
If ρ is a supercuspidal representation of GLm (F ) and ` ≥ 1, then Z u (ρ, `) is bydefinition the unique irreducible quotient of ρν (1−`)/2 × ρν (3−`)/2 × · · · × ρν (`−1)/2 . It
is an essentially square-integrable representation of GLm` (F ). All essentially squareintegrable representations of GLn (F ) arise in this way, for some decomposition n = m`.
If ρ0 is a supercuspidal representation of GLm (D), we can write JL(ρ0 ) ∼= Z u (ρ, s) foru0
some supercuspidal representation ρ and integer s ≥ 1. Then Z (ρ , `) is by definition
the unique irreducible quotient of ρ0 ν s(1−`)/2 × ρ0 ν s(3−`)/2 × · · · × ρ0 ν s(`−1)/2 . It isan essentially square-integrable representation of GLm` (D). All essentially squareintegrable representations of GLn (D) arise in this way, for some decomposition n = m`(a result of Tadić, cf. [Bad08, §2.4]). Moreover, JL(Z u (ρ0 , `)) ∼= Z u (ρ, `s) [Bad08, §3.1].
If π is a smooth representation of GLn (D) let πU denote its (unnormalized) Jacquet
module. The following lemma was proved earlier, cf. Remark 4.1.4.
Lemma 6.2.1. Suppose that π is an admissible representation of GLn (D) over C.
Then the natural map pU : π → πU induces an isomorphism π I(1) → (πU )Z∩I(1) .
The following results will be needed in Section 6.3.
Lemma 6.2.2. Suppose that Π is an irreducible generic smooth representation of
GLnd (F ) over C that is essentially unitarizable and such that the representation π :=| LJGLn (D) |(Π) of GLn (D) is non-zero. If π I(1) 6= 0, then there exist irreducible representations ρ01 , L. . . , ρ0n of D× /D(1) such that π is a subquotient of ρ01 × · · · × ρ0n andnrecF (Π)|WF ∼= i=1 recF (JL(ρ0i ))|WF .
Proof. After a twist we may assume that Π is unitarizable. As Π is moreover generic,we know that Π ∼= σ1 ν α1 × · · · × σr ν αr for some square-integrable σi of GLni (F ) andreal numbers αi ∈ (− 21 , 21 ) satisfying αi + αr+1−i = 0 and σi = σr+1−i if αi 6= 0 (see e.g.
[HT01, Lemma I.3.8]). Since | LJGLn (D) |(Π) 6= 0 by assumption, it follows that d | ni
for all i and π = | LJGLn (D) |(Π) ∼= σ10 ν α1 ×· · ·×σr0 ν αr , where σi0 is the square-integrable
representation of GLni /d (D) such that JL(σi0 ) ∼= σi . (See [Bad08, §3.5].) Let n0i := ni /d.
From π I(1) 6= 0 and Lemma 6.2.1 it follows that the supercuspidal support of π is atame representation of Z (up to conjugacy), so each σi0 is of the form Z u (ρ00i , n0i ), where
ρ00i is an irreducible representation of D× /D(1). We write JL(ρ00i ) ∼= Z u (ρi , ei ) with ρiu0∼irreducible supercuspidal, so σi = Z (ρi , ei ni ). In particular, π is a subquotient of the

40

FLORIAN HERZIG, KAROL KOZIOL, AND MARIE-FRANCE VIGNÉRAS

00 αi +ei ((n0i −1)/2−j) . On the other hand, Π is1≤i≤r,0≤j≤n0i −1 ρi νN0a subquotient of the normalized induction of 1≤i≤r,0≤j≤ei n0 −1 ρi ν αi +(ei ni −1)/2−j . As

normalized induction ofNirecF (Π)|WF only depends on the supercuspidal support of Π (see the paragraph before
Thm. VII.2.20 in [HT01]), we obtainM
α +(e n0 −1)/2−jrecF (Π)|WF ∼| · |F i i i
recF (ρi )|WF .
=1≤i≤r,0≤j≤ei n0i −1∼ Lei −1 | · |(ei −1)/2−k recF (ρi )|W . Denoting by ρ0 , . . . , ρ0
Similarly, recF (JL(ρ00i ))|WF =
n
1
F
F
k=0
00 ν αi +ei ((n0i −1)/2−j) in any order, a straightforward computation
the representations
ρiL
confirms that ni=1 recF (JL(ρ0i ))|WF ∼= recF (Π)|WF .
We now recall a result of Bushnell–Henniart concerning explicit functorial transfersof irreducible representations of D× /D(1). An admissible tame pair (E/F, ζ) consists ofan unramified extension of degree f dividing d, and a tamely ramified smooth characterζ : E × → C× such that all Gal(E/F )-conjugates of ζ are distinct. In that case, after
choosing an F -embedding of E into D (which is unique up to conjugation by D× ),B := ZD (E) is a central simple E-algebra of dimension e2 , where e := d/f . Define asmooth character Λ : B × (1 + mD ) → C× by declaring it to be ζ ◦ NrdB on B × and×trivial on 1 + mD . Then define πD (ζ) := IndD
B × (1+mD ) Λ is an irreducible representationof D× /D(1) (of dimension f ).
Proposition 6.2.3.
(i) Any irreducible representation of D× /D(1) is isomorphic to πD (ζ) for some
admissible tame pair (E/F, ζ).
(ii) The element $ ∈ F acts as the scalar ζ($)e on πD (ζ).
(iii) If (E/F, ζ) is an admissible tame pair, thene(f −1)

W
recF (JL(πD (ζ))) ∼= Spe (IndWFE (ηEζ)),

where ηE is the unramified quadratic character of E × .
We recall that the special Weil–Deligne representation Spe (σ), for σ an irreducible
e−1
Le−1
−k
.
representation of WF , is indecomposable and satisfies Sp (σ)|W ∼σ| · | 2
=
e

F

k=0

F

Proof. For (i), see [BH11, §1.5]. Part (ii) follows from the definition. Part (iii) is themain result of [BH11].

6.3. On lifting non-supersingular Hecke modules. Let H := H(GLn (D), I(1))
the corresponding pro-p Iwahori–Hecke algebra over Z [Vig16] and for a commutative
ring R let HR := H⊗R. Similarly we define HZ := H(Z, Z ∩I(1)) and HZ,R := HZ ⊗R.
Note that the pro-p Iwahori subgroup Z ∩ I(1) is normal in Z. All Hecke modules wewill consider are right modules. A finite-dimensional HQp -module is said to be integral
if it arises by base change from a HZp -module that is finite free over Zp .
Let W (1) := N /Z ∩ I(1), Λ(1) := Z/Z ∩ I(1), and define monoids
Z + := {diag(δ1 , . . . , δn ) ∈ Z : valD (δ1 ) ≥ · · · ≥ valD (δn )}and Λ(1)+ := Z + /Z ∩ I(1).

EXISTENCE OF SUPERSINGULAR REPRESENTATIONS

41

We recall that H has an Iwahori–Matsumoto basis Tw for w ∈ W (1) and a Bernsteinbasis Ew for w ∈ W (1), which in fact depends on a choice of spherical orientation. We
choose our spherical orientation such that Ew = Tw for w ∈ Λ(1)+ . (This is possible
by [Vig16, Ex. 5.30]. It is the opposite of our convention in §2.3.) Similarly, HZ has
basis TwZ for w ∈ Λ(1).
For w ∈ W (1) we have integers qw ∈ q Z>0 , as recalled in §2.3. (Note that our basealcove C is the one fixed pointwise by I(1).)
Lemma 6.3.1. For z = diag(δ1 , . . . , δn ) ∈ Z with δi ∈ D× we haveqz = q d

P

i<j | valD (δi )−valD (δj )|.

Proof. As the Iwahori–Hecke algebra has equal parameters q d we deduce that qz =q d`(z) , where ` is the length function relative to the alcove C. By using the action of thefinite Weyl group N /Z and the first length formula in [Vig16, Cor. 5.11], we may assume
that z ∈ Z + . By [Vig16, §3.9] we then have qz = (I(1)zI(1) : I(1))= (I(1) : I(1) ∩PzI(1)z −1 ) = (U0 : zU0 z −1 ), where U0 := U ∩ I(1). Hence qz = q d i<j (valD (δi )−valD (δj )) ,as required.

Let W0 ∼= Sn denote the Weyl group of T . Recall from [Vig17, §5, §1.3] that A0 (ΛT )is the free module with basis Eµ($) for µ ∈ ΛT := X∗ (T ) and that the central subalgebraP
ZT := A0 (ΛT )W0 of H has a basis consisting of the sums µ Eµ($) with µ running over
the W0 -orbits in X∗ (T ). For I ⊂ {1, . . . , n} let EI := EµI ($) , where µI ∈ X∗P(T ) ∼= Zn isdefined by µI,i = 1 if i ∈ I and µI,i = 0 otherwise. For 1 ≤ i ≤ n let Zi := I,|I|=i EI .
By induction and [Vig16, Cor. 5.28] we see that the algebra ZT is generated by Z1 ,. . . , Zn−1 , Zn±1 .
The following lemma follows from [Vig17, Prop. 6.9].
Lemma 6.3.2. A finite-dimensional HFp -module M is supersingular if and only if theaction of Zi on M is nilpotent for all 1 ≤ i ≤ n − 1.
Lemma 6.3.3. There exists a unique injective algebra homomorphism θe : HZ,Qp →e Z ) = Tw for all w ∈ Λ(1)+ . We have
H such that θ(TQp(6.3.4)w

2

EI = q d (P)e Z(|I|+12 ) θ(Ti∈I i−µI ($) ).

Proof. The first assertion follows from [OV18, §2.5.2, Rk. 2.20]. We claim that for anyµ ∈ X∗ (T ),(6.3.5)2Eµ($) = q dPr<s:µr <µs (µs −µr )e Z ),
θ(Tµ($)

which implies (6.3.4) by taking µ = µI .
Note that X∗ (T )+ = {µ ∈ X∗ (T ) : µ1 ≥ · · · ≥ µn }. If µ ∈ X∗ (T )+ , then µ($) ∈ Z +and hence Eµ($) = Tµ($) and formula (6.3.5) holds. In general, choose µ0 ∈ X∗ (T )+such that µ + µ0 ∈ X∗ (T )+ . Then formula (6.3.5) follows easily from the following three
−1
Z
Z
1/2 E
assertions: (1) Tµ($)
TµZ0 ($) = Tµ($)µ
0 ($) ; (2) Eµ($) Eµ0 ($) = (qµ($) qµ0 ($) qµ($)µ0 ($) )µ($)µ0 ($)
in the notation of [Vig16, §4.4], where we take the positive square root; and (3)Lemma 6.3.1. Assertion (1) is clear and assertion (2) is [Vig16, Cor. 5.28].
The following simple and presumably well-known lemma will be used below.

42

FLORIAN HERZIG, KAROL KOZIOL, AND MARIE-FRANCE VIGNÉRAS

Lemma 6.3.6. Suppose that ρ : WF → GLn (Qp ) is a smooth representation. Then forany γ ∈ WF the valuations of the eigenvalues of ρ(γ) depend only on the image of γ in
WF /IF ∼= Z.
Proof. Fix a geometric Frobenius element FrobF ∈ WF , and let v1 ≤ · · · ≤ vn denotethe valuations of the eigenvalues of ρ(FrobF ). We need to show that the eigenvaluesof ρ(FrobrF g) have valuations rv1 ≤ · · · ≤ rvn for any g ∈ IF . As ρ(IF ) is finite andnormalized by ρ(FrobF ), we see that ρ(FrobF )`r and ρ(IF ) commute for some ` ≥ 1, so
ρ(IF ) preserves the generalized eigenspaces of ρ(Frob`rF ). Hence the valuations of the`reigenvalues of ρ(FrobF g) are independent of g ∈ IF , and the claim follows by passing
to `-th powers.
∼

We now fix an isomorphism ı : Qp −→ C.
Proposition 6.3.7. Suppose that Π is an irreducible generic smooth representationof GLnd (F ) over C that is essentially unitarizable and such that the representation
π := | LJGLn (D) |(Π) of GLn (D) is non-zero. Suppose that ı−1 (π I(1) ) is a non-zero
integral HQp -module with non-supersingular reduction, and let v1 ≤ · · · ≤ vnd denotethe valuations of the eigenvalues of a geometric Frobenius on ı−1 (recF (Π)). Then there
exists 1 ≤ j ≤ n − 1 such that
jd
X
i=1

vi = −

d2 j(n − j)val(q).
2

Proof. Step 1: We compute the action of Z1 , . . . , Zn on the Hecke module ı−1 (π I(1) )and show in particular that it is scalar.
Note by Lemma 6.2.2 that π I(1) is a subquotient of (ρ01 × · · · × ρ0n )I(1) for someGL (D)
irreducible representations ρ0i of D× /D(1), and ρ01 × · · · × ρ0n ∼= IndB n (ρ01 ν d(n−1)/2 ⊗· · · ⊗ ρ0n ν −d(n−1)/2 ) (unnormalized induction). By [OV18, Prop. 4.4] we have
(6.3.8)ı−1 (ρ01 × · · · × ρ0n )I(1) ∼= ı−1 (ρ01 ν d(n−1)/2 ⊗ · · · ⊗ ρ0n ν −d(n−1)/2 )Z∩I(1) ⊗H

e HQp ,

Z,Qp ,θ

where we used the homomorphism θe of Lemma 6.3.3.
By Proposition 6.2.3(i) we can write ρ0i ∼= πD (ζi ) for some admissible tame pair
(Fi /F, ζi ). We let fi := [Fi : F ] and ei := d/fi . Let ζi0 := ı−1 (ζi ). From equations(6.3.4), (6.3.8) and Proposition 6.2.3(ii) we deduce that Zj acts on ı−1 (π I(1) ) as thescalarPX 2PY
2 ((n+1)j/2−
d ( i∈I i−(j+1
)
d
i)
0
−e
)
i
i∈I
2
λj :=
q
q
ζi ($)(6.3.9)
i∈I

|I|=j


X 2
Y
−d2 (2j )d (n−1)j/2
0
−ei
=q
q
ζi ($)
.
|I|=j

i∈I

Step 2: We complete the proof. By assumption, the Hecke module ı−1 (π I(1) ) is×
integral, so λi ∈ Zp for all i and λn ∈ Zp . Moreover, as the reduction of ı−1 (π I(1) ) is×

non-supersingular we deduce by Lemma 6.3.2 that λn−j ∈ Zp for some 1 ≤ j ≤ n − 1.

EXISTENCE OF SUPERSINGULAR REPRESENTATIONS

43

From now on assume for convenience that the ζi0 are ordered such that the sequenceQ2val(ζi0 ($)−ei ) is non-increasing. Consider the polynomial ni=1 (1−q d (n−1)/2 ζi0 ($)−ei X).
By (6.3.9) its Newton polygon is defined by the points (i, val(λi ) + d2 2i val(q)) for×
0 ≤ i ≤ n. From λn−j ∈ Zp , λi ∈ Zp , and the convexity of the quadratic functionx(x − 1)/2 we deduce that (n − j, d2 n−jval(q)) is a vertex of the Newton polygon. It
2follows for the sum of the largest j root valuations that  jXnn−j
d2 (n−1)/2 0−ei
2(6.3.10)val(qζi ($) ) = d−val(q).
22
i=1

Again by convexity we obtain the root valuation bounds(6.3.11)(6.3.12)2val(q d (n−1)/2 ζi0 ($)−ei ) ≥ d2 (n − j) val(q)val(qd2 (n−1)/2∀i ≤ j,ζi0 ($)−ei ) ≤ d2 (n − j − 1) val(q)∀i > j.

From Lemma 6.2.2 and Proposition 6.2.3(iii) we see that
recF (Π)|WF ∼
=

n eM
i −1
M
i=1 k=0e (f −1)

i i
F
IndW
WF (ηFii(e −1)/2−kζi )| · |F i.

If FrobF denotes a geometric Frobenius of WF , then FrobfFi is a geometric Frobeniusei (fi −1) 0Fζi ) have valuationof WFi . We see that all eigenvalues of FrobF on IndW
WF (ηFi
i

fi
1
1
0
0
fi val(ζi (FrobF )) = fi val(ζi ($)). Hence, for i ≤ j and 0 ≤ k ≤ ei − 1 all eigenvaluesei (fi −1) 0(e −1)/2−kFζi )| · |F ihave valuationof FrobF on IndW
WFi (ηFi

e − 1e − 11
1iival(ζi0 ($)) −− k val(q) ≤ val(ζi0 ($)ei ) +val(q)fi2d2
n − 1dd(2j − n)<d− (n − j) val(q) + val(q) =val(q),
22
2
where we used (6.3.11) and that ei − 1 < d. Similarly, for i > j and 0 ≤ k ≤ ei − 1
ei (fi −1) 0(e −1)/2−kF
we find that the eigenvalues of FrobF on IndW
ζi )| · |F ihave valuation
WF (ηFiigreater than d(2j−n)val(q). Therefore, from (6.3.10) we deduce that
2

 X
j eX
j
i −1
e − 1X
1i
0val(ζi ($)) −− k val(q) =val(ζi0 ($)ei )
vi =
fi
fi
2
i=1
i=1
i=1 k=0

  

n−j
n
j(n − 1)d2 j(n − j)= d2−+val(q) = −val(q).
2
2
2
2

jd
X


6.4. A reducibility lemma.
Let F0 denote the maximal absolutely unramified
intermediate field of F/Qp . The following lemma generalizes [EGH13, Prop. 4.5.2],
which dealt with regular crystalline Galois representations.
Lemma 6.4.1. Suppose that ρ : ΓF → GLn (Qp ) is a de Rham Galois representation.
Let v1 ≤ · · · ≤ vn denote the valuations of the eigenvalues of a geometric Frobenius

44

FLORIAN HERZIG, KAROL KOZIOL, AND MARIE-FRANCE VIGNÉRAS

element acting on WD(ρ), and for each embedding τ : F → Qp let hτ,1 ≤ · · · ≤ hτ,nP
P P
denote the τ -Hodge–Tate weights of ρ. Then ji=1 vi ≥ [F : F0 ]−1 ji=1 τ :F →Qp hτ,ifor any 0 ≤ j ≤ n.
Suppose that hτ,1 < hτ,n for some τ and that for some 1 ≤ j ≤ n − 1 we have
Pj
Pj P
−1
i=1 vi = [F : F0 ]i=1
τ :F →Qp hτ,i . Then ρ is reducible.
Proof. We first choose E/Qp a sufficiently large finite subextension of Qp /Qp , so thatin particular ρ can be defined over E and all embeddings τ have image containedin E. Choose F 0 /F a finite Galois extension over which ρ becomes semistable. Let
D := Dst (ρ|ΓF 0 ) be the covariantly associated free F00 ⊗Qp E-module, equipped withactions of ϕ, N , Gal(F 0 /F ), where F00 denotes theLmaximal absolutely unramified
intermediate field of F 0 /Qp . As usual, we write D ∼= σ:F 0 →E Dσ . Fix any embedding
00

σ0 : F00 → E and let f 0 := [F00 : Qp ]. Note that ϕf acts linearly on D and stabilizes
each Dσ .
0
By construction of WD(ρ) and Lemma 6.3.6, the eigenvalues of ϕf on Dσ0 have0
valuations rv1 ≤ · · · ≤ rvn , where r := [F00 : F0 ]. For any 0 ≤ j ≤ n, choose a ϕf 0
stable E-subspace Dσ0 0 ⊂ Dσ0 of dimension j such that the eigenvalues of ϕf on Dσ0 0
have valuations rv1 ≤ · · · ≤ rvj . Then Dσ0 0 is also N -stable, since N ϕ = pϕN . Now foreach σ : F00 → E choose the unique E-subspace
Dσ0 ⊂ Dσ that agrees with our choiceL00of Dσ0 when σ = σ0 and such that D := σ:F 0 →E Dσ0 is ϕ-stable. Then D0 is stable0
under the actions of F00 ⊗Qp E, ϕ, N . As in the proof of [EGH13, Prop. 4.5.2] (see also
[E:Q ] P
the proof of [BS07, Prop. 5.1]) we now compute that tN (D0 ) = [F0 :Qpp ] ji=1 vi and thatP
[E:Q ] P
tH (D0 ) ≥ [F :Qpp ] ji=1 τ :F →E hτ,i . By weak admissibility of D we have
(∗j )

j
X

−1

vi ≥ [F : F0 ]jXXhτ,i ,

i=1 τ :F →E

i=1

proving the first claim (with equality when j = 0 or j = n).
Now suppose that equality holds in (∗j ) for some 1 ≤ j ≤ n − 1. If vj = vj+1 thenmonotonicity of the hτ,i and (∗j+1 ) imply that equality holds in both (∗j−1 ), (∗j+1 )and that hτ,j = hτ,j+1 for all τ . Thus, by modifying j, we may assume without loss ofgenerality that equality holds in (∗j ) and that vj < vj+1 (as hτ,1 < hτ,n for some τ , byassumption).
0
Let D0 be the sum of all generalized ϕf -eigenspaces in the E-vector space D whose
corresponding eigenvalues have valuation at most rvj . As vj < vj+1 we see that D0 isa free F00 ⊗Qp E-module of rank j, stable under the actions of ϕ, N , and Gal(F 0 /F ).
Equality in (∗j ) gives that tH (D0 ) = tN (D0 ), so ρ admits a j-dimensional subrepresentation.
Remark 6.4.2. The lemma can fail when hτ,1 = hτ,n for all τ . For example, let F/Qp be×a quadratic extension and χ : ΓF → Qp a potentially unramified character that doesΓQ

not extend to ΓQp . Then IndΓFp χ is irreducible and de Rham with all Hodge–Tateweights equal to 0 (since it is potentially unramified). In particular, v1 = v2 = 0.
×d×
Concretely, via local class field theory, we can take F = Q 2 and χ : Q→ Q tamepand non-trivial on µp+1 (Qp2 ).

p2pEXISTENCE OF SUPERSINGULAR REPRESENTATIONS

45

6.5. On base change and descent for compact unitary groups. The purposeof this section is to discuss base change and descent results for compact unitary groups
that go slightly beyond those in [Lab11], namely allowing that the unitary group is
non-quasisplit at some finite places. The proofs will be provided by Sug Woo Shin in
Appendix A.
Suppose that Fe/Fe+ is a CM extension of number fields with Fe+ 6= Q and G a
unitary group over Fe+ such that(i) G/Fe is an inner form of GLnd ;(ii) G(Feu+ ) is compact for any place u | ∞ of Fe+ ;(iii) G is quasi-split at all finite places that are inert in Fe/Fe+ .
Let c denote the complex conjugation of Fe/Fe+ . Let ∆+ (G) denote the set of finite
places of Fe+ where G is not quasi-split. This is a finite set of places that split or ramify
in Fe. Let ∆(G) denote the set of places of Fe lying over a place of ∆+ (G).
Proposition 6.5.1. Suppose that π is a (cuspidal ) automorphic representation ofG(AFe+ ). Then there exists a partition n = n1 + · · · + nr and discrete automorphic∼ crepresentations Πi of GLni d (AFe ) satisfying Π∨
i = Πi such that Π := Π1  · · ·  Πr isa weak base change of π. More precisely, at every finite split place v = wwc of Fe+ wehave | LJG(Few ) |(Πw ) ∼= πv as representations of G(Few ) ∼= G(Fev+ ), and at infinity thecompatibility is as in [Lab11, Cor. 5.3].
Proposition 6.5.2. Suppose that Fe/Fe+ is unramified at all finite places and that Π isa cuspidal automorphic representation of GLnd (AFe ) such that Π∨ ∼= Πc , Π∞ is cohomological, and Πw is supercuspidal for all w ∈ ∆(G) (in particular | LJG(Few ) |(Πw ) 6= 0).
Then there exists a (cuspidal ) automorphic representation π of G(AFe+ ) such that at
every finite split place v = wwc of Fe+ we have | LJG(Few ) |(Πw ) ∼= πv as representations+of G(Few ) ∼= G(Fev ).
6.6. Supersingular representations of GLn (D). We now prove the existence ofsupersingular (equivalently, supercuspidal) representations of GLn (D) and PGLn (D).
Theorem 6.6.1. Suppose that C is algebraically closed of characteristic p. For anysmooth character ζ : F × → C × there exists an irreducible admissible supercuspidal
C-representation of GLn (D) with central character ζ. In particular, there exists anirreducible admissible supercuspidal C-representation of PGLn (D).
Corollary 6.6.2. If C is any field of characteristic p, then PGLn (D) admits an irreducible admissible supercuspidal representation over C.
The proof uses Galois representations associated to automorphic representations oncertain unitary groups. We now make a few relevant definitions in preparation for the
proof.
∼As in §6.3 we fix an isomorphism ı : Qp −→ C. Recall that if Fe/Fe+ is a CM
extension of number fields and Π is a regular algebraic cuspidal polarizable automorphic
representation of GLn (AFe ) (in the sense of [BLGGT14b, §2.1]) we have an associatedsemisimple potentially semistable p-adic Galois representation rp,ı (Π) : ΓFe → GLn (Qp )46

FLORIAN HERZIG, KAROL KOZIOL, AND MARIE-FRANCE VIGNÉRAS

that satisfies and is determined by local-global compatibility with Π at all finite places[BLGGT14b, Thm. 2.1.1], [BLGGT14a].
Suppose that Fe+ 6= Q and that G is a unitary group over Fe+ as in §6.5. If π is an
automorphic representation of G(AFe+ ), then its weak base change Π = Π1  · · ·  Πrof Proposition 6.5.1 is regular algebraic and each Πi is polarizable. By the Moeglin–
Waldspurger classification of the discrete spectrum and the previous paragraph it follows that Π has an associated semisimple potentially semistable p-adic Galois representation rp,ı (π) = rp,ı (Π) : ΓFe → GLnd (Qp ) that satisfies and is determined by local-global
compatibility with π at all finite places of Fe that split over Fe+ and are not contained in∆(G). (We note that the Chebotarev density theorem shows that the set of Frobenius
elements at places w of Fe that split over Fe+ is dense in ΓFe .) In particular, if Π is not
cuspidal, then rp,ı (π) is reducible.
Proof of Theorem 6.6.1.
Step 0: We show that it suffices to prove the theorem when C = Fp .
×
Given a smooth character ζ : F × → C × we can define ζ 0 : F × → Fp by extending×ζ|O× (which is of finite order and hence takes values in Fp ) arbitrarily. If Theorem 6.6.1
F

holds over Fp , there exists an irreducible admissible supercuspidal Fp -representation πof GLn (D) with central character ζ 0 . Then by Step 3 of the proof of Proposition 3.2.1
there exists an irreducible admissible supercuspidal C-representation π 0 of GLn (D)with central character ζ 0 . As C is algebraically closed, a suitable unramified twist ofπ 0 has central character ζ.
We will assume from now on that C = Fp .
Step 1: We find a CM field Fe with maximal totally real subfield Fe+ 6= Q and a placev | p of Fe+ such that(i) Fe/Fe+ is unramified at all finite places;(ii) any place of Fe+ that divides p splits in Fe;(iii) Fe+ ∼= F;v

and a cyclic totally real extension L+ /Fe+ of degree nd in which v is inert.
By Krasner’s lemma we can find a totally real number field H, a place u of H, and∼an isomorphism Hu −→ F . Now we apply [Hen83, Lemma 3.6] and its proof to findfinite totally real extensions L+ /Fe+ /H and a place v of Fe+ above u such that L+ /Fe+is cyclic of degree nd, Fev+ = Hu , and v is inert in L+ . (We briefly recall the proof:
pick a monic polynomial Q of degree nd over F whose splitting field is the unramified
extension of degree nd. Then let L+ be the splitting field of a monic polynomial P
over H that is u-adically very close to Q and let Fe+ be the decomposition field of someplace above u. We can use sign changes of P at real places to ensure that L+ is totallyreal.)
Now pick any totally imaginary quadratic extension Fe/Fe+ in which any place dividing p splits. By [CHT08, Lemma 4.1.2] we can find a finite solvable Galois totallyreal extension K + /Fe+ that is linearly disjoint from L+ /Fe+ , such that v splits in K + ,and such that for any prime v 0 of Fe+ that ramifies in Fe and any prime w0 of K + abovev 0 the extension Kw+0 /Fev+0 is isomorphic to the extension Fev0 /Fev+0 . Then we can replace
Fe/Fe+ by K + Fe/K + , L+ by K + L+ , and v by any place of K + lying above v to ensure

EXISTENCE OF SUPERSINGULAR REPRESENTATIONS

47

that, without loss of generality, Fe/Fe+ is unramified at all finite places. (In particular,
we can always achieve Fe+ 6= Q in this way.)
We let w denote a place of Fe lying over v and fix an isomorphism of topological∼fields Few −→ F . We let L := L+ Fe and let c denote the unique complex conjugation ofL.
Step 2: Letting v1 - p denote any place of Fe+ that is inert in L+ and splits in Fe, we
now find a unitary group G over Fe+ such that(i) G/Fe is an inner form of GLnd ;(ii) G(Feu+ ) is compact for any place u | ∞ of Fe+ ;(iii) G(Few ) ∼= GLn (D);(iv) G is quasi-split at all finite places not contained in {v, v1 }.
Let G∗ denote the unique quasi-split outer form of GLnd over Fe+ that splits over Fe.
By [Clo91, §2] we can find an inner form G of G∗ that satisfies all the above conditions.
(If nd is odd we do not need the auxiliary place v1 . If nd is even we use v1 to ensure
our local conditions can be globally realized.)
The set ∆+ (G) (defined in §6.5) contains v if d > 1 and is contained in {v, v1 }. Any
place of ∆(G) is inert in L and splits over Fe+ , and the set ∆L (G) of places of L lying
above ∆(G) is in canonical bijection with ∆(G).
For any finite place v 0 6∈ ∆+ (G) of Fe+ that splits as v 0 = w0 w0c in Fe we obtain an∼
isomorphism ιw0 : G(Fev+0 ) = G(Few0 ) −→ GLnd (Few0 ) that is unique up to conjugacy.
Moreover, c ◦ ιw0 and ιw0c differ by an outer automorphism of GLnd (Few0c ). We also fix∼
an isomorphism ιw : G(Fev+ ) = G(Few ) −→ GLn (D). (It is canonical, up to conjugacy,
by condition (i).)××
Step 3: We find an algebraic Hecke character χ : A×
L /L → C with associated×
potentially crystalline p-adic Galois representation ψ = rp,ı (χ) : ΓL → Qp (cf. [CHT08,
Lemma 4.1.3]) such that(i) ψψ c = ε−(nd−1) ;WFe0(ii) for any place w0 ∈ ∆L (G) the induced representation IndWLw χw0 is irrew0
ducible;Γ(iii) the representation r := IndΓFLe ψ has regular Hodge–Tate weights, i.e., for eachΓ
κ0 : Fe → Qp the nd integers HTκ0 (Ind Fe ψ) are pairwise distinct;
ΓLΓe(iv) the restriction r|ΓFe to ΓFew of the reduction r ∼= IndΓFL ψ is irreducible.
w

We first introduce some notation. Let ∆p denote the places w0 of L that divide p.
Note that, by construction, any place w0 ∈ ∆L (G) ∪ ∆p splits over L+ , i.e. w0 6= w0c .
Let SK := Homcts (K, Qp ) for any topological field K of characteristic zero and Sk :=Hom(k, Fp ) for any field k of characteristic p.
×
Our strategy is to carefully choose continuous characters θw0 : ΓLw0 → Qp for anyc )|w0 ∈ ∆L (G) ∪ ∆p that satisfy (θw0 θw= ε−(nd−1) |IL 0 and are potentially crys0c IL
w0w
0talline when w ∈ ∆p . We then deduce by [BLGGT14b, Lemma A.2.5(1)] that there×
exists a character ψ : ΓL → Qp such that ψψ c = ε−(nd−1) and ψ|IL 0 = θw0 |IL 0 for
w

w

48

FLORIAN HERZIG, KAROL KOZIOL, AND MARIE-FRANCE VIGNÉRAS

all w0 ∈ ∆L (G) ∪ ∆p . In particular, ψ is potentially crystalline, and we let χ be theassociated algebraic Hecke character. It follows that condition (i) holds.
×d×
For any w0 ∈ ∆ (G) we can choose a smooth character ζ 0 : Γab ∼→ Q such=LLw

that the Gal(Lw0 /Few0 )-conjugates of ζw0 |O×

Lw0

w0

p

are pairwise distinct. (For example, we

Lw0

×
can take a faithful character of kL× 0 and inflate it to OL.) We may assume withoutw
w0
c0
loss of generality that ζw ζw0c = 1.
Now suppose that w0 ∈ ∆p . Suppose that we are given any integers λκ (κ ∈ SL )×
cr : Γ
satisfying λκ + λκc = nd − 1 for all κ ∈ SL . Let θw0
Lw0 → Qp be any crystallinecr ) = λ for all κ ∈ Scharacter with HTκ (θw0κ
Lw0 ⊂ SL . Without loss of generality, bycr (θ cr )c = ε−(nd−1) .
our constraint on the λκ , we may assume that θw
0w0c
0
For w ∈ ∆L (G) ∪ ∆p define



ζw0
cr ζ 0θw0 := θw
0 w

 cr
θw 0

if w0 ∈ ∆L (G) − ∆p ;if w0 ∈ ∆L (G) ∩ ∆p ;if w0 ∈ ∆p − ∆L (G).

This completes the construction of a potentially crystalline character ψ and its associated algebraic Hecke character χ. By construction, for any w0 ∈ ∆L (G) the character ıζw0 |IL 0 corresponds to χw0 |O× under the local Artin map. Therefore, since thew

Lw 0

Gal(Lw0 /Few0 )-conjugates of ζw0 |O×

are pairwise distinct, we deduce that condition (ii)Lw 0holds.
Finally, we will choose the integers λκ (κ ∈ SL ) so that conditions (iii) and (iv) hold.
Note that condition (iii) is equivalent to the condition(iii0 ) for any κ0 ∈ SFe the nd integers {λκ : κ ∈ SL , κ|Fe = κ0 } are pairwise distinct.
First choose the λκ for those κ ∈ SL that do not induce either of the places w, wc
on L so that condition (iii0 ) holds for any κ0 ∈ SFe not inducing either of the places w,
wc on Fe. It remains to choose the λκ for those κ that induce the place w on L (since
the remaining λκ are determined by the condition λκ + λκc = nd − 1 for all κ), i.e. forκ ∈ SLw .
ΓeTo choose the λκ for κ ∈ SL , we note that r|Γ ∼= Ind Fw (ψ|Γ ) is irreducible if
w

ew
F

ΓL w

Lw

and only if the Gal(Lw /Few )-conjugates of ψ|ΓLw are pairwise distinct, or equivalently ifithe characters ψ|qIL (0 ≤ i ≤ nd − 1) are pairwise distinct. (Recall that q = #kF .) Wew∼ θcr ζw |I if d > 1 or ψ|I ∼have ψ|I == θcr |I otherwise. By [GHS18, Cor. 7.1.2],
Lw

w

Lw

Lw

w

Lw

cr |
noting our opposite conventions concerning Hodge–Tate weights, we have θw
ILw =
Q
×
×
× σ
−b
σ
→ Fp under localσ∈SkL ωσ , where ωσ corresponds to the character OLw  kLw −w
Pclass field theory and bσ := κ∈SL :κ=σ λκ . Fix any σ ∈ SkLw and s ∈ Z. Then wew

ican choose the λκ for κ ∈ SLw so that ψ|ILw = ωσs . By taking s so that the ωσsq
(i = 0, . . . , nd − 1) are pairwise distinct (taking, for example, s = 1), condition (iv)holds. Finally, we can ensure that condition (iii0 ) holds for all κ0 ∈ SFew while keeping

EXISTENCE OF SUPERSINGULAR REPRESENTATIONS

49

r|IFe unchanged by varying the λκ (for κ ∈ SLw ) modulo q nd − 1. This completesw
Step 3.
Step 4: Using automorphic induction and descent we define an automorphic repreΓ
sentation π 0 of G(AFe+ ) with associated Galois representation r = IndΓFLe ψ.
Let Π00 denote the automorphic induction of χ with respect to the cyclic extension
L/Fe. It is an automorphic representation of GLnd (AFe ) that is parabolically induced
from a cuspidal representation. (For the functoriality of automorphic induction incyclic extensions we refer to [Hen12], which shows in particular that it is compatible
with local automorphic induction at all places. Note the results of [Hen12] apply tounitary representations, but by twisting they continue to hold for twists of unitary
representations.)
We claim that Π00 is cuspidal. This follows from [Hen12], Theorems 2, 3, and Proposition 2.5, provided that the Hecke characters {χσ : σ ∈ Gal(L/Fe)} are pairwise distinct.
Equivalently, the Galois characters {ψ σ : σ ∈ Gal(L/Fe)} are pairwise distinct, whichΓin turn is equivalent to the condition that IndΓFLe ψ is irreducible. This is a consequenceof condition (iv) in Step 3, so Π00 is cuspidal.
(nd−1)/2−(nd−1)Let Π0 := Π00 ⊗ | det | e. By condition (i) in Step 3 we have χχc = | · |L
,
F
0
∨
0c
0
∼
hence (Π ) = Π . On the other hand, Π∞ is cohomological by [Clo90, Lemma 3.14],
as it is regular by condition (iii0 ) in Step 3. It follows that Π0 is regular algebraic andpolarizable in the sense of [BLGGT14b, §2.1], so we have an associated Galois representation rp,ı (Π0 ). By local-global compatibility at unramified places and Chebotarev
Γe
we deduce that rp,ı (Π0 ) ∼= Ind F ψ.
ΓLFor w0 ∈ ∆(G) the local factor Π0w0 is supercuspidal, as recFe 0 (Π0w0 ) is irreducible byw
condition (ii) in Step 3. It follows from what we recalled in §6.2 that | LJG(Fe 0 ) |(Π0w0 ) 6=w
0.
By Proposition 6.5.2 we deduce that Π0 descends to a (cuspidal) automorphic representation π 0 of G(AFe+ ), such that for all finite places v 0 6∈ ∆+ (G) of Fe+ that split
as v 0 = w0 w0c in Fe we have πv0 0 ∼= Π0w0 as representations of G(Fev+0 ) ∼= GLnd (Few0 ). WeΓededuce that rp,ı (π 0 ) ∼= IndΓFL ψ.
Step 5: We use the automorphic representation π 0 to define an irreducible admissible
Fp -representation σ of G(Fev+ ) ∼= GLn (D).
Q
Fix a maximal compact open subgroup Kp of v0 |p G(Fev+0 ). If U is any compact
open subgroup of Kp G(A∞,p) and W is any Zp [Kp ]-module, we let S(U, W) be theFe+
Zp -module of functions f : G(Fe+ )\G(A∞ ) → W such that f (gu) = u−1 f (g) for allpFe+g ∈ G(A∞) and u ∈ U (where up denotes the projection of u to Kp ).
Fe+
Using the compactness of G at infinity, we see as in [EGH13, Lemma 7.1.6] thatQ
there exists a Qp -algebraic representation Walg of v0 |p G(Fev+0 ) over Qp such thatlimU S(U, Walg ) contains ı−1 π 0∞ as G(A∞)-representation. Choose a Kp -invariantFe+−→◦ in W◦Zp -lattice Walg
alg and let Walg := Walg ⊗Zp Fp .
Q
Pick a compact open subgroup U = v0 -∞ Uv0 of G(A∞e+ ) such thatF(i) (π 0∞ )U 6= 0;

50

FLORIAN HERZIG, KAROL KOZIOL, AND MARIE-FRANCE VIGNÉRAS(ii) there exists a place v 0 - p∞ of Fe+ such that Uv0 contains no element of finite
order otherQthan the identity;(iii) the group v0 |p Uv0 is contained in Kp and acts trivially on Walg .
Q
Note that condition (ii) implies that for any compact open subgroup U 0 = Up0 v0 -p∞ Uv0
with Up0 ≤ Kp we have S(U 0 , W) ∼= W ⊕s as Zp -modules for some s ≥ 1 depending only0
on U . In particular,∼S(U 0 , W) ⊗Zp R −→ S(U 0 , W ⊗Zp R)(6.6.3)for any Zp -algebra R (see e.g. [EGH13, §7.1.2]). We will apply this with R = Qp and
R = Fp .
Let P denote the set of places w0 - p of Fe that split over a place v 0 of Fe+ not containedin ∆+ (G), and are such that Uv0 is a maximal compact subgroup of G(Fev+0 ). For eachsuch w0 we conjugate the isomorphism ιw0 of Step 2 so that ιw0 (Uv0 ) = GLnd (OFe 0 ).
w
Note that the set P has finite complement in the set of places of Fe that split over Fe+ .
(i)

Let TP denote the commutative polynomial Zp -algebra in the variables Tw0 for w0 ∈ P
and 0 ≤ i ≤ nd, acting on any S(U, W) as double coset operators as in [EGH13, §7.1.2].
Note that the ring TP acts by scalars on (ı−1 π 0∞ )U inside S(U, Walg ) and stabilizes the◦ ). Therefore there exists a unique maximal ideal m of TP with
Zp -lattice S(U, Walg
residue field Fp such that (ı−1 π 0∞ )U ⊂ S(U, Walg )m .
Applying (6.6.3) and localizing at m we obtain that S(U, Walg )m 6= 0. Then
S(U, Fp )m ⊗Fp Walg ∼= S(U, Walg )m 6= 0,
Qwhere the isomorphism uses condition (iii) on U . Writing U v :=v 0 6=v Uv 0 andvvv
S(U , Fp ) := limU S(U Uv , Fp ), we have S(U , Fp )m 6= 0. This is a non-zero admissible−→ vsmooth representation of G(Fe+ ) ∼= GLn (D), using the isomorphism ιw of Step 2. Letv

σ be an irreducible (admissible) GLn (D)-subrepresentation of S(U v , Fp )m , which exists
by the proof of Lemma 9.9 in [Her11] or [HV12, Lemma 7.10].
Step 6: We show that σ is supersingular, or equivalently, supercuspidal.
By [OV18, Thm. 3] it suffices to show that the HFp -module σ I(1) is supersingular,
where I(1) denotes the pro-p Iwahori subgroup of GLn (D) ∼= G(Fev+ ) defined in §6.3.
In fact, we will even show that (S(U v , Fp )m )I(1) ∼= S(U v · I(1), Fp )m is supersingular.
Assume by contradiction that this is false, so one of the operators Zj for 1 ≤ j ≤ n − 1
has a nonzero eigenvalue λj on S(U v · I(1), Fp )m .
Again from (6.6.3) we know that S(U v · I(1), Zp ) ⊗Zp R ∼= S(U v · I(1), R) for R = Qp
and R = Fp . By applying [EGH13, Lemma 4.5.1] (a version of the Deligne–Serrelemma) with A = TP [Zj ], M = S(U v · I(1), Zp ), n the maximal ideal of A generated
by m and Zj − λj , we deduce that there exists a homomorphism θ : A → Zp such×that the θ-eigenspace of S(U v · I(1), Qp ) is non-zero, ker(θ|TP ) = m, and θ(Zj ) ∈ Zp .
By [EGH13, Lemma 7.1.6], there exists an automorphic representation π of G(AFe+ )satisfyingv(i) (ı−1 π ∞ )U ·I(1) has a non-trivial θ-eigenspace;(ii) π∞ is trivial.

EXISTENCE OF SUPERSINGULAR REPRESENTATIONS
I(1)It follows from (i) that ı−1 πv

51

6= 0 is an integral HQp -module whose reduction is nonI(1)supersingular. (A priori, we get that (ı−1 πv )⊕s is integral for some s ≥ 1, but thenwe can project to any factor. Note that any finitely generated submodule of a finite
free Zp -module is free.)
By local-global compatibility and [CHT08, Cor. 3.1.2], for any w0 ∈ P the charPi
0 i(i−1)/2 T (i) X nd−i modulo m,acteristic polynomial of r(Frobw0 ) equals nd
i=0 (−1) (Nw )w0where Frobw0 denotes a geometric Frobenius element at w0 . The same is true forrp,ı (π), as ker(θ|TP ) = m, and hence we deduce by the Chebotarev density theorem
that rp,ı (π) ∼= r.
By Proposition 6.5.1 we obtain an automorphic representation Π of GLnd (AFe ) withassociated Galois representation rp,ı (Π) lifting r such that | LJG(Fe 0 ) |(Πw0 ) ∼= πv0 for all
w
0
+
0
0
0c
finite places v of Fe that split as v = w w in Fe. As r is irreducible by construction
we know that Π is cuspidal. In particular, Πw0 is essentially unitarizable and genericfor each finite place w0 of Fe. Let v1 ≤ · · · ≤ vnd denote the valuations of the eigenvaluesof a geometric Frobenius on ı−1 (recF (Πw )). From Proposition 6.3.7 (applied to Πw )we deduce that there exists 1 ≤ j ≤ n − 1 such thatjd
X

(6.6.4)vi = −

i=1

d2 j(n − j)val(q).
2

Note that the infinitesimal character of Π is the same as that of the trivial representation. By [BLGGT14b, Thm. 2.1.1] we deduce that HTτ (rp,ı (Π)|ΓFe ) = {0, 1, . . . , nd−1}w(1−nd)/2F−ss∼e
for all τ : Fw → Qp and that ıWD(rp,ı (Π)|Γ )). To= recF (Πw ⊗ | det |ew
FFgether with (6.6.4) it follows thatjdX

d2 j(n − j)val(q) + jd val(q (nd−1)/2 ) =vi0 = −
2

i=1

 
jd
val(q),
20 denote the valuations of the eigenvalues of a geometric Frobenius
where v10 ≤ · · · ≤ vnd
on WD(rp,ı (Π)|ΓFe ). By Lemma 6.4.1, noting that val(q) = [F0 : Qp ], it follows thatwrp,ı (Π)|ΓFe is reducible, which contradicts that its reduction r|ΓFe is irreducible byw
w
Step 3.
Step 7: We fix the central character.
×
Suppose we are given a smooth character ζ : F × → Fp . As in Step 0 it is enoughto construct an irreducible admissible supercuspidal representation such that OF× acts
via ζ|O× .
F
Note that σ has a central character χσ , as it is irreducible and admissible. We
claim that χσ |O× = det(r|IFe ) · εnd(nd−1)/2 under the local Artin map. The centralFwcharacter of the GLn (D)-representation ı−1 πv in Step 6 lifts χσ and is equal to thecentral character of ı−1 Πw . (This equality follows from the definition of LJ in [Bad08,§2.7], noting that Πw is generic and hence fully induced from an essentially squareintegrable representation.) By local-global compatibility at p (cf. Step 6) the latter character equals WD(det rp,ı (Π)|ΓFe )|IFe on OF× , under the local Artin map. As
w

w

52

FLORIAN HERZIG, KAROL KOZIOL, AND MARIE-FRANCE VIGNÉRAS

rp,ı (Π)|ΓFe has parallel Hodge–Tate weights 0, 1, . . . , nd − 1, we have det rp,ı (Π)|IFe =ww

ε−nd(nd−1)/2 · WD(det rp,ı (Π)|ΓFe )|IFe and hence deduce the claim.
ww
It thus suffices to show that in Step 3 above we can choose r such that det(r|IFe ) isw
any prescribed character that is extendable to ΓFew . Let us fix any κ ∈ SkLw and writeiψ|ILw = ωκs for some s ∈ Z. Then the condition that the ψ|qIL (i = 0, 1, . . . , nd − 1)w
are pairwise distinct means:nd−1) ∀` | nd, ` < nd.
(mod qq` −1Q
qis0On the other hand, det(r|IFe ) = nd−1
i=0 ψ|IL = ωκ0 , where κ ∈ SkFe is the restrictions 6≡ 0(6.6.5)

w

×

w

w

of κ to kFew . As any character ΓFew → Fp restricts to a power of ωκ0 on inertia, we canprescribe det(r|IFe ) if and only if we can choose s in any residue class modulo q − 1.
w

nd −1
Since qq` −1≥ q + 1 for any ` | nd, ` < nd, it follows that we can pick any s in theinterval [1, q − 1], completing the proof.

Proof of Corollary 6.6.2. Going back to Step 5 of the proof of Theorem 6.6.1, it isclear that the representation S(U v , Fp )m 6= 0 is defined over a finite field (as r is), andhence so is its irreducible subrepresentation σ. This proves the corollary when C is asufficiently large finite field of characteristic p. We conclude by Proposition 3.2.1. (F. Herzig) Department of Mathematics, University of Toronto, 40 St. George Street,
Room 6290, Toronto, ON M5S 2E4, Canada
Email address: herzig@math.toronto.edu(K. Koziol) Department of Mathematics, University of Michigan, Ann Arbor, MI 481091043
Email address: kkoziol@umich.edu(M.-F. Vignéras) Institut de Mathématiques de Jussieu, 4 place Jussieu, 75005 Paris,
FranceEmail address: vigneras@math.jussieu.fr

Appendix A. Base change
Sug Woo Shin1
In this appendix we will prove Propositions 6.5.1 and 6.5.2.
We need a character identity for the Jacquet–Langlands correspondence. We fix
compatible Haar measures on GLnd (F ) and GLn (D) in the sense of [Kot88, p.631].
We say that f ∈ Cc∞ (GLn (D)) and f ∗ ∈ Cc∞ (GLnd (F )) are associated, or that f ∗ isa transfer of f , if the orbital integral identity Oδ (f ) = Oδ∗ (f ∗ ) holds for every regularsemisimple elements δ ∈ GLn (D) and δ ∗ ∈ GLnd (F ) with the same characteristic
polynomial. (We use the same Haar measures on the centralizers of δ and δ ∗ in GLn (D)and GLnd (F ), respectively, to compute the orbital integrals.) A well-known fact, provenin [DKV84], is that every f ∈ Cc∞ (GLn (D)) admits a transfer in Cc∞ (GLnd (F )). (Thisis a special case of the Langlands–Shelstad transfer.) Let e(GLn (D)) ∈ {±1} denote
the Kottwitz sign [Kot83]. Explicitly e(GLn (D)) = (−1)nd−n .
1Supported by NSF grant DMS-1802039.

Base change

53

Proposition A.0.1. Let π ∗ be an irreducible unitarizable representation of GLnd (F ).
For every associated pair f ∈ Cc∞ (GLn (D)) and f ∗ ∈ Cc∞ (GLnd (F )), we havetr π ∗ (f ∗ ) = e(GLn (D)) · tr | LJGLn (D) |(π ∗ ) (f ).
Proof. This follows from [Bad07, Prop 3.3] and the Weyl integration formula [DKV84,
A.3.f] for GLn (D) and GLnd (F ).
We assume that the CM extension Fe/Fe+ and the unitary group G over Fe+ are asin Section 6.5.
Write G∗ for a quasi-split inner twist of G over Fe+ (with an isomorphism between∗
G and G over an algebraic closure of Fe+ ). By convention, every trace considered on
p-adic or adelic points of G∗ over Fe (as opposed to over Fe+ ) will mean the twisted trace
relative to the action of Gal(Fe/Fe+ ) on ResFe/Fe+ G∗ (with the Whittaker normalization),unless specified otherwise.
Proof of Proposition 6.5.1. This proposition is implied by [Lab11, Cor. 5.3] except possibly the relation | LJG(Few ) |(Πw ) ∼= πv .2 We elaborate on this point. Thus we assume
v = wwc as in the proposition. We will omit the subscript for | LJ | when there is little
danger of confusion.
Let S be a finite set of places of Fe+ containing all infinite places as well as all finite
place where either π or G is ramified. Denote by Sf the subset of finite places in S.
In particular Sf ⊃ ∆+ (G). For an irreducible admissible representation σ of G(AFe+ )unramified outside S, we write BC(σ S ) = ΠS to mean that the local unramified basechange of σu is Πu at all places u ∈/ S. (The unramified base change is defined via theSatake transform.) Using the Langlands parametrization at archimedean places, wewrite BC(σ∞ ) = Π∞ to mean that the local base change of σ∞ is Π∞ .
For each finite place u and fu ∈ Cc∞ (G(Feu+ )) let fu∗ ∈ Cc∞ (G∗ (Feu+ )) denote a transfer.
∗ e∗e+There exists φu ∈ Cc∞ (GQ (F ⊗Fe+ Fu )) whose
Q base change transfer is fu by [Lab11,
Lem. 4.1]. Write fSf := u∈Sf fu and φSf := u∈Sf φu .
Let Πv := Πw ⊗Πwc be the v-component of Π, which is a representation of G∗ (Fe ⊗Fe+
Fev+ ). Let πv∗ := Πw via the isomorphism G∗ (Fev+ ) ∼= G∗ (Few ). Then we have the following
character identities, where tr Πv (φv ) means the twisted trace by abuse of notation:(A.0.2)tr Πv (φv ) = tr πv∗ (fv∗ ) = e(G(Fev+ )) · tr (| LJ |(πv∗ )) (fv ).

The first equality holds by the same computation as for [Rog90, Prop. 4.13.2 (a)]. The
second equality is Proposition A.0.1. On the other hand, the trace formula argumentof [Lab11, Thm. 5.1] shows
X(A.0.3)m(σ) tr σSf (fSf ) = c · tr ΠSf (φSf ),σ
2In fact this assertion is implicit in [Lab11, Cor. 5.3] where it reads “Aux places non ramifiées ou

décomposées la correspondance σv 7→ πv est donnée par le changement de base local.” However whenv = wwc the author introduced the notion of local base change (§4.10 of op. cit.) only when U is a
general linear group at v (in his notation). We need the case when U is a nontrivial inner form of ageneral linear group at v.

54

Sug Woo Shinwith a constant c and the automorphic multiplicity m(σ) ∈ Z≥0 , where the sum runs
over σ such that BC(σ S ) = ΠS and BC(σ∞ ) = Π∞ . Again the trace
Q on the righthand side is the twisted trace. Since (A.0.3) holds for each f ∞ = u-∞ fu (and fu∗
and φu constructed from fu at each u as above), we choose fu to be the characteristic
function on a sufficiently small compact open subgroup of G(Feu+ ) at u ∈ Sf \{v}. Thentr σu (fu ) ≥ 0, so we obtainXn(Πv , σ) tr σv (fv ) = c0 · tr Πv (φv ), with n(Πv , σ) ≥ 0,σwhere c0 is a new constant and the sum runs over σ such that BC(σ S ) = ΠS , BC(σ∞ ) =Π∞ , and tr σu (fu ) 6= 0 at every u ∈ Sf \{v}. Notice that σ = π contributes to the sumwith n(Πv , π) > 0, by choice of fu at u ∈ Sf \{v}. By choosing a suitable fv we deducethat c0 6= 0. Substituting (A.0.2) we obtainXn(Πv , σ) tr σv (fv ) = c0 · e(G(Fev+ )) · tr (| LJ |(πv∗ )) (fv ),σwith the sum running over the same set of σ. Since the sum is not identically zero,| LJ |(πv∗ ) is irreducible (rather than 0). By linear independence of characters of G(Fev+ ),we deduce that the coefficients on the left-hand side are zero unless σv ∼= | LJ |(πv∗ ).
∗∼Since n(Πv , π) > 0, we must have πv = | LJ |(πv ), noting that no cancellation takes
place in the sum as the coefficients are non-negative.
Proof of Proposition 6.5.2. The proposition would follow from [Lab11, Thm. 5.4] but
we need some care since our G is not quasi-split3; we also need some more information at
split places. Thus we sketch the trace formula argument. Again we drop the subscriptfrom | LJ |.
The argument of [Lab11, Thm. 5.4] shows the identity (adapted to our notation)
X(A.0.4)m(σ) tr σ(f ) = tr Π(φ)σ

Qwith the functions φ = u φu on G∗ (AFe ) and f = u fu on G(AFe+ ) as in the proof
there, where the sum runs over automorphic representations σ of G(AFe+ ) with multiplicity m(σ) whose weak base change is Π. The right-hand side is interpreted as thetwisted trace by the convention mentioned earlier.
The key point to show is that the right-hand side does not always vanish. There is a
subtlety when G is not quasi-split, because not every test function φ may be allowed in(A.0.4). The potential problem is that a base change transfer of φu at u from G∗ (Feu )to G∗ (Feu+ ) is not in the image of endoscopic transfer from G(Feu+ ) to G∗ (Feu+ ). We makea choice of test functions avoiding this problem.
At ∞ one does the same as in Labesse’s proof so that tr Π∞ (φ∞ ) 6= 0. At finite
places u, we recall that fu and φu are related as follows: writing fu∗ for a transfer of fu
Q

3Contrary to the assumption on U above [Lab11, Thm. 5.4] that U is quasi-split at all inert places,

it seems the assumption ought to be that U is quasi-split at all finite places. We believe that “Lesecond membre étant non identiquement nul” (in the proof of [Lab11, Thm. 5.4], between the second
and third displays) is not always true, e.g. if Πw is a principal series representation at a non-quasi-split
place that splits in Fe. (See the third paragraph of the current proof.) If it were true, we could deduce
Proposition 6.5.2 directly from [Lab11, Thm. 5.4].

Base change

55

from G(Feu+ ) to G∗ (Feu+ ), the functions fu∗ and φu are associated in the sense of [Lab11,
4.5]. There is no problem when u ∈/ ∆+ (G) as G and G∗ are isomorphic outside ∆+ (G);more precisely we choose φu on G(Fe ⊗Fe+ Fev+ ) such thattr Πu (φu ) 6= 0
and choose fu to be a base change transfer to G(Fev+ ) (which exists by [Lab11, Lem. 4.1],
where it is called an “associated” function). At each v = wwc ∈ ∆+ (G), choose fv andlet fv∗ be a transfer. Write πv∗ := Πw via the chosen isomorphism G∗ (Few ) ∼= G∗ (Fev+ ).
Then by Proposition A.0.1,
tr πv∗ (fv∗ ) = e(G(Fev+ )) · tr (| LJ |(πv∗ )) (fv ).
Note that | LJ |(πv∗ ) is irreducible (i.e. nonzero) since πv∗ is supercuspidal by assumption.
If we choose fv such that tr (| LJ |(πv∗ )) (fv ) 6= 0 then the above identity tells us thattr πv∗ (fv∗ ) 6= 0. Choosing φv to be a function associated with fv∗ (such a φv exists byeither [Lab11, Lem. 4.1]), we have as in (A.0.2),
tr Πv (φv ) = tr πv∗ (fv∗ ) 6= 0.
We have exhibited a choice of f and φ above such that (A.0.4) is valid with theright-hand side non-vanishing. Therefore there exists some π on the left-hand side
contributing with positive multiplicity. Let S be the set of places of Fe+ containing all
infinite places and the finite places where G and Π are ramified. Write Sf for the subset
of finite places in S. As we are free to choose φu in the unramified Hecke algebra at eachu ∈ Sf , we may assume that π S is unramified with BC(π S ) = ΠS . The nonvanishingof tr π∞ (f∞ ) tells us that BC(π∞ ) = Π∞ . Thus (A.0.4) is reduced to a formula of theform (A.0.3), with π contributing nontrivially to the sum. Arguing as in the proof ofthe preceding proposition, we deduce that | LJ |(πv∗ ) ∼= πv .
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(S. W. Shin) Department of Mathematics, University of California, Berkeley, 901 Evans
Hall, Berkeley, CA 94720, USA / Korea Institute for Advanced Study, Dongdaemun-gu,
Seoul 130-722, Republic of Korea
Email address: sug.woo.shin@berkeley.edu