Journal of the Institute of Mathematics of Jussieu
http://journals.cambridge.org/JMJ
Additional services for Journal
of the Institute of Mathematics of Jussieu:
Email alerts: Click here
Subscriptions: Click here
Commercial reprints: Click here
Terms of use : Click here
THE PRO-p-IWAHORI HECKE ALGEBRA OF A REDUCTIVE
p-ADIC GROUP III (SPHERICAL HECKE ALGEBRAS AND
SUPERSINGULAR MODULES)
Marie-France Vigneras
Journal of the Institute of Mathematics of Jussieu / FirstView Article / August 2015, pp 1 - 38
DOI: 10.1017/S1474748015000146, Published online: 03 June 2015
Link to this article: http://journals.cambridge.org/abstract_S1474748015000146
How to cite this article:
Marie-France Vigneras THE PRO-p-IWAHORI HECKE ALGEBRA OF A REDUCTIVE p-ADIC
GROUP III (SPHERICAL HECKE ALGEBRAS AND SUPERSINGULAR MODULES). Journal of the
Institute of Mathematics of Jussieu, Available on CJO 2015 doi:10.1017/S1474748015000146
Request Permissions : Click here
Downloaded from http://journals.cambridge.org/JMJ, IP address: 81.194.27.167 on 01 Sep 2015
J. Inst. Math. Jussieu (2015), 1–38
c Cambridge University Press 2015
doi:10.1017/S1474748015000146
1
THE PRO-p-IWAHORI HECKE ALGEBRA OF A REDUCTIVE
p-ADIC GROUP III (SPHERICAL HECKE ALGEBRAS AND
SUPERSINGULAR MODULES)
MARIE-FRANCE VIGNERAS
UMR 7586, Institut de Mathematiques de Jussieu, 4 place Jussieu,
Paris 75005, France (vigneras@math.jussieu.fr)
(Received 26 May 2014; accepted 30 March 2015)
Abstract Let R be a large field of characteristic p. We classify the supersingular simple modules of the
pro- p-Iwahori Hecke R-algebra H of a general reductive p-adic group G. We show that the functor
of pro- p-Iwahori invariants behaves well when restricted to the representations compactly induced
from an irreducible smooth R-representation ⇢ of a special parahoric subgroup K of G. We give an
almost-isomorphism between the center of H and the center of the spherical Hecke algebra H(G, K , ⇢),
and a Satake-type isomorphism for H(G, K , ⇢). This generalizes results obtained by Ollivier for G split
and K hyperspecial to G general and K special.
Keywords: group theory and generalizations; number theory
2010 Mathematics subject classification: Primary 20C08
Secondary 22E50
Contents
1
Introduction
2
The characters of h and Haff
10
3
Distinguished representatives of W0 \W
13
4
h-eigenspace in ⌘ ⌦h H
16
5
Centers
22
6
Supersingular H-modules
24
7
Pro- p-Iwahori invariants and compact induction
31
References
2
37
M.-F. Vigneras
2
1. Introduction
Let p be a prime number, let F be a finite extension of Q p or F p ((T )), and let G be the
group of rational points of a connected reductive F-group.
1.1.
The smooth representations of G over an algebraically closed field C of characteristic p
have been the subject of many investigations in recent years, in the modulo p Langlands
program. The pro- p-Iwahori invariant functor V 7 ! V I (1) relates the representations of
G to the modules of the pro- p-Iwahori Hecke C-algebra H = HC (G, I (1)) studied in
[13–15]. The I (1)-invariant functor and the theory of H-modules play an increasingly
important role in the representation theory of G modulo p. They are the key to
the proof of the change of weight in the recent classification of irreducible smooth
C-representations of G in terms of supersingular ones (a forthcoming work by Abe
et al. [1]). The supersingular smooth irreducible C-representations ⇡ of G and their
I (1)-invariant remain mysterious, but the supersingular simple H-modules are classified
in this paper, and the supersingularity of ⇡ I (1) and of ⇡ are related. A variant
of the modulo p Langlands program seems to exist for H-modules. Grosse-Kloenne
[5] constructed a functor from finite-dimensional HC (G L(n, Q p ), I (1))-modules to
finite-dimensional smooth C-representations of GalQ p , inducing a bijection between the
simple supersingular HC (G L(n, F), I (1))-modules of dimension n and the irreducible
smooth C-representations of Gal F (the absolute Galois group of F) of dimension n as in
[9, 14].
In this paper, we prove that the I (1)-invariant functor behaves well when restricted
to compactly induced representations c-IndG
K ⇢, where ⇢ is an irreducible smooth
C-representation of a special parahoric subgroup K of G. The vector space ⇢ I (1) has
dimension 1, and the pro- p-Iwahori Hecke C-algebra h = HC (K , I (1)) of K acts on
I (1) is isomorphic to ⌘ ⌦ H, and the
⇢ I (1) by a character ⌘. The H-module (c-IndG
h
K ⇢)
G
spherical algebra EndC G (c-Ind K ⇢) is isomorphic to the algebra EndH (⌘ ⌦h H). This paper
is devoted to the study of the modules ⌘ ⌦h H and of the spherical Hecke algebras
EndH (⌘ ⌦h H). In the last section, we transfer our results from H to the group G using
the I (1)-invariant functor.
Let ⇢ be an irreducible smooth C-representation of K , and let ⌘, ⌘1 be two arbitrary
characters of h. We obtain the following:
(i) Isomorphisms
I (1)
(c-IndG
' ⇢ I (1) ⌦h H,
K ⇢)
I (1)
EndC G (c-IndG
⌦h H).
K ⇢) ' EndH (⇢
(ii) A Satake-type isomorphism for the spherical Hecke algebra H(h, ⌘) = EndH (⌘ ⌦h
H).
(iii) A basis of the space of intertwiners HomH (⌘1 ⌦h H, ⌘ ⌦h H).
(iv) An almost-isomorphism from the center of H to the center of H(h, ⌘) (an
isomorphism between finite index affine subalgebras).
(v) The classification of the supersingular simple H-modules.
The pro- p-Iwahori Hecke algebra of a reductive p-adic group III
3
When G is split and K hyperspecial, Ollivier proved (i), (ii), (iv) and (v). We follow
her method. The alcove walk bases of H and the product formula [12, 15] allow us to
simplify her method and to extend it to G general and K special. Analogs of 2, 3 were
proved for G in [6, 7] and 5 for G remains a wide-open question.
In the rest of this introduction, we consider the content of 2, 3, 4, 5.
After [13, 14], a generalization of HC (G, I (1)) was introduced in [12] when G is split,
and in [15] for G general, in order to study it. This is an algebra H R (qs , cs̃ ) over a
commutative ring R with two sets of parameters (qs ), (cs̃ ). The properties of this algebra
are often proved by reduction to (qs ) = (1) (this changes the parameters (cs̃ )), and
transferred to H R (0, cs̃ ) by specialization to (qs ) = (0). The algebra H R (qs , cs̃ ) contains
a natural finite-dimensional subalgebra h R (qs , cs̃ ).
In 1.2 and 1.3, we recall the basic properties of H R (qs , cs̃ ) used in this work and
the dictionary between h R (qs , cs̃ ), H R (qs , cs̃ ) and H R (K , I (1)), H R (G, I (1)) [15, 16].
Theorems 1.2, 1.3, 1.4, and 1.5 are proved for h R (0, cs̃ ), H R (0, cs̃ ), and are given in 1.4.
They apply to the algebras H R (K , I (1)), H R (G, I (1)) when R has characteristic p.
1.2.
Let W = (6, 1, , 3, ⌫, W, Z k , W (1)) be data consisting of the following:
(i) a reduced root system 6 of basis 1 associated with the finite Weyl Coxeter system
(W0 , S) of an affine Weyl Coxeter system (W aff , S aff ) acting on a real vector space
V of dual of basis 1, with a W0 -invariant scalar product;
(ii) three commutative groups,  and 3 finitely generated, and Z k finite;
(iii) a group W = W aff o  = 3 o W0 which is a semi-direct product of subgroups in two
di↵erent ways,  acting on (W aff , S aff ) and W0 on 3. The length ` and the Bruhat
order 6 of (W aff , S aff ) extend trivially to W = W aff o ;
(iv) a W0 -equivariant homomorphism ⌫ : 3 ! V such that the action of W aff on V
and the action of 3 on V by translation v 7 ! v + ⌫( ) for 2 3, v 2 V , extend to
an action of W by affine automorphisms permuting the set of affine hyperplanes
H = {Ker(↵ + n), | ↵ + n 2 6 aff = 6 + Z};
(v) a system of the representatives of W0 in 3:
+
+
3+ := {µ 2 3 | ⌫(µ) 2 D },
where D = {x 2 V | 0 6 ↵(x), ↵ 2 1} is the dominant closed Weyl chamber;
(vi) an extension 1 ! Z k ! W (1) ! W ! 1.
Notation. The inverse image in W (1) of a subset X of W is denoted by X (1), and w̃
denotes an element of W (1) of image w 2 W . For c 2 R[Z k ], the conjugate of c by w̃
depends only on w, and is denoted w • c := w̃cw̃ 1 . The dominant Weyl chamber D+ =
{x 2 V | 0 <
alcove C+ is the connected component
S↵(x), ↵ 2 1} is open. The dominantaff,+
+
D \ (V
of positive affine roots is the set of
H 2H H ) of vertex 0 2 V . The set 6
2 6 aff positive on C+ . The action of W on V defines by functoriality an action of W
on 6 aff .
M.-F. Vigneras
4
We will often suppose that 3 contains a subgroup 3T satisfying the following.
F
(T1) 3 = y2Y 3T y for a finite set Y .
(T2) 3T is W0 -stable.
(T3) There exists a central subgroup 3̃T of 3(1) normalized by W0 (1) such that the
'
quotient map 3(1) ! 3 induces a group isomorphism 3̃T ! 3T sending w̃µ̃w̃ 1
to wµw 1 if w̃ 2 W0 (1) lifts w 2 W0 and µ̃ 2 3̃T lifts µ 2 3T .
Let (qs̃ , cs̃ )s̃2S aff (1) ) be a set of elements in R ⇥ R[Z k ] satisfying qs̃ 0 = qs̃ , cs̃ 0 = w • cs̃ if
s̃ 0 = w̃ s̃ w̃ 1 2 S aff (1), w̃ 2 W (1), and qt s̃ = qs̃ , ct s̃ = tcs̃ if t 2 Z k . As qs̃ depends only on
the image s 2 S aff of s̃, we denote also qs̃ = qs .
There is a unique R-algebra H = H R (W, qs , cs̃ ), free of basis (Tw̃ )w̃2W (1) , with product
satisfying
(i) the braid relations:
Tw̃ Tw̃0 = Tw̃w̃0 ,
if w̃, w̃0 2 W (1), `(w) + `(w 0 ) = `(ww 0 ),
(1)
allowing one to identify R[(1)] to a subalgebra of H;
(ii) the quadratic relations:
Ts̃ Ts̃⇤ = qs s̃ 2 ,
if s̃ 2 S aff (1), Ts̃⇤ = Ts̃
cs̃ .
(2)
This is called the Iwahori–Matsumoto presentation of H R (W, qs , cs̃ ).
The R-submodule of basis (Tw̃ )w̃2W0 (1) is a finite subalgebra h = h R (W, qs , cs̃ ).
The R-submodule of basis (Tw̃ )w̃2W aff (1) is a subalgebra Haff . The R-algebra Haff is an
algebra like H with  trivial, and H is isomorphic to the twisted tensor product
x ⌦ y 7 ! x y : Haff ⌦tR[Z k ] R[(1)] ! H
(3)
of its subalgebras R[(1)] and Haff . The algebra H admits an involutive R-automorphism
◆, equal to the identity on R[(1)] and such that [15, Proposition 4.23]
◆(Ts̃ ) :=
Ts̃⇤
for s 2 S aff .
(4)
All the orientations that we consider are spherical [15]. For the orientation o associated
to an (open) Weyl chamber Do , the o-positive side of the affine hyperplane Ker(↵ + n) is
the set of x 2 V where ↵(x) + n > 0, if ↵ 2 6 takes positive values on Do . The dominant
orientation o, denoted by o+ , is associated to the dominant Weyl chamber D+ , and the
anti-dominant orientation, denoted by o , to the anti-dominant Weyl chamber D+ =
D . The orientation associated to the Weyl chamber w 1 (Do ), w 2 W0 , is denoted by
o • w. For w 2 W of projection w0 2 W0 , the orientation o • w0 is also denoted by o • w.
We have o • = o for 2 3. We set
Soaff := {s 2 S aff | C+ is in the o-positive side of Hs },
So := S \ Soaff ,
(5)
where Hs is the affine hyperplane of V fixed by s and C+ the dominant alcove
(Notation). There exists a unique set of bases (E o (w̃))w̃2W (1) of H, parameterized by
The pro- p-Iwahori Hecke algebra of a reductive p-adic group III
5
the orientations o, satisfying [15, § 5.3]
E o (s̃) := Ts̃ if s 2 S aff
Soaff ,
and the product formula, for w̃, w̃ 0 2 W (1),
E o (w̃)E o•w (w̃ 0 ) = E o (w̃w̃0 )
In particular, for ˜ , ˜ 0 2 3(1),
E o ( ˜ )E o ( ˜ 0 ) = E o ( ˜ ˜ 0 )
E o (s̃) := Ts̃⇤ if s 2 Soaff ,
(6)
if `(w) + `(w 0 ) = `(ww0 ).
(7)
if ⌫( ), ⌫( 0 ) belong to a same closed Weyl chamber.
(8)
We have E o ( ) = T when ⌫( ) 2 Do .
The basis (E o (w̃))w̃2W (1) is called an alcove walk basis; the alcove walk bases generalize
the integral Bernstein bases defined in [11, 14].
The R-submodule of basis (E o ( ˜ )) ˜ 23(1) is a subalgebra Ao of H containing the
+
˜
subalgebra A+
o of basis (E o ( )) ˜ 23+ (1) , isomorphic to R[3 (1)].
If qs = 0 for all s 2 S aff , then for w̃, w̃0 2 W (1) such that `(w) + `(w 0 ) > `(ww0 ) we have
E o (w̃)E o•w (w̃ 0 ) = 0; in particular, E o ( ˜ )E o ( ˜ 0 ) = 0 if ˜ , ˜ 0 2 3(1), and ⌫( ), ⌫( 0 ) do not
belong to the same closed Weyl chamber.
1.3.
Let F be a local field of finite residue field k with q elements and of characteristic p, and
p F a generator of the maximal ideal of the ring of integers O F of F. Let G, T, Z , and N be
respectively the F-rational points of a connected reductive F-group, a maximal F-split
subtorus, its centralizer, and its normalizer. Let C+ be an open alcove of the semi-simple
+
apartment of G defined by T , let x0 be a special vertex of the closed alcove C , and let
I, I (1), K , be respectively the Iwahori subgroup of G fixing C+ , its pro- p-Sylow subgroup,
and the parahoric subgroup of G fixing x0 .
We associate to G, T, Z , N , I, I (1), K the data
(W = (6, 1, , 3, ⌫, W, Z k , W (1)); (qs , cs̃ )),
and a group 3T , satisfying the properties given in § 1.2 with R = Z, as follows.
The apartment defined by T identifies with a Euclidean real vector space V . The set
S aff of orthogonal reflections with respect to the walls of C+ generates an affine Coxeter
system (W aff , S aff ), given by a based reduced root system (6, 1). The action of N on the
apartment transfers to an action on V . The subgroup Z acts by translations (z, x) 7 ! x +
⌫ Z (z), (z, x) 2 Z ⇥ V , for an homomorphism ⌫ Z : Z ! V satisfying ↵ ⌫ Z (t) = ↵(t) for
t 2 T and ↵ in the root system 8 of T in G. There is a surjective map ↵ 7 ! e↵ ↵ : 8 ! 6,
where e↵ is a positive integer for all ↵ 2 8.
Let T0 := T \ K (the maximal compact subgroup of T ), Z 0 := K \ Z (the parahoric
subgroup of Z ), and let Z 0 (1) be the pro- p-Sylow subgroup of Z 0 . Then
3T := T /T0 ,
3 := Z /Z 0 ,
W0 := N /Z ,
3(1) := Z /Z 0 (1),
W := N /Z 0 ,
Z k := Z 0 /Z 0 (1),
W (1) := N /Z 0 (1).
The homomorphism ⌫ Z and the action of N on V are trivial on Z 0 . They induce an
homomorphism ⌫ : 3 ! V and an action of W on N . The monoid 3+ represents the
M.-F. Vigneras
6
orbits of W0 in 3 [7, 6.3] and the double cosets K \G/K . The groups W, W (1) represent
the double cosets I \G/I, I (1)\G/I (1). The group  is the W -stabilizer of the alcove C+ .
We denote by w̃ an element of W (1) of image w in W , and we call w̃ a lift of w.
For s 2 S aff , let K s be the parahoric subgroup of G fixing the face of C+ fixed by s. The
quotient of K s by its pro- p-radical is the group G s,k of rational points of a k-reductive
connected group of rank 1. The image of I (1) in G s,k is the group Us,k of rational points of
the unipotent radical of a k-Borel subgroup Z k Us,k of opposite group Z k U s,k . It is known
that s admits a lift n s 2 N \ K s of image in G s,k belonging to the group hUs,k , U s,k i
generated by Us,k [ U s,k . The image of n s in W (1) is called an admissible lift of s. We set
Z k,s = Z k \ hUs,k , U s,k i.
For s 2 S aff , s̃ an admissible lift of s, and t 2 Z k , let
X
qs = [I n s I : I ] is a power of q, cs := (qs 1)|Z k,s | 1
z,
z2Z k,s
P
and ct s̃ = z2Z k,s cs̃ (z)t z, for positive integers cs̃ (z) = cs̃ (z 1 ) of sum qs 1, constant on
each coset modulo {xs(x) 1 | x 2 Z k }, and cs̃ ⌘ cs mod p as in [15, Theorem 2.2].
The cocharacter group X ⇤ (T ) of T is isomorphic to 3T and embeds in 3(1) by the
map µ 7 ! µ( p F ) 1 : X ⇤ (T ) ! Z followed by the quotient maps of Z onto 3 and 3(1).
Remembering the sign in the definition of ⌫,
µ 2 3+
T , ↵(µ( p F )) 2 O F
for all ↵ 2 1.
We identify µ with its image in 3T , and µ̃ denotes its image in 3(1).
For a commutative ring R, the pro- p-Iwahori Hecke R-algebra H R (G, I (1)) is
isomorphic to the algebra H R (qs , cs̃ ) associated to this data.
The pro- p-Iwahori Hecke R-algebra H R (K , I (1)) of K is a subalgebra of H R (G, I (1))
isomorphic to the finite subalgebra h(qs , cs̃ ) of H.
The Iwahori Hecke R-algebra H R (G, I ) is an algebra H associated to the same data
except that Z k = {1}, W (1) = W, cs = qs 1.
The group G is split , T = Z ) 3T = 3. The group G is quasi-split , Z is the
F-points of an F-torus ) 3(1) is commutative. The group G is semi-simple , Ker ⌫ is
finite )  is finite and ⌫ is injective on 3T .
The quotient of K by its pro- p-radical K (1) is the group G k of k-rational points of a
connected reductive k-group. The images in G k of T0 , Z 0 , I , and I (1) are the groups
Tk , Z k , Bk , and Uk of k-rational points of a maximal k-split torus, its centralizer (a
k-torus), a Borel k-subgroup containing the maximal k-split torus, and its unipotent
radical.
The finite Hecke algebras H R (K , I (1)) and H R (G k , Uk ) are isomorphic.
The condition qs = 0 for all s 2 S aff means that the characteristic of R is p. Then,
X
ct s̃ = |Z k,s | 1
t z,
z2Z k,s
and the irreducible smooth R-representations ⇢ of K are trivial on K (1); they identify
with the irreducible R-representations of G k , in bijection with the characters of
H R (G k , Uk ) by the Uk -invariant functor ⇢ 7! ⇢ Uk for R as in 1.4.
The pro- p-Iwahori Hecke algebra of a reductive p-adic group III
7
1.4.
For the remainder of this article, unless otherwise specified, we are in the setting of § 1.2
with qs = 0 for all s 2 S aff , and R is a field containing a root of unity of order the exponent
of Z k .
Notation. We denote by Ẑ k the group of R-characters of Z k . For a character
character ⌘ of h, and a character 4 of Haff , we set
S aff := {s 2 S aff | (cs̃ ) 6 = 0},
S⌘ := {s 2 S | ⌘(Ts̃ ) 6 = 0},
2 Ẑ k , a
S := S aff \ S,
(9)
aff := {s 2 S aff | 4(T ) 6 = 0}.
S4
s̃
(10)
These sets are independent of the choice of the lift s̃ of s. For (w̃, ) 2 W (1) ⇥ Ẑ k we
denote by w 2 Ẑ k the character w (t) = (w̃t w̃ 1 ) for t 2 Z k . The subgroup generated
by a subset X of a group is denoted by hX i. For 2 3 we set
1 := {↵ 2 1 | ↵ ⌫( ) = 0},
S := {s↵ | ↵ 2 1 }.
(11)
We recall from § 1.2 the R-algebra h associated to the finite Coxeter system (W0 , S)
and the extension 1 ! Z k ! W0 (1) ! W0 ! 1, of basis (Tw̃ )w̃2W0 (1) satisfying the braid
relations and the quadratic relations Ts̃ (Ts̃ cs̃ ) = 0 for s̃ 2 S(1).
Theorem 1.1 (The characters of h). (a) The characters ⌘ of h are in bijection with the
pairs ( , J ), where 2 Ẑ k and J ⇢ S , = ⌘| Z k , and J = S⌘ .
(b) For any ⌘, there exists an orientation o such that the equivalent properties S⌘ =
S \ So , ⌘(E o (s̃)) = 0, for all s 2 S, hold true. We set o := ⌘.
(c) For two characters ⌘1 , ⌘ of h, there exists an orientation o such that ⌘1 = (
o if and only if
S⌘ \ S 1 = S⌘1 \ S .
1 )o , ⌘
=
For a reduced decomposition of w̃ = s̃1 . . . s̃r of W (1), the element cw̃ = cs̃1 . . . cs̃r
of R[Z k ] does not depend on the choice of the reduced decomposition [15,
Propositions 4.13(ii) and 4.22].
Theorem 1.2 (A basis of the intertwiners). Let ⌘1 , ⌘ be two characters of h of restrictions
to Z k .
1,
(a) ⌘1 is contained in ⌘ ⌦h H (is a submodule) if and only if
1
(b) For
=
,
S⌘1 \ S = S⌘ \ S ,
for some
2 3+ .
2 3+ satisfying (a), there exists a non-zero H-intertwiner
X
8 ˜ : 1 ⌦ 1 7 ! 1 ⌦ E ˜ : ⌘1 ⌦h H ! ⌘ ⌦h H, E ˜ :=
1 (cw̃0 )
w0 2Y
w
1
⌦ T˜ w̃0 ,
where Y = {w0 2 hS 1 S⌘1 i | 1 0 = 1 , `( w0 ) = `( ) `(w0 )}, and w̃0 is a lift of
w0 ; note that 1 (cw̃0 ) 1 ⌦ T˜ w̃0 does not depend on the choice of the lift.
(8 ˜ ), for 2 3+ satisfying (a), is a basis of HomH (⌘1 ⌦h H, ⌘ ⌦h H).
M.-F. Vigneras
8
(c) If o satisfies (d) and
8o, ˜
(8o, ˜ ), for
2 3+ satisfies (a), there exists a non-zero H-intertwiner
: 1 ⌦ 1 7 ! 1 ⌦ E o ( ˜ ) : ( 1 )o ⌦h H ! o ⌦h H.
2 3+ satisfying (a), is a basis of HomH ((
We note that 1 (cw̃0 )
of w0 2 Y . We set
1⌦T
˜ w̃0
1 )o ⌦h H, o ⌦h H).
2 ⌘ ⌦h H does not depend on the choice of the lift w̃0
3 := { 2 3 |
= }, resp. 3+ := 3+ \ 3 .
(12)
P
The idempotent e := |Z k | 1 t2Z k (t) 1 t of R[Z k ] is central in R[3 (1)], and the
R-linear map
⌦ R[Z k ] R[3 (1)] ! e R[3 (1)] 1 ⌦ ˜ 7 ! e ˜ ( 2 3 )
(13)
is an isomomorphism. Any R-algebra A with a basis (a ˜ )
a ˜ a ˜ 0 = (t)a ˜ 00
for , 0 ,
00
23+
satisfying
2 3+ , t 2 Z k , ˜ ˜ 0 = t ˜ 00 ,
(14)
is canonically isomorphic to the algebra e R[3+ (1)] with its natural basis (e ˜ ) 23+ .
˜
For an orientation o, the R-submodule A+
o, of basis (E o ( )) ˜ 23+ (1) is a subalgebra
˜
+ is an R-algebra with a basis
of H. The algebra ⌦ R[Z k ] A+
o, of basis (1 ⌦ E o ( ))
23
satisfying (14).
A spherical Hecke algebra is the algebra of H-intertwiners of a right H-module ⌘ ⌦h H
induced from a character ⌘ of h, by analogy with the reductive p-adic groups
H(h, ⌘) := EndH (⌘ ⌦h H).
Theorem 1.2 with ⌘1 = ⌘ becomes the following.
Theorem 1.3 (A Satake-type isomorphism for the spherical algebra). (a) A basis of
the spherical Hecke algebra H(h, ⌘) is (8 ˜ ) 23+ , where
X
8 ˜ : 1 ⌦ 1 7 ! 1 ⌦ E ˜ : ⌘ ⌦h H ! ⌘ ⌦h H, E ˜ :=
(cw̃0 ) ⌦ T˜ w̃0 ,
Y = {w0 2 hS
S⌘ i |
w0
= , `( w0 ) = `( )
w0 2Y
`(w0 )}.
(b) Let o be an orientation such that ⌘ = o . For 2 3+ , there exists an injective
h-intertwiner
8o, ˜ : 1 ⌦ 1 7 ! 1 ⌦ E o ( ˜ ) : ⌘ ⌦h H ! ⌘ ⌦h H.
(8o, ˜ ) 23+ is a basis of the spherical Hecke algebra H(h, ⌘) satisfying (14), inducing
an isomorphism
H(h, ⌘) ' e R[3+ (1)].
W (1)
We suppose now that 3T exists. The center Z of H is the algebra Ao
W (1)-invariants of Ao , and is a free R-module of basis
X
E(C̃) =
Eo ( ˜ )
˜ 2C̃
of
(15)
(E(C̃) is independent of the choice of o) for all finite conjugacy classes C̃ of W (1).
We denote by C̃(µ) the W (1)-conjugacy class of µ̃ for µ 2 3+
T . The R-subspace of
The pro- p-Iwahori Hecke algebra of a reductive p-adic group III
9
basis (E(C̃(µ)))µ23+ is a central subalgebra ZT of H which has better properties
T
than Z.
A central element x 2 Z induces naturally a H-intertwiner of ⌘ ⌦h H:
8x : 1 ⌦ h 7 ! 1 ⌦ xh = 1 ⌦ hx
for h 2 H.
(16)
It is straightforward to check that 8x belongs to the center Z(⌘, h) of H(⌘, h). The
R-subspace of basis (8 E(C̃(µ)) )µ23+ is a central subalgebra Z(⌘, H)T of the spherical
T
algebra H(⌘, h).
Theorem 1.4 (Almost-isomorphism between the centers of H and H(⌘, h)). We suppose
that 3T exists. Let ⌘ be a character of h.
(a) ZT is a finitely generated central R-subalgebra of H, and H is a finitely generated
ZT -module. This is also true for (Z(⌘, H)T , H(⌘, h)) instead of (ZT , H).
(b) 8 E(C̃(µ)) = 8o,µ̃ for µ 2 3+
T and any orientation o such that ⌘ =
The linear map µ̃ 7 ! 8 E(C̃(µ)) :
R[3̃+
T]
o.
! Z(⌘, H)T is an algebra isomorphism.
(c) The map x 7 ! 8x : Z ! Z(⌘, H) restricts to an isomorphism ZT ! Z(⌘, H)T .
We prove (a) over any commutative ring R.
We transfer these results to the group G. The spherical Hecke algebra H R (G, K , ⇢) =
End RG c-IndG
K ⇢ of an irreducible smooth representation ⇢ of K with H R (K , I (1)) acting
I
(1)
by ⌘ on ⇢
is isomorphic to H(⌘, h) by the pro- p-Iwahori invariant functor. We denote
by Z R (G, K , ⇢)T the algebra corresponding to Z(⌘, H)T . We denote by H R (Z + , Z 0 , )
the R-algebra of elements in the Hecke algebra H R (Z + , Z 0 , ) with support contained in
the monoid Z + of z 2 Z with ⌫ Z (z) dominant.
From Theorem 1.3 we obtain an algebra isomomorphism
So : H R (G, K , ⇢) ! H R (Z + , Z 0 , )
for each orientation o such that ⌘ =
independent of the choice of o,
o.
(17)
This isomorphism restricts to an isomorphism,
ST : Z R (G, K , ⇢)T ! H R (T + , T0 , ).
(18)
Let ⇡ be a smooth R-representation of G such that Hom R (⇢, ⇡) contains a
Z R (G, K , ⇢)T -eigenvector A of eigenvalue ⇠ , seen as an homomorphism 3̃+
T ! R
(Theorem 1.4). From Theorem 1.4, for v 2 ⇢ I (1) non-zero and µ 2 3+
T,
⇠(µ̃)A(v) = A(v)E o (µ̃) = A(v)E(C̃(µ)).
Theorem 1.5 (Supersingularity in G and in H). The eigenvalue ⇠ of the
Z R (G, K , ⇢)T -eigenvector A 2 Hom R (⇢, ⇡) is supersingular if and only if the submodule
A(v)H of ⇡ I (1) is supersingular.
We recall that an homomorphism 3̃+
T ! R is called supersingular if it vanishes
on the non-invertible elements, and that a simple right H-module M is called
supersingular if M E(C̃) = 0 for all finite conjugacy classes C̃ in W (1) with positive length
[13, Definition 1]. This is equivalent to M E(C̃(µ)) = 0 for all non-invertible µ 2 3̃+
T.
M.-F. Vigneras
10
In a forthcoming article, we will study the parabolic induction for H-modules; we hope
to prove that the isomorphism So (17) is the Satake isomorphism of [7] for a good choice
of o such that ⌘ = o (this was proved by Ollivier [10, Theorem 5.5]), when G is split with
a simply connected derived group, and K is hyperspecial; as Z = T , we have So = ST ,
and that an irreducible smooth admissible representation ⇡ is supersingular if and only
if ⇡ I (1) contains a supersingular module (this was proved by Ollivier for G = G L(n, F)
and P G L(n, F) [11, Theorem 5.26]).
Finally, we classify the supersingular simple finite-dimensional H-modules (proved by
Ollivier when G is split, and K is hyperspecial [11, Corollary 5.15]).
For a character 4 of Haff , the R-subalgebra H4 of H generated by Haff and the
(1)-fixator of 4,
(1)4 := {u 2 (1) | 4(uhu
1
) = 4(h) for h 2 Haff },
is identified by (3) with the twisted tensor product Haff ⌦ R[Z k ] R[(1)4 ] ! H4 . For a
simple finite-dimensional R-representation of (1)4 equal to 4 on Z k , let
M(4, ) := (4 ⌦ ) ⌦H4 H
(19)
be the right H-module induced from the right H4 -module 4 ⌦ . The induced module
M(4, ) is finite dimensional. Two pairs (41 , 1 ), (42 , 2 ) are called conjugate by an
element u 2 (1) if
41 (uhu
1
) = 42 (h),
1 (uvu
1
)=
2 (v)
for (h, v) 2 Haff ⇥ u
1
4 (1)u.
The affine Coxeter system (W aff , S aff ) is the direct product of the irreducible affine Coxeter
systems (Wiaff , Siaff )16i6r associated to the irreducible components (6i , 1i )16i6r of the
based reduced root system (6, 1). The R-submodule of basis (Tw̃ )w̃i 2W aff (1) is a subalgebra
i
Hiaff of Haff . The algebras Hiaff are called the irreducible components of Haff .
Theorem 1.6 (Supersingular simple modules). (a) The characters 4 of Haff are in
aff
bijection with the pairs ( , J ), where 2 Ẑ k and J ⇢ S aff , = 4| Z k , and J = S4
aff
aff
(10). When S4 = S , 4 is called a sign character, and the character 4 ◆ (4) is
called a trivial character.
(b) A character 4 of Haff is supersingular if and only if it is not a sign or trivial
character on each irreducible component of Haff .
(c) A finite-dimensional right H-module is supersingular if and only if it is isomorphic
to M(4, ), where 4 is a supersingular character of Haff and
is a simple
finite-dimensional R-representation of (1)4 equal to 4 on Z k .
(d) M(41 ,
1)
' M(42 ,
2)
if and only if (41 ,
1 ), (42 , 2 )
2. The characters of h and Haff
Proposition 2.1. A simple h-module has dimension 1.
are (1)-conjugate.
The pro- p-Iwahori Hecke algebra of a reductive p-adic group III
11
Proof. The finite-dimensional R-algebra h is generated by Z k and Ts̃ for all s 2 S. By
the hypothesis on R (§ 1.4), a right simple h-module is finite dimensional and contains
an eigenvector v of Z k . Following the argument of [4, Theorem 6.10], we choose w in the
finite group W0 of maximal length such that vTw̃ 6 = 0, and we show that RvTw̃ is h-stable.
RvTw̃ is stable by Tt , because Tw̃ Tt = (w • t)Tw̃ for t 2 Z k .
RvTw̃ is stable by Ts̃ , because
– if `(ws) = `(w) + 1, vTw̃ Ts̃ = vTws̃ and by the hypothesis on w, vTw̃ s̃ = 0;
– if `(ws) = `(w) 1, Tw̃ Ts̃ = Tw̃s̃ 1 Ts̃2 = Tw̃s̃ 1 cs̃ Ts̃ = Tws̃ 1 Ts̃ cs̃ = (w • cs̃ )Tw̃ . We used
that Ts̃ and cs̃ commute.
Proposition 2.2. The characters ⌘ of h are in bijection with the pairs ( , J ), where
and J ⇢ S (9), by the recipe
⌘| Z k = ,
S⌘ = {s 2 S | ⌘(Ts̃ ) 6 = 0} = J.
We have ⌘(Ts̃ ) = (cs̃ ) if s 2 J .
The characters 4 of Haff are in bijection with the pairs ( , J ), where
J ⇢ S aff , by the recipe
4| Z k = ,
2 Ẑ k
2 Ẑ k and
aff
S4
= {s 2 S aff | 4(Ts̃ ) 6 = 0} = J.
We have 4(Ts̃ ) = (cs̃ ) if s 2 J .
The set J is independent of the choice of the lift s̃ of s. We call ( , J ) the parameters
aff )
of the character. The restriction to h of the character 4 of Haff with parameters ( , S4
aff
is the character of parameters ( , S4 \ S).
Proof. The proposition follows from the Iwahori–Matsumoto presentation in both cases.
If ⌘| Z k = , we have
⌘(Ts̃ )(⌘(Ts̃ )
(cs̃ )) = 0
for s 2 S. We can replace ⌘, S by 4, S aff .
The involutive automorphism ◆ of H (4) has the property for s 2 S that
⌘(Ts̃ ) = 0 , ⌘ ◆(Ts̃ ) = ⌘(cs̃ ).
The same holds for (4, S aff ) instead of (⌘, S).
Lemma 2.3. Let ⌘ be a character with parameters ( , S⌘ ) of h. Then ⌘ ◆ is a character
of h with parameters ( , S
S⌘ ). We can replace ⌘, S, h by 4, S aff , Haff .
Let o be an orientation. We recall the notation (5), (6), (9), (10).
Lemma 2.4. Let ⌘ be a character of h with parameters ( , S⌘ ). Then S⌘ = S \ So ,
⌘(E o (s̃)) = 0 for all s 2 S. When this holds true, we denote ⌘ = o .
We can replace (⌘, h, S, o ) by (4, Haff , S aff , oaff ).
M.-F. Vigneras
12
Proof. We compare the values of E o (s̃) and ⌘(Ts̃ ) for s 2 S:
E o (s̃) = Ts̃ , s 2 S
= Ts̃
So ,
cs̃ , s 2 So ,
⌘(Ts̃ ) = 0 , s 2 S
S⌘ ,
= (cs̃ ) 6 = 0 if s 2 S⌘ .
We see that
if s 2 S
if s 2 S
S , then ⌘(E o (s̃)) = ⌘(Ts̃ ) = (cs̃ ) = 0;
S⌘ , then ⌘(E o (s̃)) = ⌘(Ts̃ ) = 0 , s 6 2 (S
if s 2 S⌘ , then ⌘(E o (s̃)) = 0 , s 2 S⌘ \ So .
S⌘ ) \ So ;
Hence we obtain the lemma for ⌘. The proof is the same for 4.
Example 2.5. For the dominant orientation o+ , Soaff+ = S, and the parameters of
of oaff
+ are ( , S ).
For the anti-dominant orientation o , Soaff = S aff S, and the parameters of
( , ;), while those of oaff are ( , S aff S ).
o+
and
o
are
Lemma 2.6. (i) Any subset of S is equal to So for some orientation o.
A character ⌘ of h of restriction to Z k is equal to o for some orientation o, and
⌘=
o
, So \ S = S⌘ .
(ii) Two R-characters ⌘1 , ⌘ of h of parameters (
for some orientation o if and only if
1 , S⌘1 ), (
, S⌘ ) are equal to (
1 )o , o
S⌘1 \ S = S⌘ \ S 1 .
In this case, ⌘1 = (
1 )o
and ⌘ =
o
, So \ (S 1 [ S ) = S⌘1 [ S⌘ .
Proof. (i) Let wo 2 W0 . For ↵ 2 1, the root in {↵, ↵} positive on wo 1 (D+ ) is equal to
↵o = ↵ if wo (↵) > 0 and ↵o = ↵ if wo (↵) < 0; hence
s↵ 2 So , wo (↵) > 0.
For a subset X of S, we have X = So for the orientation o = o+ • wo of Weyl chamber
Do = wo 1 (D+ ), where wo is the longest element of the group hS X i (w = 1 if S = X ).
(ii) So \ S 1 = S⌘1 and So \ S = S⌘ imply that So \ S 1 \ S = S⌘1 \ S = S⌘ \ S 1 . If
S⌘1 \ S = S⌘ \ S 1 , then So \ (S 1 [ S ) = S⌘1 [ S⌘ implies that So \ S 1 = S⌘1 and So \
S = S⌘ .
Definition 2.7. A character of h not vanishing on Ts̃ for all s 2 S is called a twisted sign
character, and its image by the involution ◆ is called a twisted trivial character.
We make the same definition for Haff , S aff replacing h, S.
The pro- p-Iwahori Hecke algebra of a reductive p-adic group III
13
The twisted sign characters ⌘ are never 0 on Tw̃ for w 2 W0 . The algebra h admits
no twisted sign or trivial characters when cs̃ = 0 for some s 2 S. They are equal to o+ ,
where 2 Ẑ k satisfies S = S.
The twisted trivial characters ⌘ vanish on Tw̃ for all w 2 W0 . They are equal to o ,
where 2 Ẑ k satisfies S = S.
The same remarks can be made for Haff , (W aff , S aff ) replacing h, (W0 , S).
3. Distinguished representatives of W0 \W
We recall a well-known lemma for the affine Coxeter system (W aff , S aff ) extended to the
group W = W aff o .
For s 2 S aff , we denote by As the unique positive affine root such that s(As ) is negative.
We have s(As ) = As [8, 1.3.11]. When s 2 S we write As = ↵s .
Lemma 3.1.
(1) For (s, w) 2 S aff ⇥ W , we have
`(ws) = 1 + `(w) , w(↵s ) > 0.
(2) For v 6 w in W and s 2 S aff , we have
(a) either sv 6 w or sv 6 sw;
(b) either v 6 sw or sv 6 sw.
Proof. We recall that W = W aff o . Let (s, u, w) 2 S aff ⇥  ⇥ W aff .
(1) We have `(uws) = `(ws), `(uw) = `(w), and `(ws) = `(w) + 1 , w(↵s ) > 0
[8, 1.13.13]. By definition (§ 1.2) an affine root is positive if and only if it is positive
on the dominant alcove C+ . As the group  normalizes C+ , it normalizes the set
of positive affine roots, in particular w(↵s ) > 0 , (uw)(↵s ) > 0.
(2) Let (v, u 0 ) 2 W aff ⇥ . By definition of the Bruhat–Chevalley partial order
[14, Ap. 2], vu 0 6 wu is equivalent to u 0 = u, v 6 w. In W aff [8, 1.3.19],
(a) either sv 6 w or sv 6 sw;
(b) either v 6 sw or sv 6 sw.
We multiply (a) and (b) by u on the right without changing 6.
Remark 3.2. As `(w) = `(w 1 ) and v 6 w , v 1 6 w 1 , in Lemma 3.1(1) we also have
`(sw) = 1 + `(w) , w 1 (↵s ) > 0, and in Lemma 3.1(2), (a) and (b) can be replaced by
(c) either vs 6 w or vs 6 ws;
(d) either v 6 ws or vs 6 ws.
We introduce now a distinguished set D of representatives of W0 \W .
Proposition 3.3. The three sets
D1 = {d 2 W | d
1 (↵)
> 0 for all ↵ 2 6 + },
M.-F. Vigneras
14
D2 = { w0 | ( , w0 ) 2 3+ ⇥ W0 , `( w0 ) = `( )
`(w0 )},
D3 = {d 2 W | `(w0 d) = `(w0 ) + `(d) for all w0 2 W0 },
are equal, and will be denoted by D. The cosets W0 d, for d 2 D, are disjoint of union W .
Proof. The set D1 is also equal to
{d 2 W | `(sd) = `(d) + 1 for all s 2 S},
(20)
because one can restrict to ↵ 2 1 in the definition of D1 and, for s 2 S, d 1 (↵s ) >
0 , `(sd) = `(d) + 1 (Remark 3.2). Let w 2 W not in D1 . There exists s 2 S with
`(sw) = `(w) 1. Then w1 = sw satisfies `(w) = 1 + `(w1 ). We reiterate, and after
finitely many steps we obtain (w0 , d) 2 W0 ⇥ D1 such that w = w0 d, `(w) = `(w0 ) + `(d).
The pair (w0 , d) is unique. Indeed, for d, d 0 in D1 with d 0 d 1 2 W0 , for all ↵ 2 1 we have
d 0 d 1 (↵) = 2 6, and d 1 (↵) = d 0 1 ( ) is positive as d 2 D1 ; hence > 0 as d 0 2 D1 .
This implies d = d 0 . We deduce that D1 is a set of representatives of W0 \W , that d 2 D1
is the unique element of minimal length in W0 d, and that D1 ⇢ D3 . This implies that
D1 = D3 .
We now compare the sets D1 and D2 . For ( , w0 ) 2 3 ⇥ W0 , we deduce from Lemma 3.1
(see [15, Corollary 5.11]) that
`( w0 ) = `( )
for all ↵ 2 6 + \ w0 (6 ).
`(w0 ) , ↵ ⌫( ) > 0
On the other hand, for all ↵ 2 6 + , ( w0 )
if
w0 1 (↵) > 0, ↵ ⌫( ) > 0
1 (↵)
or
(21)
= w0 1 (↵) + ↵ ⌫( ) is positive if and only
w0 1 (↵) < 0, ↵ ⌫( ) > 0
(22)
[15, (36)]. Comparing (21) and (22), we deduce that D1 = D2 .
Remark 3.4. (i) The distinguished set Daff of representatives of W0 \W aff given by
Proposition 3.3 applied to W aff is equal to Daff = D \ W aff , and D = Daff .
(ii) The distinguished set D of representatives of W0 \W aff can be inductively
constructed: it is the set of w0 2 D for 2 3+ and w0 2 W0 , such that w0 = 1
or w0 has a reduced decomposition w0 = s1 . . . sr (si 2 S), such that
`( s1 . . . si+1 ) = `( s1 . . . si )
1
for 1 6 i 6 r.
Note that s 2 D , ↵s ⌫( ) > 0 when s 2 S.
We denote by w1 the unique element of maximal length in the finite Weyl group W0 .
Lemma 3.5. Let
maximal length,
, µ 2 3+ . The double W0 -coset W0 W0 has a unique element w of
w = w1 ,
`(w ) = `(w1 ) + `( )
and
6 µ , w 6 wµ .
The set W0 W0 \ D is equal to D( ) = { w0 | w0 2 W0 , `( w0 ) = `( )
`(w0 )}.
The pro- p-Iwahori Hecke algebra of a reductive p-adic group III
15
Proof. The coset W0 d of d 2 D contains a unique element of maximal length, equal to
w1 d, `(w1 d) = `(w1 ) + `(d). For 2 3+ , the set D \ W0 W0 contains a unique element
of maximal length, equal to (Remark 3.4(ii)). Hence W0 W0 contains a unique element
w of maximal length, equal to w1 and `(w ) = `(w1 ) + `( ). As wµ = w1 µ, `(wµ ) =
`(w1 ) + `(µ), the equivalence 6 µ , w1 6 w1 µ is clear. We have D( ) = W0 \ D
(Proposition 3.3), and µ 2 W0 W0 , µ = w w 1 for some w 2 W0 , µ = , as 3+
represents the orbits of W0 in 3 [7, 6.3].
Lemma 3.6. Let ( , w0 ) 2 3+ ⇥ W0 , d = w0 2 D, and let µ 2 3+ .
(1) For s 2 S aff , ds 6 2 D , dsd
1
2 S ) `(ds) = `(d) + 1.
(2) For s 2 S and ds 2 D, we have `(ds) = `(d) + 1 , `(w0 s) = `(w0 )
(w, d 0 )
(3) For
2 W0 ⇥ D, we have d 6
wd 0
)d6
1.
d 0.
(4) For s 2 S such that ds 2 D, we have d 6 µ ) ds 6 µ.
(5) We have d 6 wµ , d 6 µ ,
6 µ.
Proof. (1) Let s 2 S aff . By (20) and Remark 3.2,
As d
1(
) > 0 for all
s((d
1
ds 6 2 D , (ds)
2
6+,
1
(↵) < 0
and dsd
(↵))) < 0 , d
1
1
2
W aff ,
(3) d 6
we obtain
wd 0
we have
(↵) = As , ↵ = d(As ) , s↵ = dsd
We have `(ds) = `(d) + 1 by Lemma 3.1(1).
(2) Let s 2 S with ds 2 D. Then
`(ds) = `(d) + 1 , `( )
for some ↵ 2 1.
`(w0 s) = `( )
and s 2 S imply that d 6
swd 0
1
.
`(w0 ) + 1 , `(w0 s) = `(w0 )
or sd 6
swd 0
1.
by Lemma 3.1(2); as d < sd,
d 6 wd 0 ) d 6 swd 0 .
If w 6 = 1, we choose s such that sw < w. Repeating the procedure, we obtain d 6 d 0 by
induction on the length of w 2 W0 .
(4) As d 6 µ, ds 6 µ or ds 6 µs by Lemma 3.1(2). When µs < µ, we obtain ds 6 µ.
Suppose that µs > µ and ds 6 µs. By Lemma 3.1(1),
`(µs) = `(mu) + 1 , µ(↵s ) = ↵s
↵s ⌫(µ) > 0 , ↵s ⌫(µ) 6 0,
, ↵s ⌫(µ) = 0 , ⌫(µ) fixed by s , µs = sµu, u 2 3 \ .
We deduce that ds 6 sµu. By (3), ds 6 µu, because ds, µu 2 D. As 3 is commutative,
ds 6 uµ. For w 2 W , there is a unique element u w 2  such that w 2 u w W aff . By the
definition of the Bruhat–Chevalley order, d 6 µ, ds 6 uµ imply that u d = u µ = uu µ . We
deduce that u = 1, ds 6 µ.
(5) The implications d 6 wµ ( d 6 µ ( 6 µ are obvious, because d 6 , µ 6 wµ .
The implication d 6 wµ ) d 6 µ follows from (3), because wµ = w1 µ (Lemma 3.5) and
µ 2 D. The implication d 6 µ ) 6 µ follows from (4) reiterated finitely many times
for s 2 S such that `(ds) = `(d) + 1 if d 6 = (Remark 3.4(ii)).
M.-F. Vigneras
16
Remark 3.7. Results similar to Proposition 3.3 and Lemma 3.6 are already in
[9, Proposition 2.5, Lemma 2.6, Proposition 2.7], [10, Lemma 2.4], [11, Proposition 1.3],
when W is the Iwahori Weyl group of a split reductive p-adic group G.
Lemma 3.8. In Lemma 3.6, for s 2 S and 1 as in (11),
ds 6 2 D , dsd
1
= w0 sw0 1 2 S , w0 (↵s ) 2 1 , w0 (↵s ) 2 6 + , w0 (↵s ) ⌫( ) = 0.
This implies that `(w0 s) = `(w0 ) + 1 and `(ds) = `(d) + 1 = `( )
`(w0 s) + 2.
Proof. By Lemma 3.6(1), ds 6 2 D , d(↵s ) = w0 (↵s ) = w0 (↵s ) w0 (↵s ) ⌫( ) 2 1 ,
w0 (↵s ) 2 1, w0 (↵s ) ⌫( ) = 0 , w0 (↵s ) 2 1 . In the proof of Lemma 3.6(1), we saw
that dsd 1 = sw0 (↵s ) = w0 sw0 1 . Note that ds 6 2 D implies that `(ds) = `(d) + 1 = `( )
`(w0 ) + 1 6 = `( ) `(w0 s). Hence `(w0 s) = `(w0 ) + 1, `(ds) = `( w0 s) = `( ) `(w0 s) +
2.
By (22), ds 2 D , ↵ ⌫( ) > 0 for all ↵ 2 6 + \ w0 s(6 ). We have
6 + \ w0 s(6 ) = (6 + \ w0 (6 ))
+
{w0 ( ↵s )}
= (6 \ w0 (6 )) [ {w0 (↵s )}
if w0 (↵s ) 2 6 ,
if w0 (↵s ) 2 6 + ,
because, for
2 6 + , we have sw0 1 ( ) < 0 if and only if
2 {w0 (↵s )} [ (w0 (6 )
{w0 ( ↵s )}), as recalled at the beginning of this section. As d 2 D, we have ↵ ⌫( ) > 0
for all ↵ 2 6 + \ w0 (6 ). We deduce that ds 6 2 D , w0 (↵s ) 2 6 + , w0 (↵s ) ⌫( ) = 0.
4. h-eigenspace in ⌘ ⌦h H
Proposition 4.1. For any choice of lift d̃ of d 2 D in D(1), the left h-module H is free of
basis (Td̃ )d2D , and the right h-module H is free of basis (Td̃ 1 )d2D .
Proof. To the set D of distinguished representatives of the right W0 -cosets in W is
F
associated a disjoint union W (1) = d2D W0 (1)d̃. Hence H admits the R-bases
(Twd̃ )w2W0 (1),d2D
and
(Td̃
1w
)w2W0 (1),d2D .
A basis of h is (Tw )w2W0 (1) . By the braid relations, Twd̃ = Tw Td̃ and Td̃
because `(wd) = `(w) + `(d).
1w
= Td̃ 1 Tw ,
P
Remark 4.2. An element of H can be written as a sum d2D h d̃ Td̃ , where h d̃ 2 h, and,
for t 2 Z k ,
h d̃ Td̃ = h t d̃ Tt d̃ = h t d̃ h t Td̃ , h d̃ = h t d̃ h t .
The monoid 3+ represents the orbits of W0 in 3, and the double (W0 , W0 )-cosets of
W , because W = 3 o W0 . The (h, h)-module H is the direct sum
M
H=
h( )
(23)
23+
of the (h, h)-submodules h( ) of R-basis (Tw )w2W0 (1) ˜ W0 (1) . We set D( ) := W0 W0 \ D.
The pro- p-Iwahori Hecke algebra of a reductive p-adic group III
17
Corollary 4.3. Let 2 3+ . The left h-module h( ) is free of basis (Td̃ )d2D( ) , and the
right h-module h( ) is free of basis (Td̃ 1 )d2D( 1 ) .
Let ⌘ be a character of h of parameters ( , S⌘ ). Let
R-basis of ⌘ ⌦h h( ) is
(1 ⌦ Td̃ )d2D( ) .
2 3+ . By Corollary 4.3, an
(24)
When the algebra H arises from a split reductive p-adic group G, Ollivier proved that
the right h-module ⌘ ⌦h h( ) has multiplicity 1 (private communication by email March
2014). This property is general, and the characters of h contained in ⌘ ⌦h h( ) admit the
following description.
Proposition 4.4. Let ⌘1 be a character of h of parameters (
⌘ ⌦h h( ) is not 0 if and only if (⌘1 , ⌘, ) satisfies
1
=
,
1 , S⌘1 ).
The ⌘1-eigenspace of
S⌘1 \ S = S⌘ \ S .
When (⌘1 , ⌘, ) satisfies these conditions, the ⌘1 -eigenspace of ⌘ ⌦h h( ) has dimension 1
and is generated by 1 ⌦ E ˜ (defined in Theorem 1.2).
P
Proof. Let E 2 ⌘ ⌦h h( ). We write (24) E = d2D( ) ad̃ ⌦ Td̃ , where ad̃ 2 R, and, for
t 2 Zk ,
ad̃ ⌦ Td̃ = at d̃ ⌦ Tt d̃ = (t)at d̃ ⌦ Td̃ , ad̃ = (t)at d̃ .
For (w, t) 2 W ⇥ Z k and a lift w̃ of w in W (1), using the notation of §§ 1.2 and 1.4,
w
(1 ⌦ Tw̃ )Tt = 1 ⌦ (w • t)Tw̃ =
(t) ⌦ Tw̃ .
(25)
Using Proposition 2.2 and (25), E is an h-eigenvector of ⌘ ⌦h h( ) with eigenvalue ⌘1 if
and only if E satisfies
X
E=
ad̃ ⌦ Td̃ 6 = 0,
(26)
E Ts̃ = 0
d2D ( ),
for s 2 S
d=
S⌘1 ,
1
E Ts̃ =
1 (cs̃ )E
for s 2 S⌘1 .
(27)
The space ⌘ ⌦h h( ) does not contain a h-eigenvector with eigenvalue ⌘1 when the set
X = {d 2 D( ), d = 1 } is empty, and the proposition is obviously true. When ⌫( ) = 0,
we have D( ) = { } by Lemma 3.5, and the proposition is true, because it is clearly true
when X = { }.
We suppose that ⌫( ) 6 = 0. For s 2 S, the set X is the disjoint union of the subsets
X 1 (s) = {d 2 X | `(ds) = `(d) + 1, ds 2 D},
X 2 (s) = {d 2 X | ds 6 2 D},
X 3 (s) = {d 2 X | `(ds) = `(d)
1}.
In ⌘ ⌦h h( ), we have
8
>
>
> 1 ⌦ Td̃ s̃
<
(1 ⌦ Td̃ )Ts̃ = 1 ⌦ Td̃ Ts̃ = ⌘(Td̃ s̃ d̃ 1 ) ⌦ Td̃
>
>
>
: (c ) ⌦ T
1
s̃
d̃
(d 2 X 1 (s))
(d 2 X 2 (s))
(d 2 X 3 (s)).
M.-F. Vigneras
18
Indeed, if `(ds) = `(d) + 1, the braid relations imply that Td̃ Ts̃ = Td̃ s̃ . If ds 6 2 D, by
Lemma 3.6, dsd 1 2 S, Td̃ s̃ = Td̃ s̃ d̃ 1 d̃ = Td̃ s̃ d̃ 1 Td̃ . If `(ds) = `(d) 1, the braid and
quadratic relations imply that Td̃ Ts̃ = Td̃ s̃ 1 Ts̃2 = Td̃ s̃ 1 cs̃ Ts̃ = d̃cs̃ d̃ 1 Td̃ s̃ 1 Ts̃ = d̃cs̃ d̃ 1 Td̃ .
Multiplying (26) by Ts̃ on the right,
X
X
X
E Ts̃ =
ad̃ ⌦ Td̃ s̃ +
⌘(Td̃ s̃ d̃ 1 )ad̃ ⌦ Td̃ +
1 (cs̃ )ad̃ ⌦ Td̃ .
d2X 1 (s)
d2X 2 (s)
d2X 3 (s)
As X 1 (s)s = X 3 (s), the expansion of E Ts̃ in the basis (24) of ⌘ ⌦h h( ) is
X
X
E Ts̃ =
⌘(Td̃ s̃ d̃ 1 )ad̃ ⌦ Td̃ +
(ad̃(s̃) 1 + 1 (cs̃ )ad̃ ) ⌦ Td̃ .
d2X 2 (s)
(28)
d2X 3 (s)
Relations (27) are equivalent to the following.
For d 2 X 2 (s),
⌘(Td̃ s̃ d̃ 1 )ad̃ = 0
if s 2 S
S⌘1 ,
⌘(Td̃ s̃ d̃ 1 )ad̃ =
1 (cs̃ )ad̃
if s 2 S⌘1 .
(29)
For d 2 X 1 (s),
0=
if s 2 S⌘1 .
1 (cs̃ )ad̃
For d 2 X 3 (s),
ad̃(s̃)
1
=
1 (cs̃ )ad̃
if s 2 S
The relations for d 2 X 3 (s) = X 1 (s)s
For d 2 X 1 (s),
ad̃ =
1 (cs̃ )ad̃ s̃
The relations associated to
S
1
S⌘1 ,
ad̃(s̃)
1
=0
if s 2 S⌘1 .
are equivalent to the following.
if s 2 S
S⌘1 ,
ad̃ = 0
if s 2 S⌘1 .
X 3 (s)) are equivalent to
[
if d 2
X 1 (s).
s2S (X 1 (s) [
ad̃ = 0
ad̃ =
1 (cs̃ )ad̃ s̃
s2S⌘1
if d 2
S
[
X 1 (s).
(30)
(31)
s2S S⌘1
As ⌫( ) 6 = 0, we have X = s2S (X 1 (s) [ X 3 (s)), because d = w0 2 D( ), w0 2 W0
(Lemma 3.5), satisfies ds 6 2 D( ) for all s 2 S if and only if w0 (↵s ) 2 6 + , w0 (↵s ) ⌫( ) = 0
for all s 2 S (Lemma 3.8), and this is equivalent to ⌫( ) = 0.
For d = w0 2 X and d̃ = ˜ w̃0 , the relations (30), (31) are equivalent to
ad̃ =
1 (cw̃0 )
1
a˜
if w0 in hS
1
S⌘1 i,
ad̃ = 0 otherwise.
(32)
With the notation E ˜ , Y introduced in Theorem 1.2, (32) implies that E = a ˜ ⌦ E ˜ . If ⌘1
is contained in ⌘ ⌦h h( ), then a ˜ 6 = 0, the multiplicity of ⌘1 is 1, and 1 = .
STo end the proof of the proposition, we show that the conditions associated to
s2S X 2 (s) on E = 1 ⌦ E ˜ are
S
S⌘1 = S
S⌘ .
(33)
The pro- p-Iwahori Hecke algebra of a reductive p-adic group III
19
Relation (29) for d 2 X 2 (s) is always true if ad̃ = 0. For E = 1 ⌦ E ˜ , we have ad̃ 6 = 0 ,
d 2 Y . By Lemma 3.8, d 2 Y \ X 2 (s) , d = w0 , where
w0 2 hS
1
S⌘1 i,
`( w0 ) = `( )
w0
1
`(w0 ),
=
1,
1
dsd
= w0 sw0 1 2 S .
For sd = dsd 1 2 S and S
s̃d = d̃ s̃ d̃ 1 , we have (cs̃d ) = (d̃cs̃ d̃ 1 ) = d (cs̃ ) =
conditions associated to s2S X 2 (s) are as follows: for all d 2 Y \ X 2 (s),
sd 2 S
S⌘
if s 2 S
S⌘1
sd 2 S⌘
and
if s 2 S⌘1 ;
1 (cs̃ ).
The
(34)
that is, s 2 S⌘1 , sd 2 S⌘ when s 2 S, d 2 Y \ X 2 (s). They are equivalent to (33); that
is, s 2 S⌘1 , s 2 S⌘ when s 2 S , because, for d 2 Y \ X 2 (s), we have sd 2 S , and
hs, S 1 S⌘1 i = hsd , S 1 S⌘1 i; hence sd 2 S⌘1 , s 2 S⌘1 .
Let ⌘, ⌘1 be two characters of h of parameters ( , S⌘ ), ( 1 , S⌘1 ), and let o, o1 be an
orientation such that ⌘ = o , ⌘1 = ( 1 )o1 .
By the decomposition (23), the h-module ⌘ ⌦h H is a direct sum of h-submodules:
M
⌘ ⌦h H =
⌘ ⌦h h( ).
(35)
23+
Proposition 4.5. The character ⌘1 of h is contained in ⌘ ⌦h H if and only if there exists
such that (⌘, ⌘1 , ) satisfies
2 3+ ,
1
=
,
S⌘1 \ S = S⌘ \ S .
The ⌘1 -eigenspace of ⌘ ⌦h H admits the R-basis (1 ⌦ E ˜ ) for all
satisfies these conditions.
such that (⌘, ⌘1 , )
For (⌘, ⌘1 , ) as in Proposition 4.5, we denote by 8 ˜ the H-intertwiner
8 ˜ : 1 ⌦ 1 7! 1 ⌦ E ˜ : ⌘1 ⌦h H ! ⌘ ⌦h H.
Corollary 4.6. An R-basis of HomH (⌘1 ⌦h H, ⌘ ⌦h H) is (8 ˜ ) for all
satisfies the conditions of Proposition 4.5.
Taking ⌘ = ⌘1 , and recalling the 3+ -fixator 3+ of
Corollary 4.7. (8 ˜ )
23+
such that (⌘, ⌘1 , )
(12), we obtain the following.
is a basis of the spherical Hecke algebra H(⌘, h).
To obtain a basis of the spherical Hecke algebra satisfying (14), for an orientation o we
construct h-eigenvectors of the form
1 ⌦ E o ( ˜ ) 2 o ⌦h H
with ˜ 2 3+ (1), where, as in § 1.2, (E o (w̃))w̃2W (1) is the alcove walk basis of H associated
to o [15, § 5.3 Corollary 5.26], and the character o of h is as in Lemma 2.4.
Lemma 4.8. Let
2 3. We have, in
1 ⌦ Eo ( ˜ )
o ⌦h H,
1 ⌦ T˜ 2
X
d
R ⌦ Td̃ ,
where d runs over the elements of D satisfying d <
1 ⌦ E o ( ˜ ) 6 = 0 is a Z k -eigenvector of eigenvalue
.
and
d
=
. If
2 3+ , then
M.-F. Vigneras
20
Proof. For t 2 Z k , we have [15, Example 5.30] E o ( ˜ )Tt = T (t) E o ( ˜ ), T˜ Tt = T (t) T˜ ; hence
1 ⌦ E o ( ˜ )Tt = (t) ⌦ E o ( ˜ ), (1 ⌦ T˜ )Tt = (t) ⌦ T˜ . With the disjoint decomposition
S
W (1) = d2D W0 (1)d̃ and the triangular decomposition of E o ( ˜ ) in the basis (Tw̃ )w̃2W (1)
of H [15, Corollary 5.26], if 1 ⌦ E o ( ˜ ) 6 = 0 is a Z k -eigenvector of eigenvalue , we have
X
X
1 ⌦ E o ( ˜ ) 1 ⌦ T˜ 2
R ⌦ Tw̃d̃ .
d2D ,
d=
w̃2W0 (1),wd<
As `(wd) = `(w) + `(d), by the braid relations, 1 ⌦ Tw̃d̃ = 1 ⌦ Tw̃ Td̃ = ⌘(Tw̃ ) ⌦ Td̃ ,
X
R(1 ⌦ Tw̃d̃ ) = R(1 ⌦ Td̃ ).
w̃2W0 (1),wd<
As d < wd for w 2 W0 , we deduce that
1 ⌦ Eo ( ˜ )
1 ⌦ T˜ 2
d2D ,
X
d=
,d<
R ⌦ Td̃ .
For 2 3+ , 1 ⌦ E o ( ˜ ) is not 0, because 3+ ⇢ D, and (1 ⌦ Td̃ )d2D is a basis of ⌘ ⌦h H
(Proposition 4.1).
Lemma 4.9. Let 2 3. Then 1 ⌦ E o ( ˜ ) 2
if and only 1 ⌦ E o ( ˜ ) 6 = 0 and
1
=
,
o ⌦h H
is a h-eigenvector of eigenvalue (
1 ⌦ E o ( ˜ )E o1 (s̃) = 0
1 )o1
for all s 2 S.
Proof. By Lemma 4.8(ii), 1 ⌦ E o ( ˜ ) is a h-eigenvector with eigenvalue ⌘1 if and only if
1 ⌦ E o ( ˜ ) 6 = 0, and 1 = , (1 ⌦ E o ( ˜ ))E o1 (s̃) = 0 for all s 2 S (Lemma 2.4). We have
(1 ⌦ E o ( ˜ ))E o1 (s̃) = 1 ⌦ E o ( ˜ )E o1 (s̃).
Lemma 4.10. Let 2 3+ . Then 1 ⌦ E o ( ˜ ) is a h-eigenvector of eigenvalue (
only if ⌘(E o (s̃)) = 0 for all s 2 S such that `( s) = 1 + `( ).
)o if and
Proof. Let s 2 S.
1, then E o ( ˜ )E o (s̃) = 0 by the product formula.
If `( s) = `( ) + 1, then E o ( ˜ )E o (s̃) = E o ( s̃) = E o (s̃ s̃ 1 s̃) = E o (s̃)E o•s (s̃
If `( s) = `( )
1
s̃).
The latter equality follows from the fact that the length is constant on a W0 -orbit in 3.
It implies that 1 ⌦ E o (s̃)E o•s (s̃ 1 s̃) = ⌘(E o (s̃)) ⌦ E o ( ˜ ). Apply Lemmas 4.8 and 4.9.
Proposition 4.11. Let
2 3+ . Then,
1 ⌦ E o ( ˜ ) is a h-eigenvector in
o ⌦h H
E ˜ is the component of 1 ⌦ E o ( ˜ ) in
of eigenvalue (
o ⌦h h(
)o , and
).
Proof. Use Lemmas 2.4 and 4.10 for the first assertion. The non-zero components of
1 ⌦ E o ( ˜ ) in the direct decomposition (35) are h-eigenvectors of eigenvalue ( )o . Apply
Proposition 4.4 and Lemma 4.8 for the second assertion.
The pro- p-Iwahori Hecke algebra of a reductive p-adic group III
21
Corollary 4.12. If o = o1 (Lemma 2.6), an R-basis of HomH (( 1 )o ⌦h H, o ⌦h H) is
(1 ⌦ E o ( ˜ )) for all such that ( o , ( 1 )o , ) satisfies the conditions of Proposition 4.5.
Proposition 4.13. For each
2 3+ , we have an injective H-intertwiner
8o, ˜ : 1 ⌦ 1 7 ! 1 ⌦ E o ( ˜ ) :
(8o, ˜ )
23+
o ⌦h H
!
o ⌦h H.
is an R-basis satisfying (14) of the spherical Hecke algebra H(
o , h).
Proof. By Corollary 4.12 and the product formula (8), (8o, ˜ ) 23+ is an R-basis of
H( o , h) satisfying (14).
If 8o, ˜ is not injective, Ker 8o, ˜ contains a simple character ⌘1 of h, and 8o, ˜ 81 = 0
for some non-zero 81 2 Endh (⌘1 ⌦h H, ⌘ ⌦h H).
P
Expanding 81 (1 ⌦ 1) = µ23+ aµ̃ ⌦ E o (µ̃), aµ̃ 2 R, in the basis (1 ⌦ E o (µ̃))µ23+ of
⌘ ⌦h H, and using the product formula E o ( ˜ )E o (µ̃) = E o ( ˜ µ̃), the decomposition of
(8o, ˜ 81 )(1 ⌦ 1) in this basis is
X
X
X
8o, ˜ (aµ̃ ⌦ E o (µ̃)) =
aµ̃ ⌦ E o ( ˜ )E o (µ̃) =
aµ̃ ⌦ E o ( ˜ µ̃).
µ23+
µ23+
µ23+
We have 81 6 = 0 , 81 (1 ⌦ 1) 6 = 0 , aµ̃ 6 = 0 for some µ 2 3+ , (8o, ˜ 81 )(1 ⌦ 1) 6 =
0 , 8o, ˜ 81 6 = 0.
Corollary 4.14. 1 ⌦ E o ( ˜ ) = 0 in
o ⌦h H
if
23
3+ .
Proof. Let 2 3 3+ . We choose µ 2 3+ not 0. Then 8o,µ̃ of Endh ⌘ ⌦h H is injective
(Proposition 4.13) and 8o,µ̃ (1 ⌦ E o ( ˜ )) = 1 ⌦ E o (µ̃)E o ( ˜ ). As µ, belong to di↵erent
closed Weyl chambers, E o (µ̃)E o ( ˜ ) = 0; hence 1 ⌦ E o ( ˜ ) = 0.
More generally, if ( o , (
non-zero H-intertwiner
1 )o ,
) satisfies the conditions of Proposition 4.5, we have the
8o, ˜ : 1 ⌦ 1 7 ! 1 ⌦ E o ( ˜ ) : (
1 )o ⌦h H
!
o ⌦h H.
An R-basis of HomH (( 1 )o ⌦ H, o ⌦ H) is (8o, ˜ ) for all such that ( o , ( 1 )o , ) satisfies
the conditions of Proposition 4.5.
We fix x1 2 3 such that 1 = x1 . For 2 3, 1 = x1 , 2 3 . We embed
HomH (⌘1 ⌦h H, ⌘ ⌦h H) into the algebra e R[3 ] (§ 1.4) by the R-linear map
S⌘1 ,⌘,x̃1 : HomH (⌘1 ⌦h H, ⌘ ⌦h H) ! e R[3 ],
8o, ˜ x̃1 7 ! e ˜
(36)
+
1
( 2 3 \ 3 x1 ),
(37)
where ˜ , x̃1 2 3(1) lift , x1 . If ⌘ = ⌘1 and x̃1 = 1, the map S⌘,⌘,1 = S⌘,⌘ embeds the
spherical Hecke algebra H(⌘, h) = EndH (⌘ ⌦h H) into the algebra e R[3 ]
S⌘,⌘ : H(⌘, h) ! e R[3 ].
(38)
M.-F. Vigneras
22
Lemma 4.15. The composition
(A, B) 7 ! B A : HomH (⌘1 ⌦h H, ⌘ ⌦h H) ⇥ EndH (⌘ ⌦h H) ! HomH (⌘1 ⌦h H, ⌘ ⌦h H),
corresponds to the product S⌘1 ,⌘,x̃1 (A B) = S⌘,⌘ (B)S⌘1 ,⌘,x̃1 (A) in e R[3 ].
Proof. For
2 3+ and
1
2 3+ ,
1
=
1 , S⌘1
\S
1
= S⌘ \ S 1 , we have
8o, ˜ 8o, ˜ 1 (1 ⌦ 1) = 8o, ˜ (1 ⌦ E o ( ˜ 1 )) = 1 ⌦ E o ( ˜ )E o ( ˜ 1 ) = 1 ⌦ E o ( ˜ ˜ 1 ),
by the product formula (8). Hence 8o, ˜ 8o, ˜ 1 = 8o, ˜ ˜ 1 and S⌘1 ,⌘,x̃1 (8o, ˜ 8o, ˜ 1 ) = e ˜ ˜ 1
(x̃1 ) 1 . As e is a central idempotent of R[3 ], we have e ˜ ˜ 1 (x̃1 ) 1 = e ˜ e ˜ 1 (x̃1 ) 1 =
S⌘,⌘ (8o, ˜ )S⌘1 ,⌘,x̃1 (8o, ˜ 1 ).
5. Centers
We make the same hypotheses as in § 1.2, and we suppose that 3T exists.
As 3̃T is central in 3(1), the action of W (1) on 3̃T factorizes through an action of W0 ,
and the R-module Ao (3T ) of basis (E o (µ̃))µ23T is a W0 -stable subalgebra of the center
Zo of Ao , for any orientation o. The quotient map 3T (1) ! 3T of splitting µ 7 ! µ̃ is
W0 -equivariant. For µ 2 3T of W0 -conjugacy class C(µ), and C̃(µ) the W0 -conjugacy class
of µ̃, the set ⌫(C(µ)) contains a single element in the dominant closed Weyl chamber,
and
W
`(µ) = 0 , ⌫(µ) = 0 , µ 2 3T 0 , C̃(µ) = µ̃.
(39)
By axiom (T1) (1.2), a W (1)-conjugacy class C̃ is finite if and only if C̃ ⇢ 3(1).
In the following theorem, R is any commutative ring.
W (1)
Theorem 5.1. The center Z of H R (qs , cs̃ ) is the algebra Ao
of W (1)-invariants of Ao ,
W
equal to the algebra Zo 0 of the Wo -invariants of the center Zo of Ao . The center Z is a
free R-module of basis (independent of the choice of the orientation o)
X
E(C̃) =
E o ( ˜ ) for C̃ running through the finite conjugacy classes of W (1).
˜ 2C̃
The involution ◆ of H satisfies, for any finite conjugacy class C̃ of W (1),
◆(E(C̃)) = ( 1)`(C) E(C̃).
(40)
The algebra ZT = Ao (3T )W0 of W0 -invariants of Ao (3T ) is a central subalgebra of H,
and a free R-module of basis (E(C̃(µ))µ23+ .
T
The ZT -modules Z and H R (qs , cs̃ ) are finitely generated.
When the ring R is noetherian, the R-algebras ZT , Z, and H R (qs , cs̃ ) are finitely
generated.
Proof. The steps of the proof are as follows.
The pro- p-Iwahori Hecke algebra of a reductive p-adic group III
23
P
(1) The center Zo of Ao is a free R-module of basis E o (c̃) = ˜ 2c̃ E o ( ˜ ) for all conjugacy
classes c̃ of 3(1).
P
(2)
E o ( ˜ ) does not depend on the orientation o, and the center Z is equal to
˜
2C̃
W (1)
Ao
for the anti-dominant orientation o .
(3) (a) The Ao (3T )W0 -module Ao (3T ) is finitely generated, and if R is noetherian the
algebra Ao (3T )W0 is finitely generated.
(b) The left Ao (3T )-module Ao is finitely generated.
(c) The left Ao -module H R (qs , cs̃ ) is finitely generated.
The theorem is proved for the pro- p-Iwahori Hecke algebra H R (G, I (1)), where the
assertions on ZT are not formulated but are implicit in the proof. Properties (1),
(2), (3)(a), (b) and (40) admit exactly the same proofs as in [16, Propositions 2.3,
2.7, Lemma 2.15 and Proposition 3.3]. The same is true for the property (3)(c)
[16, Lemma 2.17], once we have strengthened the finiteness property [14, 1.6.3],
[16, Lemma 2.16]. This is done in Lemma 5.3 below. As in [16, added in proof], this
is a variant of the finiteness of the set of minimal elements in a subset L of Zn (n > 0)
[12, Lemma 4.2.18].
Let L be a group isomorphic to Zn . For a = (ai ), b = (bi ) 2 Zn , we write b 6 a if |ai | =
|bi | + |ai bi | for all i. We write b < a if a 6 = b, b 6 a; we say that a 2 L is minimal if
b 2 L , b 6 a implies that b = a.
Lemma 5.2.
(1) Let a 2 L. There exists b 2 L minimal such that b 6 a.
(2) The set L min of minimal elements in L is finite.
Proof. We have |ai | = |bi | + |ai bi | , bi = 0 or ai bi > 0, |bi | 6 |ai |.
(1) If a is not minimal in L, we choose b < a and we reiterate. The processes stops after
finitely many steps, because b < a implies that |bi | 6 |ai | for 1 6 i 6 n, and |bi | 2 N.
(2) Suppose that L min is infinite. If the set {ai | a 2 L min } is finite, ai is constant
for a in an infinite subset of L min . If the set {ai | a 2 L min } is infinite, L min contains
a sequence (a(m))m2N such that (a(m)i )m2N is strictly increasing positive or strictly
decreasing negative. Hence L min contains a sequence (a(m))m2N such that, for all 1 6
i 6 n, (a(m)i )m2N is either constant, or strictly increasing positive or strictly decreasing
negative. For all i in the non-empty set where (a(m)i )m2N is not constant, we have
a(m)i a(m + 1)i > 0, |a(m)i | < |a(m + 1)i | for all m 2 N. Hence a(m) < a(m + 1) for all
m 2 N. This contradicts the minimality of the a(m).
F
By axiom (T1), W = (y,w0 )2Y ⇥W0 3T yw0 . For (y, w0 ) 2 Y ⇥ W0 , let
P
L(y, w0 ) = { È(w) = (` (w))
26 +
| w 2 3T yw0 },
where `(w) =
26 + |` (w)| and ` (w) as in [15, Propositions 5.7 and 5.9]. By
Lemma 5.2, the set L(y, w0 )min is finite. Let X ⇤ (y, w0 ) be a finite subset of 3T such
that
L(y, w0 )min = { È(w) | w 2 X ⇤ (y, w0 )yw0 }.
M.-F. Vigneras
24
Let X be the finite subset
S
`(w) = `(ww 0
(y,w0 )2Y ⇥W0
1
) + `(w 0 )
X ⇤ (y, w0 )y of 3. We have
for w, w0 2 3w0 ,
È(w 0 ) 6 È(w),
[16, Proof of Lemma 2.16(18)]. This implies the following.
Lemma 5.3. For any ( , w0 ) 2 3 ⇥ W0 there exists x 2 X such that
x
1
2 3T ,
1
`( w0 ) = `( x
) + `(xw0 ).
For a central element x of H, the H-intertwiner
8x : 1 ⌦ h 7 ! 1 ⌦ xh = 1 ⌦ hx
is central in H(
o , h)
for h 2 H.
(41)
by Proposition 4.13 and
8x 8o, ˜ (1 ⌦ 1) = 8x (1 ⌦ E o, ˜ ) = 1 ⌦ x E o, ˜
= 1 ⌦ E o, ˜ x = 8o, ˜ (1 ⌦ x) = 8o, ˜ 8x (1 ⌦ 1).
We denote by Z(
o , h)
the center of H(
o , h).
The homomorphism
x 7 ! 8x : Z ! Z(
(42)
o , h)
may be not injective or not surjective.
Proposition 5.4.
8o,µ̃ .
(1) For µ 2 3+
T , we have 1 ⌦ E(C̃(µ)) = 1 ⌦ E o (µ̃) and 8 E(C̃(µ)) =
(2) (8o,µ̃ )µ23+ is a basis, independent of o, satisfying (14) of a central subalgebra
T
ZT (⌘, h) of the spherical algebra H(⌘, h), and H(⌘, h) is a finitely generated
ZT (⌘, h)-module.
Proof. (1) From Corollary 4.14,
1 ⌦ E(C̃(µ)) =
X
˜ 2C̃(µ)\3+ (1)
1 ⌦ E o ( ˜ ) in
o ⌦ H.
+
For µ 2 3+
T we have C̃(µ) \ 3 (1) = {µ̃}. Hence 1 ⌦ E(C̃(µ)) = 1 ⌦ E o (µ̃) and 8 E(C̃(µ)) =
8o,µ̃ .
(2) The canonical isomorphism H(⌘, h) ! e R[3+ ] associated to the basis (8o, ˜ ) 23+
+
(Proposition 4.13) sends ZT (⌘, h) to e R[3+
T ], and e R[3 ] is a finitely generated
e R[3+
T ]-module.
6. Supersingular H-modules
We make the same hypotheses as in § 1.2 and we suppose that 3T exists. We construct
di↵erent filtrations of H which are all equivalent when the ring R is noetherian.
Lemma 6.1. The R-module Fo,n of basis {E o (w̃) | w̃ 2 W (1), `(w) > n} for n 2 N is a right
ideal of H, for any orientation o.
The pro- p-Iwahori Hecke algebra of a reductive p-adic group III
25
Proof. We have Fo,n H ⇢ Fo,n , because, for w̃ 2 W (1), a basis of H is (E o•w (w̃0 ))w̃0 2W (1) ,
and E o (w)E o•w (w̃ 0 ) = E o (w̃w̃0 ) if `(w) + `(w 0 ) = `(ww0 ) and 0 otherwise.
The length is constant on the projection C in W of a finite W (1)-conjugacy class C̃,
and is denoted by `(C̃) = `(C).
Lemma 6.2. The R-module Z`>0 of basis E(C̃) for the finite W (1)-conjugacy classes C̃ of
positive length is an ideal of the center Z of H, stable by the involutive R-automorphism
◆ (4).
Proof. Let C̃1 , C̃2 be two finite W (1)-conjugacy classes. They are contained in 3(1). By
the product formula,
X
E(C̃1 )E(C̃2 ) =
aC̃ E(C̃),
(43)
C̃
where C̃ runs over finite conjugacy classes with `(C) = `(C1 ) + `(C2 ). The stability by ◆
follows from (40).
It is more convenient to replace the center Z of H by the central subalgebra ZT of
basis (E(C̃(µ)))µ2X ⇤+ (T ) which admits better properties.
Lemma 6.3. We have
ZT = RT
where RT is the algebra of basis (Tµ̃ )
W
µ23T 0
ZT,`>0 ,
W
, isomorphic to R[3T 0 ], and ZT,`>0 is the
ideal of ZT of basis (E(C̃(µ)))µ23+ ,`(µ)>0 .
T
The algebras RT and ZT,`>0 are stable by the involutive automorphism ◆.
Proof. The proof is straightforward.
The R-module FT,o,n of basis (E o (µ̃))µ23T ,`(µ)>n is contained in Fo,n and contains
(ZT,`>0 )n .
Proposition 6.4. When R is noetherian, the filtrations of H
((ZT,`>0 )n H)n2N ,
((Z`>0 )n H)n2N ,
(FT,o,n )n2N H,
(Fo,n )n2N ,
are equivalent.
We have (ZT,`>0 )n H ⇢ (Z`>0 )n H ⇢ Fo,n . The last inclusion uses the product formula,
the equality Ẽ(C) = Ẽ o (C), and that (E o (w))w2W (1) is a basis of H. The noetherianity of
R is used only for the proof (Lemma 6.7) of the property (which implies the proposition):
for n 2 N there exists n 0 2 N such that Fo,n 0 ⇢ (ZT,`>0 )n H.
This property follows from the next three lemmas.
Lemma 6.5. E(C̃(µ))n E o (µ̃) = E o (µ̃n+1 ) for µ 2 3T and n > 0.
M.-F. Vigneras
26
Proof. By the product formula, E(C(µ̃))E o (µ̃) = E o (µ̃2 ), because µ̃ is the only element
of C̃(µ) sent by ⌫ in the same closed Weyl chamber as ⌫(µ). By induction on n,
E(C̃(µ))n+1 E o (µ̃) = E(C̃(µ))E(C̃(µ))n E o (µ̃) = E(C̃(µ))E o (µ̃n+1 )
= E(C̃(µ))E o (µ̃)E o (µ̃n ) = E o (µ̃2 )E o (µ̃n ) = E o (µ̃n+2 ).
Lemma 6.6. There exists a positive integer a such that, for any positive integer n,
if µ 2 3T satisfies `(µ) > na.
n
E o (µ) 2 ZT,`>0
Ao
Proof. Let D be a closed Weyl chamber. We choose µ1 , . . . , µr in 3T
⌫(µ1 ), . . . , ⌫(µr ) generate the monoid ⌫(3T ) \ D. We show that
W
3T 0 such that
n
E o (µ) 2 ZT,`>0
Ao ,
if µ 2 3T , ⌫(µ) 2 D and `(µ) > n(`(µ1 ) + · · · + `(µr )). Clearly, this implies the lemma.
Let µ = µn1 1 . . . µrnr u with u 2 (3T )W0 , n 1 , . . . , nr in N. We have `(µi ) 6 = 0 for 1 6 i 6 r
and `(µ) = n 1 `(µ1 ) + · · · + nr `(µr ). Changing the numerotation, we suppose that n 1 > n,
and obtain
E o (µ) = E o (µ1 )n 1 h, h = E o (µ2 )n 2 . . . E o (µr )nr Tu 2 Ao .
By Lemma 6.5,
n
ZT,`>0
Ao .
E o (µ1 )n 1 = E(C̃(µ1 ))n 1
1E
o (µ1 ).
Hence
E o (µ) 2 E(C̃(µ1 ))n Ao ⇢
Lemma 6.7. When R is noetherian, for every positive integer n > 0 there exists a positive
integer n 0 > 0 such that Fo,n 0 ⇢ (ZT,`>0 )n H.
Proof. By Lemma 5.3, we can choose a finite subset X ⇢ 3 such that, for ( , w0 ) 2
3 ⇥ W0 , we have `( w0 ) = `( x 1 ) + `(xw0 ) for some x 2 X with µ = x 1 2 3T . By
the product formula , E o ( w0 ) = E o (µ)E o (xw0 ). If
`( w0 ) > n 0 = na + max{`(xw) | (x, w) 2 X ⇥ W0 },
we have `(µ) > na. Taking a as in Lemma 6.6, E o (µ) 2 (ZT,`>0 )n A0 ; hence E o ( w0 ) 2
(ZT,`>0 )n H. As ( , w0 ) was arbitrary, we get the lemma.
aff as F , with W (1) replaced by W aff (1). The isomorphism (3) restricts
We define Fo,n
o,n
to an isomorphism
aff
Fo,n
⌦ R[Z k ] R[(1)] ' Fo,n .
(44)
The based root system (8, 1) is the finite disjoint union of irreducible based root systems
(8i , 1i ) for 1 6 i 6 r , the Coxeter affine Weyl group (W aff , S aff ) is the product of the
irreducible Coxeter affine Weyl groups (Wiaff , Siaff ), and W aff (1) is an extension
Y
1 ! Z k ! W aff (1) !
Wiaff ! 1.
i
The algebras Hiaff defined by (8i , 1i ) identify with the subalgebras of basis (Tw )w2W aff (1)
of Haff , called the irreducible components of Haff .
i
The pro- p-Iwahori Hecke algebra of a reductive p-adic group III
Lemma 6.8. The filtrations of Haff
aff
(Fo,n
)n2N ,
✓X
i
aff
Fi,o,n
Haff
◆
27
n2N
are equivalent.
Proof. The length of wi 2 Wiaff seen as an element of (Wiaff , Siaff ) or of (W aff , S aff ) is the
same; hence
aff
aff
Fi,o,n
⇢ Fo,n
.
P
For w 2 W aff of components wi 2 Wiaff , we have `(w) = i `(wi ) and E o (w) =
Q
i E o (wi ) by the product formula, and the factors E o (wi ) commute. If `(w) > nr , at
least one component wi satisfies `(wi ) > n; hence
X
aff
aff
Fo,n
⇢
Fi,o,n
Haff .
i
Proposition 6.9. Let M be a right H-module, and let o be an orientation. The following
properties are equivalent.
(1) There exists a positive integer n such that MFo,n = 0.
(2) There exists a positive integer n such that M(Z`>0 )n = 0.
(3) There exists a positive integer n such that M(ZT,`>0 )n = 0.
(4) There exists a positive integer n such that MFT,o,n = 0.
aff = 0.
(5) There exists a positive integer n such that MFo,n
aff = 0 for 1 6 i 6 r .
(6) There exists a positive integer n such that MFi,o,n
aff = 0, because the action
Proof. The isomorphism (44) shows that MFo,n = 0 , MFo,n
of (1) is invertible. Applying Proposition 6.4 and Lemma 6.8, the properties are
equivalent.
Definition 6.10. A right H-module M is called supersingular if it is not 0 and satisfies
the properties of Proposition 6.9.
For future reference, we present the properties of the supersingular right H-modules
M deduced easily from Proposition 6.9 and Lemma 6.3, as a proposition. For a right
H-module M, we have the right H-module ◆(M), equal to M with h 2 H acting by ◆(h).
Proposition 6.11. (1) The category of supersingular right H-modules is stable by
subquotients, by extensions, and by finite sums.
(2) A right H-module is supersingular if and only it is supersingular as a right
Haff -module.
(3) A right H-module generated by a supersingular right Haff -submodule is
supersingular.
28
M.-F. Vigneras
(4) A right Haff -module is supersingular if and only if it is supersingular as a right
Hiaff -module for all the irreducible components Hiaff of Haff .
(5) A right H-module M is supersingular if and only if ◆(M) is supersingular.
(6) A simple right H-module M is supersingular if and only if MZ`>0 = 0 ,
MZT,`>0 = 0.
The properties in (vi) are also equivalent to MFT,o,1 = 0. See Remark 6.16.
The classification of the supersingular simple H-modules reduces to the classification
of the supersingular characters of Haff . For the algebra H(G, I (1)), this was a conjecture
for G = G L(n, F) [13] proved in [11, Proposition 5.10] for G split.
Proposition 6.12. A supersingular right H-module M contains a character of Haff .
Proof. A non-zero element of M generates a right h-module containing a character of
h (Proposition 2.1). We choose a h-eigenvector v 2 M of eigenvalue ⌘. Let ( , S⌘ ) be
the parameters of ⌘ (Proposition 2.2). As M is supersingular, there exists a positive
integer n such that MFo,n = 0. We choose d 2 D of maximal length satisfying v E o (d̃) 6 = 0
(Proposition 3.3). We show that v E o (d̃) is a Haff -eigenvector. Let (t, s) 2 Z k ⇥ S aff .
We have v E o (d̃)Tt = vTdtd 1 E o (d̃) = (dtd 1 )v E o (d̃) = d (t)v E o (d̃).
For the computation of v E o (d̃)Ts̃ , we distinguish three cases.
(1) `(ds) = `(d) 1. Then E o (d̃) = Tt E o (d̃ s̃)E o (s̃), where t 2 Z k , t d̃ s̃ 2 = d̃.
If E o (s̃) = Ts̃ cs̃ , we have E o (s̃)Ts̃ = (Ts̃ cs̃ )Ts̃ = 0.
If E o (s̃) = Ts̃ , we have E o (s̃)Ts̃ = Ts̃2 = cs̃ Ts̃ = cs̃ E o (s̃); as E o (d̃ s̃)cs̃ = (ds • cs̃ )E o (d̃ s̃) =
d • cs̃ E o (d̃ s̃), we deduce that v E o (d̃)Ts̃ = 0 or (d • cs̃ )v E o (d̃) = d (cs̃ )v E o (d̃).
(2) `(ds) = `(d) + 1 and ds 2 Z k D. Either E o (d̃)Ts̃ = E o (d̃)E o (s̃) = E o (d̃ s̃) or
E o (d̃)Ts̃ = E o (d̃)(E o (s̃) + cs̃ ) = E o (d̃ s̃) + (d • cs̃ )E o (d̃). By the maximality of `(d),
v E o (d̃ s̃) = 0 and v E o (d̃)Ts̃ = 0 or (d • cs̃ )v E o (d̃) = d (cs̃ )v E o (d̃).
(3) `(ds) = `(d) + 1 and ds 6 2 Z k D. Let sd 2 S such that d̃ s̃ = s̃d d̃ (Lemma 3.8). Either
E o (d̃)Ts̃ = E o (d̃)E o (s̃) = E o (d̃ s̃) = E o (s̃d d̃) = E o (s̃d )E o (d̃) or E o (d̃)Ts̃ = E o (d̃)(E o (s̃) +
cs̃ ) = E o (d̃ s̃) + E o (d̃)cs̃ = (E o (s̃d ) + d • cs̃ )E o (d̃). Hence v E o (d̃)Ts̃ = ⌘(E o (s̃d ))v E o (d̃) or
⌘(E o (s̃d ) + d • cs̃ )v E o (d̃) = (⌘(E o (s̃d )) + (d • cs̃ ))v E o (d̃) = (⌘(E o (s̃d )) + d (cs̃ ))v E o (d̃).
The compatibility of supersingularity for H and Haff (Proposition 6.9) and
Proposition 6.12 imply the following.
Corollary 6.13.
(1) A simple supersingular right Haff -module has dimension 1.
(2) A simple right H-module is supersingular
if and only if it contains a supersingular character of Haff ;
if and only if any simple right Haff -submodule is a supersingular character of
Haff .
The classification of the supersingular characters of Haff , given in Theorem 6.15
after technical Lemma 6.14, follows from the classification of the characters of Haff
(Proposition 2.2). The classification was done for H(G, I (1)) in [13] for G = G L(n, F)
and in [11, Lemma 5.11 and Theorem 5.13] for G split.
The pro- p-Iwahori Hecke algebra of a reductive p-adic group III
29
Let 4 be a character of Haff ,
a character of Z k , and o an orientation such that
4|h = o (Lemma 2.4). Let wo 2 W0 such that the Weyl chamber of o is wo 1 (D+ ). For
a subset J of S, let w J be the longest element of the subgroup of W0 generated by J .
Lemma 6.14.
(1) 4(E(C̃(µ)) = 4(E o (µ̃)) for µ 2 3+
T.
(2) If S aff S = {s0 } and 2 3+ has positive length, we have
(i) `(s0 ) = 1 + `( );
(ii) E o (s̃0 ) = Ts̃0 , wo (↵0 ) 2 6 + , where ↵0 is the highest root of 6 + ;
(iii) E o ( ˜ ) = Ts̃ E o•s0 (s̃ 1 ˜ )) if wo (↵0 ) 2 6 + ;
0
0
(iv) w J (↵0 ) 2 6 + , J 6 = S.
Proof. (1) The character ⇠ factorizes through the canonical homomorphism
h 7 ! 1 ⌦ h : Haff ! ⇠ |h ⌦h Haff ,
and 1 ⌦ E(C̃(µ)) = 1 ⌦ E o (µ̃) in o ⌦h Haff by Proposition 5.4.
(2) The hypothesis means that the root system 6 is irreducible. The highest positive
root ↵0 2 6 + has the following well-known properties: ↵0 + 1 is a simple affine root and
s0 = s ↵0 +1 , 0 < ↵0 (x) + 1 < 1 for x 2 C+ .
(i) `(s0 ) = 1 + `( ) , C+ and C+ + ⌫( ) are not on the same side of Ker( ↵0 + 1)
[15, Example 5.4]. This is equivalent to ↵0 (x + ⌫( )) + 1 = ↵0 (x) + 1 ↵0 ⌫( ) is
negative for x 2 C+ , ↵0 ⌫( ) > 1, which is true, because ↵0 ⌫( ) 2 N>0 as 2 3+
has positive length [15, Corollary 5.11].
(ii) By (6), E o (s̃0 ) = Ts̃0 , C+ is on the o-negative side of Ker( ↵0 + 1). By
[15, Definition 5.16], this means that ↵0 is o-negative, because ↵0 + 1 is positive on
C+ . The root ↵0 is o-negative if and only if ↵0 is positive on the Weyl chamber wo 1 (D+ )
of o. This is true if and only if wo (↵0 ) 2 6 + .
(iii) For any orientation o, E o ( ˜ ) = E o (s̃0 )E o•s0 (s̃0 1 ˜ ) by the product formula and
`( ) = 1 + `(s0 ) (i). Apply (ii).
P
P
(iv) Let SP
= J [ J 0 . We P
have ↵0 = ( s2J n s ↵s ) + ( s2J 0 n s ↵s ) with n s 2 N>0 , and
w J (↵0 ) = ( s2J n s ↵s ) + ( s2J 0 n s w J (↵s )). If J 0 = ;, then w J (↵0 ) 6 2 6 + . If J 0 6 = ;, for
any s 2 J 0 , the root w J (↵s ) is positive and does not belong to the group generated
by J . The decomposition of w J (↵0 ) on the basis (↵s )s2S has a positive coefficient; i.e.,
w J (↵0 ) 2 6 + .
Theorem 6.15. A character of Haff is supersingular if and only if its restriction to each
irreducible component of Haff is not a twisted sign or trivial character.
Proof. The involutive automorphism ◆ of Haff respects supersingularity and exchanges
a twisted sign character and a twisted trivial character (Definition 2.7). For s 2 S aff
and a character ⇠ of Haff , ⇠ vanishes on Ts or ◆(Ts ) (Proposition 2.2). Let µ 2 3+
T of
positive length. We have ⇠(E(C̃(µ))) = ⇠(E o (µ̃)) for any orientation o (Lemma 6.14) and
E o (µ̃) = Tµ when o is dominant [15, Example 5.30].
M.-F. Vigneras
30
(i) A twisted sign character is not 0 on Tw for all w 2 W (1) of positive length; hence it
is not 0 on E(C̃(µ)), and it is not supersingular. Applying ◆, a twisted trivial character
is not supersingular.
(ii) It remains to prove that, when Haff is irreducible, i.e., S aff S = {s0 }, a character
⇠ of Haff di↵erent from a twisted sign or trivial character is supersingular.
Applying ◆, it suffices to prove it when ⇠(Ts̃0 ) = 0. The set J = S {s 2 S | ⇠(Ts̃ ) 6 = 0}
is di↵erent from S, because ⇠ is not a twisted sign character. Let o be the orientation
of Weyl chamber w J 1 (D+ ). By Lemma 2.6, the restriction of ⇠ to h is of the form o ,
because So = {s 2 S | ⇠(Ts̃ ) 6 = 0} (5). Applying Lemma 6.14, we obtain, for any µ 2 3+
T
of positive length,
E o (s0 ) = Ts̃0 ,
E o (µ̃) = Ts̃0 E o•s0 ((s̃0 )
1
µ̃),
⇠(E(C(µ̃))) = ⇠(E o (µ̃)) = 0.
Hence ⇠ is supersingular.
Remark 6.16. We can complete Proposition 6.11(6): a simple H-module M is
supersingular if and only if MFT,o,1 = 0. This follows from Corollary 6.13 and part (ii)
in the proof of Theorem 6.15.
Cli↵ord’s theory studies classically the induction of representations from normal
subgroups. We give a “Cli↵ord’s theory style” proposition to describe the simple
finite-dimensional H-modules containing a character of Haff , as in [13, Proposition 3],
[11, Lemma 5.12] for the algebra H(G, I (1)) when G is split.
Let R be a field, and let A be an R-algebra of the form A = J B, where J is an ideal of
A and B a subalgebra of A equal to the R-algebra R[G] of a group G.
Let 4 : J ! R be a character of J with a G-fixator G 4 = {g 2 G | 4g = 4} of 4 of
finite index in G, where 4g is the character j 7 ! 4g ( j) = 4(g jg 1 ) of J .
Let V be a finite-dimensional right R[G 4 ]-module, where the group J \ G acts by
4| J \G . For g 2 G, we denote by V g the right R[g 1 G 4 g]-module V , where g 1 hg acts
by h for h 2 G 4 .
We extend V to a right A4 = J R[G 4 ]-module, where J acts by 4, denoted by 4 ⌦ V .
We induce 4 ⌦ V to a right A-module
I (4, V ) = (4 ⌦ V ) ⌦ A4 A.
Proposition 6.17. Let R, A, J, G, 4, V be as above. We suppose V to be simple. We have
the following.
(i) I (4, V ) is finite dimensional and is a simple right A-module.
(ii) A finite-dimensional simple right A-module containing 4 as a J -module is
isomorphic to I (4, V ) for some V .
g
g
(iii) I (41 , V1 ) ' I (42 , V2 ) if and only if (42 , V2 ) = (41 , V1 ), for some element g 2 G.
Proof [11, Lemma 5.12]. 4 ⌦ V is finite dimensional and is a simple A4 -module, because
its restriction
to the subalgebra R[G 4 ] satisfies these properties. The left A4 -module
L
A = g2G 4 \G A4 g is free of finite rank. The restriction of I (4 ⌦ V ) to J is isomorphic
The pro- p-Iwahori Hecke algebra of a reductive p-adic group III
31
L
L
L
to a direct sum dim R V g2G 4 \G 4g , and I (4, V ) = g2G 4 \G (4g ⌦ V g ) is equal to the
direct sum of all the conjugates of 4 ⌦ V by G. The dimension of I (4 ⌦ V ) is finite, equal
to [G : G 4 ] dim R V . The restriction
to J of a non-zero A-submodule of I (4 ⌦ V ) contains
L
a submodule isomorphic to g2G 4 \G 4g ; hence its 4-isotypic component is not 0. The
4-isotypic component of I (4 ⌦ V ) is the simple A4 -module 4 ⌦ V . Therefore I (4 ⌦ V )
is a simple A-module.
Let M be a finite-dimensional simple right A-module with a non-zero 4-isotypic
component as a J -module. The 4-isotypic component is an A4 -module of the form
4 ⌦ V 0 for some finite-dimensional right R[G 4 ]-module V 0 . The non-zero R[G 4 ]-module
V 0 contains a simple submodule V . The module 4 ⌦ V is isomorphic to an A4 -submodule
of M, and I (4 ⌦ V ) is isomorphic to an A-submodule of M. As M is simple, M = I (4, V ).
The restriction of I (4 ⌦ V ) to J shows that I (4 ⌦ V ) determines the G-orbit of 4,
the 4-isotypic part of I (4 ⌦ V ) determines V , and the 4g -isotypic part of I (4 ⌦ V )
is 4g ⌦ V g for g 2 G. This implies that I (41 , V1 ) ' I (42 , V2 ) if and only if (42 , V2 ) =
g
g
(41 , V1 ), for some g 2 G.
We can apply Proposition 6.17 to the R-algebra A = H, its ideal J = Haff , the
group G = (1), an arbitrary character 4 of Haff , and a finite-dimensional simple right
R[(1)]-module V such that Z k acts on V by the character 4| Z k . As a subgroup of  of
finite index acts trivially on V , the fixator (1)4 of 4 has a finite index in (1).
Corollary 6.13, Theorem 6.15, and Proposition 6.17 imply the following.
Theorem 6.18. The supersingular simple finite-dimensional right H-modules are
isomorphic to the H-modules I (4, V ), where
(i) 4 is a character of Haff di↵erent from a twisted sign or trivial character on each
irreducible component of Haff ,
(ii) V is a simple finite-dimensional right R[(1)4 ]-module, where Z k acts by 4| Z k .
Two H-modules I (41 , V1 ), I (42 , V2 ) are isomorphic if and only if the pairs
(41 , V1 ), (42 , V2 ) are (1)-conjugate.
7. Pro- p-Iwahori invariants and compact induction
We use the notation of 1.3, and R is as in 1.4. The algebras H and h denote the
pro- p-Iwahori Hecke algebra H R (G, I (1)) and H R (K , I (1)).
Let ⇢ be an irreducible smooth R-representation of K , let v 2 ⇢ I (1) not 0, let ⌘ be the
character of h on ⇢ I (1) , let be the restriction of ⌘ to Z k , and let o be an orientation
such that ⌘ = o (Lemma 2.4).
We show that the pro- p-Iwahori invariant functor behaves well on compact induced
representations of G, generalizing the results of Ollivier [10, Corollary 3.14] proved when
G is split.
By Cabanes [3, Theorem 2], the I (1)-invariant functor ⇢ 7 ! ⇢ I (1) gives an equivalence
– from the category of finite-dimensional R-representations ⇢ of K trivial on K (1),
such that ⇢ and its dual ⇢ ⇤ are generated by I (1);
– to the category of finite-dimensional right h-modules.
32
M.-F. Vigneras
Remark 7.1. For n 2 N \ K of image w 2 Wo (1), the action on ⇢ I (1) of the basis element
Tw 2 h is
X
1
vTw =
v = ⌘(Tw )v.
2I (1)\I (1)n I (1)
The action of Z k on ⇢ I (1) arising from the action of Z 0 ⇢ I normalizing I (1) and the
action of Z k embedded in the Hecke algebra h on ⇢ I (1) are inverse from each other.
Let
c-IndG
K ⇢
be the compactly induced representation of G by right translations on the space of
compactly supported functions f : G ! V (⇢) satisfying f (k1 g) = ⇢(k1 ) f (g) for k1 2 K
and g 2 G. Let
I (1)
[1, v] K 2 (c-IndG
K ⇢)
be the function of support K and value v at 1. The representation c-IndG
K ⇢ is generated
by [1, v] K , and dim R ⇢ I (1) = 1.
Proposition 7.2. The H-equivariant linear map
I (1)
⇢ I (1) ⌦h H ! (c-IndG
K ⇢)
1 ⌦ 1 7 ! [1, v] K
is an isomorphism.
We explain the strategy of the proof, which reduces the proposition to the next lemma.
The disjoint union of W into W0 -cosets corresponds to a disjoint union of G into
(K , I )-cosets. A (K , I )-coset is equal to a (K , I (1))-coset. We have
[
[
G=
KdI =
K d̃ I (1),
(45)
d2D
d2D
where, for d in the distinguished set D of representatives of W0 \W (Proposition 3.3),
K d I = K d̃ I (1) denotes the double coset K n d̃ I = K n d̃ I (1), d̃ 2 D(1) lifting d, and n d̃ 2 N
I (1) is the direct sum
lifting d̃, with n 1 = 1. The space (c-IndG
K ⇢)
M
I (1)
dI
(c-IndG
=
(c-Ind K
⇢) I (1)
(46)
K ⇢)
K
d2D
I (1) with support contained in K d I , for d 2 D.
of the subspaces of functions in (c-IndG
K ⇢)
The pro- p-Iwahori Hecke algebra is the direct sum
M
H=
hTd̃
(47)
d2D
of the left h-modules hTd̃ of functions in H with support contained in K d I , for d 2 D.
I (1) of
We denote by ⌘ the character of h on ⇢ I (1) , and by f d̃ the function in (c-IndG
K ⇢)
support K d I with f (n d̃ ) = v, for d 2 D. We have f 1 = [1, v] K . The proposition follows
from the following lemma.
The pro- p-Iwahori Hecke algebra of a reductive p-adic group III
Lemma 7.3.
(i) For d 2 D, we have K (1)(K \ n d̃ I (1)n
(ii) A basis of
(c-IndG
K
⇢) I (1)
(iii) f d̃ = f 1 Td̃ for d 2 D.
1
d̃
is ( f d̃ )d2D .
33
) = I (1).
I (1) of eigenvalue ⌘.
(iv) f 1 is a h-eigenvector in (c-IndG
K ⇢)
Proof. (1) We denote by I 0 the subgroup of I (1) generated by U \ I = U \ K and U \ I .
We have I (1) = Z 0 (1)I 0 and Z 0 (1) = K (1) \ Z 0 . The lemma follows from the inclusion
U \ I ⇢ n d̃ I 0 n
1
d̃
,
because K (1)(K \ n d̃ I (1)n 1 ) = K (1)(K \ n d̃ I 0 n 1 ) is a pro- p-subgroup of K and I (1) =
d̃
d̃
K (1)(U \ I ) is a pro- p-Sylow subgroup of K . The group U \ I is the product of the
groups U↵,0 = U↵ \ K for all ↵ in the set 8r+ed of positive reduced roots of associated
root subgroup U↵ . By Proposition 3.3 and § 1.3, d 1 (e↵ ↵) is positive on C+ . As e↵ is a
positive integer, d 1 (↵) is positive on C+ . By [15, §§ 3.3 and 3.5], n 1 U↵,0 n d̃ = Ud 1 (↵) .
As d
1 (↵)
is positive on C+ , Ud
1 (↵)
K d I (1)
(2) By (46) and f n d̃ 2 (c-Ind K
K n d̃ I (1)
(c-Ind K
⇢ I 0 . Hence U↵,0 ⇢ n d̃ I 0 n
⇢) I (1) , it suffices to prove that the dimension of
K d I (1)
because k f (n d̃ ) = f (kn d̃ ) = f (n d̃ n
⇢
1
d̃
d̃
d̃
.
⇢) I (1) is 1. The value at n d̃ gives a linear map
(c-Ind K
K \n d̃ I (1)n
1
1
d̃
⇢) I (1) ! ⇢
K \n d̃ I (1)n
1
,
d̃
kn d̃ ) for k 2 K . The map is clearly injective, and
= ⇢ I (1) , because ⇢ is trivial on K (1) and (1). As dim R ⇢ I (1) = 1, we have
K n I (1)
dim R (c-Ind K d̃
⇢) I (1)
= 1.
(3) We show that the support of the function f 1 Tn d̃ is contained in K d I (1) and that
the value at n d̃ of f 1 Tn d̃ is v.
For g 2 G, we have
X
1
f 1 Tg =
f1,
2I (1)\I (1)g I (1)
1 f (x) = f (x
1 ) for x 2 G. The support of f is K , and the support of f T is
and
1
1
1
1 g
contained in K g I (1).
In particular, the support of the function f 1 Tn d̃ is contained in K d I (1). We have
X
X
( f 1 Tn d̃ )(n d̃ ) =
f 1 (n d̃ 1 ) =
f 1 (u).
2I (1)\I (1)n d̃ I (1)
u2(K \n d̃ I (1)n
1
d̃
)/(I (1)\n d̃ I (1)n
1
d̃
)
By (1), this is equal to f 1 (1) = v.
(4) For k 2 K , the support of the function f 1 Tk is contained in K , and
X
X
1
( f 1 Tk )(1) =
f1( 1) =
f 1 (1) = ⌘(Tk )v.
2I (1)\I (1)k I (1)
Therefore f 1 Tk = ⌘(Tk ) f 1 for k 2 K .
2I (1)\I (1)k I (1)
M.-F. Vigneras
34
2 3, the isomorphism of Proposition 7.2 restricts to a right h-module
Remark 7.4. For
isomorphism
⇢ I (1) ⌦h h( ) ! (c-Ind K
K
K
⇢) I (1) .
Proposition 7.5. Let ⇢1 , ⇢ be two irreducible smooth R-representations of K . The
I (1)-invariant map
G
G
I (1)
I (1)
Hom RG (c-IndG
, (c-IndG
)
K ⇢1 , c-Ind K ⇢) ! HomH ((c-Ind K ⇢1 )
K ⇢)
is an isomorphism.
To explain the strategy of the proof, we recall the adjunction isomorphisms
K (1)
Hom RG (c-IndG
),
K ⇢1 , ⇡) ' Hom R K (⇢1 , ⇡) = Hom R K (⇢1 , ⇡
I (1)
HomH (⇢1
I (1)
⌦h H, ⇡ I (1) ) ' Homh (⇢1
, ⇡ I (1) ),
for any smooth R-representation ⇡ of G. The I (1)-invariant map
G
I (1)
Hom RG (c-IndG
, ⇡ I (1) )
K ⇢1 , ⇡) ! HomH ((c-Ind K ⇢1 )
is an isomorphism if and only if the I (1)-invariant map
I (1)
Hom K (⇢1 , ⇡ K (1) ) ! Homh (⇢1
, ⇡ I (1) )
(48)
is an isomorphism, by Proposition 7.2. The map (48) is injective, because ⇢ I (1) generates
⇢, but it is not surjective in general. The proposition says that the map (48) is surjective
if ⇡ = c-IndG
K ⇢.
The dominant monoid 3+ represents the cosets K \G/K (see 1.3). The anti-dominant
monoid 3 has the same property and is more convenient now. The representation of K
on c-IndG
K ⇢ is a direct sum
M
K
c-IndG
c-Ind K
⇢,
K
K ⇢ =
23
K
c-Ind K
K
where
⇢ is the space of functions in c-IndG
K ⇢ with support in K K . We will
prove that, for all 2 3 , the I (1)-invariant map
Hom K (⇢1 , (c-Ind K
K
K
I (1)
⇢) K (1) ) ! Homh (⇢1
, (c-Ind K
K
K
⇢) I (1) )
(49)
is an isomorphism. A representation of K trivial on K (1) generated by its I (1)-invariant
vectors identifies with a representation of the finite reductive group G k generated by
K
its Uk -invariant vectors (using the notation of § 1.3). We describe (c-Ind K
⇢) K (1)
K
as a representation of G k . Let z 2 Z lifting . We have K z K = K K and by [7,
Proposition 6.13] the group
K = K (1)(K \ z 1 K z)
is the inverse image in K of a parabolic subgroup Pk = Mk Nk of G k containing Bk ,
S
of unipotent radical Nk equal to the image in G k of h ↵28+ ,↵ ⌫(z)<0 U↵,0 i as ⌫(z) is
anti-dominant and h↵, zi = h↵, ⌫(z)i in the notation of [7, 6.11]; it is a parahoric
The pro- p-Iwahori Hecke algebra of a reductive p-adic group III
35
subgroup of G of pro- p-radical K (1) = K (1)(K \ z 1 K (1)z). Let ⇢z be the representation
of K \ z 1 K z on the space V (⇢) of ⇢ such that ⇢z (k) = ⇢(zkz 1 ). The map f 7 ! :
zK
k 7 ! f (zk) : Ind K
⇢ ! Ind K
⇢ is a K -equivariant isomorphism. It restricts to a
K
K \z 1 K z z
K -equivariant isomorphism
zK
(Ind K
⇢) K (1) ! (Ind K
K
K \z
1Kz
K (1)\z
⇢z ) K (1) = Ind K
K (⇢z
where the natural representation of K \ z
1K z
K (1)\z
on ⇢z
1Kz
K (1)\z
⇢z
1Kz
),
is extended to a
1Kz
representation of K trivial on K (1). The representation
of K identifies
to the representation ⇢zNk of Pk on the space V (⇢ Nk ) of ⇢ Nk such that ⇢z (m) = ⇢(zmz 1 )
S
K (1)\z 1 K z
for m in the group M0 = hZ 0 , ↵28,↵ ⌫(z)=0 U↵,0 i. The representation Ind K
)
K (⇢z
identifies to IndGPkk (⇢zNk ). The representation ⇢zNk of Pk is irreducible [2]. The Uk -invariant
functor
HomG k (⇢1 , IndGPkk (⇢zNk )) ! Homh (⇢1Uk , (IndGPkk (⇢zNk ))Uk )
(50)
is an isomorphism, by Cabanes’s equivalence recalled at the beginning of this section,
because IndGPkk (⇢zNk ) and its contragredient are generated by their Uk -invariant vectors.
This is a general property proved in the next lemma.
Lemma 7.6. Let ⌧ be an irreducible R-representation of Pk trivial on Nk . The
representation IndGPkk ⌧ of G k and its contragredient are isomorphic to a subrepresentation
Gk
and to a quotient of IndU
1. In particular, they are generated by their Uk -invariant
k
vectors. Their socle and their heads are multiplicity free.
Proof. A representation of G k is generated by its Uk -invariant vectors if and only if it is
Gk
a quotient of a direct sum of representations isomorphic to IndU
1.
k
Gk
The representation IndGPkk ⌧ is a quotient of IndU
1, because it is generated by a
k
Uk -invariant vector (a function in IndGPkk ⌧ of support Pk with non-zero value in ⌧ Uk \Mk ).
The inflation of ⌧ to Pk is contained in IndUPkk 1. By transitivity of the induction, IndGPkk ⌧
Gk
is contained in IndU
1.
k
The contragredient representation (IndGPkk ⌧ )⇤ is a subrepresentation and a quotient of
Gk
Gk
IndU
1, because IndU
1 is isomorphic to its contragredient, the contragredient permutes
k
k
the irreducible R-representations of Mk , and it commutes with the parabolic induction.
Gk
Gk
The socle of a subrepresentation of IndU
1 is contained in the socle of IndU
1.
k
k
Gk
The socle of IndU
1 is multiplicity free, because dim ⇢Uk = 1, and by adjunction
k
Gk
HomG k (⇢, IndU
1) ' HomUk (⇢Uk , 1) for any irreducible R-representation ⇢ of G k of
k
Uk -coinvariants ⇢Uk .
The contragredient of the socle is the head of the contragredient.
With (51) and the I (1)-invariant functor (Proposition 7.5 for ⇢1 = ⇢), we transfer our
results on the spherical algebra H(⌘, h) to the spherical algebra H R (G, K , ⇢), which is
the convolution algebra of compactly supported functions
: G ! End R (V (⇢)) satisfying
(k1 gk2 ) = ⇢(k1 ) (g)⇢(k2 ) for k1 , k2 2 K , g 2 G.
M.-F. Vigneras
36
It is isomorphic to the algebra End RG c-IndG
K ⇢ by the map sending
RG-intertwiner E of c-IndG
⇢
defined
by
K
E ( f 1 )(g) = (g)(v)
to the
(g 2 G).
(51)
The spherical Hecke R-algebra H R (G, K , ⇢) admits a natural basis [7, 7.3] (F ˜ )
where
F ˜ has support K K
and
23+ ,
F ˜ ( ˜ )(v) = v.
(52)
The basis (F ˜ ) 23+ does not satisfy (14) in general. The basis (52) for the spherical Hecke
algebra H R (Z , Z 0 , ) is denoted by (⌧ ˜ ) 23 ,
⌧ ˜ has support Z 0
and
⌧ ˜ ( ˜ )(v) = v.
The basis (52) for the central spherical Hecke subalgebra H R (T, T0 , ⇢ I (1) ) is (⌧µ̃ )µ23T ,
and the H R (T, T0 , ⇢ I (1) )-module H R (Z , Z 0 , ⇢ I (1) ) is finitely generated. We denote by
H R (T + , T0 , ⇢ I (1) ) ⇢ H R (Z + , Z 0 , ⇢ I (1) ) the subalgebras of bases (⌧µ̃ )µ23+ and (⌧ ˜ ) 23+ .
T
The basis (⌧ ˜ ) 23+ satisfies (14).
Theorem 7.7. The R-algebras
H R (G, K , ⇢) ' End RG c-IndG
K ⇢ ' EndH (⌘ ⌦h H) = H(⌘, h)
are isomorphic via (51) and the I (1)-invariant functor (Proposition 7.5).
The basis (F ˜ ) 23+ of H R (G, K , ⇢) (52) corresponds to the basis (E ˜ ) 23+ of H(⌘, h)
(Proposition 4.4).
The basis ( o, ˜ ) 23+ of H R (G, K , ⇢) corresponding to the basis (8o, ˜ ) 23+ of H(⌘, h)
(Proposition 4.13) satisfies (14).
For µ 2 3+
T , µ̃ = o,µ̃ does not depend on the choice of o.
( µ̃ )µ23+ is a basis of a central subalgebra Z R (G, K , ⇢)T of H R (G, K , ⇢), and
T
H R (G, K , ⇢) is a finitely generated Z R (G, K , ⇢)T -module (Proposition 5.4).
Remark 7.8. The RG-endomorphism of c-IndG
K ⇢ corresponding to µ̃ sends [1, v] K to
[1, v] K E o (µ̃) for any orientation o such that ⌘ = o (Propositions 7.2 and 4.13).
We denote by A+
o,T the R-algebra of basis (1 ⌦ E o (µ̃))µ23+ .
T
Corollary 7.9. We have an R-algebra isomorphism
(
o, ˜ ) 23+
7 ! (⌧ ˜ )
23+
So
: H R (G, K , ⇢) ! H R (Z + , Z 0 , )
ST
restricting to an isomorphism Z R (G, K , ⇢)T ! H R (T + , T0 , ) independent of o. We
have the R-algebra isomorphisms
ST
+
+
ZT ! Z R (G, K , ⇢)T ! H R (T + , T0 , ) ! A+
o,T ! R[3̃T ] ! R[3T ]
(E(C̃(µ)))µ23+ ! (
T
µ̃ )µ23+
T
! (⌧µ̃ )µ23+ ! (E o (µ̃))µ23+ ! (µ̃)µ23+ ! (µ)µ23+ .
T
T
T
T
When the group G is split, (Z + , Z 0 ) = (T + , T0 ) and Z R (G, K , ⇢)T = H R (G, K , ⇢).
The pro- p-Iwahori Hecke algebra of a reductive p-adic group III
37
Theorem 1.5 in § 1 follows from Corollary 7.9 and the next proposition. The
+
R-characters ⇠ of 3+
T identify with the characters of the R-algebras isomorphic to R[3̃T ]
in Corollary 7.9. We write
for µ 2
3+
T.
⇠(⌧µ̃ ) = ⇠(E(C̃(µ))) = ⇠(
µ̃ )
= ⇠(E o (µ̃)) = ⇠(µ̃) = ⇠(µ)
Let ⇡ be a smooth R-representation of G. We suppose that ⇡| K contains ⇢.
Proposition 7.10. Let A 2 Hom R K (⇢, ⇡) be non-zero, and let µ 2 3+
T . We have
(A
µ̃ )(v)
= A(v)E o (µ̃) = A(v)E(C̃(µ)).
In particular, if A is a Z R (G, K , ⇢)T -eigenvector in Hom R K (⇢, ⇡) of eigenvalue ⇠ ,
⇠(µ̃)A(v) = A(v)E o (µ̃) = A(v)E(C̃(µ)).
Proof. By the adjunction isomorphism, A and A µ̃ correspond to the RG-intertwiners
c-IndG
K ⇢ ! ⇡ sending [1, v] K to A(v) and to A(v)E o (µ̃) (Remark 7.8). We deduce that
(A µ̃ )(v) = A(v)E o (µ̃).
I (1) !
The H-isomorphism (c-IndG
o ⌦h H of Proposition 7.2 sends [1, v] K E(C̃(µ))
K ⇢)
to 1 ⌦ E(C̃(µ)). By Proposition 5.4, 1 ⌦ E(C̃(µ)) = 1 ⌦ E 0 (µ̃). Hence [1, v] K E(C̃(µ)) =
I (1) ! ⇡ I (1) corresponding to A
[1, v] K E o (µ̃). Applying the H-intertwiner (c-IndG
K ⇢)
sending [1, v] K to A(v), we deduce that A(v)E o (µ̃) = A(v)E(C̃(µ)).
If A is a Z R (G, K , ⇢)T -eigenvector in Hom R K (⇢, ⇡) of eigenvalue ⇠ (Corollary 7.9), we
have A µ̃ = ⇠( µ̃ )A for µ 2 3+
T (Theorem 7.7).
For J ⇢ 1, we denote by µ J an element of 3+
T such that ↵ ⌫(µ J ) > 0 for all ↵ 2 1
J.
Remark 7.11. Let ⇠ be an R-character of 3+
T . The character ⇠ is called supersingular if
it satisfies the following three equivalent properties.
+
(1) ⇠(µ) = 0 for all µ 2 3+
T non-invertible in 3T .
(2) ⇠(µ J ) = 0 for any J 6 = 1.
(3) For some n > 1, ⇠(µ) = 0 for all µ 2 3+
T with `(µ) > n.
In Proposition 7.10, the eigenvalue ⇠ of A is supersingular if and only if the module
A(v)H is supersingular (Definition 6.10).
Acknowledgements. I am grateful to N. Abe, G. Henniart, and R. Ollivier for helpful
discussions. I thank the organizer of the conferences in Mumbai (D. Prasad) and in
Berlin (E. Grosse-Kloenne) for their interest and for giving me the opportunity to present
this work, and the Institut de Mathématiques de Jussieu for its support. I thank the
anonymous referee for having read my article so carefully and for his/her comments.
References
1.
2.
N. Abe, G. Henniart, F. Herzig and M.-F. Vignéras, A classification of admissible
irreducible modulo p representations of reductive p-adic groups, preprint, 2014.
M. Cabanes, Irreducible modules and Levi supplements, J. Algebra 90 (1984), 84–97.
38
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
M.-F. Vigneras
M. Cabanes, A criterion of complete reducibility and some applications, in
Représentations linéeaires des groupes finis, (ed. M. Cabanes), Astérisque, pp. 181–182.
(1990) 93–112.
M. Cabanes and M. Enguehard, Representation theory of finite reductive groups
(Cambridge University Press, 2004).
E. Grosse-Klönne, From pro- p-Iwahori Hecke modules to ( , 0)-modules, preprint,
2013.
G. Henniart and M.-F. Vigneras, Comparison of compact induction with parabolic
induction, Pac. J. Math. 260(2) (2012), 457–495.
G. Henniart and M.-F. Vigneras, A Satake isomorphism for representations modulo
p of reductive groups over local fields, J. Reine Angew. Math. 701 (2015), 33–75.
S. Kumar, Kac–Moody groups, their flag varieties and representation theory, Progr. Math.
204 (2002).
R. Ollivier, Parabolic Induction and Hecke modules in characteristic p for p-adic G L n ,
ANT 4(6) (2010), 701–742.
R. Ollivier, An Inverse Satake isomorphism in characteristic p, Selecta Math., preprint,
2012.
R. Ollivier, Compatibility between Satake and Bernstein isomorphisms in characteristic
p, preprint, 2013.
N. A. Schmidt, Generische pro- p-algebren (Dilpomarbeit, Berlin, 2009).
M.-F. Vignéras, On a numerical Langlands correspondence modulo p with the
pro-p-Iwahori Hecke ring, Math. Ann. 331(3) 1432–1807. Erratum 333(3) (2005), 699–701.
M.-F. Vignéras, Algèbres de Hecke affines génériques, J. Represent. Theory 10 (2006),
1–20.
M.-F. Vignéras, The pro- p-Iwahori Hecke algebra of a reductive p-adic group I, preprint,
2013.
M.-F. Vignéras, The pro- p-Iwahori Hecke algebra of a reductive p-adic group II,
Muenster J. Math. 7 (2014), 363–379.