The pro-p Iwahori Hecke algebra of a reductive p-adic group V (Parabolic induction) Stronger results or better redaction Corrections

The pro-p Iwahori Hecke algebra of a reductive p-adic group V (Parabolic induction) Stronger results or better redaction

Corrections

Vign´eras Marie-France June 28, 2016

Abstract

We give basic properties of the parabolic induction and coinduction functors associated to R-algebras modelled on the pro-p-Iwahori-Hecke R-algebras H ^R (G) and H ^R (M ) of a reductive p-adic group G and of a Levi subgroup M when R is a commutative ring. We show that the parabolic induction and coinduction functors are faithful, have left and right adjoints that we determine, respect finitely generated R-modules, and that the induction is a twisted coinduction.

Contents

1 Introduction 1

2 Levi algebra 5

2.1 Monoid W M

^✏

. . . . 5

2.2 An anti-involution ⇣ . . . . 7

2.3 ✏-alcove walk basis . . . . 8

2.4 w 0 -twist . . . . 10

2.5 Distinguished representatives of W 0 modulo W M,0 . . . . 12

2.6 H as a left ✓( H M

⁺

)-module and a right ✓

^⇤

( H M )-module . . . . 14

3 Induction and coinduction 15 3.1 Almost localisation of a free module . . . . 15

3.2 Induction and coinduction . . . . 16

4 Parabolic induction and coinduction from H ^M to H 19 4.1 Basic properties of the parabolic induction and coinduction . . . . 19

4.2 Transitivity . . . . 19

4.3 w 0 -twisted induction = coinduction . . . . 21

4.4 Transitivity of the coinduction . . . . 23

5 24

1 Introduction

We give basic properties of the parabolic induction and coinduction functors associated

to R-algebras modelled on the pro-p-Iwahori-Hecke R-algebras H ^R (G) and H ^R (M ) of a

reductive p-adic group G and of a Levi subgroup M when R is a commutative ring. We

show that the parabolic induction and coinduction functors are faithful, have left and right adjoints that we determine, respect finitely generated R-modules, and that the induction is a twisted coinduction.

When R is an algebraically closed field of characteristic p, Abe [Abe, Section 4] proved that the induction is a twisted coinduction, when he classified the simple H ^R (G)-modules in term of supersingular simple H ^R (M )-modules. In two forthcoming articles [OV] and [AHHV2], we will use this paper to compute the images of an irreducible admissible R- representation of G by the basic functors: invariants by a pro-p-Iwahori subgroup, left or right adjoint of the parabolic induction.

Let R be a commutative ring and let H be a pro-p Iwahori Hecke R-algebra, associated to a pro-p Iwahori Weyl group W (1) and parameter maps S !

^q

R, S(1) !

^c

R[Z k ] [Vig1,

§ 4.3], [Vig4].

For the reader unfamiliar with these definitions, we recall them briefly. The pro- p Iwahori Weyl group W (1) is an extension of an Iwahori Weyl group W by a finite commutative group Z k , X (1) denotes the inverse image in W (1) of a subset X of W , the Iwahori Weyl group contains a normal affine Weyl subgroup W ^{af f} , S is the set of all affine reflections of W ^{af f} , q is a W -equivariant map S ! R, W acting by conjugation on S and trivially on R, c is a W (1) ⇥ Z k -equivariant map S(1) ! R[Z k ], W (1) acting by conjugation and Z k by multiplication on both sides.

The Iwahori Weyl group is a semidirect product W = ⇤ o W 0 where ⇤ is the (com- mutative finitely generated) subgroup of translations and W 0 is the finite Weyl subgroup of W ^{af f} .

Let S ^{af f} be a set of generators of W ^{af f} such that (W ^{af f} , S ^{af f} ) is an affine Coxeter system and (W 0 , S := S ^{af f} \ W 0 ) is a finite Coxeter system. The Iwahori Weyl group is also a semidirect product W = W ^{af f} o ⌦ where ⌦ denotes the normalizer of S ^{af f} in W . Let ` denote the length of (W ^{af f} , S ^{af f} ) extended to W and then inflated to W (1) such that ⌦ ⇢ W and ⌦(1) ⇢ W (1) are the subsets of length 0 elements.

Let w ˜ 2 W (1) denote a fixed but arbitrary lift of w 2 W .

The subset S ⇢ W ^{af f} of all affine reflections is the union of the W ^{af f} -conjugates of S ^{af f} and the map q is determined by its values on S ^{af f} , the map c is determined by its values on any set ˜ S ^{af f} ⇢ S ^{af f} (1) of lifts of S ^{af f} in W (1).

Definition 1.1. The R-algebra H associated to (W (1), q, c) and S ^{af f} is the free R-module of basis (T w ˜ ) w ˜

2

W (1) and relations generated by the braid and quadratic relations:

T w ˜ T w ˜

⁰

= T w ˜ w ˜

⁰

, T _{˜ s} ² = q(s)(˜ s) ² + c(˜ s)T s ˜ , for all w, ˜ w ˜

⁰

2 W (1) with `(w) + `(w

⁰

) = `(ww

⁰

) and all s ˜ 2 S ^{af f} (1).

By the braid relations, the map R[⌦(1)] ! H sending ˜ u 2 ⌦(1) to T u ˜ identifies R[⌦(1)]

with a subring of H containing R[Z k ]. This identification is used in the quadratic relations.

The isomorphism class of H in independent of the choice of S ^{af f} .

Let S M be a subset of S. We recall the definitions of the pro-p Iwahori Weyl group W M (1), the parameter maps S M

qM

! R, S M (1)

^cM

! R[Z k ] and S _M ^{af f} given in [Vig4].

The set S M generates a finite Weyl subgroup W M,0 of W 0 , W M := ⇤ o W M,0 is a subgroup of W , W M (1) is the inverse image of W M in W (1), S M (1) = S(1) \ W M (1), q M is the restriction of q to S M , and c M is the restriction of c to S M (1). The subgroup W _M ^{af f} := W ^{af f} \ W M ⇢ W M is an affine Weyl group and S _M ^{af f} denotes the set of generators of W _M ^{af f} containing S M such that (W _M ^{af f} , S _M ^{af f} ) is an affine Coxeter system.

Definition 1.2. For S M ⇢ S, the R-algebra H ^M associated to (W M (1), q M , c M ) and S ^{af f} _M is called a Levi algebra of H .

Let (T _w ^M _˜ ) w ˜

2

W

M

(1) denote the basis of H ^M associated to (W M (1), q M , c M ) and S _M ^{af f}

and ` M the length of W M (1) associated to S _M ^{af f} .

Remark 1.3. When S M = S, H ^M = H . When S M = ; , H ^M = R[⇤(1)].

In general when S M 6 = S, S _M ^{af f} is not W M \ S ^{af f} , and H ^M is not a subalgebra of H ; it embeds in H only when the parameters q(s) 2 R for s 2 S ^{af f} are invertible.

As in the theory of Hecke algebras associated to types, one introduces the subalgebra H ⁺ M ⇢ H ^M of basis (T _w ^M _˜ ) w2W ˜

_M+

(1) associated to the positive monoid W M

⁺

:= { w 2 W M | w(⌃ ⁺ ⌃ ⁺ _M ) ⇢ ⌃ ^{af f,+} } where ⌃ M ⇢ ⌃ are the reduced root systems defining W _M ^{af f} ⇢ W ^{af f} , the upper index indicates the positive roots with respect to S ^{af f} , S _M ^{af f} , and ⌃ ^{af f} is the set of affine roots of ⌃. One chooses an element ˜ µ M central in W M (1), in particular of length ` M (˜ µ M ) = 0, lifting a strictly positive element µ M in ⇤ M

⁺

:=

⇤ \ W _M

⁺

. The element T _{µ ˜} ^M

_M

of H ^M is invertible of inverse T ^M

˜

µ

_M¹

but in general T µ ˜

M

is not invertible in H .

Theorem 1.4. (i) The R-submodule H M

⁺

of basis (T _w ^M _˜ ) w ˜

2

W

_M+

(1) is a subring of H ^M , called the positive subalgebra of H ^M .

(ii) The R-algebra H ^M = H ^M

⁺

[(T _{µ ˜} ^M

_M

) ¹ ] is a localization of H ^M

⁺

at T _{µ ˜} ^M

_M

.

(iii) The injective linear map H ^M !

^✓

H sending T _w ^M _˜ to T w ˜ for w ˜ 2 W M (1) restricted to H M

⁺

is a ring homomorphism.

(iv) As an ✓( H ^M

⁺

)-module, H is the almost localization of a left free ✓( H ^M

⁺

)-module V ^M

⁺

at T µ ˜

M

.

The theorem was known in special cases. The part (iv) means that H is the union over r 2 N of

r V ^M

⁺

:= { x 2 H | T _{µ ˜} ^r

_M

x 2 V ^M

⁺

} , V ^M

⁺

= _d2

^M

_W

₀

✓( H ^M

⁺

)T d ˜ .

Here ^M W 0 is the set of elements of minimal lengths in the cosets W M,0 \ W 0 and ˜ d 2 W (1) is an arbitrary lift of d. The theorem admits a variant for the subalgebra H ^M ⇢ H ^M associated the negative submonoid W _M , inverse of W _M

⁺

, for the linear map H ^M

^✓

!

^⇤

H sending (T _w ^M _˜ )

^⇤

to T _w

^⇤

_˜ for ˜ w 2 W M (1) [Vig1, Prop. 4.14], and with left replaced by right in (iv): H ^M = H M [T _{µ ˜} ^M

_M

], ✓

^⇤

restricted to H M is a ring homomorphism, the right

✓

^⇤

( H ^M )-module H is the almost localisation at T _{µ ˜}

^⇤ 1 M

of a right free ✓

^⇤

( H ^M )-module V M

^⇤

of rank | W M,0 | ¹ | W 0 | , meaning that H is the union over r 2 N of

r V M

^⇤

:= { x 2 H | x(T _{µ ˜}

^⇤ 1

M

) ^r 2 V M

^⇤

} , V M

^⇤

:= X

d

2

W

₀^M

T _d

^⇤

_˜ ✓

^⇤

( H ^M ).

Here W ₀ ^M is the inverse of ^M W 0 .

For a ring A, let Mod A denote the category of right A-modules and A Mod the category of left A-modules. Given two rings A ⇢ B, the induction ⌦ ^A B and the coinduction Hom A (B, ) from Mod A to Mod B are the left and the right adjoint of the restriction Res ^B _A . The ring B is considered as a left A-module for the induction, and as a right A-module for the coinduction.

The property (iv) and its variant describe H as a left ✓( H M

⁺

)-module and as a right

✓

^⇤

( H M )-module. The linear maps ✓ and ✓

^⇤

identify the subalgebras H M

⁺

, H M of H ^M with the subalgebras ✓( H M

⁺

), ✓

^⇤

( H M ) of H .

Definition 1.5. The parabolic induction and coinduction from Mod

_HM

to Mod

_H

are the functors I

_H^H_M

= ⌦

HM+

,✓ H and I

^HHM

= Hom

_HM

,✓

^⇤

( H , ).

We show:

Theorem 1.6. The parabolic induction I

_H^H_M

is faithful, transitive, respects finitely gen- erated R-modules, admits a right adjoint Hom

_HM+,✓

( H ^M , ).

If R is a field, the right adjoint functor respects finite dimension.

The transitivity of the parabolic induction means that for S M ⇢ S M

⁰

⇢ S, I

_H^H_M

= I

_H^H_M0

I

_H^H_M^M0

: Mod

_HM

! Mod

_HM0

! Mod

_H

.

Let w 0 denote the longest element of W 0 , S w

0

(M ) the subset w 0 S M w 0 of S, w ^M ₀ :=

w 0 w M,0 where w M,0 is the longest element of W M,0 . A lift ˜ w ^M ₀ 2 W 0 (1) of w ₀ ^M defines an R-algebra isomorphism

H ^M ! H w

0

(M ) , T _w ^M _˜ 7! T _w ^w _˜

M⁰

^{(M )}

0

w( ˜ ˜ w

₀^M

)

¹

for ˜ w 2 W M (1), (1)

inducing an equivalence of categories Mod

_HM

w

˜

^M₀

! Mod

_{Hw0 (M)}

, of inverse ˜ w ^w ₀

⁰

^{(M )} defined by the lift ( ˜ w ^M ₀ ) ¹ 2 W 0 (1) of w ₀ ^w

⁰

^{(M )} = (w ₀ ^M ) ¹ .

Definition 1.7. The w 0 -twisted parabolic induction and coinduction from Mod

_HM

to Mod

_H

are the functors I

_H^H_w

0 (M)

w ˜ ^M ₀ and I

^HHw0 (M)

w ˜ ^M ₀ .

Modulo equivalence, these functors do not depend on the choice of the lift of w ^M ₀ used for their construction.

Theorem 1.8. The parabolic induction (resp. coinduction) is equivalent to the w 0 -twisted parabolic coinduction (resp. induction):

I

^HHM

' I

_H^H_w

0 (M)

w ˜ ^M ₀ , I

_H^H_M

' I

^HHw0 (M)

w ˜ ^M ₀ .

Using that the coinduction admits a left adjoint and that the induction is a twisted coinduction, one proves:

Theorem 1.9. The parabolic induction I

_H^H_M

admits a left adjoint equivalent to w ˜ ^w ₀

⁰

^{(M )} ( ⌦

Hw0 (M)

,✓

^⇤

H ^w

0

(M) ) : Mod

_H

! Mod

_{Hw0 (M)}

! Mod

_HM

When R is a field, the left adjoint functor respects finite dimension.

The coinduction satisfies the same properties as the induction:

Corollary 1.10. The coinduction I

^HHM

is faithful, transitive, respects finitely generated R-modules, admits a left and a right adjoint. When R is a field, the left and right adjoint functors respect finite dimension.

Note that the induction and the coinduction are exact functors, as they admit a left and a right adjoint. A localization functor is exact hence also the left adjoint of the induction and of the coinduction.

We prove Theorem 1.4 in chapter 2, Theorem 1.6 in chapter 3.2, Theorem 1.8, Theorem 1.9 in chapter 3.2.

Remark 1.11. One cannot replace ( H , H ^M , H ⁺ M ) by ( H , H ^M , H M ) to define the induction I

_H^H_M

.

When no non-zero element of the ring R is infinitely p-divisible, is the parabolic in- duction functor Mod

_HM

^I

HMH

! Mod

_H

fully faithful ? The answer is yes for the parabolic induction functor Mod

¹

_R (M ) ^Ind

G

!

P

Mod

¹

_R (G) when M is a Levi subgroup of a parabolic subgroup P of a reductive p-adic group G and Mod

¹

_R (G) the category of of smooth R- representations of G [Vig2, Theorem 5.3].

This paper is influenced by discussions with Rachel Ollivier, Noriyuki Abe, Guy Hen-

niart and Florian Herzig, and by our work in progress on representations modulo p of

reductive p-adic groups and their pro-p Iwahori Hecke algebras. I thank them, and the

Institute of Mathematics of Jussieu, the University of Paris 7 for providing a stimulating

mathematical environment.

2 Levi algebra

We prove Theorem 1.4 and its variant on the subalgebra H

^✏

_M ⇢ H M , its image in H , on H M as a localisation of H

^✏

_M and on H as an almost left localisation of ✓(H ⁺ _M ), and almost left localisation of ✓

^⇤

(H _M ).

2.1 Monoid W M

^✏

Let S M ⇢ S and ✏ 2 { +, } . To S ^{af f} is associated a submonoid W M

^✏

⇢ W M defined as follows.

Let ⌃ denote the reduced root system of affine Weyl group W ^{af f} , V the real vector space of dual generated by ⌃, ⌃ ^{af f} = ⌃ + Z the set of affine roots of ⌃ and H = { Ker V ( ) | 2 ⌃ ^{af f} } the set of kernels of the affine roots in V . We fix a W 0 -invariant scalar product on V . The affine Weyl group W ^{af f} identifies with the group generated by the orthogonal reflections with respect to the affine hyperplanes of H.

Let A denote the alcove of vertex 0 of (V, H) such that S ^{af f} is the set of orthogonal reflections with respect to the walls of A and S is the subset associated to the walls containing 0. An affine root which is positive on A is called positive. Let ⌃ ^{af f,+} denote the set of positive affine roots, ⌃ ⁺ := ⌃ \ ⌃ ⁺ _{af f} , ⌃ ^{af f,} := ⌃ ^{af f,} , ⌃ := ⌃ ⁺ .

Let M denote the set of positive roots ↵ 2 ⌃ ⁺ such that Ker ↵ is a wall of A and the orthogonal reflection s

↵

of V with respect to Ker ↵ belongs to S M , ⌃ M ⇢ ⌃ the reduced root system generated by M , ⌃

^✏

_M := ⌃ M \ ⌃

^✏

_{af f} .

Definition 2.1. The positive monoid W _M

⁺

⇢ W M is { w 2 W M | w(⌃ ⁺ ⌃ ⁺ _M ) ⇢ ⌃ ^{af f,+} } . The negative monoid W _M := { w 2 W M | w ¹ 2 W _M

⁺

} is the inverse monoid.

It is well known that the finite Weyl group W M,0 is the W 0 -stabilizer of ⌃

^✏

⌃

^✏

_M . This implies

W M

^✏

= ⇤ M

^✏

o W M,0 where ⇤ M

^✏

:= ⇤ \ W M

^✏

.

Let ⇤ !

^⌫

V denote the homomorphism such that 2 ⇤ acts on V by translation by ⌫ ( ).

Lemma 2.2. ⇤ M

^✏

= { 2 ⇤ | ( ⌫)( ) 0 for all 2 ⌃

^✏

⌃

^✏

_M } .

Proof. Let 2 ⇤. By definition, 2 ⇤ M

⁺

if and only if ( ) is positive for all 2

⌃ ⁺ ⌃ ⁺ _M . We have ( ) = ⌫( ). The minimum of the values of on A is 0[Vig1, (35)]. So (v ⌫ ( )) 0 for 2 ⌃ ⁺ ⌃ ⁺ _M and v 2 A is equivalent to ( ⌫)( ) 0 for all 2 ⌃ ⁺ ⌃ ⁺ _M .

When S M ⇢ S M

⁰

⇢ S, we have the inclusion ⌃

^✏

_M ⇢ ⌃

^✏

_M

0

, the inverse inclusion

⌃

^✏

⌃

^✏

_M ⇢ ⌃

^✏

⌃

^✏

_M

0

, and the inclusions W M ⇢ W M

⁰

and W M

^✏

⇢ W _M

^✏0

.

Remark 2.3. Set D

^✏

:= { v 2 V | (v) 0 for 2 ⌃

^✏

} and ⇤

^✏

:= ( ⌫) ¹ ( D

^✏

). The antidominant Weyl chamber of V is D and the dominant Weyl chamber is D ⁺ . Careful:

[Vig3, § 1.2 (v)] uses a di↵erent notation: ⇤

^✏

= (⌫) ¹ ( D

^✏

).

The Bruhat order  of the affine Coxeter system (W ^{af f} , S ^{af f} ) extends to W : for w 1 , w 2 2 W ^{af f} , u 1 , u 2 2 ⌦, w 1 u 1  w 2 u 2 if u 1 = u 2 and w 1  w 2 [VigRT, Appendice].

We write w < w

⁰

if w  w

⁰

and w 6 = w

⁰

for w, w

⁰

2 W . Careful: the Bruhat order  ^M on W M associated to (W _M ^{af f} , S ^{af f} _M ) is not the restriction of  when S _M ^{af f} is not contained in S ^{af f} [Vig4].

Remark 2.4. The basic properties of (W ^{af f} , S ^{af f} ) extend to W : (i) If x  y for x, y 2 W and s 2 S ^{af f} ,

sx  (y or sy), xs  (y or ys), (x or sx)  sy, (x or xs)  ys

[Vig3, Lemma 3.1, Remark 3.2].

(ii) W = t

2⇤^✏

W 0 W 0 [HV1, 6.3 Lemma].

(iii) For 2 ⇤ ⁺ , W 0 W 0 admits a unique element of maximal length w = w 0 where w 0 is the unique element of maximal length in W 0 , and `(w ) = `(w 0 ) + `( ) [Vig3, Lemma 3.5].

(iv) For 2 ⇤ ⁺ , { w 2 W | w  w } t ^µ

2⇤⁺

,µ



W 0 µW 0 [Vig3, Lemma 3.5].

Remark 2.5. { w 2 W | w  w } is a union of (W 0 , W 0 )-classes only if , µ 2 ⇤ ⁺ , µ  w 0

implies µ  . I see no reason for this to be true.

Lemma 2.6. The monoid W M

^✏

is a lower subset of W M for the Bruhat order  ^M : for w 2 W M

^✏

, any element v 2 W M such that v  ^M w belongs to W M

^✏

.

Proof. [Abe, Lemma 4.1].

An element w 2 W admits a reduced decomposition in (W, S ^{af f} ), w = s 1 . . . s r u with s i 2 S ^{af f} , u 2 ⌦. As in [Vig1], we set for w, w

⁰

2 W ,

q w := q(s 1 ) . . . q(s r ), q w,w

⁰

:= (q w q w

⁰

q _ww ¹

0

) ^1/2 . (2)

This is independent of the choice of the reduced decomposition. For w, w

⁰

2 W M and s i 2 S _M ^{af f} , u 2 ⌦ M , let q M,w , q M,w,w

⁰

denote the similar elements. They may be di↵erent from q w , q w,w

⁰

.

Lemma 2.7. We have S _M ^{af f} \ W M

^✏

⇢ S ^{af f} and q w,w

⁰

= q M,w,w

⁰

if w, w

⁰

2 W M

^✏

. In particular, ` M (w) + ` M (w

⁰

) ` M (ww

⁰

) = `(w) + `(w

⁰

) `(ww

⁰

), if w, w

⁰

2 W M

^✏

. Proof. [Abe, Lemma 4.4 and proof of lemma 4.5].

An element 2 ⇤ M

^✏

such that all the inequalities in (2.2) are strict is called strictly positive if ✏ = +, and strictly negative if ✏ = +. We choose

a central element µ ˜ M of W M (1) lifting a stricty positive element µ M of ⇤.

We set ˜ µ M

⁺

:= ˜ µ M and ˜ µ M := ˜ µ _M ¹ . The center of the pro-p Iwahori Weyl group W M (1) is the set of elements in the center of ⇤(1) fixed by the finite Weyl group W M,0 [Vig2].

Hence ˜ µ M

^✏

is an element of the center of ⇤(1) fixed by W M,0 and ⌫ (µ M

^✏

) > 0 for all 2 ⌃

^✏

⌃

^✏

_M . We have ⌫ (µ M

^✏

) = 0 for 2 ⌃ M . The length of µ M

^✏

is 0 in W M , and is positive in W when S M 6 = S.

Let H ^M

^✏

denote the R-submodule of the Iwahori Hecke R-algebra H ^M of M of basis (T _w ^M _˜ ) w ˜

2

W

M✏

(1) , and H ^M !

^✓

H (resp. H

^{M ✓}

!

^⇤

H ) the linear map sending T _w ^M _˜ to T w ˜ ( resp. T

_w^M,_˜ ^⇤

to T

_w^⇤_˜

) for ˜ w 2 W M (1).

The proof of the properties (i), (ii), (iii) of Theorem 1.4 and its variant are as follows:

1. H ^M

^✏

is a subring of H ^M , because T _w ^M _˜ T _w ^M _˜

0

is a linear combination of elements T v ˜

such that v  ^M ww

⁰

[Vig1].

2. ✓(T _w ^M _˜

₁

T _w ^M _˜

₂

) = T w ˜

1

T w ˜

2

and ✓

^⇤

((T _w ^M _˜

₁

)

^⇤

(T _w ^M _˜

₂

)

^⇤

) = T _w

^⇤

_˜

₁

T _w

^⇤

_˜

₂

for w 1 , w 2 2 W M

^✏

for w 1 , w 2 2 W M

^✏

. This follows from the braid relations if ` M (w 1 ) + ` M (w 2 ) = ` M (w 1 w 2 ) be- cause `(w 1 )+`(w 2 ) = `(w 1 w 2 ) (Lemma 2.7). If w 2 = s 2 S _M ^{af f} with ` M (w 1 ) 1 = ` M (w 1 s) this follows from the quadratic relations

T w ˜

1

T s ˜ = T w ˜

1

˜ s

¹

(q(s)(˜ s) ² + T s ˜ c(˜ s)) = q(s)T w ˜

1

s ˜ + T w ˜

1

c(˜ s), T _w

^⇤

_˜

₁

T _{s ˜}

^⇤

= q(s)T _w

^⇤

_˜

₁

_{˜ s} T _w

^⇤

_˜

₁

c(˜ s),

s 2 S ^{af f} , `(w 1 ) 1 = `(w 1 s) (Lemma 2.7) and q(s) = q M (s), c(˜ s) = c M (˜ s) [Vig4]. In

general the formula is proved by induction on ` M (w 2 ) [Abe, 4.1]. The proof of [Abe,

Lemma 4.5] applies.

We have ✓

^⇤

(T

_w^M_˜

) = T

_w^M_˜

for w 2 W

M,0

because for s 2 S

M

,

✓

^⇤

(T

_˜s^M

) = ✓

^⇤

(T

_˜s^M,⇤

+ c

^M_˜s

) = T

_s˜^⇤

+ c

s˜

= T

s˜

.

3. H ^M = H ^M

^✏

[(T _{µ ˜} ^M

_M✏

) ¹ ], because for w 2 W M there exists r 2 N such that µ

^✏r

_M w 2 W M

^✏

.

Remark 2.8. If the parameters q(s) are invertible in R, then H ^M

⁺

!

^✓

H extends uniquely to an algebra homomorphism H ^M , ! H , sending T _{µ ˜} ^M

✏r

M

w ˜ to T _{µ ˜}

_M

^r

✏

T w ˜ for ˜ w 2 W M

⁺

(1), r 2 N .

Remark 2.9. The trivial character 1 : H ! R of H is defined by

1 (T w ˜ ) = q w ( ˜ w 2 W (1)).

When H is the Hecke algebra of the pro-p-Iwahori subgroup of a reductive p-adic group G, H acts on the trivial representation of G by 1 . Note that the restriction of the trivial character of H ^M to ✓( H ^M

⁺

) is not equal to 1 ✓ when ` M (µ M ) = 0, `(µ M ) 6 = 0.

2.2 An anti-involution ⇣

The R-linear bijective map

H !

^⇣

H such that ⇣(T w ˜ ) = T _{w ˜}

¹

for w ˜ 2 W (1), (3)

is an anti-involution when ⇣(h 1 h 2 ) = ⇣(h 2 )⇣(h 1 ) for h 1 , h 2 2 H because ⇣ ⇣ = id. For S M ⇢ S, let H

^⇣M

! H ^M denote the linear map such that ⇣(T _w ^M _˜ ) = T _w ^M _˜

1

for ˜ w 2 W M (1).

Lemma 2.10. 1. The following properties are equivalent:

(i) ⇣ is an anti-involution,

(ii) ⇣(c(˜ s)) = c (˜ s)

¹

for ˜ s 2 S ^{af f} (1),

(iii) ⇣ c = c ( ) ¹ where S(1) !

^c

R[Z k ] is the parameter map.

2. If ⇣ is an anti-involution then ⇣ M is an anti-involution.

Proof. Let ˜ w = ˜ s 1 . . . ˜ s

`(w)

u ˜ be a reduced decomposition, ˜ s i 2 S ^{af f} (1), u ˜ 2 W (1), `(˜ u) = 0 and let ˜ s 2 S ^{af f} (1). Then,

⇣(T w ˜ ) = T ( ˜ w)

¹

= T (˜ u)

¹

T _{s ˜}

¹

`(w)

. . . T _{˜ s}

¹

1

= ⇣(T u ˜ )⇣(T s ˜

`(w)

) . . . ⇣(T ˜ s

1

), (⇣(T s ˜ )) ² = T _{˜ s} ²

1

= q(s)˜ s ² + c(˜ s ¹ )T _{˜ s}

¹

Tthe map ⇣ is an anti-automorphism if and only if ⇣(c(˜ s)) = c(˜ s ¹ ) for ˜ s 2 S ^{af f} (1). This is equivalent to ⇣ c = c ( ) ¹ because S(1) is the union of the W (1)-conjugates of S ^{af f} (1), c is W (1)-equivariant and ⇣ commutes with the conjugation by W (1).

If c satisfies (iii), its restriction c M to S M (1) satisfies (iii).

Lemma 2.11. When H = H (G) is the pro-p Iwahori Hecke R-algebra of a reductive p-adic group G, ⇣ is an anti-involution.

Proof. Let s 2 S, ˜ s an admissible lift and t 2 Z k . Then c(˜ s) is invariant by ⇣ [Vig1, Prop.4.4] If u 2 U

^⇤

for = ↵ + r 2 ^{af f} red , then u ¹ 2 U

^⇤

and m

↵

(u) ¹ = m

↵

(u ¹ ).

Hence the set of admissible lifts of s is stable by the inverse map. As the group Z k is commutative, we have

(⇣ c)(t˜ s) = ⇣(tc(s)) = t ¹ c(s) = c(s)t ¹ = c((t˜ s) ¹ .

From now on, we suppose that ⇣ is an anti-involution. We recall the involutive auto- morphism [Vig1, Prop. 4.24]

H !

^◆

H such that ◆(T w ˜ ) = ( 1)

^`(w)

T _w

^⇤

_˜ for w ˜ 2 W (1), and [Vig1, Prop. 4.13 2)]:

T _{˜ s}

^⇤

:= T s ˜ c(˜ s) for ˜ s 2 S ^{af f} (1), T _w

^⇤

_˜ := T _{˜ s}

^⇤₁

. . . T _{s ˜}

^⇤_r

T u ˜ for ˜ w 2 W (1) (4)

of reduced decomposition ˜ w = ˜ s 1 . . . ˜ s

`(w)

u. ˜

Remark 2.12. We have ⇣(T _w

^⇤

_˜ ) = T _{( ˜}

^⇤

_w)

1

for w ˜ 2 W (1), ⇣ and ◆ commute, ⇣ M ( H ^M

^✏

) = H M

^✏

, and ✓ ⇣ M = ⇣ ✓, ✓

^⇤

⇣ M = ⇣ ✓

^⇤

.

2.3 ✏-alcove walk basis

We define a basis of H associated to ✏ 2 { +, } and an orientation o of (V, H), that we call an ✏-alcove walk basis associated to o.

For s 2 S ^{af f} , let ↵ s denote the positive affine root such that s is the orthogonal reflection with respect to Ker ↵ s . For an orientation o of (V, H), let D ^o denote the corre- sponding (open) Weyl chamber in (V, H), A o the (open) alcove of vertex 0 contained in D ^o , and o.w the orientation of Weyl chamber w ¹ (D o ) for w 2 W . We recall [Vig1]:

Definition 2.13. The following three properties determine uniquely elements E o ( ˜ w) 2 H for any orientation o of (V, H) and w ˜ 2 W (1). For w ˜ 2 W (1), ˜ s 2 S ^{af f} (1), u ˜ 2 ⌦(1):

E o (˜ s) =

( T ˜ s if ↵ s is negative on A o , T _{˜ s}

^⇤

= T s ˜ c(˜ s) if ↵ s is positive on A o , (5)

E o (˜ u) = T u ˜ , (6)

E o (˜ s)E o.s ( ˜ w) = q s,w E o (˜ s w). ˜ (7)

They imply, for w

⁰

2 W, 2 ⇤:

E o ( ˜ w

⁰

)E o.w

⁰

( ˜ w) = q w

⁰

,w E o ( ˜ w

⁰

w), ˜ E o (˜)E o ( ˜ w) = q ,w E o (˜ ˜ w).

(8)

We recall that acts on V by translation by ⌫( ). The Weyl chamber D ^o of the orientation o is characterized by:

E o (˜) = T ˜ when ⌫( ) belongs to the closure of D ^o . (9)

The alcove walk basis of H associated to o is (E o ( ˜ w)) w ˜

2

W (1) [Vig1]. The Bernstein basis (E( ˜ w)) w ˜

2

W(1) is the alcove walk basis associated to the antidominant orientation of Weyl chamber D Remark 2.3. By (5) and (9), the Bernstein basis satisfies

E( ˜ w) = T w ˜ for w 2 ⇤ ⁺ [ W 0 , E( ˜ w) = T _w

^⇤

_˜ for w 2 ⇤ .

The alcove walk basis (E

o⁺

( ˜ w))

w˜2W(1)

associated to the dominant orientation of Weyl chamber D

⁺

satisfies similar relations with T

_w^⇤_˜

permuted with T

w˜

:

E

_o⁺

( ˜ w) = T

_w^⇤_˜

for w 2 ⇤

⁺

[ W

0

, E

_o⁺

( ˜ w) = T

w˜

for w 2 ⇤ .

Definition 2.14. The ✏-alcove walk basis (E _o

^✏

( ˜ w)) w2W ˜ (1) of H associated to o is E _o

^✏

( ˜ w) :=

( E o ( ˜ w) if ✏ = +,

⇣(E o ( ˜ w ¹ )) if ✏ = .

(10)

Lemma 2.15. The elements E _o ( ˜ w) for any orientation o of (V, H ) and w ˜ 2 W (1) are determined by the following properties. For w ˜ 2 W (1), s ˜ 2 S ^{af f} (1), u ˜ 2 ⌦(1):

E _o (˜ s) = E o (˜ s), E _o (˜ u) = E o (˜ u), (11)

E _o.s ( ˜ w)E _o (˜ s) = q w,s E _o ( ˜ w˜ s).

(12)

They imply for w

⁰

2 W, 2 ⇤:

E _o.w

₀ 1

( ˜ w)E _o ( ˜ w

⁰

) = q w,w

⁰

E _o ( ˜ w w ˜

⁰

), E _o ( ˜ w)E _o (˜) = q w, E _o ( ˜ w˜).

(13) Proof.

E _o (˜ s) = ⇣(E o ((˜ s) ¹ )) = E o (˜ s),

E _o ( ˜ w˜ u) = ⇣(E o (( ˜ w˜ u) ¹ )) = ⇣(E o ((˜ u) ¹ ( ˜ w) ¹ )) = ⇣(T (˜ u)

¹

E o (( ˜ w) ¹ ))

= ⇣(E o (( ˜ w) ¹ ))T u ˜ = E _o ( ˜ w)T u ˜ ,

E _o.s ( ˜ w)E _o (˜ s) = ⇣(E o.s (( ˜ w) ¹ ))⇣(E o ((˜ s) ¹ )) = ⇣(E o ((˜ s) ¹ )E o.s (( ˜ w) ¹ ))

= q s,w

¹

⇣(E o ((˜ s) ¹ ( ˜ w) ¹ )) = q w,s ⇣(E o (( ˜ w˜ s) ¹ )) = q w,s E _o ( ˜ w˜ s).

We used that q w = q w

¹

implies q _w

¹

1

,w

₂¹

= (q _w

¹

1

q _w

¹

2

q _w ¹

1

1

w

₂¹

) ^1/2 = (q w

1

q w

2

q _w

₂

¹ _w

₁

) ^1/2 = q w

2

,w

1

for w 1 , w 2 2 W .

The ✏-alcove walk bases satisfy the the triangular decomposition:

E _o

^✏

( ˜ w) T w ˜ 2 X

˜

w

⁰2

W (1), w ˜

⁰

< w ˜

RT w ˜

⁰

. (14)

Remark 2.16. We will denote E

+

( ˜ w) = E

o⁺

( ˜ w) and E ( ˜ w) = E

_o+

( ˜ w) as in [Abe] and call (E

✏

( ˜ w))

w˜2W(1)

the lower ✏-Bernstein basis of H (the upper ✏-Bernstein basis will be the usual Bernstein basis).

Similarly, we will denote by (E

_M^✏

( ˜ w))

w˜2WM(1)

and (E

_✏^M

( ˜ w))

w˜2WM(1)

the upper and lower ✏-Bernstein bases associated to the dominant orientation for (V

M

, H

M

); here V

M

ise the real vector space of dual generated by ⌃

M

with a W

M,0

-invariant scalar product and H

M

the corresponding set of affine hyperplanes.

Lemma 2.17. For ✏, ✏

⁰

2 { +, } and any orientation o M of (V M , H M ), (E

^✏

_o

⁰_M

( ˜ w)) w ˜

2

W

M✏

(1)

is a basis of H ^M

^✏

.

When q(s) = 0 [Abe, Lemma 4.2].

Proof. A basis of H ^M

^✏

is (T _w ^M _˜ ) w2W ˜

M✏

(1) . As w < M w

⁰

and w

⁰

2 W M

^✏

implies w 2 W M

^✏

(Lemma 2.6), the triangular decomposition (14) implies that (E _o

^✏0_M

( ˜ w)) _{w2W ˜}

_M✏

(1) is a basis of H ^M

^✏

.

Lemma 2.18. The ✏-Bernstein basis satisfies E

^✏

( ˜ w) = T w ˜ if w 2 ⇤

^✏

[ W 0 and E

^✏

( ˜ w) = T _w

^⇤

_˜ if w 2 ⇤

^✏

. The basis (E

✏

( ˜ w)) satisfies similar relations with T

_w^⇤_˜

permuted with T

w˜

: E

✏

( ˜ w) = T

_w^⇤_˜

if w 2 ⇤

^✏

[ W

0

and E ( ˜ w) = T

w˜

if w 2 ⇤

^✏

.

Proof. We described E

⁺

( ˜ w) and E

+

( ˜ w) for w 2 ⇤

⁺

[ ⇤ [ W

0

before Definition 2.14 and we have:

E ( ˜ w) = ⇣(E( ˜ w

¹

)) =

( ⇣(T

_w^⇤_˜ 1

) = T

_w^⇤_˜

(w 2 ⇤

⁺

)

⇣(T

w˜ ¹

) = T

w˜

(w 2 ⇤ [ W

0

) E ( ˜ w) = ⇣(E

_o⁺

( ˜ w

¹

)) =

( ⇣(T

_w^⇤_˜ 1

) = T

_w^⇤_˜

(w 2 ⇤ [ W

0

)

⇣(T

w˜ ¹

) = T

w˜

(w 2 ⇤

⁺

).

The upper and lower ✏-Bernstein bases are compatible with the linear embeddings ✓ and ✓

^⇤

of H

^M

into H :

Proposition 2.19. We have ✓(E

^✏

_M ( ˜ w)) = E

^✏

( ˜ w), ✓

^⇤

(E

^M_✏

( ˜ w)) = E

✏

( ˜ w) for w ˜ 2 W

M⁺

(1) [ W

M

(1).

This generalizes [Ollivier10, Prop. 4.7], [Ollivier14, Lemma 3.8], [Abe, Lemma 4.5].

Proof. It suffices to prove the proposition when the q(s) are invertible. Let ˜ w 2 W (1).

We write ˜ w = ˜ ˜ u = ˜ 1 (˜ 2 ) ¹ u ˜ with u 2 W 0 , and 1 , 2 in ⇤

^✏

. We have for any orientation o of (V, h)

E o (˜ 1 )E o ((˜ 2 ) ¹ ) = q

₁

_,

¹

2

E o (˜), E o (˜ 2 )E o ((˜ 2 ) ¹ ) = q

₂

_,

¹

2

= q

2

, E o (˜ 1 )E((˜ 2 ) ¹ )E o (˜ u) = q

₁

_,

¹

2

q ,u E o ( ˜ w).

Then, E o ( ˜ w) = q

2

(q

₁

_,

¹

2

q ,u ) ¹ E o (˜ 1 )E o (˜ 2 ) ¹ E o (˜ u). Applying Lemma 2.18 to the orientations o of Weyl chamber D

^±

we obtain:

E( ˜ w) = q

2

(q

₁

_,

¹

2

q ,u ) ¹

( T ˜

₁

T _˜ ¹

2

T u ˜ if ✏ = + T _˜

^⇤

1

(T _˜

^⇤

2

) ¹ T u ˜ if ✏ = (15)

and similar formulas for E

+

( ˜ w) with T

_w^⇤_˜

permuted with T

w˜

. We suppose now w 2 W M

^✏

, that is 2 ⇤ M

^✏

, u 2 W M,0 . Note ⇤

^✏

⇢ ⇤ M

^✏

and q M, ,u = q ,u (Lemma 2.7).

Suppose w 2 W M

⁺

. Then E M ( ˜ w) = q M,

2

(q _M,

₁

_,

¹

2

q ,u ) ¹ T _˜ ^M

1

(T _˜ ^M

2

) ¹ T _{u ˜} ^M and

✓(E M ( ˜ w)) = q M,

2

(q _M,

₁

_,

¹

2

q ,u ) ¹ T ˜

₁

T _˜ ¹

2

T u ˜

= q M,

2

(q _M,

₁

_,

¹

2

q ,u ) ¹ q

₂

¹ q

₁

_,

¹

2

q ,u E( ˜ w) = q M,

2

(q _M,

₁

_,

¹

2

q

2

) ¹ q

₁

_,

¹

2

E( ˜ w).

The triangular decomposition of E M ( ˜ w) and E( ˜ w) implies q M,

2

(q _M,

₁

_,

¹

2

q

₂

) ¹ q

₁

_,

¹

2

=

1. Hence for w 2 W M

⁺

we have ✓(E M ( ˜ w)) = E( ˜ w), and by the same arguments

✓

^⇤

(E

^M₊

( ˜ w)) = E

+

( ˜ w).

Suppose w 2 W

_M

. We write ˜ w = ˜ ˜ w

0

with ˜ 2 ⇤(1) M

1

-negative and s 2 w ˜

0

2 W

M1,0

. We have E( ˜ w) = q

,w0

T

_˜^⇤

T

w˜0

and E

M

( ˜ w) = q

^M_,w0

T

_˜^M,⇤

T

w˜0

with q

,w0

= q

^M_,w0

(Lemma 2.7). Applying the homomorphism H

M₁

!

✓

H we obtain ✓(E

M

( ˜ w)) = E( ˜ w).

The same arguments show that ✓

^⇤

(E

₊^M

( ˜ w)) = E

+

( ˜ w).

Suppose w 2 W

M⁺

[ W

M

. We proved that ✓(E

M

( ˜ w)) = E( ˜ w) and ✓

^⇤

(E

₊^M

( ˜ w)) = E

+

( ˜ w), i.e. that E

o

( ˜ w) is the image of E

_o^M

( ˜ w) by ✓ and ✓

^⇤

when o is the orientation of Weyl chamber dominant or anti-dominant. Using E _o ( ˜ w) = ⇣(E o (( ˜ w) ¹ )) and that ⇣ ✓ =

✓ ⇣ M , ⇣ ✓

^⇤

= ✓

^⇤

⇣ M (Remark 2.12), this implies that E _o ( ˜ w) is the image of E _M,o ( ˜ w) by

✓ and ✓

^⇤

, as E _o ( ˜ w) = (⇣ ✓)(E M,o (( ˜ w) ¹ )) = (✓ ⇣ M )(E M,o (( ˜ w) ¹ )) = ✓(E _M,o ( ˜ w)).

2.4 w 0 -twist

Let S M ⇢ S, w 0 denote the longest element of W 0 and S w

0

(M ) = w 0 S M w 0 ⇢ w 0 Sw 0 = S.

The longest element w M,0 of W M,0 satisfies w M,0 (⌃

^✏

_M ) = ⌃ _M

^✏

, and w M,0 (⌃

^✏

⌃

^✏

_M ) =

⌃

^✏

⌃

^✏

_M . The longest element w w

0

(M),0 of W w

0

(M ),0 is w 0 w M,0 w 0 .

Let w ^M ₀ := w 0 w M,0 . Its inverse ^M w 0 := w M,0 w 0 is w ^w ₀

⁰

^{(M )} and w ^M ₀ (⌃

^✏

_M ) = ⌃

^✏

_w

₀

_{(M )} . This implies that w ^M ₀ (⌃ ^{af f,✏} _M ) = ⌃ ^{af f,✏} _w

₀

_(M) . Indeed the image by w ^M ₀ of the simple roots of

⌃ M is the set of simple roots of ⌃ w

0

(M) , and this remains true for the simple affine roots

which are not roots. Note that the irreducible components ⌃ M,i of ⌃ M have a unique

highest root a M,i , and that the a M,i + 1 are the simple affine roots of ⌃ which are not

roots. We have w ₀ ^M ( a M,i + 1) = w 0 w M,0 ( a M,i + 1) = w 0 (a M,i ) + 1. The irreducible components of ⌃ _w

₀

_(M) are the w 0 (⌃ M,i ) and w 0 (a M,i ) is the highest root of w 0 (⌃ M,i ).

We deduce:

w ₀ ^M S _M ^{af f} (w ^M ₀ ) ¹ = S _w ^{af f}

₀

_{(M )} , w ^M ₀ W _M,0 ^{af f} (w ₀ ^M ) ¹ = W _w ^{af f}

₀

_(M,)0 , w ^M ₀ W M,0 (w ^M ₀ ) ¹ = W w

0

(M,)0 . We have ⇤ = w ^M ₀ ⇤(w ₀ ^M ) ¹ and w ^M ₀ ⇤

^✏

_M (w ₀ ^M ) ¹ = ⇤ _w

₀^✏

_(M) . Recalling W M = ⇤ o W M,0 , W M

^✏

= ⇤ M

^✏

o W M,0 and the group ⌦ M of elements which stabilize A M , we deduce:

w ^M ₀ W M (w ₀ ^M ) ¹ = W w

0

(M) , w ^M ₀ ⌦ M (w ₀ ^M ) ¹ = ⌦ w

0

(M) , w ^M ₀ W M

^✏

(w ^M ₀ ) ¹ = W _w

₀^✏

_(M) . (16)

Let ⌫ M denote the action of W M on V M and A M the dominant alcove of (V M , H M ).

The linear isomorphism V M

w

^M₀

! V w

0

(M ) , h ↵, x i = h w ₀ ^M (↵), w ^M ₀ (x) i for ↵ 2 ⌃ M , satisfies

w ^M ₀ ⌫ M (w) = ⌫ w

0

(M ) (w ^M ₀ w(w ^M ₀ ) ¹ ) w ₀ ^M for w 2 W M .

It induces a bijection H M ! H w

0

(M ) sending A M to A w

0

(M ) , a bijection D M 7! w ₀ ^M (D M ) between the Weyl chambers, a bijection o M 7! w ^M ₀ (o M ) between the orientations such that D _w

^M

0

(o

M

) = w ^M ₀ (D o

M

).

Proposition 2.20. Let w ˜ ^M ₀ 2 W 0 (1) be a lift of w ^M ₀ . The R-linear map H ^M ! ^j H ^w

0

(M ) , T _w ^M _˜ 7! T _w ^w _˜

M⁰

^{(M )}

0

w( ˜ ˜ w

^M₀

)

¹

for w ˜ 2 W M (1),

is a R-algebra isomorphism sending H ^M

^✏

onto H ^w

0

(M )

^✏

and respecting the ✏

⁰

-alcove walk basis

j(E _o

^✏0_M

( ˜ w)) = E _w

^✏0M

0

(o

M

) ( ˜ w ₀ ^M w( ˜ ˜ w ^M ₀ ) ¹ ) for w ˜ 2 W M (1), for any orientation o M of (V M , H M ) and ✏, ✏

⁰

2 { +, } .

Proof. The proof is formal using the properties given above the proposition and the characterization of the elements in the ✏

⁰

-alcove walks bases given by (5), (6), (7) if

✏

⁰

= + and (11), (12) if ✏

⁰

= .

We study now the transitivity of the w 0 -twist. Let S M ⇢ S M

⁰

⇢ S. We have the subset w M

⁰

,0 S M w M

⁰

,0 = S w

M0,0

(M) of S and we associate to the conjugation by a lift

˜

w M

⁰

,0 of w M

⁰

,0 in W (1) an isomorphism H ^{M j} !

⁰

H w

M0,0

(M ) similar to H ^M ! ^j H w

0

(M) in Proposition 2.20. We will show that j factorizes by j

⁰

.

We have w ^M ₀ = w ^M ₀

⁰

w _M ^M

0

, where w ^M _M

0

:= w M

⁰

,0 w M,0 (equal to w ^M ₀ if S = S M

⁰

), W w

_M0,0

(M ) = w ^M _M

0

W M (w ^M _M

0

) ¹ , W w

0

(M ) = w ^M ₀

⁰

W w

_M0,0

(M ) (w ^M ₀

⁰

) ¹ = w ₀ ^M W M (w ^M ₀ ) ¹ .

For S M

1

⇢ S M

⁰

, let W _M

^✏,M0

1

⇢ W M

1

denote the submonoid associated to S _M ^{af f}

0

as in Definition 2.1 (the pair (⌃ ⁺ ⌃ ⁺ _M

₁

, ⌃ ^{af f,+} ) is replaced by the pair (⌃ ⁺ _M

0

⌃ ⁺ _M

₁

, ⌃ ^{af f,+} _M

0

)).

We note that:

W _w

_M0,0

_{(M )}

^✏,M0

= w ^M _M

0

W M

^✏

(w _M ^M

0

) ¹ ,

W w

0

(M )

^✏

= w ₀ ^M

⁰

W _w

_M0,0

_{(M )}

^✏,M0

(w ^M ₀

⁰

) ¹ = w ^M ₀ W M

^✏

(w ^M ₀ ) ¹ .

Let ˜ w ^M ₀ , w ˜ ^M ₀

⁰

, w ˜ ^M _M

0

in W 0 (1) lifting w ^M ₀ , w ^M ₀

⁰

, w ^M _M

0

and satisfying ˜ w ^M ₀ = ˜ w ^M ₀

⁰

w ˜ _M ^M

0

. The algebra isomorphisms

H ^{M j} !

⁰

H w

M0,0

(M ) , H ^M

⁰

^j !

⁰⁰

H w

0

(M

⁰

) , H ^M ! ^j H w

0

(M )

defined by ˜ w ^M _M

0

, w ˜ ₀ ^M

⁰

, w ˜ ₀ ^M respectively, as in Proposition 2.20, send the ✏-subalgebra to the ✏-subalgebra and are compatible with the ✏

⁰

-Bernstein bases. We cannot compose j

⁰

with the map j

⁰⁰

defined by ˜ w ₀ ^M

⁰

, but we can compose j

⁰

with the bijective R-linear map defined by the conjugation by ˜ w ^M ₀

⁰

in W (1):

H ^w

M0,0

(M ) k

⁰⁰

! H ^w

0

(M) , T _w ^w _˜

^M0,0

^{(M )} 7! T ^w

⁰

^{(M )}

˜

w

^M0₀

w( ˜ ˜ w

₀^M0

)

¹

for ˜ w 2 W w

_M0,0

(M) (1).

Proposition 2.21. j = k

⁰⁰

j

⁰

and k

⁰⁰

is an R-algebra isomorphism respecting the ✏- subalgebras and the ✏-Bernstein bases: k

⁰⁰

( H w

M0,0

(M)

^✏

) = H w

0

(M)

^✏

and k

⁰⁰

(E _w

^✏_M0,0

_(M) ( ˜ w)) = E _w

^✏₀

_(M) ( ˜ w ^M ₀

⁰

w( ˜ ˜ w ₀ ^M

⁰

) ¹ ) for ✏ 2 { +, } , w 2 W w

_M0,0

(M) .

Proof. The relations between the groups W

_⇤

and W

_⇤^✏

imply obviously that j = k

⁰⁰

j

⁰

and that k

⁰⁰

respects the ✏-subalgebras.

k

⁰⁰

is an algebra isomorphism respecting the ✏

⁰

-Bernstein bases because j, j

⁰

are algebra isomorphisms respecting the ✏

⁰

-Bernstein bases and k

⁰⁰

= j (j

⁰

) ¹ .

2.5 Distinguished representatives of W 0 modulo W M,0

The classical set ^M W 0 of representatives on W M,0 \ W 0 is equal to M D ₁ = M D ₂ where [Carter, 2.3.3]

M D 1 := { d 2 W 0 | d ¹ (⌃ ⁺ _M ) 2 ⌃ ⁺ } , (17)

M D 2 := { d 2 W 0 | `(wd) = `(w) + `(d) for all w 2 W M,0 } . (18)

The properties of ^M W 0 used in this article that we are going to prove are probably well known. Note that the classical set of representatives of W 0 \ W is studied in [Vig3], that + can be replaced by ✏ 2 { +, } in the definition of M D 1 , that ^M w 0 = w M,0 w 0 2 ^M W 0

and that ^M W 0 \ S = S S M .

Taking inverses, we get the classical set W ₀ ^M of representatives on W 0 /W M,0 equal to D M,1 = D M,2 , where

D M,1 := { d 2 W 0 | d(⌃ ⁺ _M ) ⇢ ⌃ ⁺ } , (19)

D M,2 := { d 2 W 0 | `(dw) = `(d) + `(w) for all w 2 W M,0 } . (20)

The length of an element of W is equal to the length of its inverse, and [Vig1, Cor. 5.10]:

for 2 ⇤, w 2 W 0 ,

`( w) = X

2⌃⁺\w(⌃⁺

)

| ⌫( ) | + X

2 w

| ⌫ ( ) + 1 | . (21)

where w := ⌃ ⁺ \ w(⌃ ). If w = s 1 . . . s

`(w)

is a reduced decomposition in (W 0 , S),

w = { ↵ s

1

} [ s 1 ( s

1

w ) and `(w) is the order of w . If w 2 W M,0 , w ⇢ ⌃ <sup>+</sup> <sub>M</sub> . Let ` ( w) denote the contribution of 2 ⌃ ⁺ to the right side of (21).

We show now that W M,0 can be replaced by W M

⁺

in (18) and by W M in (20) (taking the inverses). It is also a variant of the equivalence `( w) < `( ) + `(w) , ⌫( ) > 0 for some 2 ^w for , w as in (21).

Lemma 2.22. (i) `(wd) = `(w) + `(d) for w 2 W M

⁺

and d 2 ^M W 0 .

`(dw) = `(d) + `(w) for w 2 W _M and d 2 W ₀ ^M .

(ii) For 2 ⇤, w 2 W M,0 , d 2 ^M W 0 , then `( wd) < `( w) + `(d) is equivalent to w( ) ⌫ ( ) > 0 and d ¹ ( ) 2 ⌃ for some 2 ⌃ ⁺ ⌃ ⁺ _M . Proof. [Ollivier10, Lemma 2.3], [Abe, Lemma 4.8].

Let 2 ⇤, w 2 W M,0 , d 2 ^M W 0 and 2 ⌃ ⁺ .

Suppose 2 ⌃ ⁺ _M . Then ` (d) = 0, d = ; because d ¹ (⌃

^✏

_M ) ⇢ ⌃

^✏

(17), and ` ( wd) =

` ( w) because w ¹ ( ) 2 ⌃

^✏

, w ¹ ( ) 2 ⌃

^✏

_M ) d ¹ w ¹ ( ) 2 ⌃

^✏

(17).

Suppose 2 ⌃ ⁺ ⌃ ⁺ _M . Then w ¹ ( ) 2 ⌃ ⁺ ⌃ ⁺ _M and ` ( w) = | ⌫ ( ) | .

The number `(d) of 2 ⌃ ⁺ ⌃ ⁺ _M such that d ¹ ( ) 2 ⌃ is equal to the number of 2 ⌃ ⁺ ⌃ ⁺ _M such that (wd) ¹ ( ) 2 ⌃ .

When 2 ⇤ M

⁺

and (wd) ¹ ( ) 2 ⌃ , then ⌫( )  0 and ` ( wd) = | ⌫( ) | + 1.

Therefore `( wd) = `( w) + `(d), which gives (i).

When 62 ⇤ ⇤ M

⁺

, `( wd) < `( w) + `(d) if and only if there exists 2 ⌃ ⁺ ⌃ ⁺ _M such that ⌫( ) > 0 and d ¹ w ¹ ( ) 2 ⌃ . This gives (ii) because 7! w ¹ ( ) is a permutation map of ⌃ ⁺ ⌃ ⁺ _M .

Lemma 2.23. (i) For 2 ⇤, w 2 W 0 , we have q = q w w

¹

, q w = q w

0

ww

0

, and

`(w 0 ) = `(w) + `(w <sup>1</sup> w 0 ) = `(w 0 w ¹ ) + `(w).

(ii) For w 2 W M,0 , we have q w = q _w

^M₀

_w(w

^M₀

₎

¹

.

Proof. (i) [Vig1, Prop. 5.13]. The length on W 0 is invariant by inverse and by conjugation by w 0 because w 0 Sw 0 = S and [Bki, VI § 1 Cor. 3].

(ii) q w = q _w

_M,0

_ww

¹

M,0

= q _w

^M₀

_w(w

^M₀

₎

¹

for w 2 W M,0 . Lemma 2.24. W ₀ ^M = W ₀ ^w

⁰

^(M) w ₀ ^M = w 0 W ₀ ^M w M,0 .

Proof. By (19), d 2 W ₀ ^M , d(⌃ ⁺ _M ) ⇢ ⌃ ⁺ , d(w ^M ₀ ) ¹ (⌃ ⁺ _w

₀

_{(M )} ) ⇢ ⌃ ⁺ , d(w ₀ ^M ) ¹ 2 W ₀ ^w

⁰

^(M) . This proves the equality W ₀ ^M = W ₀ ^w

⁰

^{(M )} w ^M ₀ . The equality W ₀ ^M = w 0 W ₀ ^M w M,0 , follows from d(w ^M ₀ ) ¹ (⌃ ⁺ _w

₀

_(M) ) ⇢ ⌃ ⁺ , w 0 dw M,0 w 0 (⌃ ⁺ _w

₀

_{(M )} ) ⇢ ⌃ , w 0 dw M,0 (⌃ _M ) ⇢

⌃ , w 0 dw M,0 2 W ₀ ^M .

Remark 2.25. W M = ⇤ o W M,0 but q w = q _w

^M₀

_w(w

^M₀

₎

¹

could be false for 2 ⇤, w 2 W M,0 such that `( w) < `( ) + `(w).

Lemma 2.26. `(w <sup>M</sup> <sub>0</sub> ) = `(w ^M ₀ d ¹ ) + `(d) for any d 2 W ₀ ^M .

Proof. For d 2 W ₀ ^M we have `(dw M,0 ) = `(d) + `(w M,0 ) by (20) and w = w <sup>M</sup> <sub>0</sub> d <sup>1</sup> satisfies w 0 = wdw M,0 and `(w 0 ) = `(w) + `(dw M,0 ). We have w ^M ₀ = w 0 w M,0 = wd and `(w ^M ₀ ) =

`(w 0 ) `(w M,0 ) = `(w) + `(d).

The Bruhat order x  x

⁰

in W 0 is defined by the following equivalent two conditions:

(i) There exists a reduced decomposition of x

⁰

such that by omitting some terms one obtains a reduced decomposition of x.

(ii) For any reduced decomposition of x

⁰

, by omitting some terms one obtains a reduced decomposition of x.

A reduced decomposition of w 2 W 0 followed by a reduced decomposition of w

⁰

2 W 0 is a reduced decomposition of ww

⁰

if and only `(ww

⁰

) = `(w)+`(w

⁰

). A reduced decomposition of d 2 W ₀ ^M cannot end by a non trivial element w 2 W M,0 .

Lemma 2.27. For w, w

⁰

2 W M,0 , d, d

⁰

2 W ₀ ^M , we have dw  d

⁰

w

⁰

if and only if there

exists a factorisation w = w 1 w 2 such that `(w) = `(w 1 ) + `(w 2 ), dw 1  d

⁰

and w 2  w

⁰

.

Proof. We prove the direction “only if” (the direction “if” is obvious). If dw  d

⁰

w

⁰

, a reduced decomposition of dw is obtained by omitting some terms of the product of a reduced decomposition of d

⁰

and of a reduced decomposition of w

⁰

. We have dw = d 1 w 2

with d 1  d

⁰

, w 2  w

⁰

and `(d 1 w 2 ) = `(d 1 ) + `(w 2 ). We have d 1 = dw 1 , w 1 := ww <sub>2</sub> <sup>1</sup> . As w, w 2 2 w M,0 and d 2 W <sub>0</sub> <sup>M</sup> we have `(dw 1 ) = `(d) + `(w 1 ) and `(dw) = `(d) + `(w).

Hence `(w 1 ) + `(w 2 ) = `(w).

Lemma 2.28. Let d

⁰

2 ^w

⁰

^{(M )} W 0 , d 2 W ₀ ^M .

(i) If there exists u 2 W M,0 , u

⁰

2 W ₀ ^M such that v = w ₀ ^M u  w = du

⁰

, then d = w ^M ₀ . (ii) d

⁰

d 2 w ₀ ^M W M,0 if and only if d

⁰

d = w ₀ ^M .

Proof. (i) As `(w) = `(d) + `(u

⁰

), we have u = u 1 u 2 with w ₀ ^M u 1  d, u 2  u

⁰

and u 1 , u 2 2 W M,0 (Lemma 2.27). We have `(w <sub>0</sub> <sup>M</sup> u 1 ) = `(w ₀ ^M ) + `(u 1 ) = `(w ^M ₀ d ¹ ) + `(d) + `(u 1 ) (Lemma 2.26). Hence d = w ^M ₀ , u 1 = 1.

(ii) If there exists u 2 W M,0 such that d = d

⁰

¹ w ^M ₀ u we have d = d

⁰

¹ w ₀ ^M because d

⁰

¹ w ₀ ^M 2 W ₀ ^M (Lemma 2.24).

2.6 H as a left ✓( H M

⁺

)-module and a right ✓ ^⇤ ( H M )-module

We prove Theorem 1.4 (iv) on the structure of the left ✓( H ^M

⁺

)-module H and its variant for the right ✓

^⇤

( H ^M )-module H . We suppose S M 6 = S.

Recalling the properties (i), (ii), (iii) of Theorem 1.4, H ^M = H M

⁺

[(T _{µ ˜} ^M

_M

) ¹ ] is the localisation of the subalgebra H M

⁺

at the central element T _{µ ˜} ^M

_M

. The algebra H M

⁺

embeds in H by ✓. Recalling (17), (18) we choose a lift ˜ d 2 W (1) for any element d in the classical set of representatives ^M W 0 of W M,0 \ W 0 . We define

V M

⁺

= X

d2

^M

W

0

✓( H M

⁺

)T d ˜ . (22)

Proposition 2.29. (i) V ^M

⁺

is a free left ✓( H ^M

⁺

)-module of basis (T d ˜ ) d

2^M

W

0

(ii) For any h 2 H , there exists r 2 N such that T _{˜ µ} ^r

_M

h 2 V ^M

⁺

. (iii) If q = 0, T µ ˜

M

is a left and right zero divisor in H .

For GL(n, F ), (ii) is proved in [Ollivier10, Prop. 4.7] for (q(s)) = (0). When the q(s) are invertible, T w ˜ is invertible in H for ˜ w 2 W (1).

Proof. (i) As ^M W 0 is a set of representatives of W M

⁺

\ W , a set of representatives of W M

⁺

(1) \ W (1) is the set { d ˜ | d 2 ^M W 0 } of lifts of ^M W 0 in W (1). The canonical bases of H ^M

⁺

and of H are respectively (T w ˜ ) ( ˜ w)

2

W

_M+

(1) and (T _{w ˜} d ˜ ) ( ˜ w,d)

2

W

_M+

(1)

⇥^M

W

0

, and T _{w ˜} d ˜ = T w ˜ T d ˜ by the additivity of lengths (Lemma 2.22).

(ii) We can suppose that h runs over in a basis of H . We cannot take the Iwahori- Matsumoto basis (T w ˜ ) w ˜

2

W (1) and we explain why. For ˜ w = ˜ w M d ˜ with ˜ w M 2 W M

⁺

(1), d 2

M W 0 we choose r 2 N such that ˜ µ ^r _M w ˜ M 2 W M

⁺

(1). By the length additivity (Lemma 2.22) T µ ˜

^r_M

w ˜ = T µ ˜

^r_M

w ˜

M

T d ˜ lies in ✓( H M

⁺

)T d ˜ , but we cannot deduce that T µ ˜

^r_M

T w ˜ lies in

✓( H M

⁺

)T d ˜ .

We take the Bernstein basis (2.18) and we suppose that q(s) = q _s is indeterminate

(but not invertible) with the same arguments as in [Ollivier10, Prop. 4.8]. Then E( ˜ d) =

T d ˜ for d 2 ^M W 0 . If we prove that E(˜ µ ^r _M w) lies in ˜ ✓( H M

⁺

)T d ˜ then E(˜ µ M ) ^r E o ( ˜ w) =

q µ

^r_M

,w E(˜ µ ^r _M w) lies also in ˜ ✓( H M

⁺

)T d ˜ . This implies T _{µ ˜} ^r

_M

E o ( ˜ w) 2 ✓( H M

⁺

)T d ˜ .

Now we prove E(˜ µ ^r _M w) ˜ 2 ✓( H M

⁺

)T d ˜ . We write ˜ w M = ˜ ˜ w M,0 , ˜ 2 ⇤(1), w ˜ M,0 2 W M,0 (1). Recalling E( ⇤ ) = T

_⇤

for ⇤ 2 W 0 (1) and the additivity of the length (Lemma 2.22),

q µ

^r_M

,w

M,0

d E(˜ µ ^r _M w) = ˜ E(˜ µ ^r _M ˜)E( ˜ w M,0 d) = ˜ E(˜ µ ^r _M ˜)T _{w ˜}

_M,0

d ˜ = E(˜ µ ^r _M ˜)T w ˜

M,0

T d ˜ ,

= q µ

^r_M

,w

M,0

E(˜ µ ^r _M w ˜ M )T d ˜

The monoid W M

^✏

is a lower subset of (W M ,  ^M ) (Lemma 2.6). The triangular decompo- sition (14) implies E M (˜ µ ^r _M w ˜ M ) 2 H ^M

⁺

. By Proposition 2.19 E(˜ µ ^r _M w ˜ M ) 2 ✓( H ^M

⁺

) and by the additivity of the length (Lemma 2.22),

q w

M,0

d = q w

M,0

q d , q µ

^r_M

w

M,0

d = q µ

^r_M

w

M,0

q d , implying q µ

^r_M

q w

M,0

d q _µ

r

¹

M

w

M,0

d = q µ

^r_M

q w

M,0

q _µ

r

¹

M

w

M,0

hence q µ

^r_M

,w

M,0

d = q µ

^r_M

,w

M,0

. (iii) We have `(µ M ) 6 = 0 and equivalently, ⌫(µ M ) 6 = 0 in V . We choose w 2 W 0 with w(⌫ (µ M ) 6 = ⌫ (µ M ). Then ⌫ (wµ M w ¹ ) = w(⌫(µ M )) and ⌫(µ M ) belong to di↵erent Weyl chambers. The alcove walk basis (E o ( ˜ w)) w ˜

2

W (1) of H associated to an orientation o of V of Weyl chamber containing ⌫(µ M ) satisfies

E o (˜ µ M ) = T µ ˜

M

, E o (˜ µ M )E o ( ˜ w˜ µ M w ˜ ¹ ) = E o ( ˜ w µ ˜ M w ˜ ¹ )E o (˜ µ M ) = 0.

(23)

The properties of the left ✓( H ^M

⁺

)-module H transfer to properties of the right ✓

^⇤

( H ^M )- module H , with the involutive anti-automorphism ⇣ ◆ of H (Remark 2.12) exchanging T w ˜ and ( 1)

^`(w)

T _{( ˜}

^⇤

_w)

1

for ˜ w 2 W (1), ✓( H ^M

⁺

) and ✓

^⇤

( H ^M ), V ^M

⁺

and

V M

^⇤

:= X

d

2

W

₀^M

T d

^⇤

˜ ✓

^⇤

( H M ), (24)

where W ₀ ^M = { d

⁰

¹ | d

⁰

2 ^M W 0 } is the set of classical representatives of W 0 /W M,0 (19), and ˜ d = ( ˜ d

⁰

) ¹ if d = d

⁰

¹ .

Corollary 2.30. (i) V M

^⇤

is a free right ✓

^⇤

( H M )-module of basis (T _d

^⇤

_˜ ) _d

₂

_W

^M

0

. (ii) For any h 2 H , there exists r 2 N such that h(T _(˜

^⇤

_µ

_M

₎

1

) ^r 2 V M

^⇤

.

(iii) If q = 0, T _{µ ˜}

^⇤ 1 M

is a left and right zero divisor in H .

3 Induction and coinduction

3.1 Almost localisation of a free module

In this chapter, all rings have unit elements.

Definition 3.1. Let A be a ring, and a 2 A a central non-zero divisor. We say that a left A-module B is an almost a-localisation of a left A-module B D ⇢ B of basis D when :

(i) D is a finite subset of B, and the map d

2

D A ! B, (x d ) ! P

x d d is injective, (ii) for any b 2 B, there exists r 2 N such that a ^r b lies in B D := P

d2D Ad.

Example 3.2. Our basic example is (A, a, B, D) = ( H M

⁺

, T µ

M

, H , (T d ˜ ) d

2^M

W

0

) (Thm.

2.29).