# Existence of supersingular representations of -adic reductive groups

###### Key words and phrases:

integral structure, admissibility, discrete cocompact subgroups, minimal representations, square integrable representations, reflection module, parabolic induction, pro- Iwahori Hecke algebra###### 2010 Mathematics Subject Classification:

primary 20C08, secondary 11F70###### Contents

- 1 Introduction
- 2 Iwahori Hecke ring
- 3 Supersingular and discrete modules
- 4 Characters
- 5 Discrete simple modules with supersingular reduction
- 6 Admissible integral structure via discrete cocompact subgroups
- 7 Supercuspidality criterion
- 8 Change of coefficient field
- 9 Reduction to an absolutely simple adjoint group

Let be a connected simple adjoint -adic group not isomorphic to a projective linear group of a division algebra ), or an adjoint ramified unitary group of a split hermitian form in variables. We prove that admits an irreducible admissible supercuspidal (= supersingular) representation over any field of characteristic .

## 1. Introduction

Throughout this paper, is a local non-archimedean field of characteristic and residue field of characteristic with elements, where is a connected reductive -group, a field (of coefficients) of characteristic and an algebraic closure of .

Recent applications of automorphic forms to number theory have imposed the study of smooth representations of on -vector spaces for not algebraically closed and often finite with . Indeed one expects a strong relation, à la Langlands, with -representations of the Galois group of - the only established case, however, is that of .

An irreducible admissible -representation of is called supercuspidal if it is not isomorphic to a subquotient of a representation parabolically induced from an irreducible admissible -representation of a Levi subgroup. In the established cases of the Langlands correspondence, they correspond to the irreducible continuous -representations of the Galois group of . All irreducible admissible -representations of are constructed from the irreducible admissible supercuspidal -representations of the Levi subgroups of using parabolic induction.

When , the finite group analogue of , where is a connected reductive group over a finite field of characteristic , admits no irreducible supercuspidal -representation [CE04, Thm.6.12].

For , irreducible admissible supercuspidal -representations have been constructed for some rank groups:

Breuil [Bre03] after the pionner work of Barthel-Livne [BL94]; this was the starting point of the Langlands -adic correspondance (Colmez and al.),

, Paskunas [Pas04],[BP12], using -equivariant coefficient systems on the adjoint Bruhat-Tits building of ,

unramified [Koz16],

unramified and [KX15].

###### Theorem 1.1.

Assume and absolutely simple adjoint not isomorphic to the projective linear group of a -dimensional central division -algebra ), or to the adjoint unitary group of a split hermitian form in variables over a ramified quadratic extension of .

Then, admits an irreducible admissible supercuspidal -representation.

The irreducible admissible supercuspidal -representation of will be a subquotient of a subrepresentation (smooth induction of the trivial -representation) for a discrete cocompact subgroup of [BH78]. The proof is local and uses a criterion of supercuspidality proved with R. Ollivier for algebraically closed [OV, Thm 3] and that we extend to any :

###### Proposition 1.2.

[Supercuspidality criterion] Assume . An irreducible admissible -representation of is supercuspidal if and only if it contains a non-zero pro--Iwahori invariant supersingular element if and only if is supersingular (all pro--Iwahori invariant elements of are supersingular).

The equivalence supercuspidal supersingular follows also from the classification [HV17, Thm.9]. Let denote an Iwahori subgroup of and the Iwahori Hecke -algebra.

###### Proposition 1.3.

Assume and as in Thm.1.1. Then, there exists a discrete cocompact subgroup of such that the -module contains a non-zero supersingular element.

To prove the theorem, we pick a non-zero supersingular element in and an irreducible quotient of the subrepresentation of generated by . Then is admissible ( is admissible and ) and contains a non-zero -invariant supersingular element, hence is supercuspidal by the supercuspidality criterion.

We explain now the meaning of supersingular, why we exclude and and how we prove Prop.1.3.

We choose a minimal parabolic -subgroup of containing a maximal -subtorus . Bruhat and Tits associated to them an affine Coxeter system , parameters for , where is the residue field of and the are integers , and a commutative group acting on the Dynkin diagram of decorated with the parameters . The diagram is the completed Dynkin diagram of a reduced root system .

The Iwahori Hecke ring of an Iwahori subgroup of the simply connected cover of the derived group of is isomorphic to the Hecke ring associated to , and the Iwahori Hecke ring is isomorphic to . To each cocharacter of is associated a central element of [Vig17]. Write . When , an element in a right -module is called supersingular if for all dominant but not dominant and large.

A simple right -module which is the -invariants of an irreducible admissible square integrable modulo center -representation of is called discrete. If is semi-simple, is discrete if and only if the simple components of its restriction to are discrete; the equivalence uses Casselman’s criterion of square integrability [Cas, §2.5] and the Bernstein Hecke elements [Vig16, Cor. 5.28].

Assume that is absolutely simple adjoint. If the type of is , the parameters are equal and for a -dimensional central division -algebra (). If the type of is and the two parameters are equal, then is the adjoint unitary group of a split hermitian form in variables over a ramified quadratic extension of [Tit79, §4].

When the type of is different from if the parameters are equal, we find a simple discrete -module with an -integral structure of reduction, modulo a maximal ideal of , a supersingular -module :

- When the type is , or with two distinct parameters, then admits a discrete character which is not the special character [Bor76, 5.9]^{1}^{1}1In Borel, the Iwahori group is the fixer of an alcove, where is the maximal compact subgroup of a minimal Levi subgroup; if is -split, . By extending or inducing such a character, we construct a simple discrete right -module of dimension with a natural -integral structure.
As we avoided the Steinberg character, the reduction modulo of the -integral structure is a supersingular -module. See Prop. 5.2.

- When the type is , the group is -split [Tit79, §4]. The image by a natural involution of of the reflection right -module of -dimension is discrete [Lus83, 4.23], has a natural -integral structure of reduction modulo a maximal ideal of a supersingular -module. See Prop. 5.3.

Now, we find such that contains a non-zero supersingular element. The existence of discrete cocompact subgroups in is ensured by [Mar91]. By the -adic version of the de George-Wallach limit multiplicity formula ([DKV84, Appendice3, Prop.] plus [Kaz86, Thm.K]), the simple discrete -module embeds in for some discrete cocompact subgroup of . All the -integral structures of have the same semi-simplification hence have supersingular reduction modulo , and is another -integral structure of of reduction modulo a -submodule of . See Prop. 6.3. The scalar extension from to of the supersingular -module is a non-zero supersingular -submodule of . This ends the proof of Prop.1.3.

A similar argument for an irreducible admissible supercuspidal complex representation of produces an integral structure with admissible reduction (Cor. 6.5).

Returning to general, Kret proved that admits irreducible admissible supercuspidal complex representations [Kre12] (if [BP16]). We extend this to any field of characteristic in §8:

###### Proposition 1.4.

[Change of coefficient field] If admits an irreducible admissible supercuspidal representation over some field of characteristic , supposed to be finite if , then admits an irreducible admissible supercuspidal representation over any field of characteristic .

We reduce the construction of an irreducible admissible supercuspidal representation of to the case of absolutely simple and adjoint in §9:

###### Proposition 1.5.

[Reduction to a simple adjoint group] Assume and algebraically closed or finite. If any connected absolutely simple adjoint -adic group admits an irreducible admissible supercuspidal -representation, then any connected reductive -adic group admits an irreducible admissible supercuspidal -representation.

Summarizing our results, we obtain:

###### Theorem 1.6.

Assume . Then admits an irreducible admissible supercuspidal -representation, if:

and (as in Thm.1.1) admit an irreducible admissible supercuspidal representation over some finite field of characteristic .

When , and should also admit an irreducible admissible supercuspidal -representation. We miss and because we do not know the integrality properties of the unramified irreducible admissible complex representations of corresponding to the integral reflections modules of the generalized affine Hecke algebras of supersingular reduction modulo . The missing cases will probably be completed by Herzig with a global method for and by Koziol with coefficient systems on the tree for .

Ackowledgements I thank Pioline, Savin and the organizers of the conference on automorphic forms and string theory in Banff (2017-10-30) for emails, discussions and their invitation who led me to look closely at unramified minimal representations corresponding to the reflection modules of the affine Hecke algebras [GS05, Thm. 12.3]. Minimal representations are important for string theorists [KPW02] and for mathematicians as they allow to construct analogues of theta series. Their relation to supersingular representations, a complete surprise for me, is at the origin of this work on a basic question raised fifteen years ago for modulo representations.

I thank also Heiermann for a discussion on discrete modules, Harder for emails on discrete cocompact subgroups, Waldspurger for reminding me the antipode and a reference, Henniart for providing the proof of Prop.9.2 b), Herzig for writing a draft of his global method for , Herzig and Koziol for working on the missing cases.

## 2. Iwahori Hecke ring

We review in this section the parts of the theory of Iwahori Hecke algebras [Vig16], [Vig14], [Vig17] appearing in the proof of the key proposition 1.3.

Let be a reductive connected -group, a maximal -split subtorus of , a minimal -parabolic subgroup of containing , and a special point of the apartment of the adjoint Bruhat-Tits buiding defined by . Associated to the triple (, , , ) are: the center of , the root system , the set of simple roots , the -centralizer of , the normalizer , the unipotent radical of , (hence ), the triples and for the simply connected covering and the adjoint group of the derived subgroup of , the natural homomorphisms , an alcove of the apartment with vertex , the Iwahori subgroups of fixing , the Iwahori Hecke ring and similarly for the simply connected and adjoint semi-simple groups.

The natural ring homomomorphism induced by is injective and we identify with a subring of . There is a canonical isomorphism where is the Hecke ring of an affine Coxeter system with parameters where is the residue field of and the are integers . The Dynkin diagram of is the completed Dynkin diagram of a reduced root system . The image of the Kottwitz morphism of acts on the Dynkin diagram decorated with the parameters , and the isomorphism extends to an isomorphism

(2.1) |

The Iwahori Hecke ring of is determined by the type of , the parameters and the group acting on the decorated Dynkin diagram . The quotient of by the fixer of in is isomorphic to a subgroup of the group of automorphisms of the decorated Dynkin diagram. The generalized affine Hecke ring is a free -module of basis for satisfying the braid and quadratic relations:

(2.2) |

where is the length on associated to extended to by for . The linear map for , where for with ,

(2.3) |

is an automorphism of .

The unique parahoric subgroup of is contained in the maximal compact subgroup . When is -split or semi-simple simply connected, . The group is isomorphic to , acts on the apartment and forms a system of representatives of the double classes of modulo . The subgroup of is commutative finitely generated of torsion acting by translation on the apartment, the quotient map splits, identifying the (finite) Weyl group of with the fixer in of a special vertex of the alcove . The semi-direct product is equal to and where . An element is called dominant (and anti-dominant), if for lifting . The dominant monoid consists of dominant elements of .

The cocharacter group of is isomorphic to by the -equivariant map for a fixed an uniformizer of . A cocharacter is dominant if is. For and large, , and is defined for any character . The fixer of the alcove in is and

.

###### Lemma 2.1.

The fixer of the alcove in is .

###### Proof.

Write as with . As fix a special point of , does also. As acts by translation on the apartment, fixes . As and fix , fixes also hence . ∎

The action of on the alcove and the embeddings of and into induce isomorphisms

(2.4) |

###### Lemma 2.2.

The subgroup of is finitely generated of finite index.

The submonoid of the dominant monoid is finitely generated of finite index.

###### Proof.

The -conjugacy class of is the -orbit of . A basis of the center of [Vig14, Thm.1.2] is

where for are the integral Bernstein elements of [Vig16, Cor. 5.28, Ex.5.30]:

(2.5) |

When are both dominant (or anti-dominant), . For , let denote the -orbit of and write . Any -orbit contains a unique dominant (resp. anti-dominant) cocharacter (resp. ), and .

The invertible elements in the dominant monoids and are the subgroups and . Only the trivial element is invertible in the dominant monoid . Write .

The generalized affine ring contains the commutative subring of -basis . When , the Bernstein elements are simply the classical elements , and the Iwahori Hecke ring is isomorphic to . In general, is not isomorphic to but the subring of basis is isomorphic to . Denote by the subrings of respective bases , , , .

###### Lemma 2.3.

is finitely generated as an -module and as a -module.

is finitely generated as an -module and as a -module.

## 3. Supersingular and discrete modules

Let be a field of characteristic and the Iwahori Hecke -algebras and , isomorphic to the affine and generalized affine Hecke -algebras and .

This section introduces the supersingular right -modules when (there are none when ) and the discrete simple right modules.

###### Definition 3.1.

Let be a non-zero right -module.
An element is called supersingular if and only if for all dominant with not dominant, and some large positive integer .
The -module is called supersingular when all its elements are supersingular^{2}^{2}2In [Vig17, Def. 6.10] there is a different definition: there exists with for all not invertible in .

###### Remark 3.2.

is supersingular if and only if for any and large . We can restrict to (or ) dominant, or anti-dominant.

###### Fact 3.3.

- A simple -module is finite dimensional, and is semi-simple as an -module.

- A -module is supersingular if and only if its restriction to is supersingular [Vig17, Cor.6.13].

- The simple supersingular -modules are^{3}^{3}3There are no non-zero supersingular modules if the characters which are not special or trivial (see the next section) when is “large” of characteristic [Vig17, Cor.6.13].

We denote by and the categories of -representations of generated by their -invariant vectors and of right -modules. The -invariant functor has a left adjoint .

###### Fact 3.4.

When , the functor induces a bijection between the isomorphism classes of the irreducible -representations of with and of the simple right -modules [Vig96, I.6.3]. When , the functors are inverse equivalences of categories (Bernstein-Borel-Casselman).

Let be an irreducible complex representation of with . We recall the classical properties of , including the Cassleman’s criterion of square integrability modulo center, before giving the definition of a discrete simple right -module.

###### Fact 3.5.

- a) is isomorphic to a subrepresentation of where is a -character of trivial on .

- b) The representation of on the -coinvariants is semi-simple, trivial on and contains a subrepresentation isomorphic to .

- c) The quotient map induces an -equivariant isomorphism for the Bernstein -algebra embedding

that is, for [Vig98, II.10.1]. Note that where is the modulus of and of image and for .

###### Definition 3.6.

A simple right -module is called discrete when it is isomorphic to for an irreducible admissible square integrable modulo center -representation of .

###### Proposition 3.7.

A simple right -module is discrete if and only if any complex character of contained in satisfies: the restriction of to is unitary and

.

###### Proof.

Let be a character trivial on . Writing for of image and noting , we have in c)

Hence is contained in if and only is contained . Apply Casselman’s criterion (Fact 3.5 d)) (the inverse of an anti-dominant element is dominant). ∎

###### Remark 3.8.

Some authors see as a left -module. One exchanges “left” and “right” by putting for . The left or right -module is called discrete if is square integrable modulo center. For left modules, the proposition holds true with anti-dominant instead of dominant.

###### Lemma 3.9.

For a character , the following properties are equivalent:

(i) is unitary and for any dominant with not dominant.

(ii) for any dominant .

(iii) for any and for any dominant .

###### Proof.

(ii) implies (i) because for any , implies when is invertible in , i.e. . Conversely, (i) implies that for and large with we have . This implies hence (ii). The arguments of the equivalence between (i) and (iii) are similar using Lemma 2.2. ∎

###### Proposition 3.10.

A simple right -module is discrete if and only if acts on by a unitary character and the simple components of restricted to are discrete.

###### Proof.

Lemma 3.9 (iii). ∎

## 4. Characters

In this section, is a field of characteristic and is absolutely simple. We give the characters which extend to . This is an exercice, already in the litterature when is the complex field [Bor76].

For distinct , the order of is finite except if the type is . In the finite case,

(4.1) |

The for and the relations (2.2), (4.1) give a presentation of . A presentation of is given by the for and the relations (2.2), (4.1) and

(4.2) |

We have a disjoint decomposition where is the intersection of with a conjugation class of and [Bor76, 3.3]:

When , we fix a numerotation such that when (except for where ) and when . We have for , for and has two elements for , and has respectively elements for . The parameters are equal on to a positive integer . The automorphism group of the completed Dynkin diagram is [IM65, 1.8]:

###### Lemma 4.1.

The automorphism group of the decorated Dynkin diagram is equal to with two exceptions: type and different parameters , type and different parameters , where .

###### Proof.

When , and are -stable except for where the non-trivial element of permutes ; fixes and if .

When , the type is , the non-trivial element of permutes ; fixes and if . ∎

###### Lemma 4.2.

A map is the restriction of a character if and only if:

when , it is constant and equal to or on each . There are characters if in .

when , its values are or . There are characters.

###### Proof.

This follows from the presentation of . When , the are invertible. When [Vig17, Prop.2.2]. ∎

The unique character with (resp. ) for all is called the trivial (resp. special) -character. If they are equal then .

A character extends to a character of if and only if for all and . When the image of in is trivial or when , any character of extends to . The extensions are not unique in general. The trivial and special characters extend, their extensions are also called trivial and special.

###### Lemma 4.3.

Assume . A character extends to a character of except for

- type , equal parameters , , when ,

- type , equal parameters , , when .

###### Proof.

Assume and not trivial. Then with three cases:

Type and equal parameters , say . Then permutes . Only the special and trivial characters of extends to .

Type and equal parameters , say . Then permutes , only the -characters of with extend to .

Type . Then, stabilizes and hence all -characters of extend to . ∎

We combine these results in a proposition:

###### Proposition 4.4.

(i) admits characters , they are all -integral, and their reduction modulo are supersingular except for the special and trivial character.

(ii) admits a -integral simple (left or right) module of supersingular reduction modulo , and restriction to , when

, type , , , and ,

, type , , , and .

(iii) Otherwise the characters of extends to -integral complex characters of , of supersingular reduction modulo if is not special or trivial.