1. Almost projectivity

Irreducible modular representations of a reductive p-adic group and simple modules for Hecke algebras.

Marie-France Vign´eras

Let R be an algebraically closed field of characteristic l / =p, let F be a local non archimedean field of residual characteristic p, and let G be the group of rational points of a connected reductive group defined over F. The two main points in the search for a classification of the irreducible R-representations of G is to try to prove that any irreducible cuspidal representation is induced from an open compact subgroup and that the irreducible representations with a given inertial cuspidal support are classified by simple modules for the Hecke algebra of a type. Over a field R which is not the complex field new serious difficulties arise and the purpose of this article is to indicate a way to avoid them. The mirabolic trick used when the group is GL(n, F ) does not generalize but our new method is general and we can extend from the complex case to R the results of Morris and Moy-Prasad for level 0 representations.

Summary Introduction

1. Almost projectivity

2. A simple criterium for almost projectivity.

3. A simple criterium for irreducibility.

4. Finite reductive group 5. Morris types of level 0.

6. Cuspidal representations of level 0.

7. Bushnell-Kutzko types.

Bibliography

Introduction

Let (K, σ) be an irreducible simple cuspidal R-type in GL(n, F ) as defined by Bushnell and Kutzko, or a cuspidal R-type of level 0 in G as defined by Morris. We consider also an irreducible extended maximal type (N, Λ). We denote by ind the compact induction. We will prove :

Theorem 1) The irreducible R-representations of G which contain (K, σ) are in bijection with the simple modules for the Hecke algebra of (K, σ) in G.

2) ind

Λ is irreducible.

It is known that the Hecke algebra of (K, σ) in G is very closed to products of affine Hecke R-algebras hence the construction of simple modules for affine Hecke R-algebras is strongly related to the construction of irreducible R-representations of G if 1) is true.

The purpose of this article is to give a simple proof of the theorem. As a consequence one can extend to R-representations the complex theory of representations of level 0 by Morris or by Moy and Prasad.

The theorem known for GL(n, F ) lead to a complete classification of the irreducible R-representations of GL(n, F ) [Vigneras1,Vigneras2]. But the proof did not transfer to a general reductive group G and the proof that we give here is simpler and general.

When the characteristic of R is banal there is nothing to do. Indeed, if the pro-order of the profinite

group K is invertible in R, the representation ind

σ of G is projective and 1) results from an easy lemma in

algebra. Let us denote by Z the center of G. The group N contains Z and N/Z is profinite. If the pro-order

of N/Z is invertible in R, the category of R-representations of N where Z acts by a given character is

semi-simple, and End

ind

Λ R implies the irreducibility of ind

Λ.

We explain now the proof of the theorem. In the general case the representation ind

σ is not projective, but we know that 1) will be true if ind

σ is almost projective [Vigneras2]. One knows that End

ind

Λ R, because the proof [Morris] [Moy-Prasad] [Bushnell-Kutzko] for complex types transfers to R-types. We denote by Irr

G the set of irreducible R-representations of G, modulo isomorphism.

The properties

1’) ind

σ is almost projective,

2’) if Λ is contained in π|

then Λ is a quotient of π|

, for any π ∈ Irr

G.

are - a priori - stronger property than 1) and 2). This is explained in the paragraph 2 and 3.

We denote by U the pro-p-radical of K. Suppose that we are in the level 0 case. The group N is the G-normalizer N

(K) of a maximal K and Λ is an irreducible representation of N

(K) which contains σ.

The main point of the article is the remark that the action of K (resp. N ) on th e U -invariant vectors of ind

σ (resp. ind

Λ) can be computed, and this allows to prove 1’) and 2’). The precise result given in the paragraphs 5,6 implies:

Proposition (Level 0) The action of K on the U -invariant vectors of ind

σ is a direct sum of irreducible representations conjugate to σ.

When K is maximal, the action of N

(K) on the U -invariant vectors of ind

Λ is isomorphic to Λ.

The properties 1’) and 2’) and the fact that End

ind

Λ = R follow from the proposition.

In the GL(n, F )-case, similar results are obtained with the non trivial Bushnell-Kutzko representations η ∈ Irr

(U ), κ ∈ Irr

(K) (see the definition in the paragraph 7). When K is maximal there exists an extension E/F with E

⊂ GL(n, F ); we have N = KE

and Λ is an extension of σ to KE

. One computes the action of K (resp. N ) on th e η-isotypic part of ind

σ (resp. ind

Λ) in the paragraph 7. The properties 1’) and 2’) and the fact that End

ind

Λ = R follow from:

Proposition (GL(n, F )) The action of K on the η-isotypic part of ind

σ is of the form κ ⊗ W where W is semi-simple with constituents isomorphic to conjugates of σ.

When (KE

, Λ) is an extended maximal Bushnell-Kutzko type, the action of KE

on the η-isotypic part of ind

Λ is isomorphic to Λ.

When G is a reductive finite group, our method applies and gives a new proof of the almost projectivity of any R-representation parabolically induced from a cuspidal irreducible representation [Geck-Hiss-Malle].

This is done in the paragraph 4.

In bothcases, H = K/U is a finite reductive group. The propositions follow from the computation of the functor

σ → σ

: Mod

H → Mod

H

such that the η-isotypic part of ind

κ ⊗ σ isomorphic to κ ⊗ σ

(in the level 0 case, η (resp. κ) is the trivial representation of U (resp. K)). This functor is the analogue of a well known functor : the parabolic induction followed by the parabolic restriction in H or in G. Parabolic groups of G are replaced by parahoric subgroups or by Bushnell-Kutzko groups and the unipotent radical by the pro-p-radical with a trick : the representation of the pro-p-radical is not trivial in the Bushnell-Kutzko case. The intersection of two Bushnell-Kutzko groups or of two parahoric subgroups have similar properties than the intersection of two parabolic subgroups and the representations κ have a coherent behaviour. The classical formula for parabolics originally due to Harish-Chandra for finite reductive groups and generalised by Casselman for p-adic groups has an analogue for open compact parahoric or Bushnell Kutzko groups.

It is pleasure to thank Alberto Arabia for his work [Arabia] which lead to the simple criterium of almost

projectivity of ind

σ, which is basic in our proof.

1 Almost-projectivity

The notion of almost projectivity was introduced by Dipper for finite groups, and generalised by Arabia to the more general notion of quasi-projectivity for modules over an algebra.

Let R be a field and let A be an R-algebra. We consider the category Mod

(A) (resp. Mod

(A)) of unital finite type left (resp. right) A-modules.

1.1 Definition A finite type unital A-module Q is called quasi-projective when for any two morphisms

Q

π

//

α

// V

in Mod

(A) with π surjective, there exists β ∈ End

Q such that α = π ◦ β.

It is called almost projective when there exists a surjective morphism π : P → Q from a projective finite type unital A-module P to Q, such that for any morphism α ∈ End

Q there exists β ∈ Hom

(Q, P ) such that α = π ◦ β.

An almost projective module is quasi-projective. The fundamental property of quasi-projective modules is the following :

1.2 Main property (Arabia [Vigneras2, appendice th. 10]) When Q is quasi-projective, the functor Hom

(Q, −) : Mod

(A) → Mod

(End

Q)

induces a bijection between

a) the isomorphism classes of simple A-modules V such that Hom

(Q, V )/ =0 and b) the isomorphism classes of simple right End

Q-modules.

This functor is not in general an equivalence of category.

2 A simple criterium ofalmost projectivity

Let G be a locally profinite group, K an open compact subgroup of G, R a field and σ an irreducible smooth R-representation of K.

We denote by

Mod

(G) the category of smooth R-representations of G.

V

the σ-isotypic part of V ∈ Mod

(G), i.e. the biggest RK -submodule of V which is isomorphic to a direct sum ⊕

σ.

The functor of compact induction ind

: Mod

(K) → Mod

(G) is exact when G has an R-Haar measure, has a right ajoint (the restriction from G to K denoted by π → π |

), respects projectivity and the property of beeing of finite type [Vigneras1 I.5.10, I.5.7, I.5.9].

2.1 Lemma Suppose that the R-representation of G Q = ind

σ

admits a K-equivariant direct decomposition Q = Q

⊕ Q

and no subquotient of Q

isomorphic to σ. Then Q is almost-projective.

We will show in the paragraphs 4,5 that the property of the lemma is verified when R is an algebraically

closed field of characteristic =p / and

- K is a parabolic subgroup of a reductive finite group over a field of characteristic p, or a parahoric subgroup of a reductive p-adic group,

- σ is trivial on the pro-p-radical of K and cuspidal,

We will show in the paragraph 7 that the property of the lemma is verified when (K, σ) is a simple Bushnell- Kutzko R-type.

This simple lemma which is basic for us was found by Arabia when G is a finite group. In this case he proved a better result [Arabia 2.4.2]: when G is a finite group, then Q is quasi-projective if and only if the σ-isotypic part of Q/Q

is zero.

2.2 Exercise Proof of the lemma.

Let f : P

→ σ be a projective cover in Mod

K (we are reduced to the case of finite groups because σ is trivial on a subgroup of finite index). We take

P := ind

P

, π = ind

f.

The isomorphism of adjunction,

Hom

(ind

P

, ind

σ) Hom

(P

, ind

σ).

is given by restriction to the RK-submodule P

isomorphic to P

of functions withsupport in K in the canonical model of ind

P

. The image of ind

f under the isomorphism of adjunction is f : P

→ σ (where σ σ is the space of functions with support in K in the canonical model of ind

σ).

The hypothesis on ind

σ and the definition of a projective cover imply that the map γ → γ ◦ f : Hom

(σ, ind

σ) → Hom

(P

, ind

σ)

is an isomorphism. Hence the map

β → β ◦ π : End

ind

σ → Hom

(ind

P

, ind

σ) is surjective (it is clearly injective). We deduce that ind

σ is almost projective.

2.3 Exercise If Q = ind

σ satisfies the simple criterium (2.1), then Q is quasi-projective.

Let α, π : ind

σ → V be two morphisms in Mod

G with π surjective. We look for β ∈ End

ind

σ suchthat α = π ◦ β . The criterium implies that there exists a simple RK-submodule W

of Q

suchthat π(W

) = α(σ). Let β

: σ → ind

σ be the RK -equivariant morphism with image W

with π ◦ β

= α|

, and β ∈ End

ind

σ the image of β

by adjunction. Then α = π ◦ β.

3 A simple criterium for irreducibility

We replace the property “compact” for K by “compact mod center”, we denote by Ind

: Mod

K → Mod

G the induction without condition on the support. This functor has a left adjoint (the restriction π → π|

from G to K) [Vign´eras1 I.5.7]. By the Mackey decomposition and by adjunction one sees that :

3.1 Lemma Let Λ ∈ Irr

K. When the space End

(ind

Λ) is finite dimensional, it is equal to Hom

(ind

Λ, Ind

Λ).

3.2 Lemma The R-representation ind

Λ is irreducible when a) End

(ind

Λ) = R

b) If Λ is contained in π |

then Λ is also a quotient of π |

, for any π ∈ Irr

G.

Proof Suppose that a) and b) are true. Let π ∈ Irr

G be a quotient of ind

Λ. By adjunction and b),

Λ is quotient of π|

. By adjunction π ⊂ Ind

Λ. Hence there is a morphism ind

Λ → Ind

Λ withimage

π. By a) and (3.1) ind

Λ = π. Hence ind

Λ is irreducible.

We will show in the paragraph 6 and 7 that the properties a) and b) of the lemma can be proved when R is an algebraically closed field of characteristic =p / and

- K is the normalizer of a maximal parahoric subgroup in a reductive p-adic group, or an extended maximal Bushnell-Kutzko group in GL(n, F ),

- the restriction of Λ to the maximal parahoric subgroup is cuspidal, or Λ is an extended maximal Bushnell-Kutzko R-type.

4 Finite reductive group Let

G the group of rational points of a reductive connected group over a finite field of characteristic p, P = M U a parabolic subgroup of G withunipotent radical U , and Levi subgroup M ,

Cusp

(G) ⊂ Irr

(G) the subset of irreducible cuspidal R-representations, σ ∈ Cusp

(M ) identified witha representation of P trivial on U .

We denote by

N

(M ) th e G-normalizer of M , W (M ) = N

(M )/M,

g

σ

(?) = σ

(g

?g) ∈ Cusp

(gP g

) for g ∈ G,

w

σ =

σ with g ∈ N

(M ) above w ∈ W (M ), W (M, σ) := {w ∈ W (M ),

σ σ}

V → V

: Mod

G → Mod

M the functor of U -invariant vectors.

4.1 Proposition [Harish-Chandra] We have

(ind

σ)

⊕

σ.

With(2.1) we get a new proof of :

4.2 Corollary [Geck-Hiss-Malle (2.3)] ind

σ satisfies the simple criterium of almost-projectivity.

Proof. As U is the p-radical of P , we have a direct P -equivariant decomposition ind

σ = (ind

σ)

⊕ W

⊕

σ ⊕ W

where W

has no non zero U-fixed vector, and W has no subquotient isomorphic to σ.

5 Morris types oflevel 0 Let

F a local non archimedean field of residual field F

with q elements and characteristic p, G the group of rational points of a reductive connected group G over F ,

P , Q two parahoric subgroups of G [BT2 5.2.4], withtheir canonical exact sequence.

1 // U // P

// P (q) // 1,

1 // V // Q

// Q (q) // 1

The kernels U , V are pro-p groups, the quotients P (q), Q (q) are the groups of rational points of connected reductive finite groups over F

. We denote by f

: Mod

P (q) → Mod

P the inflation along f

, and f

the inverse (on the representations of P trivial on U ).

5.1 Definition We call parahoric induction the functor of compact induction : ind

= ind

◦f

: Mod

P(q) → Mod

G

and parahoric restriction the functor of U -invariants :

res

= f

◦ res

: Mod

G → Mod

P (q).

(where res

V = V

∈ Mod

P ).

The composite functor

T

= res

◦ ind

: Mod

P (q) → Mod

Q(q) can be computed, as in the parabolic case, using the Mackey decomposition.

We change notations. Let

T a maximal split torus of G (more precisely the group of rational points of this torus), N = N

(T ) th e G-normalizer or T (the N of the introduction is no more used), A the appartment in the semi-simple Bruhat-Tits building of G defined by T . x, y ∈ A suchthat G

= P , G

= Q .

We denote G

= U , G

= V , G

(q) = P (q), G

(q) = Q (q), f

= f

, f

= f

.

5.2 Lemma Let z be a point in the building, G

the corresponding parahoric sugroup of G and f

: G

→ G

(q) the canonical surjection. Then f

(G

∩G

) is a parabolic subgroup of G

(q) with unipotent radical f

(G

∩ G

).

Proof [Morris]. We reduce easily to the case treated by Morris where x, x

∈ A are in the closure of a chamber and z = nx

, using the following properties :

a) gG

g

= G

for g ∈ G [T 2.1], [BT2 5.2.4], [BT1 6.2.10], b) the Bruhat decomposition

G = G

N G

where G

is the Iwahori subgroup fixing a point o in a chamber C of A [BT1 7.2.6, 7.3.4] [SS page 105], c) G

⊃ G

when x is contained in the closure of the facet containing z [BT 5.2.4],

d) given a point a and a chamber C of closure C in the building, there exists g ∈ G suchthat ga ∈ C [T 1.7, 2.1]

By d) we can suppose z = g

x

where x, x

belong to the closure of a chamber C of A and g

∈ G.

By b) we can write g

= b

nb where b

, b ∈ G

where o ∈ C and n ∈ N . By c) G

⊂ G

∩ G

hence bx

= x

, b

x = x. Witha) we get G

∩ G

= b

(G

∩ G

)b

. The lemma for G

∩ G

[Morris] implies the lemma for G

∩ G

.

The quotient of the parabolic subgroup by its unipotent radical in the lemma (5.2) is denoted by (G

∩ G

)

(q) = f

(G

∩ G

)/f

(G

∩ G

)

Withthis notation (G

)

(q) = G

(q). The groups G

(q) are isomorphic when x belongs to an orbit of G in the buiding (property a)) and the conjugation by g ∈ G induces an equivalence of categories

σ →

σ : Mod

G

(q) → Mod

G

(q)

and more generally an equivalence of categories

Mod

(G

∩ G

)

(q) → Mod

(G

∩ G

)

(q)

because Mod

(G

∩ G

)

(q) is identified via f

to the category of R-representations of G

∩ G

trivial on G

∩ G

and on G

∩ G

, and Mod

(G

∩ G

)

(q) is identified via f

to the category of R-representations of G

∩ G

trivial on G

∩ G

and on G

∩ G

.

Withthe notations P , Q we denote (G

∩ G

)

(q) = ( P ∩ Q )

(q).

5.3 Definition The functor

F

: Mod

P (q) → Mod

( P ∩ g

Q g)

(q) → Mod

(g P g

∩ Q )

(q) → Mod

Q (q) is the composite of

a) the f

(P ∩ g

Vg)-invariants b) the conjugation by g

c) the parabolic induction.

5.4 Proposition T

⊕

F

.

Proof. By the Mackey formula, we have a Q -equivariant decomposition ind

σ ⊕

ind

(f

σ)(g

?g).

Taking the V invariants we get the proposition.

When the functor F

does not vanishon Cusp

P (q), then f

(P ∩ g

Vg) = {1} i.e

(5.4.1) f

(P ∩ g

Qg) = P (q).

When there exists σ ∈ Mod

P (q) suchthat F

(σ) admits a cuspidal non zero representation as a sub-module or as a quotient, then f

(gU g

∩ Q) = {1} i.e.

(5.4.2) f

(g P g

∩ Q ) = Q (q).

The equations (5.4.1) and (5.4.1) are not independant.

5.5 Lemma Let x, z be two points in the building with corresponding parahoric subgroups G

, G

. If f

(G

∩ G

) = G

(q) then the three following properties are equivalent :

(5.5.i) f

(G

∩ G

) = G

(q),

(5.5.ii) the order of G

(q) is less or equal to the order of G

(q), (5.5.iii) G

(q) G

(q) (G

∩ G

)/(G

∩ G

).

If x is a vertex, then f

(G

∩ G

) = G

(q) is equivalent to z = x.

Proof. f

(G

∩ G

) = G

(q) is equivalent to G

∩ G

= G

∩ G

. When this holds, we have (G

∩ G

)/(G

∩ G

) f

(G

∩ G

) ⊂ G

(q)

(G

∩ G

)/(G

∩ G

) → f

(G

∩ G

) = G

(q)

where the map is the canonical surjection. We denote by a, b, c the number of elements of G

(q), (G

∩ G

)/(G

∩ G

), G

(q). Then a ≤ b ≤ c. They are equal if and only if a = c. This proves that (5.5.ii) and (5.5.iii) are equivalent. If b = c, then G

(q) is a quotient of G

(q). As the relation x ∼ z given by

f

(G

∩ G

) = G

(q), f

(G

∩ G

) = G

(q)

is symetric, we deduce that G

(q) is a quotient of G

(q). As the groups are finite, they are isomorphic.

Hence a = b = c. This proves that (5.5.i) and (5.5.iii) are equivalent.

The last part the lemma follows from the following facts :

Let C be a chamber of A and let ∆ = {α

, . . . , α

} be the basis of the affine roots of G associated to C [T 1.8]. For x in the closure C of C we denote by ∆

⊂ ∆ the proper subset of α

with α

(x) = 0 and by W

the group generated by all reflexions s

for α ∈ ∆

.

- Let x, y ∈ C and n ∈ N. Suppose that the image w of n in the affine Weyl group W is of minimal lengthin W

wW

. Then f

(G

∩ G

) = G

(q) implies w∆

⊃ ∆

[Morris].

- x ∈ C is a vertex if and only if ∆

has n elements. Then y ∈ C is equal to x is and only if ∆

= ∆

. If y / =x then there is no element w ∈ W suchthat w∆

⊃ ∆

. Otherwise s

∆

= ∆

where ∆ = ∆

∪ { α } but α ∈ ∆

and α/ ∈ s

∆.

On an orbit of G the relation x ∼ z defined above is equivalent to f

(G

∩ G

) = G

(q)

because the groups G

(q) G

(q) are isomorphic on an orbit and (5.5). The set of g ∈ G with x ∼ gx is a disjoint union of double classes G

nG

for n in some set of representatives N (x) ⊂ N. For n ∈ N (x) withimage w ∈ W and σ ∈ Mod

G

(q) the isomorphism class of

σ depends only on w and we denote by

n

σ =

σ and W (x) the image of N (x) in N . We define W (x, σ) := {w ∈ W (x),

σ σ}. When P = G

we replace x by P in the notation. We deduce from (5.5) and (5.4) :

5.6 Corollary Let σ ∈ Cusp

P (q). The U -invariants vectors of ind

σ is isomorphic to

⊕

σ.

As in (4.2) this implies the existence of a P -equivariant decomposition ind

σ = ⊕

σ ⊕ W

where W has no subquotient isomorphic to σ. From (2.1) we deduce that ind

σ satisfies the simple criterium of almost projectivity.

6 Cuspidal representations oflevel 0

We suppose that x is a vertex (not necessarily special). The parahoric subgroup G

is maximal , th e G-normalizer P

which is the fixator of x, is an open compact mod center subgroup of G, and is the set of g ∈ G such f

(G

∩ G

) = G

(q) by (5.5). Let Λ ∈ Irr

P

trivial on G

and restriction to G

identified by f

to a cuspidal representation of G

(q).

6.1 Proposition (ind

Λ)

is an R-representation of P

isomorphic to Λ.

Proof. The functor

Λ → (ind

Λ)

: Mod

P

→ Mod

P

/G

is a direct sum

(ind

Λ)

= ⊕

F

(Λ) of functors F

: Mod

P

→ Mod

P

/G

composite of

- the invariants by P

∩ g

G

g

- the conjugation by g

- the induction from (P

∩ gP

g

)/(G

∩ gP

g

) to P

/G

.

The cuspidality of Λ implies that if the P

∩g

G

g

-invariants vectors are not 0 then G

∩g

G

g

= G

∩ g

G

g

, i.e. g ∈ P

. Then (ind

Λ)

= F

(Λ) = Λ and the proposition is proved.

The simple criterium for irreduciblity (3.2) is easily deduced from this proposition. By adjunction (6.1) implies

End

(ind

Λ) = R and if π ∈ Irr

G is a quotient of ind

Λ (6.1) implies

π

Λ.

In Mod

P

, π

is a direct factor of π|

and the simple criterium for irreduciblity is satisfied. Hence ind

Λ is irreducible.

7 Bushnell-Kutzko simple types in GL(n, F )

We suppose G = GL

(V ) GL(n, F ) for an F -vector space V of dimension n.

We fix β ∈ GL

(V ) suchthat - the algebra E = F(β) is a field

- k

(β) < 0 [Bushnell-Kutzko 1.4.5, 1.4.13, 1.4.15] [Bushnell-Kutzko2 4.1]. We will not use k

(β) and we do not recall its definition.

We denote by q

the number of elements of the residual field of E and by B

= GL

(V ) the centralizer of E

in GL

(V ). If d[E : F] = n then B

GL(d, E).

We consider the Bushnell Kutzko group associated associated to a “defining sequence” for β and a parahoric subgroup P in B

[Bushnell-Kutzko 2.4.2, 3.1.8, 3.1.14]. The group does not depend on the defining sequence [Bushnell-Kutzko 3.1.9 (v)] and is denoted J = J (β, P ). The Bushnell-Kutzko groups, called the BK-groups, have the following properties [Bushnell-Kutzko 1.6.1]:

7.1 Lemma Let P be a parahoric subgroup of B

with the canonical exact sequence

1 // U // P

// P (q

) // 1.

The BK-group J = J(β, P ) is an open compact subgroup of G normalized by E

with pro-p radical J

, with

(1) J = J

P, J ∩ B

= P , J

∩ B

= U .

The canonical surjection f

given by the lemma

1 // J

// J

// P(q

) // 1.

has the property

(2) f

(J ∩ K) = f

( P ∩ Q )

because f

(J ∩ K) = f

(J ∩ K ∩B

) and J ∩ K ∩ B

= P ∩Q. By (2), the properties of parahoric subgroups

of B

seen in the paragraphs 5 and 6 transfer to the Bushnell-Kutzko groups associated to β.

7.2 We consider irreducible Bushnell Kutzko (or BK) representations η

∈ Irr

J

, κ

∈ Irr

J attached to a fixed endo-class Θ [Bushnell-Kutzko 5.1.8, 5.2.1 and 5.2.2] [Bushnell-Kutzko2 4.3]. We do not recall the definitions. The BK-representations satisfy :

(3) The restriction of κ

to J

is η

and κ

is normalised by E

.

When η

is fixed there is some choice for κ

but only by multiplication by a character trivial on J

and normalized by E

. We consider different functors. We use the definition of the paragraph 5 for the parahoric subgroup P of B

.

7.3 Definition a) The κ

-inflation functor

σ → κ

⊗ f

σ : Mod

P (q

) → Mod

J

which induces an equivalence of categories between M od

P (q

) and the η

-isotypic R-representations of J . b) The compact κ

-induction functor :

ind

: Mod

P (q

) → Mod

G

composite of the κ

-inflation Mod

P(q

) → Mod

J by the compact induction ind

: Mod

J → Mod

G.

c) The κ

-restriction functor

res

: Mod

G → Mod

P (q

) π → σ

given by the η

-invariants π → π

= κ

⊗ f

σ : Mod

G → Mod

J followed by the inverse of the κ

-inflation.

d) When K is the BK-group J(β,Q) attached to another parahoric Q of B

, the functor T

= res

◦ ind

: Mod

P (q

) → Mod

Q(q

).

We will prove (7.7) that the functor T

is equal to the functor T

E),P(qE)

associated to the parahoric subgroups P , Q of B

(5.1) and already described (5.3) (5.4).

7.4 Remark For g ∈ G, we have J (gβg

, g P g

) = gJ(β, P )g

.

This is not immediately apparent that the elaborate definition of J is G-equivariant.

Let L = (L

)

be a strictly decreasing periodic lattice chain of O

-modules in V suchthat P is the set of f ∈ GL

(V ) with f (L

) ⊂ L

for all i ∈ Z. We consider

- the period e of L (the smallest integer n suchthat L

= p

L

for all i ∈ Z), - th e L -valuation v of β (the biggest integer n suchthat βL

⊂ L

for all i ∈ Z).

- the herditary order End

L of End

(V ) associated to L seen as a chain of O

-modules (the F - endomorphisms f suchthat f (L

) ⊂ L

for all i ∈ Z).

- s : End

V → End

V the tame corestriction map relative to E/F [Bushnell-Kutzko 1.3.3].

We take g ∈ G and we replace (β, L) by (gβg

, gL). It is easy to see what happens to the various

objects and numbers introduced above. First we see that (P , End

L) is replaced by gP g

, g(End

L)g

,

that the period e and the valuation v do not change. Then looking at the definition of k

(β ) [Bushnell-

Kutzko 1.3.5], we see that it does not change, and finally if c

: End

(V ) → End

(V ) is the natural

isomorphism f (x) → gf(g

xg)g

, we see on the definition [Bushnell-Kutzko 1.3.3] that c

◦ s is a tame

corestriction map relative to gEg

/F .

These remarks imply that a defining sequence ( A

, n, r

, γ

) for ( A , n, r, β) gives a defining sequence (g A

g

, n, r

, gγ

g

) for (g A g

, n, r, gβg

) with the definitions [Bushnell-Kutzko 2.4.2]. We deduce from [Bushnell-Kutzko 3.1.8] that J (gβg

, g P g

) = gJ(β, P )g

.

It seems to me that the construction of η

and κ

is G-equivariant [Bushnell-Kutzko 3.5, 5.7] but I didnt check all the details.

7.5 The three κ

-functors (7.3) do not determine the group J neither the representation κ

. In fact we will not keep (J, κ

).

Let P

be a maximal parahoric subgroup of B

and let P

be a minimal parahoric subgroup of B

contained in P

. We suppose P

⊂ P ⊂ P

. Let η

∈ Irr

J

, κ

∈ Irr

J

be the BK-representations associated to (β, Θ). We consider

J

= J

P , J

= J

U , η

= κ

|

, κ

= κ

|

It is clear that (1) hence (2), (3) are satisfied for (J

, J

, η

, κ

).

Let L

= (L

p

) be a O

-lattice chain in End

V suchthat P

= GL

(L

) is the set of g ∈ GL

(V ) with gL

⊂ L

. We denote GL

(L

) the set of g ∈ GL

(V ) with gL

p

⊂ L

p

for all i ∈ Z. The open compact subgroup A = GL

(L

)P of G has a pro-p-radical A

= GL

(L

)U and satisfies (1). By construction J, J

are contained in A and J

= A

∩ J, J

= A

∩ J

. Moreover [Bushnell-Kutzko 5.2.5] proved that

η

:= ind

η

ind

η

are isomorphic and irreducible, and one may suppose (or we twist κ

by a character normalised by E

) that κ

:= ind

κ

ind

κ

are isomorphic and irreducible. This implies :

7.6 Lemma The κ

, κ

, κ

-functors are equal, the compact BK-induction and BK-restriction functors associated to κ

, κ

, κ

are equal.

Hence the functor T

= T

can be computed using κ

, κ

. By the Mackey formula, we have a K

-equivariant decomposition (where we forget f

hence σ instead of f

σ)

ind

κ

⊗ σ ⊕

ind

1 maxQ

J_max¹ Q∩gJ_max¹ Pg⁻¹

(κ

⊗ σ)(g

?g).

We compute the η

-isotypic part, i.e the κ

|

-isotypic part. We recall [Bushnell-Kutzko 5.1.8 page 160, 5.2.7 page 170] :

dim

Hom

max∩gJ_max¹ g⁻¹

(η

,

η

) = dim

Hom

(κ

,

κ

) is equal to 1 for g ∈ J

B

J

, and is equal to 0 when g / ∈J

B

J

.

The terms in g / ∈ J

B

J

give no contribution to the η

-isotypic part of ind

κ

⊗ σ and Q\ B

/ P = J

Q\ J

B

J

/J

P

Using the property (2) for f

(J

∩ gK

g

) when g ∈ B

we deduce withthe same proof and notations of (5.3) and (5.4) :

7.7 Proposition T

⊕

F

E)gP(qE)

T

E),P(qE)

.

With(5.6) we get :

7.8 Corollary When σ ∈ Cusp

P (q

), then res

(ind

κ

⊗ σ) = ⊕

σ.

As J

is a pro-p-group, the restriction of ind

κ

⊗ σ to J

is semi-simple, and its η

-isotypic part is a direct factor. We deduce (3.2) that ind

κ ⊗ σ satisfies the simple criterium of almost projectivity.

The representation κ

extends to J

E

by Clifford theory. An R-representation Λ ∈ Irr

J

E

which extends κ ⊗ σ with σ ∈ Cusp

P

(q

) is called a maximal extended Bushnell-Kutzko type. We prove as in (6.1) :

7.9 Proposition The η

-isotypic part of ind

Λ is an R-representation of J

E

isomorphic to Λ.

We deduce from (7.9) that ind

Λ is irreducible. The proof of the almost projectivity of ind

κ ⊗ σ and of the irreducibility of ind

Λ given here is simple and more general than the proofs given in [Vigneras1, Vigneras2].

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