The right adjoint of the parabolic induction

The right adjoint of the parabolic induction

January 10, 2014

Abstract

We extend the results of Emerton on the ordinary part functor to the category of the smooth representations over a general commutative ring R, of a general reductivep-adic group G (F-points of a reductive connected F-group over a local non archimedean field of residual characteristic p). In Emerton’s work,F =Qp, R is a complete artinian local Zp-algebra having a finite residual field, and the representations are admissible. We show:

The (smooth) parabolic induction functor admits a right adjoint. The center-locally finite part of the (smooth) right adjoint is equal to the (admissible) right adjoint of the admissible parabolic induction functor whenRis noetherian. The (smooth or admissible) parabolic induction functor is fully faithful when 0 is the only infinitelyp-divisible element in R.

Contents

1 Introduction 1

2 Review on adjunction between grothendieck abelian categories 3

3 The category Mod^∞_R(G) 6

3.1 Mod^∞_R(G) is grothendieck . . . 6 3.2 Admissibility andz-finiteness . . . 6 4 The right adjoint R^G_P of Ind^G_P : Mod^∞_R(M)→Mod^∞_R(G) 7 5 Ind^G_P is fully faithful if R_p−ord={0}. 9 6 The z-locally finite parts ofR^G_P and ofR^{P P}_P are equal 10 7 The Hecke description of R^{P P}

P : Mod^∞_R(P)→Mod^∞_R(M) 12

8 The right adjoint Ord_P of Ind^G_P : Mod^adm_R (M)→Mod^adm_R (G) 14

1 Introduction

LetRbe a commutative ring, letF be a local non archimedean field of finite residual field of characteristic p, let G be a reductive connected F-group. Let P,P be two opposite parabolicF-subgroups of unipotent radicalN,Nand Levi subgroupM=P∩Pof center Z(M). The groups of F-points are denoted by the same letter but not in bold. The parabolic induction functor Ind^G_P : Mod^∞_R(M) → Mod^∞_R(G) between the categories of

smoothR-representations ofM and ofGis the right adjoint of theN-coinvariant functor, and respects admissibility.

For any (R, F, G), we show that Ind^G_P admits a right adjointR^G_P.

WhenRis noetherian, we show that theZ(M)-locally finite part ofR^G_P respects admis- sibility, hence is the right adjoint of the functor Ind^G_P between admissibleR-representations.

Ifpis invertible inRwe show that theN-coinvariant functor respects admissibility, hence is the left adjoint of the functor Ind^G_P between admissibleR-representations.

When 0 is the only infinitelyp-divisible element inR, we show that the counit of the adjoint pair (−_N,Ind^G_P), is an isomorphism, hence Ind^G_P is fully faithful and the unit of the adjoint pair (Ind^G_P, R_P^G) is an isomorphism.

The results of this paper have already be used [HV] in the comparison between parabolic and compact induction for smooth representations over an algebraically closed field of characteristicpforany(F,G), following the arguments of Herzig when the char- acteristic of F is 0 and Gis split. The comparison is a basic step in the classification of the non-supersingular admissible irreducible representations ofG(work in progress with Abe, Henniart, and Herzig, see also Ly’s work [Ly] for GL(n, D) where D/F is finite dimensional division algebra).

When pis invertible in R, it was known that Ind^G_P has a right adjoint. When R is the field of complex numbers, Casselman for admissible representations and Bernstein in general proved that the right adjoint is equal to theN-coinvariant functor multiplied by the modulus of P (“the Bernstein second adjointness”), and stressed in his unpublished notes the incredible power of this innocent statement. A proof was published by Bushnell.

Both proofs rely on the property that the category Mod_C(G) is noetherian. Conversely, Dat [Dat] proved that the Bernstein second adjointess implies the noeheriannity of Mod_R(G) and prove it assuming the existence of certain idempotents (that he constructed using the theory of types for linear groups, classical groups ifp6= 2, and groups of semi-simple rank 1). Under the same hypothesis onG, Dat proved also thatN-coinvariant functor respects admissibility. See [Dat].

When F = Qp and R is a complete artinian local Zp-algebra having finite residual field, Emerton [Emerton] showed that Ind^G_P restricted to admissible representations has a right adjoint equal to the ordinary part functor Ord_P. Studying its derived functors he showed that theN-coinvariant functor respects admissibility [Emerton2].

In section 2 we give precise definitions and references to the litterature on adjoint functors and on grothendieck abelian locally small categories.

The existence of a right adjoint of Ind^G_P : Mod^∞_R(M)→Mod^∞_R(G) is proved using that Mod^∞_R(G) is a grothendieck abelian locally small category and that Ind^G_P is an exact func- tor commuting with small direct sums, in sections 3 and 4. When the ringRis noetherian, this method cannot be applied to the functor Ind^G_P : Mod^adm_R (M)→Mod^adm_R (G) because the category of smooth admissible R-representations is not grothendieck. It is not even known if it is an abelian category when the characteristic ofF ispandpis not invertible in R.

In section 5, we assume that 0 is the only infinitelyp-divisible element inR; we show the vanishing of the N-coinvariants of ind^{P gP}_P whenP gP 6=P and that the counit of the adjunction (−_N,Ind^G_P) is an isomorphism; the general arguments of section 2 imply that the unit of the adjunction (Ind^G_P, R^G_P) is an isomorphism and that Ind^G_P is fully faithful.

WhenRis noetherian, Ind^G_P : Mod^adm_R (M)→Mod^adm_R (G) is also obviously fully faithful.

The set P P is open dense in G. The partial compact induction functor ind^{P P}_P : Mod^∞_R(M)→Mod^∞_R(P) admits a right adjointR^{P P}_P by the general method of section 2.

Letzbe an arbitrary element of the center Z(M) ofM strictly contractingN. We prove

that thez-locally finite parts ofR^G_P(V) and ofR^{P P}_P (V) forV ∈Mod^∞_R(G), are isomorphic in section 6.

The right adjointR^{P P}

P : Mod^∞_R(P)→Mod^∞_R(M) of the functor ind^{P P}_P is explicit. It is the smooth part of the functor Hom_R[N](C_c^∞(N, R),−). In section 7, following Casselman and Emerton, we give another description ofR^{P P}

P . Choosing an arbitrary open compact subgroupN0ofN, the submonoidM⁺of elements ofM contractingN0acts onV^N0by the Hecke action. We have the smooth induction functor Ind^M_M+: Mod^∞_R(M⁺)→Mod^∞_R(M).

We show that R^{P P}

P : Mod^∞_R P →Mod^∞_R(M) is equal to the functorV 7→ Ind^M_M+(V^N0).

TheZ(M)-locally finite part of this functor is the Emerton’s ordinary part functor OrdP : Mod^∞_R P →Mod^∞_R(M).

In section 8 we assume thatRis noetherian and we show that OrdP(V) is admissible whenV is an admissibleR-representation ofG. Therefore the parabolic induction functor Ind^G_P : Mod^adm_R M → Mod^adm_R G admits a right adjoint equal to the functor Ord_P : Mod^adm_R G → Mod^adm_R M. If pis invertible in R, we prove that VN is admissible when V is an admissible R-representation ofG. Therefore the admissible parabolic induction functor Ind^G_P admits a left adjoint equal to the admissible N-coinvariant functor.

I thank Guy Henniart for his comments on the preceeding version of this article cor- recting some inaccuracies.

2 Review on adjunction between grothendieck abelian categories

In a small category, the collection of objects and the morphisms Hom(A, B) between two objects A and B form sets. In a locally small category only the morphisms Hom(A, B) between two objects AandB must form a set. The categorySetis locally small but not small. A category which is not locally small is big.

LetI be a small category,C,Dlocally small categories, C^op the locally small opposite category ofC andD^C the category of functorsC → D. The categoriesSet^Cop,Set^C are big ([KS] Def. 1.4.2). A contravariant functorC → Dis a functor C^op→ D.

Proposition 2.1. ([KS] Def. 1.2.11, Cor. 1.4.4)

The covariant Yoneda functor : C 7→Hom(−, C) :C →Set^Cop and the contravariant Yoneda functor : C7→Hom(C,−) :C →Set^C are fully faithful.

A functor F ∈ Set^C is called representable when there exists C ∈ C such that F ' Hom(C,−). A functorF∈Set^Copis called representable when there existsC∈ Csuch that F 'Hom(−, C). In both cases, the objectCwhich is unique modulo unique isomorphism is called a representative of the functorF.

A functorF :I → C defines functors lim−→F :C →Set C7→Hom_CI(F, ctC), lim

←−F :C^op→Set C7→Hom_CI(ctC, F), where ct_C :I → C is the constant functor defined byC∈ C.

When the functor lim

−→F ∈ Set^C is representable, a representative is called the injec- tive limit (or colimit or direct limit) of F and denoted also by lim

−→F. We have natural isomorphisms ([ML] III.4 (2), (3)), for C∈ C,

Hom_CI(F, ctC)'Hom_C(lim

−→F, C).

When the functor lim

←−F ∈ Set^Cop is representable, a representative is called the projec- tive limit (or inverse limit or limit ) of F and denoted also by lim

←−F. We have natural

isomorphisms, for C∈ C,

Hom_CI(ct_C, F)'Hom_C(C,lim

←−F).

One says that (F(i))_i∈I is an inductive (resp. projective) system inC indexed byI(resp.

I^op)) and one writes lim

−→(F(i))_i∈I or lim

←−(F(i))_i∈Iop for the object lim

−→F or lim

←−F. Example 2.2. 1) A set of objects (C_i)_i∈I of C indexed by a set I can be viewed as a functor F : I → C where I is identified with a discrete category (the only morphisms are the identities). When they exist, lim

−→F =⊕_i∈IC_i is the direct sum, or coproduct, or disjoint uniont_i∈IC_i, and lim

←−F=Q

i∈IC_i is the direct product.

2) WhenI has two objects with two parallel morphisms other than the identities,F is nothing but two parallels arrowsC1

−→g

−→f

C2inC. When they exist, lim−→F is the cokernel of (f, g) and lim

←−F is the kernel of (f, g) ([KS] D´ef. 2.2.2).

3) When they exist, it is possible to construct the inductive (resp. projective) limit of a functor F :I 7→ C, using only coproduct and cokernels (resp. products and kernels) ([KS] Prop. 2.2.9). If Ob(I) and Hom(I) denote the set of objects and of morphisms of the small categoryI, and ifs:σ(s)→τ(s) withσ(s), τ(s)∈Ob(I) fors∈Hom(I), then

lim−→F is the cokernel off, g:⊕_s∈Hom(I)F(σ(s))−→−→^g

f

⊕_i∈Ob(I)F(i), (1)

where f, gcorrespond respectively to the two morphisms id_F(σ(s), F(s), fors∈Hom(I), lim←−F is the kernel of Y

i∈Ob(I)

F(i)−→−→^g

f

Y

s∈Hom(I)

F(σ(s)),

wheref, g are deduced from the morphisms id_F(τ(s), F(s) :F(τ(s))×F(σ(s))−→−→^g

f

F(τ(s) fors∈Hom(I).

LetF :C 7→ Dbe a functor. For U ∈ D, we have the category C_U whose objects are the pairs (X, u) withX ∈ C, u:F(X)→U. We say thatF is right exact if the category CU is filtrant for any U ∈ D, and that F is left exact if the functor F^op : D^op → C^op is right exact ([KS] 3.1.1, 3.3.1).

Proposition 2.3. Let a functorF :C 7→ D.

1) WhenCadmits finite projective limits, F is left exact if and only it commutes with finite projective limits. In this case, F commutes with the kernel of parallel arrows.

2) When C admits small projective limits, F is left exact and commutes with small direct products, if and only if F commutes with small projective limits.

3) The similar statements hold true for right exact functors, inductive limits, small direct sums, and cokernels.

Proof. 1) ([KS] Prop. 3.3.3, Cor. 3.3.4).

2) If F preserves small projective limits, F is left exact and preserves small direct products (Example 2.2 1)). Conversely, from (??), a left exact functor which commutes wit small direct products preserves small projective limits because it commutes with the kernel of the parallel arrows by 1).

3) ReplaceC byC^op.

Let F : C → D and G : D → C be the functors. Then (F, G) is a pair of adjoint functors, or F is the left adjoint of G, or G is the right adjoint if F, if their exists an isomorphism of bifunctors fromC^op× C toSet

Hom_D(F(.), .)'Hom_C(., G(.)),

called the adjunction isomorphism ([KS] Def. 1.5.2). The functorF determines the func- tor Gup to unique isomorphism and Gdetermines F up to unique isomorphism ([KS]

Thm. 1.5.3). For X ∈ C, the image of the identity id_F(X) in Hom_D(F(X), F(X)) by the adjunction isomorphism is a morphismX 7→G◦F(X). Similarly, for Y ∈ D, the image of id_G(Y) is a morphismF◦G(Y)→Y. The morphisms are functorial inX and Y. We have constructed morphisms of functors, called the unit and the counit :

: 1_C →G◦F, η:F◦G→1_D. Proposition 2.4. Let(F, G)be a pair of adjoint functors.

F is fully faithful iff the unit: 1→G◦F is an isomorphism.

Gis fully faithful iff the counit η:F◦G→1 is an isomorphism.

FandGare fully faithful iffF is an equivalence (fully faithful and essentially surjective ([KS] Def. 1.2.11,1.3.13)) iff Gis an equivalence. In this caseF andGare quasi-inverse one to each other.

Proof. ([KS] Prop. 1.5.6).

Proposition 2.5. Let (F, G)be a pair of adjoint functors. Then F is right exact andG is left exact.

Proof. ([KS] Prop. 3.3.6)

LetAbe a locally small abelian category. A generator ofAis an objectE ofAsuch that the functor Hom(E,−) :A →Set is faithful (i.e. any object ofA is a quotient of a small direct sum ⊕iE).

IfAadmits small inductive limits, the functor between abelian categories F 7→lim

−→F:A^I→ A is additive and right exact.

The category I is filtrant when it is not empty, for any objects i, j of I there exist morphisms i → k, j → k, and for any parallel morphisms f, g : i → j there exists a morphismh→ksuch thath◦f =h◦g([KS] Def. 3.1.1).

Definition 2.6. ([KS] D´ef. 8.3.24)The locally small abelian categoryAis called grothendieck if it admits a generator, small inductive limits, and the small filtered inductive limits are exact.

Example 2.7. The locally small abelian category of left modules over a ring is grothendieck.

Proof. ([KS] Ex. 8.3.25).

Proposition 2.8. A grothendieck abelian locally small category admits small projective limits.

Proof. ([KS] Prop. 8.3.27).

Proposition 2.9. Let a functor F : A → C where A is a grothendieck abelian locally small category. The following properties are equivalent:

1)F admits a right adjoint,

2)F commutes with small inductive limits,

3)F is right exact and commutes with small direct sums.

Proof. This is a combination of Prop. 2.3 with ([KS] Thm. 5.2.6, Prop. 5.2.8, Prop. 5.2.9).

3 The category Mod

(G)

Let R be a commutative ring, let G be a secound countable locally profinite group (for instance, for a parabolic subgroup of a reductive group), and let (Kn)_n∈N be a strictly decreasing sequence (Kn)_n∈Nof pro-p-open subgroups ofG, with trivial intersection. We suppose Rn normal inR0for alln.

The category ModR(G) ofR-representations ofGis the grothendieck abelian locally small category of leftR[G]-modules (Ex. 3).

LetV ∈ModR(G). A vector inV is called smooth when it is fixed by an open subgroup ofG. The set of smooth vectors ofV is aR[G]-submodule ofV, equal toV^∞=∪n∈NV^Kn where V^Kn is the submodule of the vectors v ∈ V fixed by Rn. When V = V^∞, V is called smooth.

The same definition applies whenGis a locally profinite monoid (its maximal subgroup is open and locally profinite).

3.1 Mod

(G) is grothendieck

The moduleCc(G, R) of functionsf :G→kwith compact support is aR[G×G]-module for the left and right translations. For n∈ N, the submodule Cc(Kn\G, R) of functions left invariant by Kn, is a smooth representation of G for the right translation. They form a strictly increasing sequence of union C_c^∞(G, R), equal to the smooth part of the R[G×G]-moduleC_c(G, R).

The subcategory Mod^∞_R(G) ⊂ Mod_R(G) of smooth R-representations of G, is an abelian locally small full subcategory.

Lemma 3.1. Mod^∞_R(G)is a grothendieck abelian locally small category.

Proof. A direct sum of smooth representations is smooth, the cokernel of two parallel arrows in Mod^∞_R(G) is smooth hence Mod^∞_R(G) admits small inductive limits (1). Small filtered inductive limits are exact because they are already exact in Mod_R(G). A generator is⊕n∈NC_c(K_n\G, R).

ForW ∈Mod^∞_R(G), V ∈ModR(G) we have Hom_R[G](W, V) = Hom_R[G](W, V^∞). The smoothification

V 7→V^∞: ModR(G)→Mod^∞_R(G)

has a left adjoint: the inclusion Mod^∞_R(G)→ModR(G) hence is left exact (Prop. 2.5). The smoothification is never right exact ([Viglivre] I.4.3) hence does not have a right adjoint (Prop. 2.5).

3.2 Admissibility and z-finiteness

A smooth R-representationV ofGis called admissible when for any compact open sub- group H of G, the R-module V^H of H-fixed elements of V is finitely generated. The category Mod^adm_R (G) does not have a generator or small inductive limits. Worse, it is not known if the quotient of an admissible representation remains admissible.

Letzbe an element in the centerZ(G) ofG.

Definition 3.2. Let V ∈Mod_R(G). An elementv∈V is called z-finite if the R-module R[z]v is contained in a finitely generatedR-submodule of V.

The subset V^z−lf of z-finite elements is a R-subrepresentation of V, called the z- locally finite part of V. WhenV =V^z−lf, the representationV is calledz-locally finite.

The category Mod^z−lf_R (G) ofz-locally finite smoothR-representations ofGis a full abelian subcategory of Mod^∞_R(G). Thez-locally finite functor

V 7→V^z−lf : Mod^∞_R(G)→Mod^z−lf_R (G) (2)

is the right adjoint of the inclusion Mod^z−lf_R (G)→Mod^∞_R(G).

Thez-locally finite part is contained in thez^±1-locally finite partV^z±1−lf =V^z−lf∩ V^z−1−lf, and in the Z(G)-locally finite part V^Z(G)−lf = ∩_z∈Z(G)V^z−lf (this is a finite intersection when Z(G) is a finitely generated monoid).

Lemma 3.3. An admissibleR-representation of GisZ(G)-locally finite.

Proof. Ifv∈V there existsn∈Nsuch thatv∈V^Kn, andV^Kn is aZ(G)-stable finitely generated R-module.

4 The right adjoint R

of Ind

: Mod

(M ) → Mod

(G)

let F be a local non archimedean field of residual characteristicp, let Gbe a reductive connectedF-group. We fix a maximalF-split subtorusSofG, and a minimal parabolic F-subgroupBofGcontainingS. LetUbe the unipotent radical ofB. TheG-centralizer ZofSis a Levi subgroup of B. We choose a parabolicF-subgroupPofGcontainingB of unipotent radicalNand LeviMcontainingZ. We denote byXthe group ofF-rational points of an algebraic group X over F, with the exception that we writeNG(S) for the group ofF-rational points of theG-normalizerNG(S) of S. LetZbe the G- centralizer ofSandW0=NG(S)/Z the finite Weyl group.

Let (Kn)n∈N be a strictly decreasing sequence of pro-p-open subgroups of G with trivial intersection, with Rn normal in R0 for all n. We write Mn = Kn ∩M, Nn = Kn∩N, Nn =Kn∩N. We suppose, as we may, that Rn =NnMnNn =NnMnNn has an Iwahori decomposition.

Lemma 4.1. The quotient spaceP\Gis compact and the quotient mapG→P\Gadmits a continuous section ϕ:P\G→G.

Proof. This is well known: G is a finite disjoint union th∈X_nP Nnh for a finite subset Xn ⊂Gfor n large enough ([SVZ] Prop. 5.3), and ϕ(P xh) =xh, x ∈ Nn, h ∈Xn is a section.

The smooth parabolic induction

Ind^G_P : Mod^∞_R(M)→Mod^∞_R(G)

sendsW ∈Mod^∞_R(M) to the smooth representation Ind^G_P(W) ofGacting by right trans- lations on the module of functions g : G → W such that f(mnxgx) = mf(x) for m ∈ M, n ∈ N, x ∈ G, x ∈ Kn where n ∈ N depends on f. We write C^∞(P\G, W) for the locally constant functions on the compact setP\Gwith values inW. The smooth parabolic induction

Ind^G_P : Mod^∞_R(M)→Mod^∞_R(G) is the right adjoint of the N-coinvariant functor

V 7→VN : Mod^∞_R(G)→Mod^∞_R(M)

([Viglivre] I.5.7 (i), I.A.3 Proposition). TheN-coinvariant functor ModR(P)→ModR(N) is the right adjoint of the inflation functor Infl^P_M : ModR(M) → ModR(P) sending a representation ofM =P/N to the natural representation ofP trivial onN.

Remark 4.2. The N-coinvariants of the inflation functor Infl^P_M is the identity functor of Mod_RM (the unit1_ModRM 7→ −_N◦Infl^P_M of the adjunction(Infl^P_M,−_N)is an isomor- phism).

Proposition 4.3. The smooth parabolic induction functorInd^G_P : Mod^∞_R(M)→Mod^∞_R(G) Ind^G_P is exact, commutes with small direct sums, and admits a right adjoint

R^G_P : Mod^∞_R(G)→Mod^∞_R(M).

Proof. By Lemma 4.1, theR-linear map

(3) f 7→f ◦ϕ:C^∞(P\G, W)→Ind^G_P(W) is an isomorphism. We have a natural isomorphism

(4) C^∞(P\G, W)'C^∞(P\G, k)⊗RW 'C^∞(P\G,Z)⊗_ZW.

The Z-module C^∞(P\G,Z) is free, because it is the union of the increasing sequence of the Z-modules (C^∞(P\G/K_n,Z)_n∈N which are free of finite rank. Hence the tensor product byC^∞(P\G,Z) is exact, and the smooth parabolic induction functor is exact.

The smooth parabolic induction commutes with small direct sums⊕_i∈IWi because a functionf ∈C^∞(P\G, W) takes only finitely many values.

Applying Prop. 2.9 and Lemma 3.1, the parabolic induction admits a right adjoint.

Remark 4.4. When pis invertible in R, Dat ([Dat] between Cor. 3.7 and Prop. 3.8) showed that

R^G_P(V) = ([Hom_R[G](C_c^∞(G, R), V)]^N)^∞ (V ∈Mod^∞_R(G)).

In this case, the modulusδP ofP is well defined, and R^G_P(V)'δPVN

whenRis the field of complex numbers (Bernstein) or whenGis a linear group, a classical group when p6= 2, or of semi-simple rank 1 [Dat].

Letg∈GandQ=PorQ=Pthe parabolic subgroup ofGopposite toP,P∩P =M. The partial compact parabolic induction functor

ind^{P gQ}_P : Mod^∞_R(M)→Mod^∞_R(Q)

associates toW ∈ M^∞_k (M) the smooth representation ind^{P gQ}_P (W) ofQby right transla- tion on the submodule of functions with support contained inP gQ,, with compact support modulo left multiplication by P (P gQis generally not closed in the compact setP\G).

Proposition 4.5. The partial compact parabolic induction functorind^{P gQ}_P is exact, com- mutes with small direct sums, and admits a right adjoint

R^{P gQ}_P : Mod^∞_R(Q)→Mod^∞_R(M).

Proof. Same proof as for the functor Ind^G_P (Prop. 4.3).

Remark 4.6. When P gP =P, the partial compact parabolic induction functor ind^P_P is the inflation functor Infl^P_M.

Lemma 4.7. W ∈ Mod^∞_R(M) is admissible if and only if Ind^G_P(W) ∈ Mod^∞_R(G) is admissible.

Proof. This is well known and follows from the decomposition ([Viglivre] I.5.6, II.2.1):

(Ind^G_PW)^Kn' ⊕_{P gKn}(Ind^{P gK}_P ⁿW)^Kn' ⊕_{P gKn}W^M∩gKng−1 (n∈N, g∈G), where the sum is finite and Ind^{P gK}_P ⁿW ⊂Ind^G_PW is theR-submodule of functions with support contained in P gK_n.

Corollary 4.8. The smooth parabolic induction restricts to a functor Ind^G_P : Mod^adm_R (M)→Mod^adm_R (G).

We will later show that this functor admits also a right adjoint when the ring is noetherian.

5 Ind

is fully faithful if R

= {0}.

Definition 5.1. Thep-ordinary partRp−ordofRis the subset ofx∈kwhich are infinitely p-divisible.

WhenR is a field,R_p−ord={0} if and only if the characteristic ofRisp.

Let Φ_G be the set of roots ofSin G. Letα∈Φ_G. We writeU_αfor the root subgroup ofGassociated toα. We haveR_p−ord6={0}if and only if there exists a Haar measure on Uαwith values inR if and only if

C_c^∞(Uα, R)U_α6= 0 (5)

([Viglivre] I (2.3.1)).

Proposition 5.2. We supposeR_p−ord={0}. TheN-coinvariants of the partial compact induction functor ind^{P gP}_P is 0if P gP 6=P.

Proof. LetW ∈Mod_RM. The action ofN on

ind^{P gP}_P (W) =C_c^∞(P\P gP, R)⊗_RW

is trivial onW and the right translation onC_c^∞(P\P gP, R). By the Bruhat decomposition, P gP =P nP for an elementn∈NG(S) of non trivial imagew∈W0. There exists a root α∈ΦG such thatUα⊂N andU_w(α)is not contained in N, and the multiplication map P gM Nα×Uα →P gP is an homeomorphism, whereNα is a subgroup of N normalized byUα. We deduce

C_c^∞(P\P gM NαUα, R)U_αN_α= (C_c^∞(P\P gM Nα)⊗RC_c^∞(Uα, R)U_α)N_α. Applying (5), we obtainC_c^∞(P\P gP, R)N = 0.

Theorem 5.3. We supposeR_p−ord={0}. Then

1. The parabolic inductionInd^G_P : Mod^∞_R(M)→Mod^∞_R(G)is fully faithful, 2. The unitid_Mod^∞

R(M)→R^G_P◦Ind^G_P of the adjoint pair(Ind^G_P, R^G_P)is an isomorphism.

3. The counit η :−N ◦Ind^G_P →idMod^∞_R(M) of the adjoint pair (−N,Ind^G_P) is an iso- morphism.

Proof. By Prop. 2.4 the 3 properties are equivalent. The counit η of the adjoint pair (−_N,Ind^G_P) is an isomorphism, because the N-coinvariant of Ind^G_P is isomorphic to the N-coinvariants of Infl^G_P. This follows from Prop. 5.2 and Remarks 4.2, 4.6.

a) Beeing a right adjoint, theN-coinvariant functor Mod^∞_R(P)→Mod^∞_R(M) is right exact.

b) The representationW inflated toMis a quotient of the restriction toPof Ind^G_P(W).

The Bruhat decomposition implies that the kernel of the quotient map Ind^G_P(W)→ W admits a finite filtration F1 ⊂ F2 ⊂ . . . ⊂ Fr of quotients ind^{P gP}_P for all double cosets P gP 6=P ofG.

6 The z-locally finite parts of R

and of R

are equal

We compare the right adjoint R^G_P : Mod^∞_R(G) → Mod^∞_R(M) of the smoooth parabolic induction Ind^G_P and the right adjointR^{P P}_P : Mod^∞_R P →Mod^∞_R M of the partial compact parabolic induction ind^{P P}_P .

Let Z(M) ⊂ S be the split center of M and let z ∈ Z(M) be an element strictly contractingN : for any open compact subgroupN0⊂N, the sequence (zⁿN0z⁻ⁿ)n∈Nis strictly decreasing of trivial intersection. We denote by

R^G,z−lf_P : Mod^∞_R G→Mod^z−lf_R M, resp.R^{P P ,z−lf}_P : Mod^∞_R P →Mod^z−lf_R M, thez-locally finite part ofR^G_P, resp.R^{P P}_P . The restriction toP is a faithful functor

Res^G_P : Mod^∞_R(G)→Mod^∞_R P .

Theorem 6.1. The functorsR^G,z−lf_P andR_P^{P P ,z−lf}◦Res^G_P are isomorphic.

Proof. We want to prove that there exists a functorial isomorphism (6) Hom_R[M](W, R^G,z−lf_P (V))→Hom_R[M](W, R^{P P ,z−lf}_P (V))

in (W, V) ∈ Mod^z−lf_R (M)×Mod^∞_R(G). The image of W by a R[M]-homomorphism in R^G_P(V), resp.R^{P P}_P (V) is contained inR^G,z−lf_P (V), resp.R^{P P ,z−lf}_P (V). Also, we can replace in (6) R_P^G,z−lf by R^G_P and R^{P P ,z−lf}_P byR^{P P}_P . Then we use the adjunctions (Ind^G_P, R^G_P) and (ind^{P P}_P , R^{P P}_P ). We are reduced to find a functorial isomorphism

(7) J : Hom_R[G](Ind^G_PW, V)→Hom_R[P](ind^{P P}_P W, V)

in (W, V) ∈ Mod^z−lf_R (M)×Mod^∞_R(G). There is an obvious candidate: the restriction of a R[G]-homomorphism ϕ : Ind^G_PW → V to the submodule ind^{P P}_P W ⊂ Ind^G_PW is a R[P]-homomorphismJ(ϕ) : ind^{P P}_P W →V. This is functorial. TwoR[G]-homomorphisms Ind^G_P(W) →V equal on Ind^{P P}_P (W) are equal because an arbitrary open subset ofP\G is a finite disjoint union of G-translates of compact open subsets ofP\P P ([SVZ] Prop.

5.3).

To study the image ofJwe introduce more notations. Forr∈N, n∈N and a non-zero elementw∈W^Mr, letfr,n,w∈ind^{P P}_P (W) be the function of supportP Nrnand equal to w onN_rn. Forg ∈G, the support of the function gf_r,n,w ∈Ind^G_P(W) is P N_rng⁻¹. We say that (g, r, n, w) is admissible when

P N_rg=P N_rn (r∈N, g∈G, n∈N , w∈W^Mr).

Let Φ∈Hom_R[P](ind^{P P}_P W, V). We will prove:

1) Φ belongs to the image ofJwhen Φ(gf_r,n,w) =gΦ(f_r,n,w) for all admissible (g, r, n, w).

2) Φ(gf_r,n,w) =gΦ(f_r,n,w) for all admissible (g, r, n, w) (it is only here that we use that W isz-locally finite).

The proof of 1), resp. 2), follows ([Emerton] 4.4.6, resp. 4.4.3).

Proof of 1). Φ belongs to the image ofJ if and only ifP

ig_iΦ(f_i) = 0 for allg₁, . . . , g_n in Gandf1, . . . , fn in ind^{P P}_P (W) such thatP

igifi= 0.

Letg₁, . . . , g_n, f₁, . . . , f_n satisfying this property. We chooser∈Nlarge enough, such that thef_iseen as elements of C_c^∞(N , W) are leftN_r-invariant with values inW^Mr. Let Y_i⊂X_r∩N such that the support off_i istn∈YiP N_rn. We have f_i|_{P N}

rn=f_r,n,fi(n). Leth∈X_r. We have P

i(g_if_i)|_{P N}

rh= 0. In the sum, the non-zero elements (g_if_i)|_{P N}

rh=g_i(f_i|_{P N}

rhgi),

are those such thatP N_rhg_i=P N_rnfor somen∈Y_i. We have alsoP

ihg_i(f_i|_{P N}

rhg_i) = 0.

This implies

X

i

hg_iΦ(f_i|_{P N}

rhg_i) = 0, under the hypothesis of the proposition. We have also P

ig_iΦ(f_i|_{P N}

rhgi) = 0.

For alli,P\Gis the finite disjoint union∪h∈XrP N_rhg_i hencef_i=P

h∈X_rf_i|_{P N}

rhg_i. We deduce

X

i

g_iΦ(f_i) = X

h∈Xr

X

i

g_iΦ(f_i|_{P N}

rhg_i) = 0.

Proof of 2). We haven⁻¹f_r,n,w=f_r,1,w, andf_r,1,w is fixed byR_r. First we reduce ton= 1 by replacing (g, n) by (gn⁻¹,1).

We choose three integersr⁰ ∈Z, r⁰⁰ ≥r, a∈N as follows. The subsetNrg⁻¹ ⊂P Nr

beeing compact, its projection onto the first factor of the isomorphismN×M×N →P N is compact and contained in Nr⁰, hence Nrg⁻¹⊂Nr⁰P. TheR-submodule generated by Φ(1r,1,w⁰) forw⁰ ∈R[z]wis contained in a finitely generatedR-submodule ofV, contained in V^Kr00. We havez^aNr⁰z^−a⊂Nr⁰⁰⊂Nr.

The function fr,1,w is the sum of its restrictions to P z^−aNrz^av = P Nrz^av for the cosets z^−aNrz^av contained inNr,

fr,1,w = X

v∈z^−aN_rz^a\Nr

(z^av)⁻¹fr,1,z^a(w).

By R[P]-linearity of Φ, we are reduced to prove, forv∈Nr,

Φ(gv⁻¹z^−afr,1,z^a(w)) =gΦ(v⁻¹z^−afr,1,z^a(w)) =gv⁻¹z^−aΦ(fr,1,z^a(w)).

We may write gv⁻¹=pnr⁰ withnr⁰ ∈Nr⁰, p∈P. We have to prove that Φ(nr⁰z^−afr,1,z^a(w)) =nr⁰z^−aΦ(fr,1,z^a(w)).

Applyingz^a, it is equivalent to prove

Φ(z^an_r⁰z^−af_r,1,za(w)) =z^an_r⁰z^−aΦ(f_r,1,za(w)).

Asf_r,1,za(w) and Φ(f_r,1,za(w)) are fixed byz^anr⁰z^−a, the equality is obvious.

7 The Hecke description of R

P

: Mod

(P ) → Mod

(M )

We choose a compact open subgroup N0 of N. The submonoid M⁺ ⊂ M contracting N0 is the set ofm∈M such thatmN0m⁻¹ ⊂N0. The inverse monoid M⁻ ⊂M is the submonoid ofM dilatingN0. The groupM⁺∩M⁻ is open inM. We recall the element z∈Z(M) strictly contracting N. We have the induction functor

I_M^M+: ModR(M⁺)→ModR(M)

sendingW ∈Mod_R(M⁺) to the representation ofM by right translations on the module of R-linear maps ψ: M →W such that ψ(mx) =mψ(x) for all m∈ M⁺, x∈M. The smooth induction functor

Ind^M_M+: Mod^∞_R(M⁺)→Mod^∞_R(M) is the smoothification of the induction functorI_M^M+.

Definition 7.1. Let V ∈ Mod^∞_R(P). The submonoid M⁺ ⊂M contracting N0 acts on V^N0 by the Hecke action(m, v)7→hN₀,m(v),

(8) hN₀,m(v) = X

n∈N0/mN0m⁻¹

nmv (m∈M⁺, v∈V^N0).

The Hecke action ofM⁺ onV^N0 is smooth because it extends the natural action of the open subgroupM⁺∩M⁻.

Theorem 7.2. The functor

(9) V 7→Ind^M_M+(V^N0) : Mod^∞_R(P)→Mod^∞_R(M) is isomorphic to R^{P P}

P .

The theorem implies that the functor R^{P P,z−lf}

P : Mod^∞_R(P)→Mod^z−lf_R (M)

is isomorphic to thez-locally finite part of the functor (9). The Emerton’s ordinary functor is theZ(M)-locally finite part of the functor (9):

Ord_P =R^{P P,Z(M}_P ^)−lf : Mod^∞_R(P)→Mod^Z(M)−lf_R (M) is isomorphic to Ord_P. Applying Thm. 6.1, we obtain:

Corollary 7.3. The functor

V 7→(Ind^M_M+(V^N0))^z−lf : Mod^∞_R(G)→Mod^z−lf_R (M) is isomorphic to R^G,z−lf_P . The functor

V 7→Ord_P(V) = (Ind^M_M+(V^N0))^Z(M)−lf : Mod^∞_R(G)→Mod^z−lf_R (M) is isomorphic to R^G,Z(M)−lf_P .

The functorR^{P P}

P is the right adjoint of

ind^{P P}_P = ind^{P N}_P 'C_c^∞(N, R)⊗R−: Mod^∞_R(M)→Mod^∞_R(P)

Lemma 7.4. The functor R^{P P}

P : Mod^∞_R(P) →Mod^∞_R(M) is the smoothification of the functor Hom_R[N](C_c^∞(N, R),−).

Thm. 7.2 follows from:

Proposition 7.5. LetV ∈Mod^∞_R(P). The map:

(10) Φ : Hom_R[N](C_c^∞(N, R), V)→I_M^M+(V^N0) f 7→Φ(f)(m) = (mf)(1N₀) (m∈M), is an isomorphism ofR[M]-modules.

Proof. ForV ∈Mod^∞_R(P),m∈M andn∈N act onf ∈Hom_k(C_c^∞(N, R), V) by mf =m◦f ◦m⁻¹:x7→mf(m⁻¹xm), nf :x7→f(xn).

The map (10) isR[M]-linear by adjunction because the map

ϕ: Hom_R[N](C_c^∞(N, R), V)→V^N0 , f 7→f(1N₀).

isR[M⁺]-linear: Form∈M⁺, f ∈Homk(C_c^∞(N, R), V), we compute (mf)(1N₀) =m◦f◦m⁻¹(1N₀) =m◦f(1_m⁻¹_N0m) =m X

n∈m⁻¹N₀m/N₀

f(1_N0n⁻¹)

=m X

n∈m⁻¹N0m/N0

nf(1N₀) = X

n∈N0/mN0m⁻¹

nmϕ(1N₀) =hN₀,m(f(1N₀)).

The map Φ (10) is injective because theR[P]-moduleC_c^∞(N, R) is generated by 1_N0, and we compute:

Φ(f)(m) = (mf)(1_N0) =m◦f◦m⁻¹(1_N0) =m◦f(1_N0(m−m⁻¹)) =m◦f(1_m−1N₀m).

If Φ(f) = 0 then f(1_m−1N₀m) = 0 for all m∈ M. As f is N-equivariant, we have also 0 =f(1_m−1N₀mn) =f(mn1N₀) for allm∈M, n∈N, hencef = 0.

We show the surjectivity of Φ. An arbitrary element ofC_c^∞(N, R) is a sum X

i

xiniz^a1N₀=X

i

xi1_zaN0z^−an⁻¹_i =X

i

xini1_zaN0z^−a =X

i

xiniz^a1N₀

where xi ∈ k, ni ∈ N, a ∈ N and the open sets z^aN0z^−an⁻¹_i are disjoint. Let ψ ∈ Ind^M_M+(V^N0). There is aR[N]-linear map

fψ:C_c^∞(N, R)→V such thatfψ(z^a1N₀) =hN₀,z^a(ψ(z^−a)) (a∈N).

We have Φ(fψ) =ψ.

Remark 7.6. (Ind^M_M+(V^N0))^z−1−lf = (I_M^M+(V^N0))^z−1−lf, for V ∈Mod^∞_R(P).

Proof. Let ϕ ∈ (I_M^M+(V^N0))^z−1−lf. We have to prove that ϕ is smooth. Let Wϕ ⊂ I_M^M+(V^N0) be a finitely generated R-module containing R[z⁻¹]ϕ. The image of Wϕ by the mapf 7→f(1) is a a finitely generatedR-submodule ofV^N0 containingϕ(z^−a) for all a∈N. The action ofM⁺onV^N0 is smooth. This implies that there exists a large integer n∈Nsuch thatMn =Kn∩M ⊂M⁺∩M⁻ fixesϕ(z^−a) for alla∈N. The elementϕis fixed by M_n because M =∪a∈NM⁺z^−a, hence two elements of I_M^M₊(V^N0) equal onz^−a for alla∈Nare equal.

Remark 7.7. Let W ∈Mod^∞_R(M⁺)and supposeM_r⊂M⁺∩M. The map f 7→f|z^Z: (I_M^M+W)^Mr →I_z^zN^Z(W^Mr)

is aR[z^Z]-isomorphism.

Proof. The map f 7→ f|_zZ : I_M^M+W → I_z^zN^ZW is a R[z^Z]-isomorphism because M =

∪a∈NM⁺z^−a. Let f ∈ I_M^M+W. For m⁺ ∈ M⁺, mr ∈Mr, a∈ Z, we have f(m⁺z^amr) = m⁺f(m_rz^a) =m⁺m_rf(z^a) because M_r ⊂M⁺ andz∈Z(M). If f is fixed byM_r, then f(z^a) is fixed byM_rfor alla∈Zbecausef(z^a) = (z^am_r) =m_rf(z^a). Conversely, iff(z^a) is fixed by M_r for alla ∈Z, then f is fixed byM_r because f(m⁺z^am_r) =m⁺f(z^a) = f(m⁺z^a) andM =∪_a∈ZM⁺z^−a.

8 The right adjoint Ord

of Ind

: Mod

(M ) → Mod

(G)

Theorem 8.1. We suppose thatRis noetherian. ForV ∈Mod^adm_R (G), the representation (I_M^M+(V^N0))^z−1−lf of M is admissible.

Proof. Let r large enough so that Mr is an open compact subgroup of M⁺∩M. Note that M_rN₀ is a group. The R-module of elements fixed by M_r in (I_M^M₊(V^N0))^z−1−lf is isomorphic to

X = (I_z^zN^N(V^N0Mr))^z−1−lf

by the mapf 7→f|_zZ (Remark 7.7). We want to show thatX is finitely generated.

The image Y of X by f 7→ f(1) is a submodule of V^N0Mr containing f(z^a) for all a∈Zandf ∈X. We haveX ⊂I_z^zN^Z(Y). We have the compact open subgroupNrMrN0. We will prove (Prop. 8.2) that

Y ⊂V^NrMrN0.

Admitting this, Y is a finitely generated R-module because V is admissible and R is noetherian. The action h_N0,z ofz onY is surjective because, forf ∈X we have f(1) = f(zz⁻¹) = hN₀,zf(z⁻¹). A surjective endomorphism of a finitely generated R-module is bijective (this is an application of Nakayama lemma [Matsumura] Thm. 2.4). Hence the action ofzonY is bijective. Hence Y 'I_z^z_N^Z(Y) is a finitely generatedR-module. AsRis noetherian,X is a finitely generatedR-module.

Proposition 8.2. We suppose R noetherian. If f ∈ (I_z^zN^Z(V^MrN0))^z−1−lf, then f(1) ∈ V^NrMrN0.

Proof. We have

V^MrN0=∪_t≥rV^NtMrN0, (11)

where NtMrN0=KtMrN0⊂Gis a compact open subgroup asMrN0⊂K0 normalizes Rt, and the sequence (NtMrN0)t≥r is strictly decreasing of intersectionMrN0.

We supposet≥r. We writen(r, t)∈Nfor the smallest integer such thatz⁻ⁿNrzⁿ⊂ Nt⊂Nrforn≥n(r, t). We show that

1)hN₀,zⁿ(V^NtMrN0)is fixed byNrMrN0whenn≥n(r, t). .

Letv∈V^NtMrN0. Letnr∈Nrand (ni)_i∈Ia system of representatives ofN0/zⁿN0z⁻ⁿ. We compute:

nrhN₀,zⁿ(v) =X

i∈I

nrnizⁿv=X

i∈I

n⁰_ibizⁿv=X

i∈I

n⁰_izⁿv, (12)