Modulo p parabolic induction of pro- p -Iwahori Hecke algebra
MMODULO p PARABOLIC INDUCTION OF PRO- p -IWAHORIHECKE ALGEBRA NORIYUKI ABE
Abstract.
We study the structure of parabolic inductions of a pro- p -IwahoriHecke algebra. In particular, we give a classification of irreducible modulo p representations of pro- p -Iwahori Hecke algebra in terms of supersingular rep-resentations. Since supersingular representations are classified by Ollivier andVign´eras, it completes the classification of irreducible modulo p representa-tions. Introduction
Let G be the group of F -valued points of a connected reductive group over a p -adic field F and K (cid:48) its open compact subgroup. Then K (cid:48) -biinvariant functionswith compact support on G forms an algebra via the convolution product. We callthis algebra the Hecke algebra. When we investigate the representation theory of G over a characteristic zero field, this algebra has an important role. One of themost important case is when K (cid:48) is an Iwahori subgroup. In this case, the categoryof representations of the Hecke algebra is equivalent to a block of the category ofsmooth representations of G .We are interested in the representations of G over a characteristic p field. In thissetting, it is natural to consider a pro- p -Iwahori subgroup since any non-zero repre-sentations of G over a characteristic p field has a non-zero vector fixed by a pro- p -Iwahori subgroup. The corresponding Hecke algebra is called a pro- p -Iwahori Heckealgebra and the structure of this algebra is studied by Vign´eras [Viga]. This struc-ture theorem is used in [AHHV14] in which we gave a classification of irreducibleadmissible modulo p representations of G in terms of supersingular representations.Another motivation to consider a pro- p -Iwahori Hecke algebra is to study Galoisrepresentations. It is conjectured by Vign´eras [Vig05] and proved by Ollivier [Oll10]that there is the “numerical modulo p Langlands correspondence” between modulo p Galois representations and modulo p representations of a pro- p -Iwahori Heckealgebra of GL n . It asserts that the number of n -dimensional modulo p Galoisrepresentations with a fixed determinant and the number of n -dimensional modulo p supersingular representations of a pro- p -Iwahori Hecke algebra of GL n with a fixedaction of the center of GL n coincide with each other. Recently, Große-Kl¨onne [GK]constructed a functor from the category of modulo p representations of a pro- p -Iwahori Hecke algebra of GL n to the category of modulo p Galois representationswhich gives a realization of the numerical modulo p Langlands correspondence.In this paper, we study the representations, especially parabolic inductions ofa pro- p -Iwahori Hecke algebra. Our main theorem is the classification of the ir-reducible modulo p representations of a pro- p -Iwahori Hecke algebra in terms ofsupersingular representations. The supersingular representations are classified byOllivier [Oll12] (split case) and Vign´eras [Vigb] (general).We state our main result. Let C be an algebraically closed field of character-istic p . Fix a pro- p -Iwahori subgroup of G and let H be the corresponding Hecke Mathematics Subject Classification. a r X i v : . [ m a t h . R T ] D ec NORIYUKI ABE algebra with coefficients in C . We consider H -modules. Here, modules always areright modules unless otherwise stated. Let P be a parabolic subgroup, M its Levisubgroup and H M a pro- p -Iwahori Hecke algebra of M . Then for a representation σ of H M , the parabolic induction I P ( σ ) is defined (Definition 4.8).The statement of the main theorem is almost the same as the group case [Abe13,AHHV14]. To state it, fix a minimal parabolic subgroup B and its Levi subgroup Z . Then we have a root system Φ and the set of simple roots Π. For α ∈ Π, let G (cid:48) α be the group generated by the root subgroups of G corresponding to ± α and put Z (cid:48) α = Z ∩ G (cid:48) α . Then for a standard Levi subgroup M ⊃ Z and a representation σ of H M we defineΠ( σ ) = { α ∈ Π | (cid:104) Π M , ˇ α (cid:105) = 0 , T Mλ is identity for any λ ∈ Z (cid:48) α } ∪ Π M , where T Mλ ∈ H M is the function supported on the double coset containing λ withrespect to the pro- p -Iwahori subgroup of M (Iwahori-Matsumoto basis) and Π M isthe set of simple roots of M . Let P ( σ ) be the corresponding parabolic subgroup.Now consider triples ( P, σ, Q ) such that • P is a parabolic subgroup with Levi subgroup M . • σ is a supersingular representation of M . • Q is a parabolic subgroup such that P ⊂ Q ⊂ P ( σ ).For such a pair, we define an H -module I ( P, σ, Q ) as follows. Let M ( σ ) ⊃ Z bethe Levi subgroup of P ( σ ). We can prove that there is the “extension” e ( σ ) of σ to H M ( σ ) (Proposition 4.16). Then for each Q (cid:48) ⊃ Q , we have an embedding I Q (cid:48) ( e ( σ )) (cid:44) → I Q ( e ( σ )). Now let I ( P, σ, Q ) = I Q ( e ( σ )) / (cid:88) Q (cid:48) (cid:41) Q I Q (cid:48) ( e ( σ )) . Then the main theorem of this paper is the following.
Theorem 1.1.
The corresponding ( P, σ, Q ) (cid:55)→ I ( P, σ, Q ) gives a bijection betweensuch triples and isomorphism classes of irreducible representations of H . The study is based on the structure theory of pro- p -Iwahori Hecke algebrasstudied by Vign´eras [Viga]. Fix a maximal split torus in Z and denote (the group of F -valued points of) it by S . Let I (1) be the pro- p -Iwahori subgroup. (It correspondsto a certain chamber in the apartment corresponding to S , see the next section forthe detail.) Then Z is the centralizer of S in G . Put (cid:102) W (1) = N G ( S ) / ( Z ∩ I (1)).Then we have the Bruhat decomposition (cid:102) W (1) = I (1) \ G/I (1). Hence there isa basis of H which is indexed by (cid:102) W (1). The basis is called Iwahori-Matsumotobasis . Similar to the affine Hecke algebra, this basis satisfies the braid relations andquadratic relations. Like the affine Hecke algebra, we have another basis { E ( (cid:101) w ) | (cid:101) w ∈ (cid:102) W (1) } , called Bernstein basis . Put Λ(1) = Z/ ( Z ∩ I (1)) ⊂ (cid:102) W (1). Then A = (cid:76) λ ∈ Λ(1) CE ( λ ) is a subalgebra. In general, A is not commutative. However,it is almost commutative and the study of representations of A is not so difficult.The basic tactics to study the representations is via the restriction to A . Henceit is important to know about irreducible representations of A . An importantinvariant of an irreducible representation X of A is its support supp X defined bysupp X = { λ ∈ Λ(1) | XE ( λ ) (cid:54) = 0 } . Then we can prove that it is the closure ofa facet (Proposition 3.6). More precisely, the following holds. Let X ∗ ( S ) be thegroup of cocharacters of S and set V = X ∗ ( S ) ⊗ R . Then the finite Weyl group W acts on V as a reflection group. Hence we have a notion of a facet in V . Wealso have a group homomorphism ν : Λ(1) → V . Let Θ be a subset of Π and putΛ Θ (1) = { λ ∈ Λ(1) | (cid:104) ν ( λ ) , α (cid:105) = 0 ( α ∈ Θ) } . For w ∈ W , fix a representative ODULO p PARABOLIC INDUCTION OF PRO- p -IWAHORI HECKE ALGEBRA 3 n w ∈ (cid:102) W (1). Let C [Λ Θ (1)] = (cid:76) τ ∈ Λ Θ (1) Cτ λ be the group algebra of Λ Θ (1). Define wχ Θ : A → C [Λ Θ (1)] by wχ Θ ( E ( λ )) = (cid:40) τ n − w λn w ( w − ( ν ( λ )) is dominant and n − w λn w ∈ Λ Θ (1)) , . Then we can prove that any irreducible representation factors through wχ Θ for some w and Θ. For the proof of our main theorem, we study the modules wχ Θ ⊗ A H .We study these modules in Section 3 by constructing intertwining operators.Let s ∈ W be a simple reflection and assume that sw > w . Then we constructa homomorphism wχ Θ ⊗ A H → swχ Θ ⊗ A H (Proposition 3.2) which is alwaysinjective (Proposition 3.12). Moreover, often it is an isomorphism. In fact, if we putΠ w = { α ∈ Π | w ( α ) > } , then Π w = Π sw implies that wχ Θ ⊗ A H → swχ Θ ⊗ A H is an isomorphism (Theorem 3.13). Moreover, we have w χ Θ ⊗ A H (cid:39) w χ Θ ⊗ A H if Π w = Π w . If Π w (cid:54) = Π sw , wχ Θ ⊗ A H → swχ Θ ⊗ A H is not an isomorphism.We can construct an intertwining operator swχ Θ ⊗ A H → wχ Θ ⊗ A H in oppositedirection and calculate the compositions.In Section 4, we give a definition and fundamental properties of parabolic in-ductions. One of the most important properties to prove our main theorem is thestructure of a parabolic induction as an A -module (Lemma 4.23). For example, if P is a minimal parabolic subgroup and P ( σ ) = P , then we can prove I P ( σ ) | A = (cid:76) w ∈ W wX for some irreducible representation X of A . In this case, using resultsin Section 3, we can prove that for any w, w (cid:48) ∈ W , we have wX ⊗ A H (cid:39) w (cid:48) X ⊗ A H .Hence if π is a submodule of I P ( σ ) and π | A contains wX for some w ∈ W , then π | A contains wX for any w ∈ W . Therefore we get π = I P ( σ ), so I P ( σ ) = I ( P, σ, P ) isirreducible. This is a part of our main theorem. Using such arguments, we proveour main theorem in Section 4.We have nothing about the relations between the representations of the groupin this paper. We hope to address this question in a future work.
Acknowledgment.
I had many discussion with Marie-France Vign´eras on thestructure of pro- p -Iwahori Hecke algebras. I thank her for reading the manuscriptand giving helpful comments. This work was supported by JSPS KAKENHI GrantNumber 26707001. 2. Notation and Preliminaries
Notation.
We will use the slightly different notation from the introduction.Our main reference is [Viga].Let F be a non-archimedean local field, O its ring of integers, κ its residue fieldand p the characteristic of κ . Let G be a connected reductive group over F . Asusual, the group of its valued points is denoted by G ( F ). Fix a maximal split torus S of G and a minimal parabolic subgroup B which contains S . Then the centralizer Z of S in G is a Levi subgroup of B . Let U be the unipotent radical of B , B = ZU the opposite parabolic subgroup of B . Take a special point from the apartmentattached to S and let K be the special parahoric subgroup corresponding to thispoint. Then there is a connected reductive group G κ over κ and the surjectivehomomorphism K → G κ ( κ ). The kernel of K → G κ ( κ ) is a pro- p group. Theimage of Z ( F ) ∩ K (resp. U ( F ) ∩ K , U ( F ) ∩ K ) is the κ -valued points of analgebraic subgroup Z κ (resp. U κ , U κ ) of G κ . Put Z κ = Z κ ( κ ). Let I (1) be theinverse image of U κ ( κ ) by K → G κ ( κ ). Then this is a pro- p -Iwahori subgroup of G ( F ). The Hecke algebra attached to ( G ( F ) , I (1)) is called a pro- p -Iwahori Heckealgebra . We study the modules of this algebra over an algebraically closed filed ofcharacteristic p . However, we will not use this algebra directly due to a technical NORIYUKI ABE reason. Instead of this algebra, we use another algebra defined by generators andrelations which will be introduced later. Under a suitable specialization, the algebrabecomes the pro- p -Iwahori Hecke algebra.The torus S gives a root datum ( X ∗ ( S ) , Φ , X ∗ ( S ) , ˇΦ) and B gives a positivesystem Φ + ⊂ Φ. Let W = N G ( F ) ( S ( F )) / Z ( F ) be the finite Weyl group and set (cid:102) W = N G ( F ) ( S ( F )) / ( Z ( F ) ∩ K ) where N G ( F ) ( S ( F )) is the normalizer of S ( F )in G ( F ). Then we have the surjective homomorphism (cid:102) W → W and its kernel Z ( F ) / ( Z ( F ) ∩ K ) is denoted by Λ. Set V = X ∗ ( S ) ⊗ Z R . Then V is identified withthe apartment corresponding to S and (cid:102) W acts on V as an affine transformations. Inparticular, Z ( F ) acts on V as a translation. Hence we have a group homomorphism ν : Z ( F ) → V . It is characterized by α ( ν ( z )) = − val( α ( z )) for α ∈ X ∗ ( S ) and z ∈ S ( F ). Since ν annihilates a compact subgroup ( V has no non-trivial compactsubgroup), ν factors thorugh Z ( F ) → Λ. The induced homomorphism Λ → V isdenoted by the same letter ν . Since we have a positive system Φ + , we have theset of simple reflections S aff ⊂ (cid:102) W . Let (cid:102) W aff be the group generated by S aff . Thenthere exists a reduced root datum Σ ⊂ V ∗ = Hom R ( V, R ) such that (cid:102) W aff is thegroup generated by the reflections with respect to { v ∈ V | α ( v ) + k = 0 } where α ∈ Σ and k ∈ Z . Since we fixed a positive system Φ + ⊂ Φ, this determines apositive system Σ + ⊂ Σ. Let ∆ ⊂ Σ + be the set of simple roots. There is a naturalbijection between ∆ and the set of simple roots in Φ + . (The set of simple roots inΦ + is denoted by Π in Section 1. We will never use Π.) More precisely, for α ∈ ∆,there exists a real number r > rα is a simple root for Φ + . Define thepositive system Σ +aff of Σ aff by Σ +aff = Σ + × Z ≥ ∪ Σ × Z > . For α ∈ Σ (resp. (cid:101) α ∈ Σ aff ), let s α ∈ W (resp. s (cid:101) α ∈ (cid:102) W ) be the corresponding reflection. An elementin v ∈ V is called dominant (resp. anti-dominant) if for any α ∈ Σ + , (cid:104) v, α (cid:105) ≥ (cid:104) v, α (cid:105) ≤ v regular if (cid:104) v, α (cid:105) (cid:54) = 0 for any α ∈ Σ.Let α ∈ Σ. Then there exists a unique element α (cid:48) ∈ Φ red ∩ R > α . The root α (cid:48) gives a unipotent subgroup of G which is denoted by U α . Let G (cid:48) α be the groupgenerated by U α ( F ) and U − α ( F ). We also denote this group by G (cid:48) s where s = s α .Notice that it is not the group of F -valued points of an algebraic subgroup ingeneral. Set Z (cid:48) α = G (cid:48) α ∩ Z ( F ) and the image of Z (cid:48) α in Λ is denoted by Λ (cid:48) α . It isalso denoted by Λ (cid:48) s where s = s α . Let ( α, k ) ∈ Σ aff and r ∈ R > such that α (cid:48) = rα .Put U ( α,k ) = U α (cid:48) + rk where the notation is in [Viga, 3.5].As usual, the length function of (cid:102) W is denoted by (cid:96) . It is defined by (cid:96) ( (cid:101) w ) = +aff ∩ (cid:101) w Σ − aff ). We have the following length formula [IM65, Proposition 1.23] for λ ∈ Λ and w ∈ W :(2.1) (cid:96) ( λw ) = (cid:88) α ∈ Σ + ,w − ( α ) > |(cid:104) ν ( λ ) , α (cid:105)| + (cid:88) α ∈ Σ + ,w − ( α ) < |(cid:104) ν ( λ ) , α (cid:105) − | . Notice that we have an embedding
W (cid:44) → (cid:102) W since we fixed a special point and (cid:102) W = W (cid:110) Λ. Put S = S aff ∩ W . This is the set of simple reflections in W and( W, S ) is a Coxeter system.Put (cid:102) W (1) = N G ( F ) ( S ( F )) / ( Z ( F ) ∩ I (1)). Then we have a bijection (cid:102) W (1) (cid:39) I (1) \ G ( F ) /I (1). Hence the pro- p -Iwahori Hecke algebra H ( G ( F ) , I (1)) has a basisparameterized by (cid:102) W (1). There is a surjective homomorphism (cid:102) W (1) → (cid:102) W . Thekernel is isomorphic to Z κ . We will regard Z κ as a subgroup of (cid:102) W (1). The kernelof (cid:102) W (1) → W is Z ( F ) / ( Z ( F ) ∩ I (1)) and it is denoted by Λ(1). The compositionΛ(1) → Λ ν −→ V is also denoted by ν . We call λ ∈ Λ(1) dominant (resp. anti-dominant, regular) if ν ( λ ) is dominant (resp. anti-dominant, regular).For s ∈ S aff , let q s be the indeterminates such that if s and s (cid:48) are conjugateto each other, then q s = q s (cid:48) . Let Z [ q s | s ∈ S aff / ∼ ] be the polynomial algebra ODULO p PARABOLIC INDUCTION OF PRO- p -IWAHORI HECKE ALGEBRA 5 with these indeterminate here ∼ means the equivalence relation defined via theconjugation by (cid:102) W . It will be denoted by Z [ q s ] for short. Via a reduced expression,we extend s (cid:55)→ q s to (cid:102) W (cid:51) (cid:101) w → q (cid:101) w ∈ (cid:81) s ∈ S aff / ∼ q Z ≥ s . By (cid:102) W (1) → (cid:102) W we have q (cid:101) w for any (cid:101) w ∈ (cid:102) W (1). Since Z κ ⊂ (cid:102) W (1) is a normal subgroup, the adjoint action of (cid:101) w ∈ (cid:102) W (1) gives a homomorphism Z [ q / s ][ Z κ ] → Z [ q / s ][ Z κ ]. Denote this action by c (cid:55)→ (cid:101) w · c . For each (cid:101) s which is a lift of s ∈ S aff , fix c (cid:101) s ∈ Z [ Z κ ] such that if (cid:101) w (cid:101) s (cid:101) w − = (cid:101) s (cid:48) for (cid:101) w ∈ (cid:102) W , then (cid:101) w · c (cid:101) s = c (cid:101) s (cid:48) and c t (cid:101) s = tc (cid:101) s for t ∈ Z κ . Then Vign´eras [Viga, 4.3]defined an algebra over Z [ q s ]. Under the specialization q s (cid:55)→ I (1) sI (1) /I (1))and a suitable choice of c (cid:101) s (which will be explained later), this algebra becomes H ( G ( F ) , I (1)). Let H Z be the scalar extension of this algebra to Z [ q / s ]. This isthe main object of this paper. Later, we will put H = H Z ⊗ Z C for a field C ofcharacteristic p . This is a C [ q / s ]-algebra.As a Z [ q / s ]-module, H Z is free and has a basis indexed by (cid:102) W (1). The basis isdenoted by { T (cid:101) w } (cid:101) w ∈ (cid:102) W (1) , namely we have H Z = (cid:76) (cid:101) w ∈ (cid:102) W (1) Z [ q / s ] T (cid:101) w . The multipli-cations in H Z is described as follows. Define (cid:96) : (cid:102) W (1) → Z ≥ by the composition (cid:102) W (1) → (cid:102) W (cid:96) −→ Z ≥ . The multiplication in H is defined as follows [Viga, 4.3]. • the braid relations: T (cid:101) w T (cid:101) w = T (cid:101) w (cid:101) w if (cid:96) ( (cid:101) w ) + (cid:96) ( (cid:101) w ) = (cid:96) ( (cid:101) w (cid:101) w ). • the quadratic relations: T (cid:101) s = q s T (cid:101) s + c (cid:101) s T (cid:101) s for (cid:101) s ∈ (cid:102) W (1) which is a lift of s ∈ S aff .In particular, Z κ (cid:51) t (cid:55)→ T t ∈ H Z gives an embedding Z [ q / s ][ Z κ ] (cid:44) → H Z .It is convenient to fix a representative n s of s ∈ S aff in (cid:102) W (1) as follows. Considerthe apartment attached to S . Then the elements in the apartment fixed by s is ahyperplane. Take a facet F contained in this hyperplane whose codimension in V isone and consider the parahoric subgroup K F attached to this facet. Then we take n s from the group generated by K F ∩ U ( F ) and K F ∩ U ( F ). Moreover, we can(and do) take n s such that { n s } s ∈ S satisfies the braid relations [Tit66]. (See also[Viga, Proposition 3.4].) Hence we define n w for w ∈ W via a reduced expressionof w .Let G (cid:48) be the group generated by (cid:83) α ∈ Σ G (cid:48) α . It is also the group generated by U ( F ) and U ( F ). Then (cid:102) W aff is the image of N G ( F ) ( S ( F )) ∩ G (cid:48) . Let (cid:102) W aff (1) bethe image of N G ( F ) ( S ( F )) ∩ G (cid:48) in (cid:102) W (1). (This group is different from the groupdenoted by W aff (1) in [Viga].) This is a normal subgroup of (cid:102) W (1) since G (cid:48) isnormal in G ( F ). Put Λ (cid:48) (1) = (cid:102) W aff (1) ∩ Λ(1). This is the image of G (cid:48) ∩ Z ( F )and it is again normal in (cid:102) W (1) since it is an intersection of normal subgroups. Weintroduce the Bruhat order on (cid:102) W (1) as follows. Let (cid:101) w , (cid:101) w ∈ (cid:102) W (1) and denote itsimage in (cid:102) W by w , w . Then (cid:101) w < (cid:101) w if and only if w < w and (cid:101) w ∈ (cid:101) w (cid:102) W aff (1).Here the order in (cid:102) W is the usual Bruhat order. Assume that for (cid:101) s ∈ (cid:102) W aff (1) whichis a lift of a simple reflection, we have c (cid:101) s ∈ C [ Z κ ∩ (cid:102) W aff (1)]. (This assumption issatisfied if c (cid:101) s comes from the group, see subsection 2.3.) Then we have T (cid:101) w T (cid:101) w ∈ (cid:80) (cid:101) v ≤ (cid:101) w , (cid:101) v ≤ (cid:101) w , (cid:101) v ≤ (cid:101) v (cid:101) v Z [ q / s ] T (cid:101) v . Remark . In [Viga, 5.3], the order is defined as follows: let (cid:101) w , (cid:101) w ∈ (cid:102) W (1) withthe image w , w in (cid:102) W respectively, then (cid:101) w < (cid:101) w if and only if w < w . If (cid:101) w < (cid:101) w in the sense of this paper, then (cid:101) w < (cid:101) w in the sense of Vign´eras. Howeverthe converse is not true.2.2. Properties of (cid:102) W aff (1) . The subgroup (cid:102) W aff (1) has the similar properties to (cid:102) W aff . The first property is the Bruhat decomposition. NORIYUKI ABE
Lemma 2.2.
We have G (cid:48) ( Z ( F ) ∩ I (1)) = (cid:96) (cid:101) w ∈ (cid:102) W aff (1) I (1) (cid:101) wI (1) .Proof. Put N = N G ( F ) ( S ( F )). We have G ( F ) = (cid:96) (cid:101) w ∈ (cid:102) W (1) I (1) (cid:101) wI (1) [Viga, Propo-sition 3.35]. Let g ∈ G (cid:48) ( Z ( F ) ∩ I (1)) and take n ∈ N such that g ∈ I (1) nI (1).We have the decomposition I (1) = ( U ( F ) ∩ I (1))( Z ( F ) ∩ I (1))( U ( F ) ∩ I (1)).Since U ( F ) , U ( F ) ⊂ G (cid:48) , I (1) is contained in G (cid:48) ( Z ( F ) ∩ I (1)). Hence we have n ∈ G (cid:48) ( Z ( F ) ∩ I (1)) ∩ N . Since Z ( F ) ∩ I (1) ⊂ N , we have G (cid:48) ( Z ( F ) ∩ I (1)) ∩ N =( G (cid:48) ∩ N )( Z ( F ) ∩ I (1)). Therefore the image of n in (cid:102) W (1) is in (cid:102) W aff (1). (cid:3) Lemma 2.3 ([AHHV14, III.16. Proposition]) . For s = s α ∈ S , we have an exactsequence → Λ (cid:48) s (1) ∩ Z κ → Λ (cid:48) s (1) ν −→ Z ˇ α → . We have the generators of (cid:102) W aff (1). Lemma 2.4.
The subgroup (cid:102) W aff (1) of (cid:102) W (1) is generated by (cid:83) s ∈ S Λ (cid:48) s (1) and { n s | s ∈ S } .Proof. Let (cid:102) W (cid:48) s (1) be the subgroup of (cid:102) W (1) generated by Λ (cid:48) s (1) and n s . Then wehave G (cid:48) s ⊂ I (1) (cid:102) W (cid:48) s (1) I (1) by the Bruhat decomposition of G (cid:48) s . Let (cid:102) W (cid:48) (1) be thegroup generated by (cid:83) s ∈ S Λ (cid:48) s (1) and { n s | s ∈ S } . Since G (cid:48) is generated by (cid:83) s ∈ S G (cid:48) s ,we have G (cid:48) ⊂ I (1) (cid:102) W (cid:48) (1) I (1). Comparing the Bruhat decomposition of G (cid:48) , we get (cid:102) W aff (1) ⊂ (cid:102) W (cid:48) (1). (cid:3) Using this lemma, we get the following property on Λ (cid:48) (1).
Lemma 2.5.
The subgroup Λ (cid:48) (1) of Λ(1) is generated by (cid:83) s ∈ S Λ (cid:48) s (1) and Λ (cid:48) (1) ∩ Z κ is generated by (cid:83) s ∈ S (Λ (cid:48) s (1) ∩ Z κ ) .Proof. Let Λ (cid:48) (1) be the subgroup generated by (cid:83) s ∈ S Λ (cid:48) s (1). We prove that thissubgroup is stable under the action of n w for w ∈ W . It is sufficient to provethat n s λn − s ∈ Λ (cid:48) s (1) λ for λ ∈ Λ(1) and s ∈ S . Let z ∈ Z ( F ) be a representativeof λ . We have n s zn − s z − = ( n s zn − s ) z − ∈ Z ( F ). Since G (cid:48) s is normalized by Z ( F ), we have n s zn − s z − = n s ( zn − s z − ) ∈ G (cid:48) s . Hence n s zn − s z − ∈ G (cid:48) s ∩ Z ( F ).Therefore we have n s λn − s λ − ∈ Λ (cid:48) s (1). Hence by the above lemma, any element in (cid:102) W aff (1) has the form λn s · · · n s l where λ ∈ Λ (cid:48) (1) and s i ∈ S . Take the minimal i such that s · · · s i < s · · · s i − . Put w = s · · · s i − . Then n w = n ws i n s i andwe have λn s · · · n s l = λn ws i n s i n s i +1 · · · n s l = λ ( n ws i n s i n − ws i ) n ws i n s i +1 · · · n s l ∈ Λ (cid:48) (1) n ws i n s i +1 · · · n s l . Therefore, we can take λ and s , . . . , s l such that s · · · s l is a reduced expression. Hence λn s · · · n s l = λn s ··· s l . It is in Λ(1) if and only if s · · · s l = 1. Therefore, Λ (cid:48) (1) = (cid:102) W aff (1) ∩ Λ(1) is Λ (cid:48) (1) .Let s , . . . , s n ∈ S such that S = { s , . . . , s n } where n = S . Since Λ (cid:48) s i (1) isnormal in Λ(1), we have Λ (cid:48) (1) = Λ (cid:48) s (1) · · · Λ (cid:48) s n (1). Let λ ∈ Λ (cid:48) (1) ∩ Z κ and take λ i ∈ Λ (cid:48) s i (1) such that λ = λ · · · λ n . Then 0 = ν ( λ ) = (cid:80) i ν ( λ i ). Take α i ∈ ∆ suchthat s i = s α i . Then ν ( λ i ) ∈ Z ˇ α i . Since { ˇ α , . . . , ˇ α n } is linearly independent, wehave ν ( λ i ) = 0. Hence λ i ∈ Λ (cid:48) s i (1) ∩ Z κ by Lemma 2.3. (cid:3) Bernstein basis.
For investigating the representations of pro- p -Iwahori Heckealgebra, another basis, called Bernstein basis is important.Let ∆ (cid:48) ⊂ Σ be a set of simple roots corresponding to a positive system of Σ.Vign´eras [Viga] defined a Z [ q / s ]-basis { E ∆ (cid:48) ( (cid:101) w ) } (cid:101) w ∈ (cid:102) W (1) of H Z . If c (cid:101) s ∈ C [ Z κ ∩ (cid:102) W aff (1)] for any (cid:101) s ∈ (cid:102) W aff (1) which is a lift of a simple affine reflection, then E ∆ (cid:48) ( (cid:101) w ) ∈ T (cid:101) w + (cid:80) (cid:101) v< (cid:101) w Z [ q s ] T (cid:101) v . (The same proof for [Viga, Corollary 5.6] is ap-plicable.) ODULO p PARABOLIC INDUCTION OF PRO- p -IWAHORI HECKE ALGEBRA 7 Remark . In fact, Vign´eras defined a basis { E o ( (cid:101) w ) | (cid:101) w ∈ (cid:102) W (1) } for any orienta-tion o . we can attache a spherical orientation o ∆ (cid:48) [Viga, Definition 5.16] and onecan consider a basis { E o ∆ (cid:48) ( (cid:101) w ) } . We put E ∆ (cid:48) ( (cid:101) w ) = E o ∆ (cid:48) ( (cid:101) w ).Here are some examples of E ∆ (cid:48) ( (cid:101) w ). (See [Viga, 5.3].) For (cid:101) w ∈ (cid:102) W (1), T (cid:101) w − isinvertible in H [ q ± / s ] and q (cid:101) w T − (cid:101) w − ∈ H . We denote this element by T ∗ (cid:101) w . Proposition 2.7.
Let ∆ (cid:48) be a set of simple roots. (1) If λ ∈ Λ(1) is dominant with respect to ∆ (cid:48) , then E ∆ (cid:48) ( λ ) = T λ . (2) If λ ∈ Λ(1) is anti-dominant with respect to ∆ (cid:48) , then E ∆ (cid:48) ( λ ) = T ∗ λ . (3) Let w ∈ W . We have E − ∆ ( n w ) = T n w and E ∆ ( n w ) = E w ( − ∆) ( n w ) = T ∗ n w .In particular, for s ∈ S , E ∆ ( n s ) = E s ( − ∆) ( n s ) = T ∗ n s = T n s − c n s . For (cid:101) w ∈ (cid:102) W (1), put (cid:101) w (∆ (cid:48) ) = w (∆ (cid:48) ) for the set of simple roots ∆ (cid:48) attached to apositive system where w ∈ W is the image of (cid:101) w in W . Proposition 2.8 ([Viga, Theorem 5.25]) . Let (cid:101) w, (cid:101) w (cid:48) ∈ (cid:102) W (1) . Then we have E ∆ (cid:48) ( (cid:101) w (cid:101) w (cid:48) ) = q − / (cid:101) w q − / (cid:101) w (cid:48) q / (cid:101) w (cid:101) w (cid:48) E ∆ (cid:48) ( (cid:101) w ) E (cid:101) w − (∆ (cid:48) ) ( (cid:101) w (cid:48) ) . In particular, if λ , λ ∈ Λ(1), then E ∆ (cid:48) ( λ ) E ∆ (cid:48) ( λ ) = q / λ q / λ q − / λ λ E ∆ (cid:48) ( λ λ ).By (2.1), we have (cid:96) ( λ λ − λ − λ ) = 0. Therefore we have q λ λ − λ − λ = 1 and q λ λ λ − q λ λ − λ − λ = q λ . Namely q λ λ λ − = q λ . Hence we get E ∆ (cid:48) ( λ ) E ∆ (cid:48) ( λ ) = q / λ q / λ λ λ − q − / λ λ E ∆ (cid:48) (( λ λ λ − ) λ )= E ∆ (cid:48) ( λ λ λ − ) E ∆ (cid:48) ( λ ) . (2.2)In this paper, we use E − ∆ ( (cid:101) w ) mainly. Set E ( (cid:101) w ) = E − ∆ ( (cid:101) w ). Using the aboveproperties, we have the following description of E ( (cid:101) w ). (One can regard this de-scription as a definition of E ( (cid:101) w ).) Take anti-dominant λ , λ ∈ Λ(1) and w ∈ W such that (cid:101) w = λ λ − n w . Then we have E ( λ λ − n w ) = q / λ λ − n w q − / λ q / λ q − / n w T λ T − λ T n w Put θ ( λ ) = q − / λ E ( λ ) ∈ H Z [ q ± / s ] for λ ∈ Λ(1). Then for anti-dominant λ ∈ Λ(1),we have θ ( λ ) = q − / λ T λ and θ ( λ λ − ) = θ ( λ ) θ ( λ ) − for any λ , λ ∈ Λ(1). Wehave E ( λn w ) = q / λn w q − / n w θ ( λ ) T n w . The element θ ( λ ) is denoted by (cid:101) E o − ∆ ( λ ) in [Viga, Lemma 5.44].2.4. Bernstein relations.
Let (cid:101) α ∈ Σ aff be an affine root. Consider the apart-ment attached to S and take a facet F of codimension one in V contained in thehyperplane on which (cid:101) α is null. Let K F be the parahoric subgroup attached to F .Then it defines a connected reductive subgroup over κ . Let G F ,κ be the group of κ -valued points of this reductive group, U F ,κ (resp. U F ,κ ) the image of K F ∩ U ( F )(resp. K F ∩ U ( F )). Define Z (cid:101) α,κ as the intersection of Z κ with the group generatedby U F ,κ and U F ,κ . Lemma 2.9.
The subgroup Z (cid:101) α,κ ⊂ Z κ does not depend on a choice of F .Proof. Let α ∈ Σ be a root corresponding to (cid:101) α , U (cid:101) α,κ (resp. U (cid:101) α,κ ) the image of U α ( F ) ∩ K F (resp. U − α ( F ) ∩ K F ) in G F ,κ . The root system of G F ,κ is {± (cid:101) α } .Hence Z (cid:101) α,κ is the intersection of Z κ with the group generated by U (cid:101) α,κ and U (cid:101) α,κ .Set r = min { n ∈ Z | α ( x ) + n ≥ x ∈ F} . We have U α ( F ) ∩ K F = U ( α,r ) [Viga, (45)]. Since α ( x ) only depends on (cid:101) α , r does not depend on a choice of F .Hence K F ∩ U α ( F ) does not depend on F . (cid:3) NORIYUKI ABE
For s ∈ S aff , take a simple affine root (cid:101) α such that s = s (cid:101) α . We take c n s comingfrom the Hecke algebra attached to ( G, I (1)) [Viga, 4.2]. We have c n s ∈ Z ≥ [ Z (cid:101) α,κ ]and c n s ≡ − ( Z (cid:101) α,κ ) − (cid:88) t ∈ Z (cid:101) α,κ T t (mod p ) . In particular, c n s (mod p ) does not depend on a choice of n s . It satisfies n s · c n s = c n s .Put n w ( λ ) = n w λn − w for w ∈ W . The following lemma is a reformulation ofthe Bernstein relations of Vign´eras [Viga, Theorem 5.38]. We refer this lemma asBernstein relations in this paper. Lemma 2.10.
Let λ ∈ Λ(1) , s = s α ∈ S . There exist µ n s ( k ) ∈ Λ (cid:48) s (1) and c n s ,k ∈ Z [ Z κ ] such that, if (cid:104) ν ( λ ) , α (cid:105) ≥ , then θ ( n s ( λ )) T n s − T n s θ ( λ ) = (cid:104) ν ( λ ) ,α (cid:105)− (cid:88) k =0 θ ( n s ( λ ) µ n s ( k )) c n s ,k = (cid:104) ν ( λ ) ,α (cid:105) (cid:88) k =1 c n s ,k θ ( µ n s ( − k ) λ ) . If (cid:104) ν ( λ ) , α (cid:105) < , then θ ( n s ( λ )) T n s − T n s θ ( λ ) = − −(cid:104) ν ( λ ) ,α (cid:105)− (cid:88) k =0 c n s , − k θ ( µ n s ( k ) λ )= − −(cid:104) ν ( λ ) ,α (cid:105) (cid:88) k =1 θ ( n s ( λ ) µ n s ( − k )) c n s , − k . These elements satisfy the following. We have c n s ,k ∈ Z [ Z ( α,k ) ,κ ] and c n s ,k ≡ − ( Z ( α,k ) ,κ ) − (cid:88) t ∈ Z ( α,k ) ,κ T t (mod p ) . We have µ n s (0) = 1 , ν ( µ n s ( k )) = k ˇ α for k ∈ Z , n − s ( µ n s ( k )) = µ n s ( − k ) for k ∈ Z ≥ , c n s , = c n s and µ n s ( − − · c n s , = c n s , − .Proof. First assume that (cid:104) ν (Λ(1)) , α (cid:105) = Z . Let λ s ∈ Λ(1) such that (cid:104) ν ( λ s ) , α (cid:105) = −
1. Then by [Viga, Theorem 5.38], if n = (cid:104) ν ( λ ) , α (cid:105) >
0, we have θ ( n s ( λ )) T n s − T n s θ ( λ ) = n − (cid:88) k =0 θ ( n s ( λ ) n s ( λ s ) k λ − ks )( λ ks · c n s ) . Take λ = λ − ns in this formula. Then(2.3) θ ( n s ( λ s ) − n ) T n s − T n s θ ( λ − ns ) = n − (cid:88) k =0 θ ( n s ( λ s ) − n + k λ − ks )( λ ks · c n s )We have θ ( n s ( λ s ) − n + k λ − ks )( λ ks · c n s ) = ( n s ( λ s ) − n + k · c n s ) θ ( n s ( λ s ) − n + k λ − ks ). Hencereplacing k with n − k , we have(2.4) θ ( n s ( λ s ) − n ) T n s − T n s θ ( λ − ns ) = n (cid:88) k =1 ( n s ( λ s ) − k · c n s ) θ ( n s ( λ s ) − k λ − n + ks ) . Since (cid:104) ν ( λ ns λ ) , α (cid:105) = 0, we have T n s θ ( λ ns λ ) = θ ( n s ( λ ns λ )) T n s [Viga, Lemma 5.34,5.35]. Hence multiplying by θ ( λ ns λ ) on the right, we get(2.5) θ ( n s ( λ )) T n s − T n s θ ( λ ) = n (cid:88) k =1 ( n s ( λ s ) − k · c n s ) θ ( n s ( λ s ) − k λ ks λ ) . ODULO p PARABOLIC INDUCTION OF PRO- p -IWAHORI HECKE ALGEBRA 9 Multiply by θ ( n s ( λ s ) n ) on the left and by θ ( λ ns ) on the right to (2.3) and (2.4).From (2.3), we have T n s θ ( λ ns ) − θ ( n s ( λ s ) n ) T n s = n − (cid:88) k =0 θ ( n s ( λ s ) k λ − ks )( λ ks · c n s ) θ ( λ ns )= n − (cid:88) k =0 ( n s ( λ s ) k · c n s ) θ ( n s ( λ s ) k λ n − ks ) . Multiplying by θ ( λ − ns λ ) on the right, if − n = (cid:104) ν ( λ ) , α (cid:105) <
0, then T n s θ ( λ ) − θ ( n s ( λ )) T n s = n − (cid:88) k =0 ( n s ( λ s ) k · c n s ) θ ( n s ( λ s ) k λ − ks λ )Similarly, from (2.4), we have T n s θ ( λ ns ) − θ ( n s ( λ s ) n ) T n s = n (cid:88) k =1 θ ( n s ( λ s ) n − k λ ks )( λ − ks · c n s ) . and, if − n = (cid:104) ν ( λ ) , α (cid:105) <
0, then(2.6) T n s θ ( λ ) − θ ( n s ( λ )) T n s = n (cid:88) k =1 θ ( n s ( λ ) n s ( λ s ) − k λ ks )( λ − ks · c n s ) . In (2.5) and (2.6), we may replace λ s with n − s ( λ s ) − since (cid:104) ν ( n − s ( λ s ) − ) , α (cid:105) = − n = (cid:104) ν ( λ ) , α (cid:105) >
0, then θ ( n s ( λ )) T n s − T n s θ ( λ ) = n (cid:88) k =1 ( λ ks · c n s ) θ ( λ ks n − s ( λ s ) − k λ )and, if − n = (cid:104) ν ( λ ) , α (cid:105) <
0, then T n s θ ( λ ) − θ ( n s ( λ )) T n s = n (cid:88) k =1 θ ( n s ( λ ) λ ks n − s ( λ s ) − k )( n − s ( λ s ) k · c n s ) . Now, for k ∈ Z ≥ , put µ n s ( k ) = n s ( λ s ) k λ − ks , µ n s ( − k ) = λ ks n − s ( λ s ) − k ,c n s ,k = λ ks · c n s , c n s , − k = n s ( λ s ) k · c n s . Then if k ∈ Z ≥ , c n s ,k ∈ Z [ λ ks · Z ( α, ,κ ] = Z [ Z λ ks ( α, ,κ ] = Z [ Z ( α,k ) ,κ ]. Similarly, wehave c n s , − k ∈ Z [ Z ( α, − k ) ,κ ]. Since n s ∈ Z κ and Z κ is commutative, n − s ( λ s ) k · c n s = n − s · ( n s ( λ s ) k · ( n s · c n s )) = n s ( λ s ) k · c n s = c n s , − k . Therefore these elements satisfythe Bernstein relations. To check the other conditions is easy.Next assume that (cid:104) ν (Λ(1)) , α (cid:105) = 2 Z . Let s (cid:48) be the reflection correspondingto the highest root in the component of Σ containing s , α (cid:48) the highest root and (cid:101) α (cid:48) = ( α (cid:48) , ∈ Σ aff . Take (cid:101) w ∈ (cid:102) W (1) such that λ s = n s (cid:101) wn s (cid:48) (cid:101) w − ∈ Λ(1) satisfies (cid:96) ( λ s ) = 2 (cid:96) ( (cid:101) w ) + 2 and ν ( λ s ) = − ˇ α [Viga, Lemma 5.15]. Put c (cid:48) = (cid:101) w · c n s (cid:48) . By thecondition, in (cid:102) W , we have s (cid:101) w ( (cid:101) α (cid:48) ) = sλ s = s ( α, − Hence we have (cid:101) w ( (cid:101) α (cid:48) ) = ( α, − n = (cid:104) ν ( λ ) , α (cid:105) ≥
0, then θ ( n s ( λ )) T n s − T n s θ ( λ ) is equal to n − (cid:88) k =0 θ ( n s ( λ ))( θ ( n s ( λ s ) k n s λ − k − s )( λ k +1 s · c (cid:48) ) + θ ( n s ( λ s ) k λ − ks )( λ ks · c n s ))and(2.7) n (cid:88) k =1 (( n s ( λ s ) − k · c (cid:48) ) θ ( n s ( λ s ) − k n s λ k − s ) + ( n s ( λ s ) − k · c n s ) θ ( n s ( λ s ) − k λ ks )) θ ( λ ) . If − n = (cid:104) ν ( λ ) , α (cid:105) < T n s θ ( λ ) − θ ( n s ( λ )) T n s is equal to n − (cid:88) k =0 (( n s ( λ s ) k · c (cid:48) ) θ ( n s ( λ s ) k n s λ − k − s ) + ( n s ( λ s ) k · c n s ) θ ( n s ( λ s ) k λ − ks )) θ ( λ )and(2.8) n (cid:88) k =1 θ ( n s ( λ ))( θ ( n s ( λ s ) − k n s λ k − s )( λ − k +1 s · c (cid:48) ) + θ ( n s ( λ s ) − k λ ks )( λ − ks · c n s )) . Replace n s with n − s and n s (cid:48) with n − s (cid:48) in (2.7). Then λ s becomes n − s ( λ s ) − , c n s becomes c n − s = n − s c n s and c (cid:48) becomes (cid:101) w · c n − s (cid:48) = c (cid:48) (cid:101) w · ( n − s (cid:48) ) = c (cid:48) ( λ − s n s λ − s n s ).Hence it gives θ ( n − s ( λ )) T n − s − T n − s θ ( λ )= n (cid:88) k =1 (( n − s ( λ s ) k · ( c (cid:48) λ − s n s λ − s n s )) θ ( n − s ( λ s ) k n − s n − s ( λ s ) − k +1 )+ ( n − s ( λ s ) k · ( n − s c n s )) θ ( n − s ( λ s ) k n − s ( λ s ) − k )) θ ( λ ) . Since λ − s n s λ − s n s = (cid:101) wn − s (cid:48) (cid:101) w ∈ Z κ , it commutes with n − s . Hence we have( n − s ( λ s ) k · ( c (cid:48) ( λ − s n s λ − s n s ))) θ ( n − s ( λ s ) k n − s n − s ( λ s ) − k +1 )= ( n − s ( λ s ) k · c (cid:48) ) θ ( n − s ( λ s ) k ( λ − s n s λ − s n s ) n − s n − s ( λ s ) − k +1 )= ( n − s ( λ s ) k · c (cid:48) ) θ ( n − s ( λ s ) k n − s ( λ − s n s λ − s n s ) n − s ( λ s ) − k +1 )= ( n − s ( λ s ) k · c (cid:48) ) θ ( n − s ( λ s ) k ( n − s λ − s n s )( n − s λ − s n s ) n − s ( λ s ) − k +1 )= ( n − s ( λ s ) k · c (cid:48) ) θ ( n − s ( λ s ) k − n − s ( λ s ) − k ) . Similarly, we have ( n − s ( λ s ) k · ( n − s c n s )) θ ( n − s ( λ s ) k n − s ( λ s ) − k )= ( n − s ( λ s ) k · c n s ) θ ( n − s ( λ s ) k n − s n − s ( λ s ) − k ) . As in the case of (cid:104) ν (Λ(1)) , α (cid:105) = Z , since Z κ is commutative, we have n − s ( λ s ) k · c (cid:48) = λ ks · c (cid:48) and n − s ( λ s ) k · c n s = λ ks · c n s . Therefore θ ( n − s ( λ )) T n − s − T n − s θ ( λ ) is equalto n (cid:88) k =1 (( λ ks · c (cid:48) ) θ ( n − s ( λ s ) k − n − s ( λ s ) − k ) + ( λ ks · c n s ) θ ( n − s ( λ s ) k n − s n − s ( λ s ) − k )) θ ( λ ) . Multiply by T n s on the left. Since n s ∈ Z κ , T n s θ ( n − s ( λ )) = θ ( n s ) θ ( n − s ( λ )) = θ ( n s n − s ( λ )) = θ ( n s ( λ )) θ ( n s ) = θ ( n s ( λ )) T n s . We also remark that T n s commuteswith λ ks · c (cid:48) and λ ks · c n s since everything is in Z [ Z κ ]. Hence, if 2 n = (cid:104) ν ( λ ) , α (cid:105) ≥ θ ( n s ( λ )) T n s − T n s θ ( λ ) is equal to n (cid:88) k =1 (( λ ks · c (cid:48) ) θ ( λ k − s n s n − s ( λ s ) − k ) + ( λ ks · c n s ) θ ( λ ks n − s ( λ s ) − k )) θ ( λ ) . The same argument for (2.8) implies that, if − n = (cid:104) ν ( λ ) , α (cid:105) <
0, then T n s θ ( λ ) − θ ( n s ( λ )) T n s is equal to n (cid:88) k =1 θ ( n s ( λ ))( θ ( λ k − s n s n − s ( λ s ) − k )( n s ( λ s ) k − · c (cid:48) ) + θ ( λ ks n − s ( λ s ) − k )( n s ( λ s ) k · c n s ) . ODULO p PARABOLIC INDUCTION OF PRO- p -IWAHORI HECKE ALGEBRA 11 Put µ n s (2 k ) = n s ( λ s ) k λ − ks , µ n s (2 k + 1) = n s ( λ s ) k n s λ − k − s ( k ≥ ,µ n s ( − k ) = λ ks n − s ( λ s ) − k , µ n s ( − k + 1) = λ k − s n s n − s ( λ s ) − k ( k ≥ ,c n s , k = λ ks · c n s , c n s , k +1 = λ k +1 s · c (cid:48) ( k ≥ ,c n s , − k = n s ( λ s ) k · c n s , c n s , − k +1 = n s ( λ s ) k − · c (cid:48) ( k ≥ . We have µ n s ( −
1) = n s n − s ( λ s ) − and c n s , = λ s · c (cid:48) . Hence µ n s ( − − · c n s , =( n − s λ s n − s λ s ) · c (cid:48) . We have n − s λ s = (cid:101) wn s (cid:48) (cid:101) w − . Hence n − s λ s n − s λ s = (cid:101) wn s (cid:48) (cid:101) w − ∈ Z κ . Therefore µ n s ( − − · c n s , = c (cid:48) = c n s , − . It is easy to check the otherconditions. (cid:3) Since n − s is also a lift of s ∈ S in (cid:102) W (1), we have the Bernstein relations for n − s . Lemma 2.11.
Put µ n − s ( k ) = µ n s ( k ) n − s and c n − s ,k = c n s ,k . Then they satisfy theBernstein relations for n − s .Proof. Assume that (cid:104) ν ( λ ) , α (cid:105) >
0. Replacing λ with n − s ( λ ) in Lemma 2.10, wehave θ ( n − s ( λ )) T n s − T n s θ ( n − s ( λ )) = (cid:104) ν ( λ ) ,α (cid:105)− (cid:88) k =0 θ ( n − s ( λ ) µ n s ( k )) c n s ,k = (cid:104) ν ( λ ) ,α (cid:105) (cid:88) k =1 c n s ,k θ ( µ n s ( − k ) n − s ( λ )) . We have θ ( n − s ( λ )) θ ( n − s ) = θ ( n − s λ ) = T n − s θ ( λ ). Since n − s , c n s ,k ∈ C [ Z κ ], thesecommute with each other. Hence multiplying by θ ( n − s ) = T n − s on the right of theboth sides, we get θ ( n − s ( λ )) T n − s − T n − s θ ( λ ) = (cid:104) ν ( λ ) ,α (cid:105)− (cid:88) k =0 θ ( n − s ( λ ) µ n s ( k ) n − s ) c n s ,k = (cid:104) ν ( λ ) ,α (cid:105) (cid:88) k =1 c n s ,k θ ( µ n s ( − k ) n − s λ ) . The same argument implies, if (cid:104) ν ( λ ) , α (cid:105) <
0, we have θ ( n − s ( λ )) T n − s − T n − s θ ( λ ) = − −(cid:104) ν ( λ ) ,α (cid:105)− (cid:88) k =0 c n s , − k θ ( µ n s ( k ) n − s λ )= − −(cid:104) ν ( λ ) ,α (cid:105) (cid:88) k =1 θ ( n s ( λ ) µ n s ( − k ) n − s ) c n s , − k . We get the lemma. (cid:3)
Let C be an algebraically closed field of characteristic p and set H = H Z ⊗ Z C .This is a C [ q / s ]-algebra. As an element of H , c n s and c n s ,k does not depend on achoice of n s and we denote it by c s and c s,k . The field C is a C [ q / s ]-algebra via q / s (cid:55)→
0. Then
H ⊗ C [ q / s ] C is isomorphic to the Hecke algebra for ( G ( F ) , I (1))with coefficients in C . The following lemma is useful for calculations in H⊗ C [ q / s ] C . Lemma 2.12.
Let λ, µ ∈ Λ(1) and α ∈ Σ + such that (cid:104) ν ( λ ) , α (cid:105) > . Assume thatthere exists k ∈ Z ≥ such that k ≤ (cid:104) ν ( λ ) , α (cid:105) and ν ( µ ) = ν ( λ ) − k ˇ α . Then we have (cid:96) ( λ ) ≥ (cid:96) ( µ ) + 2 min { k, (cid:104) ν ( λ ) , α (cid:105) − k } . Moreover, equality holds if and only if k = 0 or k = (cid:104) ν ( λ ) , α (cid:105) or w ( α ) is simple for some w ∈ W such that n w ( λ ) is dominant. The elements appeared in the Bernstein relations (e.g. λµ n s ( − k )) satisfies thecondition of µ in this lemma. Proof.
Since (cid:96) ( µ ) = (cid:96) ( n s ( µ )) and ν ( n s ( µ )) = ν ( λ ) − ( (cid:104) ν ( λ ) , α (cid:105) − k )ˇ α , replacing µ with n s ( µ ) if necessary, we may assume k ≤ (cid:104) ν ( λ ) , α (cid:105) /
2. Take v ∈ W suchthat ν ( n v ( µ )) is dominant. We may assume v ( α ) > (cid:104) ν ( n v ( µ )) , v ( α ) (cid:105) = (cid:104) ν ( µ ) , α (cid:105) = (cid:104) ν ( λ ) , α (cid:105) − k ≥
0. Set ρ = (1 / (cid:80) β ∈ Σ + β . Then we have (cid:96) ( µ ) = (cid:96) ( n v ( µ ))= (cid:88) γ ∈ Σ + |(cid:104) ν ( n v ( µ )) , γ (cid:105)| = (cid:88) γ ∈ Σ + (cid:104) ν ( n v ( µ )) , γ (cid:105) = (cid:88) γ ∈ Σ + (cid:104) ν ( n v ( λ )) , γ (cid:105) − k (cid:88) γ ∈ Σ + (cid:104) v (ˇ α ) , γ (cid:105)≤ (cid:88) γ ∈ Σ + |(cid:104) ν ( n v ( λ )) , γ (cid:105)| − k (cid:88) γ ∈ Σ + (cid:104) v (ˇ α ) , γ (cid:105) = (cid:96) ( n v ( λ )) − k (cid:104) v (ˇ α ) , ρ (cid:105) = (cid:96) ( λ ) − k (cid:104) v (ˇ α ) , ρ (cid:105) Since v (ˇ α ) ∈ Σ + , we have (cid:104) v (ˇ α ) , ρ (cid:105) ≥
1. Hence we have (cid:96) ( µ ) ≤ (cid:96) ( λ ) − k .Assume k (cid:54) = 0. By the above argument, if the equality holds, then n v ( λ ) is domi-nant and v ( α ) is simple. Conversely, assume that there exists v ∈ W such that n v ( λ )is dominant and v ( α ) is simple. We have (cid:104) ν ( n v ( µ )) , v ( α ) (cid:105) = (cid:104) ν ( µ ) , α (cid:105) = (cid:104) ν ( λ ) , α (cid:105) − k ≥
0. If β ∈ ∆ \ { v ( α ) } , then (cid:104) ν ( n v ( µ )) , β (cid:105) = (cid:104) ν ( n v ( λ )) , β (cid:105) − k (cid:104) v ( α ) , β (cid:105) ≥ n v ( µ ) is dominant. By the above argument, the equality holds. (cid:3) One application of the above lemma and the Bernstein relations is the following“the simple Bernstein relations at q = 0”. (See [Viga, Corollary 5.53].) Lemma 2.13.
Let λ ∈ Λ(1) , s = s α ∈ S . In H ⊗ C , we have the following. If (cid:104) ν ( λ ) , α (cid:105) > , then E ( n s ( λ ))( T n s − c s ) = T n s E ( λ ) . If (cid:104) ν ( λ ) , α (cid:105) < , then E ( n s ( λ )) T n s = ( T n s − c s ) E ( λ ) . Proof. If (cid:104) ν ( λ ) , α (cid:105) >
0, by the Bernstein relations, E ( n s ( λ )) T n s − T n s E ( λ ) = (cid:104) ν ( λ ) ,α (cid:105)− (cid:88) k =0 q / λ q − / n s ( λ ) µ ns ( k ) E ( n s ( λ ) µ n s ( k )) c s,k . The lemma follows from the following claim in this case.
Claim.
We have q / λ q − / n s ( λ ) µ ns ( k ) ∈ C [ q / s ]. If k (cid:54) = 0, then under C [ q / s ] → C , q / λ q − / n s ( λ ) µ ns ( k ) goes to 0. Proof of Claim.
In the above equation, the left hand side is in H . The images ofby ν of ( n s ( λ ) µ n s ( k )) k are distinct by Lemma 2.10. Recall that { E ( w ) } w ∈ (cid:102) W (1) isa C [ q / s ]-basis of H . Therefore q / λ q − / n s ( λ ) µ ns ( k ) ∈ C [ q / s ]. Hence there exists n s ∈ Z ≥ for s ∈ S aff / ∼ such that q / λ q − / n s ( λ ) µ ns ( k ) ∈ (cid:81) s q n s / s . We have (cid:80) s ∈ S aff / ∼ n s = (cid:96) ( λ ) − (cid:96) ( n s ( λ ) µ n s ( k )). By the above lemma, the right hand side is grater than orequal to 2 k . Hence if k (cid:54) = 0, there exists s such that n s >
0. The claim follows. (cid:3)
ODULO p PARABOLIC INDUCTION OF PRO- p -IWAHORI HECKE ALGEBRA 13 The same argument implies the lemma if (cid:104) ν ( λ ) , α (cid:105) < (cid:3) If (cid:104) ν ( λ ) , α (cid:105) = 0, we have T n s θ ( λ ) = θ ( n s ( λ )) T n s ∈ H [ q ± / s ]. We have a slightlymore general properties. If k = 0, it is [AHHV14, IV.17. Lemma]. Lemma 2.14.
Let λ ∈ Λ , α ∈ ∆ , s = s α and k ∈ Z ≥ . Assume that (cid:104) ν ( λ ) , α (cid:105) = 0 .Then we have θ ( µ n s ( − k )) c s, − k θ ( λ ) = θ ( n s ( λ )) θ ( µ n s ( − k )) c s, − k in H [ q ± / s ] . Inparticular c s θ ( λ ) = θ ( n s ( λ )) c s and hence ( T n s − c s ) E ( λ ) = E ( n s ( λ ))( T n s − c s ) in H .Proof. Take λ from the center of Λ(1) such that (cid:104) ν ( λ ) , α (cid:105) < − k . Put n = −(cid:104) ν ( λ ) , α (cid:105) . Since T n s θ ( λ ) = θ ( n s ( λ )) T n s , we have n (cid:88) l =0 θ ( n s ( λ ) µ n s ( − l )) c s, − l θ ( λ ) = ( T n s θ ( λ ) − θ ( n s ( λ )) T n s ) θ ( λ )= T n s θ ( λ ) θ ( λ ) − θ ( n s ( λ )) T n s θ ( λ )= θ ( n s ( λ )) T n s θ ( λ ) − θ ( n s ( λ )) θ ( n s ( λ )) T n s = θ ( n s ( λ ))( T n s θ ( λ ) − θ ( n s ( λ )) T n s )= n (cid:88) l =0 θ ( n s ( λ )) θ ( n s ( λ ) µ n s ( − l )) c s, − l = n (cid:88) l =0 θ ( n s ( λ )) θ ( n s ( λ )) θ ( µ n s ( − l )) c s, − l . Hence we have(2.9) n (cid:88) l =0 θ ( µ n s ( − l )) c s, − l θ ( λ ) = n (cid:88) l =0 θ ( n s ( λ )) θ ( µ n s ( − l )) c s, − l . This is an equality in (cid:76) λ (cid:48) ∈ Λ(1) Z θ ( λ (cid:48) ). We have θ ( µ n s ( − l )) c s, − l θ ( λ ) , θ ( n s ( λ )) θ ( µ n s ( − l )) c s, − l ∈ (cid:77) ν ( λ (cid:48) )= ν ( λ ) − l ˇ α C [ q ± / s ] θ ( λ (cid:48) ) . Hence projecting the both sides of (2.9) to (cid:76) ν ( λ (cid:48) )= ν ( λ ) − k ˇ α C [ q ± / s ] θ ( λ (cid:48) ), we getthe first statement of the lemma. Putting k = 0, we have c s θ ( λ ) = θ ( n s ( λ )) c s .Hence the second statement follows. (cid:3) These formulas are very simple. However, it is too simple even for studying rep-resentations over C . So later we use the original Bernstein relations (Lemma 2.10).For specializing q s →
0, Lemma 2.12 and an argument in the proof of Lemma 2.13is useful and it will be used later.Finally, we prove the following lemma which will be used later. This lemma isdiscovered in the study of [AHHV14].
Lemma 2.15 ([AHHV14, IV.24. Proposition]) . Let α ∈ ∆ and put s = s α . Thegroups Z ( α, ,κ and Z ( α, − ,κ generates Λ (cid:48) s (1) ∩ Z κ . In particular, for a non-trivialcharacter ψ : Λ (cid:48) s (1) ∩ Z κ → C × we have ψ ( c s c s, − ) = 0 .Proof. Replacing G with the algebraic group generated by U α and U − α , we mayassume that G has a semisimple F -rank 1. Put s = s ( α, , s = s ( α, − . Thenwe have S aff = { s , s } and the map s (cid:55)→ n s , s (cid:55)→ n s extends to the map (cid:102) W aff → G (cid:48) such that the braid relations hold. (Since the semisimple F -rankis 1, a reduced expression of any element in (cid:102) W aff is unique.) We denote thismap by (cid:101) w (cid:55)→ n (cid:101) w . Let Z (resp. Z ) be the inverse image of Z ( α, ,κ (resp. Z ( α, − ,κ ) in Z ( F ) ∩ K and H the group generated by Z and Z . Put H = (cid:83) w ∈ (cid:102) W aff I (1) n w H I (1). Since we have (cid:102) W aff (1) = { n w | w ∈ (cid:102) W aff } Z κ , we have G (cid:48) ( Z ( F ) ∩ I (1)) = (cid:83) (cid:101) w ∈ (cid:102) W aff (1) I (1) (cid:101) wI (1) = (cid:83) w ∈ (cid:102) W aff I (1) n w ( Z ( F ) ∩ K ∩ G (cid:48) ) I (1) = H ( Z ( F ) ∩ K ∩ G (cid:48) ). By relations between the Iwahori-Matsumoto basis (braid re-lations and quadratic relations), (cid:80) (cid:101) w ∈ H C T (cid:101) w ⊂ H Z ⊗ C is a subalgebra where weregard C as a Z [ q / s ]-algebra via q / s (cid:55)→ I (1) sI (1) /I (1)) / . By the definitionof the convolution product, the coefficient of T (cid:101) w (cid:101) w in T (cid:101) w T (cid:101) w is non-zero. Hence H is a subgroup.Let ψ : Z ( F ) ∩ K ∩ G (cid:48) → C × be a character which is trivial on H . We prove ψ is trivial. It follows that H = Z ( F ) ∩ K ∩ G (cid:48) . By the Bruhat decomposition, H ∩ ( Z ( F ) ∩ K ∩ G (cid:48) ) ⊂ H , on which ψ is trivial. Hence we can define a map ψ (cid:48) : G (cid:48) ( Z ( F ) ∩ I (1)) = H ( Z ( F ) ∩ K ∩ G (cid:48) ) → C × by ψ (cid:48) ( ht ) = ψ (cid:48) ( t ) for h ∈ H and t ∈ Z ( F ) ∩ K ∩ G (cid:48) .We prove ψ (cid:48) is a character. Let h , h ∈ (cid:83) w ∈ (cid:102) W aff n w H and t , t ∈ Z ( F ) ∩ K ∩ G (cid:48) .Then t normalizes I (1) and h normalizes Z ( F ) ∩ K ∩ G (cid:48) . Hence I (1) h I (1) t I (1) h I (1) t = I (1) h I (1) h I (1)( h − t h ) t . Since H is a subgroup, I (1) h I (1) h I (1) ⊂ H . Therefore on the above set, the valueof ψ (cid:48) is ψ (( h − t h ) t ). We prove ψ ( h − t h ) = ψ ( t ). We may assume h = n s i where i = 0 or 1. We prove t − h − t h ∈ Z i . Put (cid:101) α = ( α, − i ) ∈ Σ aff . Let F be afacet which we used when we defined Z (cid:101) α,κ . Then we have a finite group G F ,κ . Let U (cid:101) α,κ and U (cid:101) α,κ be as in the proof of Lemma 2.9 and let G (cid:48) s i ,κ be the subgroup of G F ,κ generated by these groups. Let t (cid:48) (resp. h (cid:48) ) be the image of t (resp. h ) in G F ,κ .Then h (cid:48) ∈ G (cid:48) s i ,κ . Since t (cid:48) normalizes this group, ( t (cid:48) ) − ( h (cid:48) ) − t (cid:48) h (cid:48) ∈ G (cid:48) s i ,κ . It isalso in Z κ . Hence ( t (cid:48) ) − ( h (cid:48) ) − t (cid:48) h (cid:48) ∈ G (cid:48) s i ,κ ∩ Z κ = Z (cid:101) α,κ . Therefore t − h t h ∈ Z i .The map ψ (cid:48) is a character.Since any character of G (cid:48) is trivial, ψ (cid:48) is a trivial character. Hence ψ is trivial. (cid:3) Intertwining operators
Construction of intertwining operators.
We use the notation in the pre-vious section. In particular, C is an algebraically closed filed of characteristic p , H = H Z ⊗ Z C and C is a C [ q / s ]-algebra via q / s (cid:55)→
0. In the rest of this paper, weinvestigate the representations of
H ⊗ C . All modules (or representations) in thispaper are right modules. Remark . Since
H ⊗ C is finitely generated as a module of its center and thecenter is finitely generated over C [Vig14, Theorem 1.2], any irreducible H ⊗ C -module is finite-dimensional.Let λ , λ ∈ Λ(1). Then we have E ( λ ) E ( λ ) = q − / λ λ q / λ q / λ E ( λ λ ) by Propo-sition 2.8. Hence A = (cid:76) λ ∈ Λ(1) C [ q / s ] E ( λ ) is a subalgebra of H . Let Λ + (1) bethe set of dominant elements in Λ(1) and C [Λ + (1)] = (cid:76) λ ∈ Λ + (1) Cτ λ the monoidalgebra of Λ + (1).In A ⊗ C , E ( λ ) E ( λ ) = q − / λ λ q / λ q / λ E ( λ λ ) is E ( λ λ ) or 0 and it is notzero if and only if (cid:96) ( λ λ ) = (cid:96) ( λ ) + (cid:96) ( λ ). By the length formula (2.1), we have (cid:96) ( λ λ ) = (cid:96) ( λ ) + (cid:96) ( λ ) if and only if ν ( λ ) , ν ( λ ) are in the same closed chamber.Therefore we get(3.1) E ( λ ) E ( λ ) = (cid:40) E ( λ λ ) ( ν ( λ ) and ν ( λ ) are in the same closed chamber) , ODULO p PARABOLIC INDUCTION OF PRO- p -IWAHORI HECKE ALGEBRA 15 in A ⊗ C . Hence the C -linear map (cid:101) χ : A ⊗ C → C [Λ + (1)] defined by (cid:101) χ ( E ( λ )) = (cid:40) τ λ ( λ ∈ Λ + (1)) , A → C [Λ + (1)]. For w ∈ W ,we define w (cid:101) χ : A → C [Λ + (1)] by ( w (cid:101) χ )( E ( λ )) = (cid:101) χ ( E ( n − w ( λ ))). (Note that thishomomorphism depends on a lift n w of w .) We will prove that, for an irreduciblerepresentation X of A ⊗ C , the action of A ⊗ C on X factors through w (cid:101) χ for some w ∈ W (Proposition 3.6). This is a motivation to consider this homomorphism.We construct intertwining operators between { w (cid:101) χ ⊗ A H} w ∈ W . When G = GL n ,they are constructed by Ollivier [Oll10, 5D1]. Proposition 3.2.
Let w ∈ W and s ∈ S such that sw > w . Then ⊗ (cid:55)→ ⊗ T ∗ n s gives a homomorphism w (cid:101) χ ⊗ A H → sw (cid:101) χ ⊗ A H .Proof. We prove 1 ⊗ ( T n s − c s ) E ( λ ) = ( w (cid:101) χ )( E ( λ )) ⊗ ( T n s − c s ) ∈ sw (cid:101) χ ⊗ A H for λ ∈ Λ(1). Take a simple root α such that s = s α . If (cid:104) ν ( λ ) , α (cid:105) = 0, then by Lemma 2.14,we have 1 ⊗ ( T n s − c s ) E ( λ ) = 1 ⊗ E ( n s ( λ ))( T n s − c s ) = ( sw (cid:101) χ )( E ( n s ( λ ))) ⊗ ( T n s − c s ) . Since sw > w , we have n sw = n s n w . Therefore we get ( sw (cid:101) χ )( E ( n s ( λ ))) = (cid:101) χ ( E ( n − w n − s n s ( λ ))) = (cid:101) χ ( E ( n − w ( λ ))) = ( w (cid:101) χ )( E ( λ )). We get the lemma in thiscase.Assume that (cid:104) α, ν ( λ ) (cid:105) >
0. Then (cid:104) ( sw ) − ( α ) , ( sw ) − ( ν ( λ )) (cid:105) >
0. Therefore n − sw ( λ ) is not dominant since ( sw ) − ( α ) <
0. Hence ( sw (cid:101) χ )( E ( λ )) = 0. Thereforewe have 1 ⊗ c s E ( λ ) = 1 ⊗ E ( λ )( λ − · c s ) = ( sw (cid:101) χ )( E ( λ )) ⊗ ( λ − · c s ) = 0. In H ⊗ C ,we have E ( n s ( λ ))( T n s − c s ) = T n s E ( λ ) by Lemma 2.13. Hence1 ⊗ ( T n s − c s ) E ( λ ) = 1 ⊗ T n s E ( λ )= 1 ⊗ E ( n s ( λ ))( T n s − c s )= ( sw (cid:101) χ )( E ( n s ( λ ))) ⊗ ( T n s − c s )= ( w (cid:101) χ )( E ( λ )) ⊗ ( T n s − c s ) . Finally, assume that (cid:104) α, ν ( λ ) (cid:105) <
0. Then (cid:104) w − ( α ) , w − ( ν ( λ )) (cid:105) <
0. Since w − ( α ) > n − w ( λ ) is not dominant. Hence ( w (cid:101) χ )( E ( λ )) = 0. Therefore it issufficient to prove that 1 ⊗ ( T n s − c s ) E ( λ ) = 0. By Lemma 2.13, we have1 ⊗ ( T n s − c s ) E ( λ ) = 1 ⊗ E ( n s ( λ )) T n s = ( sw (cid:101) χ )( E ( n s ( λ ))) ⊗ T n s = ( w (cid:101) χ )( E ( λ )) ⊗ T n s = 0 . (cid:3) The homomorphism obtained in the above lemma is denoted by Φ sw,w : w (cid:101) χ ⊗ A H → sw (cid:101) χ ⊗ A H . More generally, if (cid:96) ( w ) = (cid:96) ( w w − ) + (cid:96) ( w ), we have a homo-morphism Φ w ,w : w (cid:101) χ ⊗ A H → w (cid:101) χ ⊗ A H defined by 1 ⊗ (cid:55)→ ⊗ T ∗ n w w − by theabove proposition and induction on (cid:96) ( w w − ).To construct the intertwining operator of the inverse direction, we need a nota-tion. For α ∈ ∆, s = s α and λ ∈ Λ(1) such that (cid:104) ν ( λ ) , α (cid:105) >
0, define d ( s, λ ) = (cid:40) n w ( λ ) is dominant and w ( α ) is simple for some w ∈ W ) , . Lemma 3.3. In C , we have d ( s, λ ) = q / λ q − s q − / λµ n − s ( − . Proof.
By the length formula (2.1) and the assumption (cid:104) ν ( λ ) , α (cid:105) >
0, we have (cid:96) ( λn − s ) = (cid:88) β ∈ Σ + \{ α } |(cid:104) ν ( λ ) , β (cid:105)| + |(cid:104) ν ( λ ) , α (cid:105) − | = (cid:88) β ∈ Σ + \{ α } |(cid:104) ν ( λ ) , β (cid:105)| + |(cid:104) ν ( λ ) , α (cid:105)| − (cid:96) ( λ ) − . Hence E ( λn − s ) = q / λ q − s θ ( λ ) T n − s . By the Bernstein relation (Lemma 2.10), wehave E ( λn − s ) = q / λ q − s θ ( λ ) T n − s = q λ q − s T n − s θ ( n s ( λ )) − (cid:104) ν ( λ ) ,α (cid:105) (cid:88) k =1 q / λ q − s q − / λµ n − s ( − k ) E ( λµ n − s ( − k )) c s, − k . Hence we have q / λ q − s q − / λµ n − s ( − ∈ C [ q s ]. The lemma follows from Lemma 2.12and an argument in the proof Lemma 2.13. (cid:3) Proposition 3.4.
Let α ∈ ∆ , s = s α and w ∈ W such that sw > w . Take λ fromthe center of Λ(1) such that w − ( λ ) is dominant and (cid:104) ν ( λ ) , α (cid:105) ≥ . Then ⊗ (cid:55)→ ⊗ ( E ( λn − s ) + d ( s, λ ) E ( λµ n − s ( − c s, − ) gives a homomorphism sw (cid:101) χ ⊗ A H → w (cid:101) χ ⊗ A H .Proof. Put d = d ( s, λ ). We have E ( λn − s ) = q / λ q − s θ ( λ ) T n − s by the proof ofLemma 3.3. Let µ ∈ Λ(1). We prove1 ⊗ ( E ( λn − s ) + dE ( λµ n − s ( − c s, − ) E ( µ )= ( sw (cid:101) χ )( E ( µ )) ⊗ ( E ( λn − s ) + dE ( λµ n − s ( − c s, − )(3.2)in w (cid:101) χ ⊗ A H . Assume that (cid:104) ν ( µ ) , α (cid:105) = 0. Since λ is in the center of Λ(1), byLemma 2.14, we have θ ( λµ n − s ( − c s, − θ ( µ ) = θ ( λ ) θ ( µ n − s ( − c s, − θ ( µ )= θ ( λ ) θ ( n − s ( µ )) θ ( µ n − s ( − c s, − = θ ( n − s ( µ )) θ ( λµ n − s ( − c s, − . Hence E ( λµ n − s ( − c s, − E ( µ ) = E ( n − s ( µ )) E ( λµ n − s ( − c s, − . Since T n − s θ ( µ ) = θ ( n − s ( µ )) T n − s , we have θ ( λ ) T n − s θ ( µ ) = θ ( λ ) θ ( n − s ( µ )) T n − s = θ ( n − s ( µ )) θ ( λ ) T n − s . Hence E ( λn − s ) E ( µ ) = E ( n − s ( µ )) E ( λn − s ). Therefore,( E ( λn − s ) + dE ( λµ n − s ( − c s, − ) E ( µ )= E ( n − s ( µ ))( E ( λn − s ) + dE ( λµ n − s ( − c s, − ) . We get the lemma in this case.Next assume that (cid:104) ν ( µ ) , α (cid:105) >
0. Then (cid:104) ( sw ) − ( α ) , ( sw ) − ( ν ( µ )) (cid:105) >
0. Since( sw ) − ( α ) = − w − ( α ) < n − sw ( µ ) is not dominant. Hence we have ( sw (cid:101) χ )( E ( µ )) =0. The right hand side of (3.2) is zero. ODULO p PARABOLIC INDUCTION OF PRO- p -IWAHORI HECKE ALGEBRA 17 By the Bernstein relation, we have E ( λn − s ) E ( µ ) = q / λ q / µ q − s θ ( λ ) T n − s θ ( µ )= q / λ q / µ q − s θ ( λn − s ( µ )) T n − s − (cid:104) ν ( µ ) ,α (cid:105) (cid:88) k =1 q / λ q / µ q − s θ ( λ ) c s,k θ ( µ n − s ( − k ) µ ) . (3.3)Since λ is in the center of Λ(1), we have q / λ q / µ q − s θ ( λn − s ( µ )) T n − s = q / λ q / µ q − s θ ( n − s ( µ )) θ ( λ ) T n − s = E ( n − s ( µ )) E ( λn − s ) . (3.4)Hence 1 ⊗ q / λ q / µ q − s θ ( λn − s ( µ )) T n − s = 1 ⊗ E ( n − s ( µ )) E ( λn − s )= ( w (cid:101) χ )( E ( n − s ( µ ))) ⊗ E ( λn s )= ( sw (cid:101) χ )( E ( µ )) ⊗ E ( λn s ) = 0 . We calculate the second term of the right hand side of (3.3). We have(3.5) q / λ q / µ q − s θ ( λ ) c s,k θ ( µ n − s ( − k ) µ ) = q / λ q / µ q − s q − / λµ n − s ( − k ) µ c s,k E ( λµ n − s ( − k ) µ )and, by the argument in the proof of Lemma 2.13, it is zero in H ⊗ C if (cid:96) ( λ ) + (cid:96) ( µ ) − (cid:96) ( λµ n − s ( − k ) µ ) − >
0. By Lemma 2.12, we have(3.6) (cid:96) ( λ ) + (cid:96) ( µ ) − ≥ (cid:96) ( λµ ) − ≥ (cid:96) ( λµ n − s ( − k ) µ ) + 2 min { k, (cid:104) ν ( λµ ) , α (cid:105) − k } − . Hence if k > (cid:104) ν ( λµ ) , α (cid:105)− k >
1, then (3.5) is zero in H⊗ C . Since k ≤ (cid:104) ν ( µ ) , α (cid:105) , (cid:104) ν ( λµ ) , α (cid:105) − k ≥ (cid:104) ν ( λ ) , α (cid:105) ≥ >
1. Hence (3.5) is zero in
H ⊗ C if k (cid:54) = 1. Since µ n s ( − − · c s, = c s, − , by Lemma 3.3, (3.5) is q / λ q / µ q − s θ ( λ ) c s, θ ( µ n − s ( − µ ) = q / λ q − s q − / λµ n − s ( − E ( λµ n − s ( − c s, − E ( µ )= dE ( λµ n − s ( − c s, − E ( µ )Hence the left hand side of (3.2) is zero and we get (3.2).Finally assume that (cid:104) ν ( µ ) , α (cid:105) <
0. By the Bernstein relations, we have E ( λn − s ) E ( µ ) = q / λ q − s q / µ θ ( λ ) T n − s θ ( µ )= q / λ q − s q / µ θ ( λn − s ( µ )) T n − s + −(cid:104) ν ( µ ) ,α (cid:105) (cid:88) k =1 q / λ q − s q / µ θ ( λn − s ( µ ) µ n − s ( − k )) c s, − k . By (3.4), q / λ q − s q / µ θ ( λn − s ( µ )) T n − s = E ( n − s ( µ )) E ( λn − s ) . The term q / λ q − s q / µ θ ( λn − s ( µ ) µ n − s ( − k )) c s, − k = q / λ q − s q / µ q − / λn − s ( µ ) µ n − s ( − k ) E ( λn − s ( µ ) µ n − s ( − k )) c s, − k (3.7) is zero if (cid:96) ( λ ) + (cid:96) ( µ ) + (cid:96) ( λn − s ( µ ) µ n − s ( − k )) − > (cid:96) ( λ ) + (cid:96) ( µ ) + (cid:96) ( λn − s ( µ ) µ n − s ( − k )) − (cid:96) ( λ ) + (cid:96) ( n − s ( µ )) + (cid:96) ( λn − s ( µ ) µ n − s ( − k )) − ≥ (cid:96) ( λn − s ( µ )) + (cid:96) ( λn − s ( µ ) µ n − s ( − k )) − ≥ { k, (cid:104) ν ( λn − s ( µ )) , α (cid:105) − k } − k ≤ −(cid:104) ν ( µ ) , α (cid:105) = (cid:104) ν ( n − s ( µ )) , α (cid:105) , we have (cid:104) ν ( λn − s ( µ )) , α (cid:105) − k ≥ (cid:104) ν ( λ ) , α (cid:105) >
1. Hence (3.7) is zero if k (cid:54) = 1. If k = 1, then (3.7) is q / λ q − s q − / λµ n − s ( − E ( n − s ( µ )) E ( λµ n − s ( − c s, − = dE ( n − s ( µ )) E ( λµ n − s ( − c s, − . by Lemma 3.3. Hence we get E ( λn − s ) E ( µ ) = E ( n − s ( µ ))( E ( λn − s ) + dE ( λµ n − s ( − c s, − ) . in H ⊗ C .On the other hand, by (2.2), we have1 ⊗ E ( λµ n − s ( − c s, − E ( µ )= 1 ⊗ E ( µ ) E ( µ − λµ n − s ( − µ )( µ − · c s, − )= ( w (cid:101) χ )( E ( µ )) ⊗ E ( µ − λµ n − s ( − µ )( µ − · c s, − ) . Since (cid:104) w − ( α ) , w − ( µ ) (cid:105) < w − ( α ) > w − ( µ ) is not dominant. Hence w (cid:101) χ ( E ( µ )) = 0. Therefore, the left hand side of (3.2) is1 ⊗ E ( n − s ( µ ))( E ( λn − s ) + dE ( λµ n − s ( − c s, − )= ( w (cid:101) χ )( E ( n − s ( µ ))) ⊗ ( E ( λn − s ) + dE ( λµ n − s ( − c s, − )= ( sw (cid:101) χ )( E ( µ )) ⊗ ( E ( λn − s ) + dE ( λµ n − s ( − c s, − )We get the proposition. (cid:3) Proposition 3.5.
Let α, s, w, λ be as in Proposition 3.4. Assume that there exist λ , λ which satisfy the same conditions for λ such that λ = λ λ . Then thecompositions of homomorphisms in Proposition 3.2 and Proposition 3.4 is given bythe multiplication of ( w (cid:101) χ )( E ( λ ) − d ( s, λ ) E ( λµ n − s ( − c s, − c s ) ∈ C [Λ + (1)] . Proof.
Put d = d ( s, λ ) and Φ = Φ sw,w . Let Ψ : sw (cid:101) χ ⊗ A H → w (cid:101) χ ⊗ A H be thehomomorphisms defined in Proposition 3.4. Then we haveΨ ◦ Φ(1 ⊗
1) = 1 ⊗ ( E ( λn − s ) + dE ( λµ n − s ( − c s, − )( T n s − c s ) . By the proof of Lemma 3.3, we have E ( λn − s ) = q / λ q − s θ ( λ ) T n − s . Therefore E ( λn − s )( T n s − c s ) = q / λ q − s θ ( λ ) T n − s ( T n s − c s ) = q / λ θ ( λ ) = E ( λ ) . Since (cid:104) ν ( λ ) , α (cid:105) = (cid:104) ν ( λ ) , α (cid:105) + (cid:104) ν ( λ ) , α (cid:105) ≥ (cid:104) ν ( λµ n − s ( − , α (cid:105) = (cid:104) ν ( λ ) , α (cid:105) − >
0. From this and by the length formula (2.1), we have q λµ n − s ( − n s = q λµ ns ( − q − s . (See the proof of Lemma 3.3.) Therefore, by Proposition 2.8, wehave E ( λµ n − s ( − T n s = q s E ( λµ n − s ( − n s ) = 0 in H ⊗ C . HenceΨ ◦ Φ(1 ⊗
1) = 1 ⊗ ( E ( λn − s ) + dE ( λµ n − s ( − c s, − )( T n s − c s )= 1 ⊗ ( E ( λ ) − dE ( λµ n − s ( − c s, − c s )= ( w (cid:101) χ )( E ( λ ) − dE ( λµ n − s ( − c s, − c s ) ⊗ . ODULO p PARABOLIC INDUCTION OF PRO- p -IWAHORI HECKE ALGEBRA 19 Next we calculateΨ ◦ Φ(1 ⊗
1) = 1 ⊗ ( T n s − c s )( E ( λn − s ) + dE ( λµ n − s ( − c s, − ) . First, by the Bernstein relation, we have T n s θ ( λ ) T n − s = θ ( n s ( λ )) T n s T n − s − (cid:104) ν ( λ ) ,α (cid:105)− (cid:88) k =0 θ ( n s ( λ ) µ n s ( k )) c s,k T n − s = θ ( n s ( λ ))( q s + c s T n − s ) − (cid:104) ν ( λ ) ,α (cid:105)− (cid:88) k =0 θ ( n s ( λ ) µ n s ( k )) c s,k T n − s = q s θ ( n s ( λ )) − (cid:104) ν ( λ ) ,α (cid:105)− (cid:88) k =1 θ ( n s ( λ ) µ n s ( k )) c s,k T n − s . Hence, using (3.4) with µ = 1, T n s E ( λn − s ) = q / λ q − s T n s θ ( λ ) T n − s = E ( n s ( λ )) − (cid:104) ν ( λ ) ,α (cid:105)− (cid:88) k =1 q / λ q − s q − / n s ( λ ) µ ns ( k ) E ( n s ( λ ) µ n s ( k )) c s,k T n − s . The term(3.8) q / λ q − s q − / n s ( λ ) µ ns ( k ) E ( n s ( λ ) µ n s ( k )) c s,k T n − s is zero if (cid:96) ( λ ) − − (cid:96) ( n s ( λ ) µ n s ( k )) > k (cid:54) = 1 , (cid:104) ν ( λ ) , α (cid:105)−
1. If k = (cid:104) ν ( λ ) , α (cid:105)−
1, then,since (cid:104) ν ( n s ( λ ) µ n s ( k )) , α (cid:105) >
0, we have (cid:96) ( n s ( λ ) µ n s ( k ) n − s ) = (cid:96) ( n s ( λ ) µ n s ( k )) − E ( n s ( λ ) µ n s ( k )) T n − s = q s E ( n s ( λ ) µ n s ( k ) n − s )by Proposition 2.8. Therefore, (3.8) is q / λ q − / n s ( λ ) µ ns ( k ) E ( n s ( λ ) µ n s ( k ) n − s ) n s · c s,k . It is zero since (cid:96) ( λ ) − (cid:96) ( n s ( λ ) µ n s ( k )) > k = 1, we have q n s ( λ ) µ ns (1) = q λn − s ( µ ns (1)) = q λµ ns ( − . Hence by Lemma 3.3, we have that (3.8)is dE ( n s ( λ ) µ n s (1)) c s, T n − s . Hence, by Lemma 2.10, 2.11, we have1 ⊗ T n s E ( λn − s ) = 1 ⊗ ( E ( n s ( λ )) − dE ( n s ( λ ) µ n s (1)) c s, T n − s )= 1 ⊗ ( E ( n s ( λ )) − dE ( n s ( λ ) µ n s (1)) T n − s ( n s · c s, ))= 1 ⊗ ( E ( n s ( λ )) − dE ( n s ( λ ) µ n − s (1)) T n s c s, − )= 1 ⊗ ( E ( n s ( λ )) − dE ( n s ( λµ n − s ( − T n s c s, − )= ( w (cid:101) χ )( E ( λ )) ⊗ − d ( w (cid:101) χ )( E ( λµ n − s ( − ⊗ T n s c s, − . By the length formula (2.1), we have (cid:96) ( λn − s ) = (cid:96) ( λ λ n − s ) = (cid:96) ( λ ) + (cid:96) ( λ n − s ).Hence E ( λn − s ) = E ( λ ) E ( λ n − s ). We have1 ⊗ c s E ( λn − s ) = 1 ⊗ c s E ( λ ) E ( λ n − s )= 1 ⊗ E ( λ )( λ − · c s ) E ( λ n − s )= ( sw (cid:101) χ )( E ( λ )) ⊗ ( λ − · c s ) E ( λ n − s ) . Since (cid:104) α, ν ( λ ) (cid:105) >
0, we have (cid:104) ( sw ) − ( α ) , ( sw ) − ( ν ( λ )) (cid:105) >
0. By the assumption,( sw ) − ( α ) <
0. Hence n − sw ( λ ) is not dominant. We have ( sw (cid:101) χ )( E ( λ )) = 0.Hence 1 ⊗ c s E ( λn − s ) = 0 . By Proposition 3.2, we have1 ⊗ ( T n s − c s ) E ( λµ n − s ( − c s, − = ( w (cid:101) χ )( E ( λµ n − s ( − ⊗ ( T n s − c s ) c s, − . From these calculations, we haveΨ ◦ Φ(1 ⊗
1) = 1 ⊗ ( T n s − c s )( E ( λn − s ) + dE ( λµ n − s ( − c s, − )= ( w (cid:101) χ )( E ( λ )) ⊗ − d ( w (cid:101) χ )( E ( λµ n − s ( − ⊗ c s, − c s = ( w (cid:101) χ )( E ( λ ) − dE ( λµ n − s ( − c s, − c s ) ⊗ . (cid:3) The modules wχ Θ ⊗ A H . For a subset Θ ⊂ ∆, set Λ Θ (1) = { λ ∈ Λ(1) |(cid:104) ν ( λ ) , α (cid:105) = 0 ( α ∈ Θ) } and put Λ +Θ (1) = Λ Θ (1) ∩ Λ + (1). Let C [Λ Θ (1)] = (cid:76) λ ∈ Λ Θ (1) Cτ λ be the group algebra. Define χ Θ : A → C [Λ Θ (1)] by χ Θ ( E ( λ )) = (cid:40) τ λ ( λ ∈ Λ +Θ (1)) , . This homomorphism factors through (cid:101) χ . By (3.1), it is easy to see that this is analgebra homomorphism.The motivation to consider χ Θ is the following lemma. For an A ⊗ C -module X , set supp X = { λ ∈ Λ(1) | E ( λ ) (cid:54) = 0 on X } . We define a new module wX by( wX )( E ( λ )) = X ( E ( n − w ( λ ))) on the same space. We have supp wX = n w (supp X ). Proposition 3.6.
Let X be an irreducible representation of A over C . Then itfactors through wχ Θ for some w ∈ W and Θ ⊂ ∆ .Proof. The statement is equivalent to • There is a closure of a facet
F ⊂ V with respect to Σ such that ν (supp X ) = ν (Λ(1)) ∩ F . • If λ ∈ supp X , then E ( λ ) is invertible on X .Since E ( µ ) is invertible for any element in µ ∈ Ker ν , we have ν ( λ ) ∈ ν (supp X ) ifand only if λ ∈ supp X .First we prove that ν (supp X ) is a subset of a closed Weyl chamber. Take λ ∈ supp X and set Y = { µ ∈ Λ(1) | ν ( µ ) is in the same closed chamber of ν ( λ ) } .Then XE ( λ ) (cid:54) = 0. Hence XE ( λ ) A = X . If µ / ∈ Y , E ( λ ) E ( µ ) = 0 and if µ ∈ Y then E ( λ ) E ( µ ) = E ( λµ ) by (3.1). Hence X = (cid:80) µ ∈Y XE ( λ ) E ( µ ) = (cid:80) µ ∈Y XE ( λµ ).Let λ (cid:48) ∈ Λ(1) such that λ (cid:48) / ∈ Y . Then E ( λµ ) E ( λ (cid:48) ) = 0 for µ ∈ Y . Hence XE ( λ (cid:48) ) = (cid:80) µ ∈Y XE ( λµ ) E ( λ (cid:48) ) = 0. Therefore λ (cid:48) / ∈ supp X . Hence ν (supp X )is contained in a closed chamber.Hence there exists w ∈ W such that supp X ⊂ n − w (Λ + (1)). Replacing X with wX , we may assume that supp X ⊂ Λ + (1). For each α ∈ ∆, take λ α from thecenter of Λ(1) which satisfies (cid:104) ν ( λ α ) , β (cid:105) = 0 for β ∈ ∆ \ { α } and (cid:104) ν ( λ α ) , α (cid:105) > { α ∈ ∆ | λ α / ∈ supp X } . We prove supp X = Λ +Θ (1).Take λ ∈ Λ + (1). There exists n, n α ∈ Z > such that (cid:104) ∆ , ν ( λ n (cid:81) α ∈ ∆ λ − n α α ) (cid:105) =0. Put µ = λ n (cid:81) α ∈ ∆ λ − n α α . Then E ( µ ) is invertible and we have E ( λ ) n = E ( µ ) (cid:81) α ∈ ∆ E ( λ α ) n α . If λ (cid:54)∈ Λ +Θ (1), namely (cid:104) ν ( λ ) , α (cid:105) (cid:54) = 0 for some α ∈ Θ, then n α (cid:54) = 0 for such α ∈ Θ. Hence E ( λ ) n = 0 on X since E ( λ α ) = 0 on X for α ∈ Θ. Hence { x ∈ X | xE ( λ ) = 0 } is not zero. Take x ∈ X such that xE ( λ ) = 0. Let µ ∈ Λ(1). Then xE ( µ ) E ( λ ) = xE ( λ ) E ( λ − µλ ) = 0 by (2.2).Hence { x ∈ X | xE ( λ ) = 0 } is A -stable and by the irreducibility of X , it is equalto X . Therefore, E ( λ ) = 0 on X . Hence supp X ⊂ Λ +Θ (1).If λ ∈ Λ +Θ (1) then n α = 0 for α ∈ Θ. If α / ∈ Θ, then E ( λ α ) is non-zero. Since λ α is in the center of Λ(1), E ( λ α ) is in the center of A . Hence E ( λ α ) is scalar on X .In particular, for α / ∈ Θ, E ( λ α ) is invertible. Since E ( λ ) n = E ( µ ) (cid:81) α/ ∈ Θ E ( λ α ) n α , E ( λ ) is invertible on X . (cid:3) By the above lemma, supp X = n w (Λ +Θ (1)) for some w ∈ W and Θ ⊂ ∆. Let W Θ be the group generated by { s α | α ∈ Θ } . Then for w ∈ W Θ , n w (Λ +Θ (1)) = Λ +Θ (1). ODULO p PARABOLIC INDUCTION OF PRO- p -IWAHORI HECKE ALGEBRA 21 Since the multiplication { w ∈ W | w (Θ) ⊂ Σ + } × W Θ → W is bijective, we cantake the above w such that w (Θ) ⊂ Σ + . We have supp X = n w (Λ +Θ (1)) if and onlyif the action of A on X factors through wχ Θ . Remark . For an irreducible A -module X , we can attache Θ ⊂ Λ(1), w ∈ W/W Θ and an irreducible module X of C [Λ Θ (1)] such that X = X ◦ wχ Θ . On the otherhand, if (Θ , w, X ) is given, then we get X = X ◦ wχ Θ . Hence irreducible A -modules are classified by such triples.The H -module wχ Θ ⊗ A H has the right action of C [Λ Θ (1)]. Hence it is a( C [Λ Θ (1)] , H )-bimodule. The first property of wχ Θ ⊗ A H is the following free-ness as a C [Λ Θ (1)]-module. Proposition 3.8.
The module wχ Θ ⊗ A H is free of finite rank as a C [Λ Θ (1)] -module. We use the following easy lemma.
Lemma 3.9.
Let X be a C [Λ +Θ (1)] -module. Assume that X has a decomposition X = (cid:76) λ ∈ Λ Θ (1) X λ such that dim X λ ≤ and τ µ ( X λ ) ⊂ X µλ . Then the module C [Λ Θ (1)] ⊗ C [Λ +Θ (1)] X is free of rank less than or equal to one.Proof. Notice that C [Λ Θ (1)] is the ring of fractions of C [Λ +Θ (1)] with respect to { τ λ | λ ∈ Λ +Θ (1) } . We may assume C [Λ Θ (1)] ⊗ X λ (cid:54) = 0 for some λ ∈ Λ Θ (1), sinceif there is no such λ , then C [Λ Θ (1)] ⊗ C [Λ +Θ (1)] X = 0. Then τ µ ( X λ ) (cid:54) = 0 for any µ ∈ Λ +Θ (1). Since dim X λ , dim X µλ ≤ τ µ gives an isomorphism X λ → X µλ . Hencea homomorphism C [Λ +Θ (1)] → X defined by f (cid:55)→ f m is injective for m ∈ X λ \ { } .Fix m ∈ X λ \{ } and consider the homomorphism C [Λ Θ (1)] → C [Λ Θ (1)] ⊗ C [Λ +Θ (1)] X induced by the above homomorphism (namely f (cid:55)→ f ⊗ m ).We prove that it is an isomorphism. It is injective since the above homomor-phism is injective and C [Λ Θ (1)] is a ring of fractions. Take λ (cid:48) ∈ Λ Θ (1) such that C [Λ Θ (1)] ⊗ X λ (cid:48) (cid:54) = 0. Take µ , µ ∈ Λ +Θ (1) such that µ λ = µ λ (cid:48) . Then wehave τ µ ( X λ ) = X µ λ = τ µ ( X λ (cid:48) ). Hence C [Λ Θ (1)] ⊗ X λ (cid:48) = C [Λ Θ (1)] τ − µ τ µ ⊗ X λ = C [Λ Θ (1)] ⊗ X λ . Since X = (cid:76) λ (cid:48) ∈ Λ Θ (1) X λ (cid:48) , we have C [Λ Θ (1)] ⊗ X = (cid:80) λ (cid:48) ∈ Λ Θ (1) C [Λ Θ (1)] ⊗ X λ (cid:48) . Hence the homomorphism is surjective. (cid:3) Proof of Proposition 3.8.
Notice that the image of wχ Θ is C [Λ +Θ (1)] and the kernelof wχ Θ is (cid:80) λ ∈ Λ(1) \ n w (Λ +Θ (1)) CE ( λ ). Hence wχ Θ ⊗ A H = C [Λ Θ (1)] ⊗ C [Λ +Θ (1)] ( H ⊗ C ) / (cid:88) λ ∈ Λ(1) \ n w (Λ +Θ (1)) CE ( λ ) H . For λ ∈ Λ(1) and v ∈ W , put σ λ,v = CE ( n w ( λ ) n v ) / (cid:88) µ ∈ Λ(1) \ n w (Λ +Θ (1)) CE ( µ ) E ( µ − n w ( λ ) n v ) . Then dim σ λ,v ≤
1. Since E ( λ ) E ( µn v ) ∈ CE ( λµn v ) for λ, µ ∈ Λ(1) and v ∈ W , wehave ( H ⊗ C ) / (cid:88) λ ∈ Λ(1) \ n w (Λ +Θ (1)) CE ( λ ) H = (cid:77) λ ∈ Λ(1) ,v ∈ W σ λ,v . Moreover, for µ ∈ Λ +Θ (1), τ µ ( σ λ ) ⊂ σ µλ . We have wχ Θ ⊗ A H = (cid:77) λ ∈ Λ Θ (1) \ Λ(1) ,v ∈ W C [Λ Θ (1)] ⊗ C [Λ +Θ (1)] (cid:77) µ ∈ Λ Θ (1) σ µλ,v . Applying the above lemma to (cid:76) µ ∈ Λ Θ (1) σ µλ,v , we get the freeness. Since H isfinitely generated as a left module over A , wχ Θ ⊗ A H is finitely generated as a C [Λ Θ (1)]-module. Hence the rank is finite. (cid:3) Next, we prove the injectivity of the intertwining operator obtained in Proposi-tion 3.2. We remark the following easy lemma.
Lemma 3.10.
Let α ∈ ∆ and λ ∈ Λ Θ (1) . An element in τ λ + (cid:80) (cid:104) ν ( µ ) ,α (cid:105) < (cid:104) ν ( λ ) ,α (cid:105) Cτ µ is not a zero-divisor in C [Λ Θ (1)] .Proof. Let E be such element and F = (cid:80) µ ∈ Λ Θ (1) c µ τ µ ∈ C [Λ Θ (1)]. Assume that F (cid:54) = 0 and put l = max {(cid:104) ν ( µ ) , α (cid:105) | c µ (cid:54) = 0 } . Then EF ∈ (cid:80) (cid:104) ν ( µ ) ,α (cid:105) = l c µ τ λµ + (cid:80) (cid:104) ν ( µ ) ,α (cid:105) < (cid:104) ν ( λ ) ,α (cid:105) + l Cτ µ . Hence EF (cid:54) = 0. (cid:3) We apply this lemma to the term appearing in Proposition 3.5. So if we cantake a suitable λ , then it is proved that the intertwining operator obtained inProposition 3.2 is injective. To take λ , we use the following lemma. Lemma 3.11.
Let w ∈ W and α ∈ ∆ such that s α w > w . If s α w (Θ) ⊂ Σ + , then w − ( α ) (cid:54)∈ Z Θ .Proof. By the condition, w − ( α ) >
0. If w − ( α ) ∈ Z Θ, then w − ( α ) ∈ Z Θ ∩ Σ + .Hence − α = s α w ( w − ( α )) ∈ s α w ( Z Θ ∩ Σ + ) ⊂ Σ + . This is a contradiction. (cid:3) Proposition 3.12.
Let w ∈ W and s ∈ S such that sw (Θ) ⊂ Σ + . Then thehomomorphism Φ sw,w : wχ Θ ⊗ A H → swχ Θ ⊗ A H obtained in Proposition 3.2 isinjective.Proof. Take α ∈ ∆ such that s = s α . By the above lemma, w − ( α ) (cid:54)∈ Z Θ.Hence we can take λ from the center of Λ(1) such that n − w ( λ ) ∈ Λ +Θ (1) and (cid:104) w − ( α ) , ν ( n − w ( λ )) (cid:105) ≥
2. Then λ = λ satisfies the condition of Proposition 3.4and 3.5. Hence we have a homomorphism swχ Θ ⊗ A H → wχ Θ ⊗ A H and the compo-sition with Φ sw,w is given in Proposition 3.5. By Proposition 3.8 and Lemma 3.10,the composition is injective. Hence Φ sw,w is injective. (cid:3) For w ∈ W , put ∆ w = { α ∈ ∆ | w ( α ) > } . For Θ (cid:48) ⊂ ∆, let w Θ (cid:48) be thelongest element in W Θ (cid:48) . Notice that ∆ w ∆ w Θ (cid:48) = Θ (cid:48) . We have Θ ⊂ ∆ w if and onlyif w (Θ) ⊂ Σ + . We prove the following theorem. Theorem 3.13.
Let w, w (cid:48) ∈ W . Assume that ∆ w = ∆ w (cid:48) and Θ ⊂ ∆ w . Then wχ Θ ⊗ A H (cid:39) w (cid:48) χ Θ ⊗ A H . We need lemmas to prove the theorem.
Lemma 3.14.
Let w ∈ W and λ ∈ Λ such that for any α ∈ Σ + , if w ( α ) < , then (cid:104) ν ( λ ) , α (cid:105) > . Then we have T − n w E ( n w ( λ )) ∈ H .Proof. We have T − n w = q − n w T ∗ n − w = q − n w E w ( − ∆) ( n − w ). Hence T − n w E ( n w ( λ )) = q − / n w q / n w ( λ ) q − / λn − w E w ( − ∆) ( λn − w ) by Proposition 2.8. By (2.1), we have (cid:96) ( λn − w ) = (cid:88) α ∈ Σ + ,w ( α ) > |(cid:104) ν ( λ ) , α (cid:105)| + (cid:88) α ∈ Σ + ,w ( α ) < |(cid:104) ν ( λ ) , α (cid:105) − | = (cid:88) α ∈ Σ + ,w ( α ) > |(cid:104) ν ( λ ) , α (cid:105)| + (cid:88) α ∈ Σ + ,w ( α ) < ( |(cid:104) ν ( λ ) , α (cid:105)| − (cid:96) ( λ ) − (cid:96) ( w ) . Hence we have q n w ( λ ) = q λ = q λn − w q n w . Therefore we get T − n w E ( n w ( λ )) = E w ( − ∆) ( λn − w ) ∈ H . (cid:3) ODULO p PARABOLIC INDUCTION OF PRO- p -IWAHORI HECKE ALGEBRA 23 Lemma 3.15 ([Oll10, Lemma 5.17, Lemma 5.18]) . Let w ∈ W such that w (cid:54) = w ∆ w ∆ w . Then there exists α ∈ ∆ such that s α w > w , w − ( α ) is not a simple rootand ∆ w = ∆ s α w .Proof. There exists α ∈ ∆ such that s α ww − w > ww − w . If s α w < w , then w − ( α ) <
0. Since w ∆ w w − ( α ) > w − ( α ) ∈ Σ − ∩ Z ∆ w . Hence α ∈ w (Σ − ∩ Z ∆ w ) ⊂ Σ − ,this is a contradiction. Therefore s α w > w . If w − ( α ) is a simple root, then w − ( α ) ∈ ∆ w . Hence w ∆ w w − ( α ) <
0. This is a contradiction. Hence w − ( α ) isnot a simple root. Therefore, for any β ∈ ∆, w ( β ) (cid:54) = ± α . Hence we have w ( β ) > s α w ( β ) >
0. Therefore ∆ w = ∆ s α w . (cid:3) Remark . From this lemma, w ∆ w Θ (cid:48) is the maximal element in { w ∈ W | ∆ w =Θ (cid:48) } . If Θ (cid:48)(cid:48) ⊃ Θ, then w ∆ w Θ (cid:48)(cid:48) < w ∆ w Θ (cid:48) . Hence w ∆ w Θ (cid:48) is also the maximal elementin { w ∈ W | ∆ w ⊃ Θ (cid:48) } . Proof of Theorem 3.13.
We may assume w (cid:48) = w ∆ w ∆ w . By Lemma 3.15, thereexists α , . . . , α l such that • s α i · · · s α l w > s α i +1 · · · s α l w . • ( s α i +1 · · · s α l w ) − ( α i ) is not simple. • w ∆ w ∆ w = s α · · · s α l w .Set s i = s α i . By Proposition 3.2, we have the homomorphism Φ w ∆ w ∆ w ,w : wχ Θ ⊗ A H → w ∆ w ∆ w χ Θ ⊗ A H defined by Φ w ∆ w ∆ w ,w (1 ⊗
1) = 1 ⊗ T ∗ n s ··· sl . This is injectiveby Proposition 3.12. We prove that Φ w ∆ w ∆ w ,w is surjective. Claim.
We have 1 ⊗ T ∗ n s ··· sl − T n w = 0 in w ∆ w ∆ w χ Θ ⊗ A H .It is sufficient to prove that Φ w ∆ ,w ∆ w ∆ w (1 ⊗ T ∗ n s ··· sl − T n w ) = 0 by Proposi-tion 3.12. Put s = s l . Then Φ w ∆ ,w ∆ w ∆ w (1 ⊗ T ∗ n s ··· sl − T n w ) = 1 ⊗ T ∗ n w ∆ w − s T n w .We have T ∗ n w ∆ w − s = E w ∆ w − s ( − ∆) ( n w ∆ w − s ) and T n w = E − ∆ ( n w ). Thereforewe have T ∗ n w ∆ w − s T n w = q / n w ∆ w − s q / n w q − / n w ∆ w − sw E w − s ( − ∆) ( n w ∆ w − s n w ). It is suf-ficient to prove that (cid:96) ( w ∆ w − s ) + (cid:96) ( w ) > (cid:96) ( w ∆ w − sw ). Since the positive root w − ( α l ) is not simple, (cid:96) ( s w − ( α l ) ) >
1. Hence (cid:96) ( w ∆ w − sw ) = (cid:96) ( w ∆ s w − ( α l ) )= (cid:96) ( w ∆ ) − (cid:96) ( s w − ( α l ) ) < (cid:96) ( w ∆ ) −
1= ( (cid:96) ( w ∆ w − ) −
1) + (cid:96) ( w )= (cid:96) ( w ∆ w − s ) + (cid:96) ( w ) . We get the claim.By the claim, 1 ⊗ T ∗ n s ··· sl − c n sl T n w = 1 ⊗ T ∗ n s ··· sl − T n w ( n − w · c n sl ) = 0. Noticethat s l w > w . Hence1 ⊗ T ∗ n s ··· sl T n w = 1 ⊗ T ∗ n s ··· sl − ( T n sl − c n sl ) T n w = 1 ⊗ T ∗ n s ··· sl − T n slw − ⊗ T ∗ n s ··· sl − c n sl T w = 1 ⊗ T ∗ n s ··· sl − T n slw . By induction on l , we get1 ⊗ T ∗ n s ··· sl T n w = 1 ⊗ T n s ··· slw = 1 ⊗ T n w ∆ w ∆ w . Hence 1 ⊗ T n w ∆ w ∆ w ∈ Im Φ w ∆ w ∆ w ,w .Let λ ∈ Λ(1) such that (cid:104) ν ( λ ) , α (cid:105) = 0 for all α ∈ ∆ w and (cid:104) ν ( λ ) , α (cid:105) > α ∈ ∆ \ ∆ w . Since Θ ⊂ ∆ w , we have λ ∈ Λ +Θ (1). Hence χ Θ ( E ( λ )) = τ λ is invertible. Assume that α ∈ Σ + satisfies w ∆ w ∆ w ( α ) <
0. Then α ∈ Σ + \ Z ≥ ∆ w .Therefore (cid:104) ν ( λ ) , α (cid:105) >
0. Hence λ satisfies the condition of Lemma 3.14. Therefore T − n w ∆ w ∆ w E ( n w ∆ w ∆ w ( λ )) ∈ H . Hence τ − λ ⊗ T n w ∆ w ∆ w T − n w ∆ w ∆ w E ( n w ∆ w ∆ w ( λ )) = τ − λ ⊗ E ( n w ∆ w ∆ w ( λ )) = 1 ⊗ w ∆ w ∆ w ,w . Since 1 ⊗ w ∆ w ∆ w χ Θ ⊗ A H , Φ w ∆ w ∆ w ,w is surjective. (cid:3) Corollary 3.17.
Let Θ ⊂ ∆ be a subset and X an irreducible C [Λ Θ (1)] -module.We regard X as an irreducible A -module via χ Θ . Set ∆( X ) = { α ∈ ∆ | (cid:104) Θ , ˇ α (cid:105) = 0 , τ λ ∈ C [Λ Θ (1)] is identity on X for λ ∈ Λ (cid:48) s α (1) } ∪ Θ . Assume that w, w (cid:48) ∈ W satisfies ∆ w ∩ ∆( X ) = ∆ w (cid:48) ∩ ∆( X ) . Then wX ⊗ A H (cid:39) w (cid:48) X ⊗ A H .Proof. First notice that if α ∈ ∆ and λ ∈ Λ (cid:48) s α (1), then ν ( λ ) ∈ R ˇ α . Hence if (cid:104) Θ , ˇ α (cid:105) = 0, (cid:104) Θ , ν ( λ ) (cid:105) = 0. Therefore λ ∈ Λ Θ (1). Hence Λ (cid:48) s α (1) ⊂ Λ Θ (1). So thedefinition makes sense.We may assume that there exists α ∈ ∆ w such that ∆ w = ∆ w (cid:48) (cid:113) { α } . ByTheorem 3.13, we may assume w = w ∆ w ∆ w and w (cid:48) = ws α . Put s = s α , α (cid:48) = w ( α ) ∈ ∆ and s (cid:48) = s α (cid:48) . By the assumption, we have α / ∈ Θ. Hence we cantake a dominant λ ∈ Λ Θ (1) from the center of Λ(1) such that (cid:104) ν ( λ ) , α (cid:105) ≥ λ = λ . Then n w ( λ ) satisfies the conditions of Proposition 3.5 for w and s (cid:48) . Hence we have a homomorphism s (cid:48) w (cid:101) χ ⊗ A H → w (cid:101) χ ⊗ A H and w (cid:101) χ ⊗ A H → s (cid:48) w (cid:101) χ ⊗ A H . These homomorphisms induce s (cid:48) wX ⊗ A H → wX ⊗ A H and wX ⊗ A H → s (cid:48) wX ⊗ A H , respectively. The compositions are induced by X (cid:51) x (cid:55)→ xχ Θ ( E ( λ ) − E ( λµ n s ( − c s c s, − ) ∈ X . Let ϕ : X → X be this homomorphism. We prove thatit is an isomorphism.By the condition, α / ∈ ∆( X ). Hence (cid:104) Θ , ˇ α (cid:105) (cid:54) = 0 or { τ λ | λ ∈ Λ (cid:48) s (1) } is not trivialon X . If (cid:104) Θ , ˇ α (cid:105) (cid:54) = 0, then λµ n − s ( − / ∈ Λ Θ (1). Hence χ Θ ( E ( λµ n − s ( − ϕ ( x ) = xχ Θ ( E ( λ )) = xτ λ on X , which is invertible.Assume that (cid:104) Θ , ˇ α (cid:105) = 0. Then λµ n s ( − ∈ Λ +Θ (1). We have ϕ ( x ) = xτ λ (1 − τ µ n − s ( − c s c s, − ), here we regard C [ Z κ ] as a subalgebra of C [Λ Θ (1)].Assume that there exists t ∈ Λ (cid:48) s (1) ∩ Z κ such that τ t is not identity on X . Namely, Y = { x ∈ X | xτ t = x for any t ∈ Λ (cid:48) s (1) ∩ Z κ } (cid:54) = X . Since Λ (cid:48) s (1) ∩ Z κ ⊂ Λ(1) is anormal subgroup, the subspace Y is C [Λ Θ (1)]-stable. Hence, by the irreducibilityof X , Y = 0. Since (cid:48) s (1) ∩ Z κ ) is prime to p , we have a decomposition X = (cid:76) χ κ X χ κ where χ κ runs characters of Λ (cid:48) s (1) ∩ Z κ . Then the trivial character doesnot appear in X . Therefore, by Lemma 2.15, c s c s, − is zero on X . Hence τ λ − τ λµ ns ( − c s c s, − = τ λ on X , which is invertible.Finally, assume that (cid:104) Θ , ˇ α (cid:105) = 0 and E ( t ) for t ∈ Λ (cid:48) s (1) ∩ Z κ acts trivially on X .Then c s = c s, − = − X . Hence ϕ ( x ) = xτ λ (1 − τ µ ns ( − ). Since the groupΛ (cid:48) s (1) is generated by Λ (cid:48) s (1) ∩ Z κ and µ n − s ( − τ µ n − s ( − is not identity. Hence τ λ (1 − τ µ n − s ( − (cid:54) = 0 on X . Since ϕ is an homomorphism between irreduciblemodules, it is an isomorphism. (cid:3) Classification theorem pro- p -Iwahori Hecke algebra of a Levi subgroup. Let M be a Levisubgroup of a standard parabolic subgroup P such that Z ⊂ M . There is a pro- p -Iwahori Hecke algebra for M . This is an algebra over C [ q / M ,s ] and under thespecialization q M ,s (cid:55)→ H M is isomorphic to the pro- p -Iwahori Hecke algebra of M with the pro- p -Iwahori subgroup I M (1) = I (1) ∩ M ( F ). Later we will construct ODULO p PARABOLIC INDUCTION OF PRO- p -IWAHORI HECKE ALGEBRA 25 an algebra homomorphism C [ q / M ,s ] → C [ q / s ] and we denote the base change ofthe pro- p -Iwahori Hecke algebra of M to C [ q / s ] by H M . Its Iwahori-Matsumotobasis is denoted by T M (cid:101) w and the Bernstein basis is denoted by E M ( (cid:101) w ). The objectsfor M is denoted with the suffix ‘ M ’. The finite Weyl group of M is denoted by W M , etc. We also write ‘ P ’ instead of M , W P = W M , etc.As in [Oll10, 2C4], an element (cid:101) w ∈ (cid:102) W M (1) is called M -positive (resp. M -negative) if (cid:101) w ((Σ + \ Σ + M ) × { } ) ⊂ Σ +aff (resp. (cid:101) w − ((Σ + \ Σ + M ) × { } ) ⊂ Σ +aff ). It iseasy to check that, for λ ∈ Λ(1) and w ∈ W M , λn w ∈ (cid:102) W M (1) is M -positive (resp. M -negative) if and only if (cid:104) ν ( λ ) , α (cid:105) ≤ (cid:104) ν ( λ ) , α (cid:105) ≥
0) for any α ∈ Σ + \ Σ + M .In particular, for w , w ∈ W M and (cid:101) w ∈ W M (1), (cid:101) w is M -positive (resp. M -negative) if and only if n w (cid:101) wn w is M -positive (resp. M -negative). The productof M -positive (resp. M -negative) elements is M -positive (resp. M -negative). Lemma 4.1. If (cid:101) w ∈ W M (1) is M -positive (resp. M -negative) and (cid:101) v ≤ (cid:101) w in (cid:102) W M (1) , then (cid:101) v is M -positive (resp. M -negative).Proof. If (cid:101) w ∈ Λ(1), then it is [Oll12, Fact ii]. In general, let λ ∈ Λ(1) and w ∈ W M such that (cid:101) w = λn w . We prove the lemma by induction on (cid:96) M ( w ). Let s ∈ S M such that ws < w . Then (cid:101) v ≤ λn w implies (cid:101) v or (cid:101) vn s is less than or equal to λn ws .Hence (cid:101) v or (cid:101) vn s is M -positive. Therefore (cid:101) v is M -positive. Since we have (cid:101) v ≤ (cid:101) w ifand only if (cid:101) v − ≤ (cid:101) w − , if (cid:101) w ∈ W M (1) is M -negative, then (cid:101) v is M -negative. (cid:3) Assume that an algebra homomorphism C [ q / M ,s ] → C [ q / s ] is given. Let H + M (resp. H − M ) be the sub C [ q / s ]-module of H M spanned by { T M (cid:101) w } where (cid:101) w ∈ (cid:102) W M (1)is M -positive (resp. M -negative). By the above lemma, H ± M is a subalgebra. Lemma 4.2.
For any set of simple roots ∆ (cid:48) , { E M ∆ (cid:48) ( (cid:101) w ) | (cid:101) w ∈ (cid:102) W M (1) is M -positive (resp. M -negative) } . is a basis of H + M (resp. H − M ).Proof. Since E M ∆ (cid:48) ( (cid:101) w ) ∈ T M (cid:101) w + (cid:80) (cid:101) v< (cid:101) w C [ q / s ] T M (cid:101) v , this follows from Lemma 4.1. (cid:3) We have the Bernstein basis { E ( (cid:101) w ) | (cid:101) w ∈ (cid:102) W (1) } . We need another basis { E − ( (cid:101) w ) | (cid:101) w ∈ (cid:102) W (1) } defined by E − ( n w λ ) = q / n w λ q − / n w T ∗ n w θ ( λ )for w ∈ W and λ ∈ Λ(1).Recall that we have an anti-involution ι of H defined by T (cid:101) w (cid:55)→ T (cid:101) w − . To see thatthis is an anti-involution, we need to check that the braid relations and quadraticrelations are preserved by ι . For the braid relations, let w , w ∈ (cid:102) W (1) such that (cid:96) ( w w ) = (cid:96) ( w ) + (cid:96) ( w ). Then we have (cid:96) (( w w ) − ) = (cid:96) ( w − ) + (cid:96) ( w − ). Hence T ( w w ) − = T w − T w − . Namely the braid relations are preserved by ι . For thequadratic relations, we have T n − s = q s T n − s + T n − s c s by the quadratic relation for n − s and c s T n − s = T n − s c s [Viga, Remark 4.9 (b)]. This means that ι preserves thequadratic relations. Lemma 4.3. E − ( (cid:101) w ) = ι ( E ∆ ( (cid:101) w − )) for (cid:101) w ∈ (cid:102) W (1) . In particular, { E − ( (cid:101) w ) | (cid:101) w ∈ (cid:102) W (1) } is a C [ q / s ] -basis of H .Proof. Let λ ∈ Λ(1) and take anti-dominant λ , λ ∈ Λ(1) such that λ = λ λ − .Then we have E ∆ ( λ − ) = E ∆ ( λ λ − ) = q − / λ q − / λ q / λ E ∆ ( λ ) E ∆ ( λ − ) by Proposition 2.8. By Proposition 2.7, we have E ∆ ( λ ) = T ∗ λ = q λ T − λ − and E ∆ ( λ − ) = T λ − . Hence we get E ∆ ( λ − ) = q / λ q − / λ q / λ T − λ − T λ − . Therefore ι ( E ∆ ( λ − )) = q / λ q − / λ q / λ T λ T − λ . By the definition of θ ( λ ) and Proposition 2.7, we have T λ = q / λ θ ( λ ) and T λ = q / λ θ ( λ ). Hence ι ( E ∆ ( λ − )) = q / λ θ ( λ ) θ ( λ ) − = q / λ θ ( λ λ − ) = q / λ θ ( λ ) . Now let w ∈ W . Then we have E ∆ ( λ − n − w ) = q − / λ q − / w q / n w λ E ∆ ( λ − ) E ∆ ( n − w )(Proposition 2.8). We have E ∆ ( n − w ) = T ∗ n − w = q n w T − n w by Proposition 2.7. Hence ι ( E ∆ ( n − w )) = q n w T − n − w = T ∗ n w . Therefore we get ι ( E ∆ (( n w λ ) − )) = q − / λ q − / n w q / n w λ ι ( E ∆ ( n − w )) ι ( E ∆ ( λ − ))= q − / n w q / n w λ T ∗ n w θ ( λ ) = E − ( n w λ ) . (cid:3) Hence { E − ( (cid:101) w ) | (cid:101) w ∈ (cid:102) W (1) } is a C [ q / s ]-basis of H . We have E ∆ ( (cid:101) w − ) ∈ T (cid:101) w − + (cid:80) (cid:101) v< (cid:101) w C [ q / s ] T (cid:101) v − . Applying ι , we get E − ( (cid:101) w ) ∈ T (cid:101) w + (cid:88) (cid:101) v< (cid:101) w C [ q / s ] T (cid:101) v = E ( (cid:101) w ) + (cid:88) (cid:101) v< (cid:101) w C [ q / s ] E ( (cid:101) v ) . By the definition, for w ∈ W and λ, λ ∈ Λ(1), we have E − ( n w λ ) E ( λ ) = q / n w λ q / λ q − / n w λλ E − ( n w λλ ) . In particular, in
H ⊗ C , we get(4.1) E − ( n w λ ) E ( λ ) = (cid:40) E − ( n w λλ ) ( (cid:96) ( n w λ ) + (cid:96) ( λ ) = (cid:96) ( n w λλ )) , . We want to construct an embedding H ± M (cid:44) → H . To do this, we have to relate { q s | s ∈ S aff / ∼} with { q M ,s | s ∈ S M , aff / ∼} . For (cid:101) α ∈ Σ aff , take a simple affine root (cid:101) β ∈ (cid:102) W (cid:101) α from the (cid:102) W -orbit through (cid:101) α and put q ( (cid:101) α ) = q s (cid:101) β . Then it does not dependon (cid:101) β . For a finite subset A ⊂ Σ aff , put q ( A ) = (cid:81) (cid:101) α ∈ A q ( (cid:101) α ). By induction on (cid:96) ( (cid:101) w ),we have q (cid:101) w = q (Σ +aff ∩ (cid:101) w (Σ − aff )). Now we define a homomorphism C [ q / M ,s ] → C [ q / s ]by q M , (cid:101) w (cid:55)→ q (Σ + M , aff ∩ (cid:101) w (Σ − M , aff )) for (cid:101) w ∈ (cid:102) W M (1). The motivation of this definitionis the following. Lemma 4.4.
Under the specialization q s (cid:55)→ I (1) sI (1) /I (1)) , we have q M , (cid:101) w (cid:55)→ I M (1) (cid:101) wI M (1) /I M (1)) .Proof. The proof of Corollary 3.31 and Proposition 3.28 in [Viga] is applicable. (cid:3)
We have the following relation between q s and q M ,s . Lemma 4.5. If (cid:101) w , (cid:101) w ∈ (cid:102) W M (1) are M -positive (resp. M -negative), then we have q − (cid:101) w (cid:101) w q (cid:101) w q (cid:101) w = q − M , (cid:101) w (cid:101) w q M , (cid:101) w q M , (cid:101) w . In particular, we have (cid:96) M ( (cid:101) w ) + (cid:96) M ( (cid:101) w ) − (cid:96) M ( (cid:101) w (cid:101) w ) = (cid:96) ( (cid:101) w ) + (cid:96) ( (cid:101) w ) − (cid:96) ( (cid:101) w (cid:101) w ) . ODULO p PARABOLIC INDUCTION OF PRO- p -IWAHORI HECKE ALGEBRA 27 Proof.
Put C ± = Σ ± aff \ Σ ± M , aff . By the definition, we have q (cid:101) w = q M , (cid:101) w q ( C + ∩ (cid:101) wC − )for (cid:101) w ∈ (cid:102) W (1). Hence it is sufficient to prove that q ( C + ∩ (cid:101) w (cid:101) w C − ) = q ( C + ∩ (cid:101) w C − ) q ( C + ∩ (cid:101) w C − ) . Define A ± = C ± ∩ ((Σ + \ Σ + M ) × Z ) and B ± = C ± ∩ ((Σ − \ Σ − M ) × Z ). Thenwe have C ± = A ± ∪ B ± . For λ ∈ Λ(1) and w ∈ W M , we have λn w ( α, k ) =( w ( α ) , n − (cid:104) w ( α ) , ν ( λ ) (cid:105) ). If w ∈ W M and α / ∈ Σ M , we have w ( α ) > α >
0. Hence if (cid:101) w ∈ W M (1), we have (cid:101) w ( A + ∪ A − ) = A + ∪ A − and (cid:101) w ( B + ∪ B − ) = B + ∪ B − . Hence C + ∩ (cid:101) wC − = ( A + ∩ (cid:101) wA − ) ∪ ( B + ∩ (cid:101) wB − ). If λn w is M -positive then for α ∈ Σ − \ Σ − M , we have (cid:104) α, ν ( λ ) (cid:105) ≥
0. Hence (cid:101) w = λn w satisfies (cid:101) w ( B − ) ⊂ B − . Therefore C + ∩ (cid:101) wC − = A + ∩ (cid:101) wA − . We also have (cid:101) w ( A + ) ⊂ A + and (cid:101) w ( A − ) ⊃ A − which will be used later.Let (cid:101) w , (cid:101) w ∈ (cid:102) W M (1) be M -positive elements. Then we have A + ∩ (cid:101) w (cid:101) w A − = ( A + ∩ (cid:101) w (cid:101) w A − ∩ (cid:101) w A + ) (cid:113) ( A + ∩ (cid:101) w (cid:101) w A − ∩ (cid:101) w A − ) . Since (cid:101) w A + ⊂ A + , we have A + ∩ (cid:101) w (cid:101) w A − ∩ (cid:101) w A + = (cid:101) w (cid:101) w A − ∩ (cid:101) w A + = (cid:101) w ( (cid:101) w A − ∩ A + ). Since A − ⊂ (cid:101) w A − , we have A + ∩ (cid:101) w (cid:101) w A − ∩ (cid:101) w A − = A + ∩ (cid:101) w A − . Hence A + ∩ (cid:101) w (cid:101) w A − = (cid:101) w ( A + ∩ (cid:101) w A − ) (cid:113) ( A + ∩ (cid:101) w A − ) . Therefore, q ( A + ∩ (cid:101) w (cid:101) w A − ) = q ( (cid:101) w ( A + ∩ (cid:101) w A − )) q ( A + ∩ (cid:101) w A − ) . Since q ( (cid:101) w ( A + ∩ (cid:101) w A − )) = q ( A + ∩ (cid:101) w A − ), we get the lemma if (cid:101) w , (cid:101) w are M -positive. Since q (cid:101) w − = q (cid:101) w and q M , (cid:101) w − = q M , (cid:101) w . the lemma holds if (cid:101) w , (cid:101) w are M -negative. (cid:3) We define a C [ q / s ]-module homomorphism j + M : H + M → H (resp. j − M : H − M →H ) by j + M ( T M (cid:101) w ) = T (cid:101) w (resp. j − M ( T M , ∗ (cid:101) w ) = T ∗ (cid:101) w )for M -positive (resp. M -negative) (cid:101) w ∈ (cid:102) W M (1). Lemma 4.6.
The C [ q / s ] -module homomorphisms j ± M are homomorphisms be-tween C [ q / s ] -algebras. We have j + M ( E M ( (cid:101) w )) = E ( (cid:101) w ) , j + M ( E M − ( (cid:101) w )) = E − ( (cid:101) w ) for M -positive (cid:101) w ∈ (cid:102) W M (1) and j − M ( E M ( (cid:101) w )) = E ( (cid:101) w ) , j − M ( E M − ( (cid:101) w )) = E − ( (cid:101) w ) for M -negative (cid:101) w ∈ (cid:102) W M (1) .Proof. We prove j + M ( T M (cid:101) w T M (cid:101) w ) = T (cid:101) w T (cid:101) w for M -positive elements (cid:101) w , (cid:101) w ∈ (cid:102) W M (1).By induction on (cid:96) M ( (cid:101) w ), we may assume (cid:96) M ( (cid:101) w ) = 0 or (cid:101) w = n s for s ∈ S M , aff . If (cid:96) M ( (cid:101) w ) = 0, or (cid:101) w = n s and (cid:101) w n s > (cid:101) w , then (cid:96) M ( (cid:101) w ) + (cid:96) M ( (cid:101) w ) − (cid:96) M ( (cid:101) w (cid:101) w ) = 0.Hence T M (cid:101) w T M (cid:101) w = T M (cid:101) w (cid:101) w . We have (cid:96) ( (cid:101) w ) + (cid:96) ( (cid:101) w ) − (cid:96) ( (cid:101) w (cid:101) w ) = 0 by Lemma 4.5.Therefore T (cid:101) w T (cid:101) w = T (cid:101) w (cid:101) w . We have j + M ( T M (cid:101) w T M (cid:101) w ) = T (cid:101) w T (cid:101) w .If (cid:101) w = n s and (cid:101) w n s < (cid:101) w , then we have T M (cid:101) w T M n s = T M (cid:101) w n − s ( q M ,s T n s − T n s c s ) = q M ,s T M (cid:101) w n s − T M (cid:101) w c s . Therefore j + M ( T M (cid:101) w T M (cid:101) w ) = q M ,s T (cid:101) w n s − T (cid:101) w c s . We have (cid:96) ( n s ) = (cid:96) M ( n s ) = 0. Hence 2 (cid:96) ( n s ) = (cid:96) ( n s ) + (cid:96) ( n s ) − (cid:96) ( n s ) = (cid:96) M ( n s ) + (cid:96) M ( n s ) − (cid:96) M ( n s ) =2 by Lemma 4.5. Therefore, (cid:96) ( n s ) = 1. Namely, s ∈ S aff . Therefore, by thedefinition of q M ,s , we have q M ,s = q s . We also have (cid:96) ( (cid:101) w ) + (cid:96) ( n s ) − (cid:96) ( (cid:101) w n s ) = (cid:96) M ( (cid:101) w )+ (cid:96) M ( n s ) − (cid:96) M ( (cid:101) w n s ) = 2. Hence by the same calculation implies T (cid:101) w T n s = q s T (cid:101) w n s − T (cid:101) w c s . Therefore j + M ( T M (cid:101) w T M (cid:101) w ) = T (cid:101) w T (cid:101) w . Hence j + M is an algebrahomomorphism. By the same argument, j − M is an algebra homomorphism.By the argument in the proof of [Oll10, Proposition 4.7], we have j + M ( E M ( (cid:101) w )) = E ( (cid:101) w ) for M -positive (cid:101) w ∈ (cid:102) W M (1) and j − M ( E M ( λ )) = E ( λ ) for M -negative λ ∈ Λ(1). If s ∈ S M , then s ∈ S . Hence j − M ( T M n s ) = j − M ( T M , ∗ n s + c s ) = T ∗ n s + c s = T n s . Therefore j − M ( T M n w ) = T n w for w ∈ W M . In other words, we have j − M ( E M ( n w )) = E ( n w ). Since j − M is an algebra homomorphism, for M -negative λ ∈ Λ(1) and w ∈ W M , we have j − M ( E M ( λn w )) = q − / M ,λ q − / M ,n w q / M ,λn w j − M ( E M ( λ ) E M ( n w ))= q − / λ q − / n w q / λn w E ( λ ) E ( n w )= E ( λn w )by Lemma 4.5. A similar argument implies j − M ( E M − ( (cid:101) w )) = E − ( (cid:101) w ) for M -negative (cid:101) w ∈ (cid:102) W M (1) and j + M ( E M − ( (cid:101) w )) = E − ( (cid:101) w ) for M -positive (cid:101) w ∈ (cid:102) W M (1). (cid:3) Remark . Let λ +0 ∈ Λ(1) be an M -positive element such that it is in the centerof (cid:102) W M (1) and (cid:104) α, ν ( λ +0 ) (cid:105) < α ∈ Σ + \ Σ + M . Then H M is a localization E M ( λ +0 ) − H + M of H + M at E M ( λ +0 ).We give a proof. First we prove that E M ( λ +0 ) is in the center of H M . Since λ +0 is in the center of (cid:102) W M (1), we have n s α λ +0 n − s α = λ +0 for any α ∈ Φ M . Hence s α ( ν ( λ +0 )) = λ +0 , namely (cid:104) α, ν ( λ +0 ) (cid:105) = 0. Therefore by the length formula (2.1), wehave the length of λ +0 as an element of (cid:102) W M (1) is zero. Hence for (cid:101) w ∈ (cid:102) W M (1), wehave E M ( λ +0 ) E M ( (cid:101) w ) = E M ( λ +0 (cid:101) w ) = E M ( (cid:101) wλ +0 ) = E M ( (cid:101) w ) E M ( λ +0 ). Therefore E M ( λ +0 ) is in the center of H M and since the length of λ +0 is zero, it is invertible.By the above argument, we have E M ( λ +0 ) E M ( (cid:101) w ) = E M ( λ +0 (cid:101) w ) for (cid:101) w ∈ (cid:102) W M (1).Moreover, for any (cid:101) w ∈ (cid:102) W M (1), there exists n ∈ Z ≥ such that ( λ +0 ) n (cid:101) w is M -positive. Therefore E M ( (cid:101) w ) = E M ( λ +0 ) − n E M (( λ +0 ) n (cid:101) w ) is in the image of thehomomorphism E M ( λ +0 ) − H + M → H M . Hence we have E M ( λ +0 ) − H + M (cid:39) H M .In particular, if σ is an H M -module, it is determined by its restriction to H + M .Since E M ( λ +0 ) is in the center of H M , if σ is irreducible, E M ( λ +0 ) acts by a scalaron σ . Hence if σ is irreducible then its restriction to H + M is also irreducible.This is also true for H − M . Namely, for λ − such that it is in the center of (cid:102) W M (1)and (cid:104) ν ( λ − ) , α (cid:105) > α ∈ Σ + \ Σ + M , we have E M ( λ − ) − H − M (cid:39) H M .4.2. Parabolic induction.Definition 4.8.
Let P = M N be a parabolic subgroup of G . For a representation σ of H M ⊗ C , define I P ( σ ) = Hom H − M ( H , σ ) . Here, H − M acts on H by the multiplication from the right through j − M .Set W M = { w ∈ W | w (∆ M ) ⊂ Σ + } . This is a complete representative of W/W M and for w ∈ W M and w ∈ W M , we have (cid:96) ( w w ) = (cid:96) ( w ) + (cid:96) ( w ). Lemma 4.9.
Let λ ∈ Λ(1) , w ∈ W M and w ∈ W M . Assume that λ is M -negative. Then (cid:96) ( n w w λ ) = (cid:96) ( n w ) + (cid:96) ( n w λ ) . In particular, T ∗ n w E − ( n w λ ) = E − ( n w w λ ) .Proof. By the length formula (2.1), we have (cid:96) ( n w w λ ) = (cid:96) ( λ ) + (cid:96) ( n w w ) − { α ∈ Σ + | ( w w )( α ) < , (cid:104) ν ( λ ) , α (cid:105) < } . We have (cid:96) ( w w ) = (cid:96) ( w ) + (cid:96) ( w ). Hence for α ∈ Σ + , ( w w )( α ) < w ( α ) < α = w − ( β ) for some β ∈ Σ + such that w ( β ) <
0. Assume that α = w − ( β ) for some β ∈ Σ + such that w ( β ) <
0. Since w ( β ) < β ∈ Σ + \ Σ + M by the definition of W M . Hence α = w − ( β ) ∈ Σ + \ Σ + M . Since λ is M -negative, (cid:104) ν ( λ ) , α (cid:105) ≥
0. Therefore, { α ∈ Σ + | ( w w )( α ) < , (cid:104) ν ( λ ) , α (cid:105) < } = { α ∈ Σ + | w ( α ) < , (cid:104) ν ( λ ) , α (cid:105) < } . ODULO p PARABOLIC INDUCTION OF PRO- p -IWAHORI HECKE ALGEBRA 29 Hence (cid:96) ( n w w λ ) = (cid:96) ( λ ) + (cid:96) ( n w w ) − { α ∈ Σ + | w ( α ) < , (cid:104) ν ( λ ) , α (cid:105) < } = (cid:96) ( λ ) + (cid:96) ( n w ) + (cid:96) ( n w ) − { α ∈ Σ + | w ( α ) < , (cid:104) ν ( λ ) , α (cid:105) < } = (cid:96) ( n w λ ) + (cid:96) ( n w ) . (cid:3) Lemma 4.10 (See [Oll10, Proposition 5.2]) . The map ϕ (cid:55)→ ( ϕ ( T n w )) w ∈ W M givesan isomorphism I P ( σ ) (cid:39) (cid:76) w ∈ W M σ as vector spaces.Proof. We may replace ϕ (cid:55)→ ϕ ( T n w ) with ϕ (cid:55)→ ϕ ( T ∗ n w ). Take λ − ∈ Λ(1) as in Re-mark 4.7. Then ϕ is extended to H E ( λ − ) − → σ . Fix v ∈ W M . Then by the Bern-stein relation and the definition of E − ( (cid:101) w ), H v = (cid:76) w ∈ W M ,λ ∈ Λ(1) CE − ( n vw λ ) ⊂H ⊗ C is right j − M ( H − M )-stable. We prove Φ : H − M (cid:51) F (cid:55)→ T ∗ n v j − M ( F ) ∈ H v induces an isomorphism Φ : H M ⊗ C (cid:39) H v E ( λ − ) − .Let w ∈ W M and λ ∈ Λ(1) such that λ is M -negative. Then by Lemma 4.9, wehave T ∗ n v E − ( n w λ ) = E − ( n vw λ ). Hence Φ is injective. Therefore Φ is injective.We prove surjectivity. Let w ∈ W M and λ ∈ Λ(1). Take n ∈ Z ≥ such that λ ( λ − ) n is M -negative. If (cid:96) ( n vw λ ( λ − ) n ) = (cid:96) ( n vw λ ) + (cid:96) (( λ − ) n ), then E − ( n vw λ ) = E − ( n vw λ ( λ − ) n ) E ( λ − ) − n = T ∗ n v E − ( n w λ ( λ − ) n ) E ( λ − ) − n ∈ Im Φ by Lemma 4.9.If (cid:96) ( n vw λ ( λ − ) n ) > (cid:96) ( n vw λ ) + (cid:96) (( λ − ) n ), then E − ( n vw λ ) E ( λ − ) n = 0. Hence, in H v E ( λ − ) − , E ( n vw λ ) = 0.Therefore ( H ⊗ C ) E ( λ − ) − = (cid:76) v ∈ W M H v E ( λ − ) − (cid:39) (cid:76) v ∈ W M ( H M ⊗ C ).Since H M = H − M E M ( λ − ) − by Remark 4.7, we have I P ( σ ) = Hom H − M ( H , σ ) =Hom H M (( H ⊗ C ) E ( λ − ) − , σ ) (cid:39) (cid:76) v ∈ W M Hom H M ( H M ⊗ C, σ ) (cid:39) (cid:76) v ∈ W M σ . (cid:3) For w ∈ W M , we denote the subspace { ϕ ∈ I P ( σ ) | ϕ ( T n v ) = 0 ( v ∈ W M \{ w } ) } by wσ . We have I P ( σ ) = (cid:76) w ∈ W M wσ . By the lemma below, wσ is A -stable. Remark . Let v ∈ W M , ϕ ∈ vσ ⊂ I P ( σ ) and λ ∈ Λ(1). From the proof of theabove lemma, if w ∈ { v ∈ W M | v < v } W M then ϕ ( E − ( n w λ )) = 0. We remarkthat if w is in this subset and w ≤ w , then w is also in this subset. (It followsfrom [Abe13, Lemma 4.20]. See also the proof of Lemma 4.13.)We can describe the A -module structure of I P ( σ ) using the above decomposition. Proposition 4.12.
Let σ be an H M ⊗ C -module. Define an A -module σ A by • σ = σ A as vector spaces. • for M -negative λ ∈ Λ(1) we have σ A ( E ( λ )) = σ ( E M ( λ )) . • for λ ∈ Λ(1) which is not M -negative, σ A ( E ( λ )) = 0 .Then ϕ (cid:55)→ ( ϕ ( T n w )) w ∈ W M gives an isomorphism I P ( σ ) | A (cid:39) (cid:76) w ∈ W M wσ A .Proof. We prove ( ϕE ( λ ))( T n w ) = ϕ ( T n w ) σ A ( E ( n − w ( λ ))) by induction on (cid:96) ( w ).If w = 1, then ( ϕE ( λ ))(1) = ϕ ( E ( λ )). If λ is M -negative, then ϕ ( E ( λ )) = ϕ (1) E M ( λ ) = ϕ (1) σ A ( E ( λ )). Assume that λ is not M -negative. Take λ − ∈ Λ(1)as in Remark 4.7. Since λ is not M -negative, ν ( λ ) and ν ( λ − ) do not belong tothe same closed chamber. Hence E ( λ ) E ( λ − ) = 0 by (3.1). Therefore ϕ ( E ( λ )) = ϕ ( E ( λ ) E ( λ − )) E M (( λ − ) − ) = 0 = ϕ (1) σ A ( E ( λ )).Assume that w (cid:54) = 1 and take α ∈ ∆ such that s = s α satisfies sw < w . Since { β ∈ Σ + | sw ( β ) < } ⊂ { β ∈ Σ + | w ( β ) < } , sw ∈ W M . Since w ( − w − ( α )) = − α < w ∈ W M , we have − w − ( α ) ∈ Σ + \ Σ + M . Assume that (cid:104) ν ( λ ) , α (cid:105) > Then by Lemma 2.13, E ( λ ) T n s = ( T n s − c s ) E ( n − s ( λ )). Hence( ϕE ( λ ))( T n w ) = ϕ ( E ( λ ) T n w )= ϕ ( E ( λ ) T n s T n sw )= ϕ (( T n s − c s ) E ( n − s ( λ )) T n sw )= ( ϕ ( T n s − c s ) E ( n − s ( λ )))( T n sw ) . Applying the inductive hypothesis to ϕ ( T n s − c s ), ( ϕ ( T n s − c s ) E ( n − s ( λ )))( T n sw ) =( ϕ ( T n s − c s ))( T n sw ) σ A ( E ( n − w ( λ ))). Since (cid:104) ν ( n − w ( λ )) , − w − ( α ) (cid:105) = −(cid:104) ν ( λ ) , α (cid:105) < n − w ( λ ) is not M -negative. Hence σ A ( E ( n − w ( λ ))) = 0. Therefore ( ϕ ( T n s − c s ))( T n sw ) σ A ( E ( n − w ( λ ))) = 0 = ϕ ( T n w ) σ A ( E ( n − w ( λ ))).If (cid:104) ν ( λ ) , α (cid:105) = 0, then E ( λ ) T n s = T n s E ( n − s ( λ )) [Viga, Lemma 5.34, 5.35]. Hence( ϕE ( λ ))( T n w ) = ϕ ( E ( λ ) T n s T n sw )= ϕ ( T n s E ( n − s ( λ )) T n sw )= ϕ ( T n w ) σ A ( E ( n − w ( λ ))) . Finally, assume that (cid:104) ν ( λ ) , α (cid:105) <
0. Then (cid:104) ν ( n − sw ( λ )) , − w − ( α ) (cid:105) = (cid:104) ν ( λ ) , α (cid:105) < n − sw ( λ ) is not M -negative. Therefore, we have ϕ ( E ( λ ) c s T n sw ) = ( ϕ ( λ · c s ) E ( λ ))( T n sw ) = ( ϕ ( λ · c s ))( T n sw ) σ A ( E ( n − sw ( λ ))) = 0. By Lemma 2.13, E ( λ )( T n s − c s ) = T n s E ( n − s ( λ )). Hence( ϕE ( λ ))( T n w ) = ϕ ( E ( λ )( T n s − c s ) T n sw )= ( ϕT n s )( E ( n − s ( λ )) T n sw )= ϕ ( T n w ) σ A ( E ( n − w ( λ ))) . (cid:3) Another description of parabolic induction.
We give a realization ofparabolic induction via a tensor product. First we find a submodule of a pro- p -Iwahori Hecke algebra of a Levi subgroup (which is not M in general) in I P ( σ ).Set P (cid:48) = n w ∆ w ∆ M P n − w ∆ w ∆ M . This is a standard parabolic subgroup of G corresponding to w ∆ w ∆ M (∆ M ) = − w ∆ (∆ M ). Let M (cid:48) be the Levi part of P .Then the map m (cid:55)→ n w ∆ w ∆ M mn − w ∆ w ∆ M is an isomorphism between M and M (cid:48) .Moreover, this map preserves the maximal split torus, the pro- p -Iwahori subgroup,the minimal parabolic subgroup, the root system, the set of simple roots etc...Since the pro- p -Iwahori Hecke algebra is defined by these data, this map inducesan isomorphism H M → H M (cid:48) . Explicitly it is given by T M (cid:101) w (cid:55)→ T M (cid:48) n w ∆ w ∆ M (cid:101) wn − w ∆ w ∆ M .We also have E M − ( (cid:101) w ) (cid:55)→ E M (cid:48) − ( n w ∆ w ∆ M (cid:101) wn − w ∆ w ∆ M ). Notice that w ∆ w ∆ M ∈ W M . Lemma 4.13.
Let v ∈ W . (1) We have W \ w ∆ w ∆ M W M = { v ∈ W M | v < w ∆ w ∆ M } W M . (2) If v < w ∆ w ∆ M , then v / ∈ w ∆ w ∆ M W M . (3) If v / ∈ w ∆ w ∆ M W M and v (cid:48) ≤ v , then v (cid:48) / ∈ w ∆ w ∆ M W M .Proof. Let w ∈ W and take w ∈ W M and w ∈ W M such that w = w w . If w < w ∆ w ∆ M , then, since W/W M (cid:39) W M , w W M ∩ w ∆ w ∆ M W M = ∅ . Hence w / ∈ w ∆ w ∆ M W M . Assume that w / ∈ w ∆ w ∆ M W M . Since w ≤ w ∆ = w ∆ w ∆ M w ∆ M and w ∆ w ∆ M ∈ W M , we have w ≤ w ∆ w ∆ M by [Abe13, Lemma 4.20]. If w = w ∆ w ∆ M ,then w ∈ w ∆ w ∆ M W M . Hence w < w ∆ w ∆ M . We get (1).Assume that v ∈ w ∆ w ∆ M W M and take v (cid:48) ∈ W M such that v = w ∆ w ∆ M v (cid:48) .Since w ∆ w ∆ M ∈ W M and v (cid:48) ∈ W M , we have (cid:96) ( w ∆ w ∆ M v (cid:48) ) = (cid:96) ( w ∆ w ∆ M ) + (cid:96) ( v (cid:48) ).Hence w ∆ w ∆ M ≤ w ∆ w ∆ M v (cid:48) = v . We get (2).We prove (3). Take v ∈ W M and v ∈ W M such that v = v v . By (1), v Let σ be an H M -module and define an H M (cid:48) -module σ (cid:48) bypulling back σ by the above isomorphism. (Namely it is given by σ (cid:48) ( E M (cid:48) − ( (cid:101) w )) = σ ( E M − ( n − w ∆ w ∆ M (cid:101) wn w ∆ w ∆ M )) for (cid:101) w ∈ (cid:102) W M (cid:48) (1) . Then w ∆ w ∆ M σ ⊂ I ( P , σ, Q ) is H + M (cid:48) -stable and isomorphic to σ (cid:48) . In particular, if σ is irreducible and an H -submodule π of I P ( σ ) has a non-zero intersection with w ∆ w ∆ M σ , then π contains w ∆ w ∆ M σ .Proof. Let w ∈ W M , λ ∈ Λ(1). Put w = w ∆ w ∆ M w ( w ∆ w ∆ M ) − and λ = n w ∆ w ∆ M ( λ ). We prove ϕ ( E − ( n w λ ) T n w ∆ w ∆ M ) = ϕ ( T n w ∆ w ∆ M E − ( n w λ )) for ϕ ∈ w ∆ w ∆ M σ Since w ∈ W M and w ∆ w ∆ M ∈ W M , we have (cid:96) ( w ∆ w ∆ M )+ (cid:96) ( w ) = (cid:96) ( w ∆ w ∆ M w ).Hence n w ∆ w ∆ M n w = n w ∆ w ∆ M w . We have w ∆ w ∆ M w = w w ∆ w ∆ M and by the sameargument implies n w w ∆ w ∆ M = n w n w ∆ w ∆ M . Hence n w ∆ w ∆ M n w = n w n w ∆ w ∆ M .Therefore n − w n w ∆ w ∆ M = n w ∆ w ∆ M n − w . Since w − ∈ W M and w ∆ w ∆ M ∈ W M ,we also have (cid:96) ( n w ∆ w ∆ M n − w ) = (cid:96) ( n w ∆ w ∆ M ) + (cid:96) ( n − w ). Hence T n w ∆ w ∆ M n − w = T n w ∆ w ∆ M T n − w . The same argument implies T n − w n w ∆ w ∆ M = T n − w T n w ∆ w ∆ M . Hencein H , we have T n w ∆ w ∆ M T n − w = T n − w T n w ∆ w ∆ M . Therefore we get T − n − w T n w ∆ w ∆ M = T n w ∆ w ∆ M T − n − w . The definition of T ∗ n w is T ∗ n w = q n w T − n − w . Hence T ∗ n w T n w ∆ w ∆ M = T n w ∆ w ∆ M T ∗ n w . Therefore, T n w ∆ w ∆ M E − ( n w λ ) − E − ( n w λ ) T n w ∆ w ∆ M ∈ C [ q ± / s ]( T n w ∆ w ∆ M T ∗ n w θ ( λ ) − T ∗ n w θ ( λ ) T n w ∆ w ∆ M )= C [ q ± / s ] T ∗ n w ( T n w ∆ w ∆ M θ ( λ ) − θ ( λ ) T n w ∆ w ∆ M )By the Bernstein relation, we have T n w ∆ w ∆ M θ ( λ ) − θ ( λ ) T n w ∆ w ∆ M ∈ (cid:88) v Keep the notation in Proposition 4.14. Then we have σ (cid:48) ⊗ H + M (cid:48) H (cid:39) I P ( σ ) .Proof. We get Φ : σ (cid:48) ⊗ H + M (cid:48) H → I P ( σ ) by Proposition 4.14. Put M (cid:48) W = { w − | w ∈ W M (cid:48) } . Then for w ∈ W M (cid:48) and w ∈ M (cid:48) W , we have (cid:96) ( w w ) = (cid:96) ( w ) + (cid:96) ( w ).By a similar argument of the proof of Lemma 4.10, the homomorphism ( x w ) w (cid:55)→ (cid:80) w x w ⊗ T ∗ n w gives an isomorphism (cid:76) w ∈ M (cid:48) W σ (cid:48) (cid:39) σ (cid:48) ⊗ H + M (cid:48) H . By the construction,Φ induces an isomorphism σ (cid:48) (cid:39) σ (cid:48) ⊗ → w ∆ w ∆ M σ ⊂ I P ( σ ).First we prove that v (cid:55)→ w ∆ w ∆ M v − gives a bijection W M → M (cid:48) W . Wehave w ∆ M w ∆ (∆ M (cid:48) ) = w ∆ M ( − ∆ M ) = ∆ M . Hence ( w ∆ w ∆ M v − ) − (∆ M (cid:48) ) ⊂ v (∆ M ) ⊂ Σ + . Therefore the map is well-defined. The inverse map is given by w (cid:55)→ w − w ∆ w ∆ M .We prove that the homomorphism Φ induces an isomorphism σ (cid:48) ⊗ T ∗ n w ∆ w ∆ M v − (cid:39) vσ for v ∈ W M . To do it, it is sufficient to prove the following: Let w ∈ M (cid:48) W and v ∈ W M . Take x ∈ σ (cid:48) and put ϕ = Φ( x ⊗ T ∗ n w ). Then we have ϕ ( T n v ) = x if wv = w ∆ w ∆ M and if not, ϕ ( T n v ) = 0.Put ϕ (cid:48) = Φ( x ⊗ ϕ ( T n v ) = ϕ (cid:48) ( T ∗ n w T n v ). We have T ∗ n w T n v = E w ( − ∆) ( n w ) E − ∆ ( n v ) = ( q − n w n v q n w q n v ) / E w ( − ∆) ( n w n v ) . Hence it is zero if (cid:96) ( w ) + (cid:96) ( v ) > (cid:96) ( wv ). Assume that (cid:96) ( w ) + (cid:96) ( v ) = (cid:96) ( wv ). Then T ∗ n w T n v = E w ( − ∆) ( n w n v ) = E w ( − ∆) ( n wv ) ∈ T n wv + (cid:80) v (cid:48) In this subsection, assume that an orthogonal de-composition ∆ = ∆ (cid:113) ∆ is given. Then we have the decomposition S = S (cid:113) S and S aff = S aff , (cid:113) S aff , . Let M (resp. M ) be the Levi subgroup correspondingto ∆ (resp. ∆ ).To formulate the classification theorem, we need the following proposition. Proposition 4.16. Let σ be an H M ⊗ C -module such that E M ( λ ) acts triviallyfor λ ∈ Λ (cid:48) M (1) . Then there exists a unique H ⊗ C -module e ( σ ) such that: • as an H − M -module, e ( σ ) = σ . • for s ∈ S aff , , T n s is zero on e ( σ ) . • T t acts trivially for t ∈ Λ (cid:48) M (1) ∩ Z κ . We call the module e ( σ ) the extension of σ .We summarize some facts which come from the orthogonal decomposition. Wehave a decomposition as Coxeter groups (cid:102) W aff = (cid:102) W M , aff × (cid:102) W M , aff . Put (cid:96) M i ( (cid:101) w ) = + M i , aff ∩ (cid:101) w (Σ − M i , aff )) for i = 1 , 2. Since Σ ± aff = Σ ± M , aff (cid:113) Σ ± M , aff and the actionof (cid:102) W preserves this decomposition, we have (cid:96) ( (cid:101) w ) = (cid:96) M ( (cid:101) w ) + (cid:96) M ( (cid:101) w ). If (cid:101) w ∈ W M i (1), then (cid:96) M i ( (cid:101) w ) is the usual length of (cid:101) w . It is easy to see that (cid:96) M i ( (cid:101) w − ) = (cid:96) M i ( (cid:101) w ) for i = 1 , 2. If (cid:96) M ( (cid:101) w ) = 0, then (cid:101) w (Σ − M , aff ) ⊂ Σ − M , aff . By (cid:96) M ( (cid:101) w − ) =0, we have (cid:101) w − (Σ − M , aff ) ⊂ Σ − M , aff , hence (cid:101) w (Σ − M , aff ) ⊃ Σ − M , aff . Therefore (cid:101) w (Σ − M , aff ) = Σ − M , aff . Hence for any (cid:101) w ∈ (cid:102) W (1), we have (cid:101) w (cid:101) w (Σ − M , aff ) = (cid:101) w (Σ − M , aff ). We get (cid:96) M ( (cid:101) w (cid:101) w ) = (cid:96) M ( (cid:101) w ). If (cid:101) w ∈ (cid:102) W M , aff (1), then its im-age in (cid:102) W is contained in (cid:102) W M , aff . The group (cid:102) W M , aff is generated by S aff , and, since the decomposition is orthogonal, (cid:102) W M , aff acts trivially on Σ M , aff .Hence (cid:96) M ( (cid:101) w ) = 0 for (cid:101) w ∈ (cid:102) W M , aff (1). Set Ω = { (cid:101) w ∈ (cid:102) W | (cid:96) ( (cid:101) w ) = 0 } . If (cid:101) w ∈ Ω, then (cid:96) M ( (cid:101) w ) + (cid:96) M ( (cid:101) w ) = (cid:96) ( (cid:101) w ) = 0. Hence (cid:96) M ( (cid:101) w ) = (cid:96) M ( (cid:101) w ) = 0.We have (cid:102) W = (cid:102) W aff Ω = (cid:102) W M , aff (cid:102) W M , aff Ω. Let Ω(1) be the inverse image ofΩ by (cid:102) W (1) → (cid:102) W . Since (cid:102) W M i , aff (1) → (cid:102) W M i , aff is surjective for i = 1 , 2, weget (cid:102) W = (cid:102) W M , aff (1) (cid:102) W M , aff (1)Ω(1). Namely, for any (cid:101) w ∈ (cid:102) W (1), we can find (cid:101) w ∈ (cid:102) W M , aff (1), (cid:101) w ∈ (cid:102) W M , aff (1) and u ∈ (cid:102) W (1) such that (cid:96) ( u ) = 0 and (cid:101) w = (cid:101) w (cid:101) w u . Since (cid:96) M ( (cid:101) w ) = (cid:96) M ( u ) = 0, we have (cid:96) M ( (cid:101) w ) = (cid:96) M ( (cid:101) w ). Wehave proved (1)–(3) of the following lemma. Lemma 4.17. Let (cid:101) w, (cid:101) v ∈ (cid:102) W (1) . (1) We have (cid:96) ( (cid:101) w ) = (cid:96) M ( (cid:101) w ) + (cid:96) M ( (cid:101) w ) . (2) If (cid:96) M i ( (cid:101) w ) = 0 , then (cid:96) M i ( (cid:101) w (cid:101) v ) = (cid:96) M i ( (cid:101) v (cid:101) w ) = (cid:96) M i ( (cid:101) v ) for i = 1 , . (3) There exists (cid:101) w ∈ (cid:102) W M , aff (1) , (cid:101) w ∈ (cid:102) W M , aff (1) and u ∈ (cid:102) W (1) such that (cid:96) ( u ) = 0 and (cid:101) w = (cid:101) w (cid:101) w u . We have (cid:96) M ( (cid:101) w ) = (cid:96) M ( (cid:101) w ) and (cid:96) M ( (cid:101) w ) = (cid:96) M ( (cid:101) w ) . In particular, there exists (cid:101) w ∈ (cid:102) W M , aff (1) and u ∈ (cid:102) W (1) suchthat (cid:96) M ( u ) = 0 and (cid:101) w = (cid:101) w u . (4) Let (cid:101) w , (cid:101) w ∈ (cid:102) W (1) such that (cid:96) M ( (cid:101) w ) = 0 and (cid:96) M ( (cid:101) w ) = 0 . Then (cid:96) ( (cid:101) w (cid:101) w ) = (cid:96) ( (cid:101) w ) + (cid:96) ( (cid:101) w ) . (5) If (cid:101) w ≤ (cid:101) v , then (cid:96) M i ( (cid:101) w ) ≤ (cid:96) M i ( (cid:101) v ) for i = 1 , For w ∈ W , λ ∈ Λ(1) and i = 1 , , we have (cid:96) M i ( λn w ) = (cid:88) α ∈ Σ + M i ,w − ( α ) > |(cid:104) ν ( λ ) , α (cid:105)| + (cid:88) α ∈ Σ + M i ,w − ( α ) < |(cid:104) ν ( λ ) , α (cid:105) − | . (7) For i = 1 , , Λ (cid:48) M i (1) is normal in (cid:102) W (1) .Proof. We may assume i = 1 in (5), (6) and (7).We prove (4). Since (cid:96) M ( (cid:101) w ) = 0, (cid:96) M ( (cid:101) w (cid:101) w ) = (cid:96) M ( (cid:101) w ). Hence (cid:96) ( (cid:101) w (cid:101) w ) = (cid:96) M ( (cid:101) w (cid:101) w ) + (cid:96) M ( (cid:101) w (cid:101) w )= (cid:96) M ( (cid:101) w ) + (cid:96) M ( (cid:101) w )= ( (cid:96) M ( (cid:101) w ) + (cid:96) M ( (cid:101) w )) + ( (cid:96) M ( (cid:101) w ) + (cid:96) M ( (cid:101) w ))= (cid:96) ( (cid:101) w ) + (cid:96) ( (cid:101) w ) . For (5), we prove the same statement in the group (cid:102) W . Let (cid:101) w, (cid:101) v ∈ (cid:102) W . Take (cid:101) w , (cid:101) v ∈ (cid:102) W M , aff , (cid:101) w , (cid:101) v ∈ (cid:102) W M , aff and u, u (cid:48) ∈ Ω such that (cid:101) w = (cid:101) w (cid:101) w u and (cid:101) v = (cid:101) v (cid:101) v u (cid:48) . Then (cid:101) w ≤ (cid:101) v implies (cid:101) w ≤ (cid:101) v , (cid:101) w ≤ (cid:101) v and u = u (cid:48) . Since (cid:101) w , (cid:101) v ∈ (cid:102) W M , aff ,if (cid:101) w ≤ (cid:101) v with respect to the Bruhat order of (cid:102) W , then (cid:101) w ≤ (cid:101) v with respect to theBruhat order of (cid:102) W M . Hence (cid:96) M ( (cid:101) w ) ≤ (cid:96) M ( (cid:101) v ). On the other hand, we have (cid:96) M ( (cid:101) w ) = (cid:96) M ( (cid:101) w ) and (cid:96) M ( (cid:101) v ) = (cid:96) M ( (cid:101) v ). Hence we get (5).We prove (6). If w ∈ W M , this is the length formula (2.1). In general, take w ∈ W M and w ∈ W M such that w = w w . Then we have (cid:96) M ( w ) = 0.Hence (cid:96) M ( λn w w ) = (cid:96) M ( λn w ). Since λn w ∈ (cid:102) W M (1), applying the lengthformula (2.1) we get (cid:96) M ( λn w ) = (cid:88) α ∈ Σ + M ,w − ( α ) > |(cid:104) ν ( λ ) , α (cid:105)| + (cid:88) α ∈ Σ + M ,w − ( α ) < |(cid:104) ν ( λ ) , α (cid:105) − | . For α ∈ Σ M , w − w − ( α ) = w − w − ( α ) = w − ( α ) since w ∈ W M and w ∈ W M . Therefore w − ( α ) = ( w w ) − ( α ) = w − ( α ). Since M (cid:48) is normal in G [AHHV14, II.7 Remark 4], we have (7). (cid:3) Now we give a proof of Proposition 4.16. For the proof, let I be the two-sidedideal of H ⊗ C generated by { T n s | s ∈ S aff , } ∪ { T t − | t ∈ Λ (cid:48) M (1) ∩ Z κ } and J the two-sided ideal of H M ⊗ C generated by { E M ( λ ) − | λ ∈ Λ (cid:48) M (1) } . ThenProposition 4.16 follows from( H − M ⊗ C ) / ( J ∩ ( H − M ⊗ C )) ∼ −→ ( H ⊗ C ) /I. We prove it. We need a lemma. ODULO p PARABOLIC INDUCTION OF PRO- p -IWAHORI HECKE ALGEBRA 35 Lemma 4.18. (1) We have I = (cid:77) (cid:101) w ∈ (cid:102) W aff (1) \ (cid:102) W (1) I ∩ (cid:77) (cid:101) v ∈ (cid:102) W aff (1) CT (cid:101) v (cid:101) w = (cid:77) (cid:101) w ∈ (cid:102) W aff (1) \ (cid:102) W (1) I ∩ (cid:77) (cid:101) v ∈ (cid:102) W aff (1) CE ( (cid:101) v (cid:101) w ) . (2) If (cid:101) w ∈ (cid:102) W M , aff (1) , then T ∗ (cid:101) w ∈ I . (3) Let λ ∈ Λ(1) , w ∈ W M , a ∈ (cid:102) W M , aff (1) and b ∈ (cid:102) W (1) such that λn w is M -negative, λn w = ab and (cid:96) M ( b ) = 0 . Then E ( λn w ) ∈ T b + (cid:80) c CT (cid:101) w . Let I (cid:48) be a such ideal. Then I (cid:48) ⊃ (cid:76) (cid:96) M ( (cid:101) w ) > CT (cid:101) w isobvious. For the reverse inclusion, we prove that (cid:76) (cid:96) M ( (cid:101) w ) > CT (cid:101) w is an ideal.Let (cid:101) w ∈ (cid:102) W (1) such that (cid:96) M ( (cid:101) w ) > (cid:101) v ∈ (cid:102) W (1). We prove T (cid:101) w T (cid:101) v ∈ (cid:76) (cid:96) M ( (cid:101) w (cid:48) ) > CT (cid:101) w (cid:48) . We may assume (cid:101) v = n s for s ∈ S aff . If (cid:101) wn s > (cid:101) w , then T (cid:101) w T n s = T (cid:101) wn s and (cid:96) M ( (cid:101) wn s ) = (cid:96) M ( (cid:101) w ) + (cid:96) M ( n s ) > (cid:96) M ( (cid:101) w ) > 0. If (cid:101) wn s < (cid:101) w ,then T (cid:101) w T n s = T (cid:101) w c s . (cid:3) Proof of Proposition 4.16. We prove H − M ⊗ C → ( H ⊗ C ) /I is surjective and the kernel is J ∩ ( H − M ⊗ C ).First we prove that the homomorphism is surjective. Let (cid:101) w ∈ (cid:102) W (1). We provethat T (cid:101) w is in the image. If (cid:96) M ( (cid:101) w ) > 0, then T (cid:101) w ∈ I . Hence we have nothing toprove. Assume that (cid:96) M ( (cid:101) w ) = 0. Since T (cid:101) w ∈ E ( (cid:101) w ) + (cid:80) (cid:101) v< (cid:101) w CE ( (cid:101) v ) and (cid:101) v < (cid:101) w implies (cid:96) M ( (cid:101) v ) = 0 by Lemma 4.17 (5), it is sufficient to prove that E ( (cid:101) w ) is in theimage. Take λ ∈ Λ(1), w ∈ W M and w ∈ W M such that (cid:101) w = λn w n w . Thenwe have E ( (cid:101) w ) = q / λn w n w q − / λn w q − / n w E ( λn w ) T n w . By the assumption, we also have (cid:96) M ( λn w n w ) = 0. We have (cid:96) M ( n w ) = 0.Hence by Lemma 4.17 (4), we have (cid:96) ( λn w ) = (cid:96) ( λn w n w )+ (cid:96) ( n − w ) = (cid:96) ( λn w n w )+ (cid:96) ( n w ). Therefore we have q λn w = q λn w n w q n w . Hence we get E ( (cid:101) w ) = q − n w E ( λn w ) T n w . Therefore E ( (cid:101) w ) T ∗ n − w = E ( λn w ). Since w ∈ W M , T ∗ n − w ∈ I by the abovelemma (2). Hence E ( (cid:101) w ) ∈ E ( λn w ) + I . By (cid:96) M ( λn w n w ) = 0 and Lemma 4.17(6), (cid:104) ν ( λ ) , α (cid:105) = 0 or 1 for α ∈ Σ + M = Σ + \ Σ + M . In particular, λn w is M -negative. Hence E ( (cid:101) w ) is in the image of the homomorphism.The two-sided ideal J is generated by { E M ( λ ) − | λ ∈ Λ (cid:48) M (1) } . If λ ∈ Λ (cid:48) M (1), then (cid:96) M ( λ ) = 0. By Lemma 4.17 (7), (cid:101) wλ (cid:101) w − ∈ Λ (cid:48) M (1) for (cid:101) w ∈ (cid:102) W M (1) and in particular, we have (cid:96) M ( (cid:101) wλ (cid:101) w − ) = 0. Hence E M ( (cid:101) w ) E M ( µ ) = E M ( (cid:101) wµ ) = E M ( (cid:101) wµ (cid:101) w − ) E M ( (cid:101) w ). Therefore (cid:80) (cid:101) w ∈ (cid:102) W M (1) ,µ ∈ Λ (cid:48) M (1) C ( E M ( µ ) − E M ( (cid:101) w ) = (cid:80) (cid:101) w ∈ (cid:102) W M (1) ,µ ∈ Λ (cid:48) M (1) C ( E M ( µ (cid:101) w ) − E M ( (cid:101) w )) is a two-sided idealand it is J . Hence if (cid:80) (cid:101) w ∈ (cid:102) W M (1) c (cid:101) w E M ( (cid:101) w ) ∈ J , then (cid:80) µ ∈ Λ (cid:48) M (1) c µ (cid:101) w = 0 for any (cid:101) w ∈ (cid:102) W M (1).Assume that (cid:80) (cid:101) w ∈ (cid:102) W M (1) c (cid:101) w E M ( (cid:101) w ) ∈ J . For each x ∈ Λ (cid:48) M (1) \ (cid:102) W M (1), fix arepresentative (cid:101) w ( x ) ∈ (cid:102) W M (1) which is M -negative. Then we have (cid:88) (cid:101) w ∈ (cid:102) W M (1) c (cid:101) w E M ( (cid:101) w )= (cid:88) x ∈ Λ (cid:48) M (1) \ (cid:102) W M (1) (cid:88) µ ∈ Λ (cid:48) M (1) c µ (cid:101) w ( x ) ( E M ( µ (cid:101) w ( x )) − E M ( (cid:101) w ( x ))) . ODULO p PARABOLIC INDUCTION OF PRO- p -IWAHORI HECKE ALGEBRA 37 Therefore ( H − M ⊗ C ) ∩ J is contained in the subspace generated by { E M ( (cid:101) w ) − E M ( (cid:101) w ) | (cid:101) w (cid:101) w − ∈ Λ (cid:48) M (1) , (cid:101) w , (cid:101) w are M -negative } .Let J (cid:48) be the kernel of H − M ⊗ C → ( H⊗ C ) /I . We prove J ∩ ( H − M ⊗ C ) ⊂ J (cid:48) . Bythe above argument, to prove it, it is sufficient to prove that E ( λ n w ) − E ( λ n w ) ∈ I if λ n w , λ n w are M -negative and λ λ − ∈ Λ (cid:48) M (1).Let µ ∈ Λ (cid:48) M (1) be a dominant element. By the length formula (2.1), we have (cid:96) ( µλn w )= (cid:88) α ∈ Σ + ,w − ( α ) > |(cid:104) ν ( µ ) , α (cid:105) + (cid:104) ν ( λ ) , α (cid:105)| + (cid:88) α ∈ Σ + ,w − ( α ) < |(cid:104) ν ( µ ) , α (cid:105) + (cid:104) ν ( λ ) , α (cid:105) − | If w − ( α ) < 0, then since w ∈ W M , we have α ∈ Σ + M . Since µ ∈ Λ (cid:48) M (1), wehave (cid:104) ν ( µ ) , α (cid:105) = 0. Therefore we have |(cid:104) ν ( µ ) , α (cid:105) + (cid:104) ν ( λ ) , α (cid:105) − | = |(cid:104) ν ( λ ) , α (cid:105) − | = |(cid:104) ν ( µ ) , α (cid:105)| + |(cid:104) ν ( λ ) , α (cid:105) − | . If w − ( α ) > α ∈ Σ + M , then again we have (cid:104) ν ( µ ) , α (cid:105) = 0. Hence |(cid:104) ν ( µ ) , α (cid:105) + (cid:104) ν ( λ ) , α (cid:105)| = |(cid:104) ν ( λ ) , α (cid:105)| = |(cid:104) ν ( µ ) , α (cid:105)| + |(cid:104) ν ( λ ) , α (cid:105)| . If α ∈ Σ + M (hence w − ( α ) > (cid:104) ν ( λ ) , α (cid:105) ≥ λ is M -negative. Since µ is dominant, we have (cid:104) ν ( µ ) , α (cid:105) ≥ 0. Hence |(cid:104) ν ( µ ) , α (cid:105) + (cid:104) ν ( λ ) , α (cid:105)| = |(cid:104) ν ( µ ) , α (cid:105)| + |(cid:104) ν ( λ ) , α (cid:105)| . Therefore we get (cid:96) ( µλn w )= (cid:88) α ∈ Σ + |(cid:104) ν ( µ ) , α (cid:105)| + (cid:88) α ∈ Σ + ,w − ( α ) > |(cid:104) ν ( λ ) α (cid:105)| + (cid:88) α ∈ Σ + ,w − ( α ) < |(cid:104) ν ( λ ) , α (cid:105) − | = (cid:96) ( µ ) + (cid:96) ( λn w ) . Hence E ( µλn w ) = E ( µ ) E ( λn w ). Since E ( µ ) = T ∗ µ ∈ I by Proposition 2.7 andthe above lemma, we have E ( µλn w ) − E ( λn w ) ∈ I . Let λ , λ be M -negative ele-ments such that λ − λ ∈ Λ (cid:48) M (1). Since Λ (cid:48) M (1) is normal in Λ(1) by Lemma 4.17(7), we also have λ λ − ∈ Λ (cid:48) M (1). Take a dominant µ ∈ Λ (cid:48) M (1) such that µ λ λ − is dominant. Then for w ∈ W M , E ( λ n w ) − E ( λ n w )= ( E ( λ n w ) − E (( µ λ λ − ) λ n w )) + ( E ( µ λ n w ) − E ( λ n w )) ∈ I Finally, we prove J (cid:48) ⊂ J ∩ ( H − M ⊗ C ). Take (cid:80) (cid:101) w ∈ (cid:102) W M (1) c (cid:101) w E M ( (cid:101) w ) ∈ J (cid:48) .Then for any (cid:101) v ∈ (cid:102) W (1), we have (cid:80) (cid:101) w ∈ (cid:102) W aff (1) c (cid:101) w (cid:101) v E ( (cid:101) w (cid:101) v ) ∈ I by the above lemma(1). Hence we may assume c (cid:101) w = 0 if (cid:101) w / ∈ (cid:102) W aff (1) (cid:101) v for a fixed (cid:101) v ∈ (cid:102) W (1). Since (cid:102) W aff (1) (cid:102) W M (1) ⊃ (cid:102) W aff (1)Λ(1) = (cid:102) W (1), we may assume (cid:101) v ∈ (cid:102) W M (1). Moreover,we may assume (cid:101) v is M -negative. Take λ (cid:48) ∈ Λ(1) and w (cid:48) ∈ W M such that (cid:101) v = λ (cid:48) n w (cid:48) . Let λ ∈ Λ(1), w ∈ W such that λn w ∈ (cid:102) W aff (1) (cid:101) v . Then λ (cid:48) n w (cid:48) n − w λ − ∈ (cid:102) W aff (1). We have λ (cid:48) n w (cid:48) n − w λ − = λ (cid:48) λ − ( λn w (cid:48) n − w λ − ) and λn w (cid:48) n − w λ − ∈ (cid:102) W aff (1)since (cid:102) W aff (1) is normal in (cid:102) W (1) (see subsection 2.1). Hence λ (cid:48) λ − ∈ (cid:102) W aff (1) ∩ Λ(1) = Λ (cid:48) (1). By Lemma 2.5, we have Λ (cid:48) (1) = Λ (cid:48) M (1)Λ (cid:48) M (1). Take λ ∈ Λ (cid:48) M (1)and λ ∈ Λ (cid:48) M (1) such that λ (cid:48) λ − = λ λ . Since λ ∈ Λ (cid:48) M (1), we have (cid:96) M ( λ ) =0. Hence E M ( λn w ) − E M ( λ λn w ) = (1 − E M ( λ )) E M ( λn w ) ∈ J . Moreover,since λ ∈ Λ (cid:48) M (1), we have (cid:104) α, ν ( λ ) (cid:105) = 0 for any α ∈ Σ + M . By the orthogonaldecomposition ∆ = ∆ (cid:113) ∆ , we have Σ + \ Σ + M = Σ + M . Hence for any α ∈ Σ + \ Σ + M , (cid:104) α, ν ( λ (cid:48) ) (cid:105) = (cid:104) α, ν ( λ λ λ ) (cid:105) = (cid:104) α, ν ( λ λ ) (cid:105) . Therefore, since λ (cid:48) is M -negative, λ λ is also M -negative. Hence we have E M ( λn w ) − E M ( λ λn w ) ∈ J ∩ ( H − M ⊗ C ). Therefore, to prove (cid:80) (cid:101) w ∈ (cid:102) W M (1) c (cid:101) w E M ( (cid:101) w ) ∈ J ∩ ( H − M ⊗ C ), it issufficient to prove that (cid:80) (cid:101) w ∈ (cid:102) W M (1) c (cid:101) w E M ( (cid:101) w ) − c λn w ( E M ( λn w ) − E M ( λ λn w )) ∈ J ∩ ( H − M ⊗ C ). Notice that λ (cid:48) ( λ λ ) − = λ ∈ Λ (cid:48) M (1). Therefore, we may assume that c λn w (cid:54) = 0 implies λ (cid:48) λ − ∈ Λ (cid:48) M (1). Or, equivalently, since Λ (cid:48) M (1) is normalin Λ(1), we may assume λ − λ (cid:48) ∈ Λ (cid:48) M (1). Hence ( λn w ) − λ (cid:48) n w (cid:48) ∈ (cid:102) W M , aff (1).Take a ∈ (cid:102) W M , aff (1) and b ∈ (cid:102) W (1) such that λ (cid:48) n w (cid:48) = a b and (cid:96) M ( b ) = 0.If c λn w (cid:54) = 0, then (cid:96) M (( λ (cid:48) n w (cid:48) ) − λn w ) = 0. Namely (cid:96) M ( b − a − λn w ) = 0. Since (cid:96) M ( b ) = 0, (cid:96) M ( a − λn w ) = 0.Put l = max { (cid:96) ( a − λn w ) | c λn w (cid:54) = 0 } . We prove (cid:80) c λn w E M ( λn w ) ∈ J ∩ ( H − M ⊗ C ) by induction on l . We have (cid:80) c λn w E ( λn w ) ∈ I . By (3) of the above lemma, wehave E ( λn w ) ∈ T a − λn w + (cid:80) b
Let P = M N be a parabolic subgroup such that ∆ M ⊃ ∆ and σ as in Proposition 4.16. Denote the extension of σ to H (resp. H M ) by e ( σ ) (resp. e M ( σ ) ). Then e ( σ ) | H − M (cid:39) e M ( σ ) | H − M .Proof. Let (cid:101) w ∈ (cid:102) W M (1) be a M -negative element and we prove e M ( σ )( T M , ∗ (cid:101) w ) = e ( σ )( T ∗ (cid:101) w ). Let λ ∈ Λ(1), w ∈ W M and w ∈ W M ∩ W M such that (cid:101) w = λn w n w .First we prove that e ( σ )( T ∗ (cid:101) w ) = e ( σ )( T ∗ λn w ) by induction on (cid:96) ( w ). Take s ∈ S such that w s < w . If λn w n w n s > λn w n w , then T ∗ λn w n w T ∗ n − s = T ∗ λn w n w s .If λn w n w n s < λn w n w , then T ∗ λn w n w = T ∗ λn w n w s T ∗ n s . By Lemma 4.18 (2), wehave e ( σ )( T ∗ n s ) = e ( σ )( T ∗ n − s ) = 1. Hence in any case, we have e ( σ )( T ∗ λn w n w ) = e ( σ )( T ∗ λn w n w s ). It completes the induction. From the same argument, we have e M ( T M , ∗ (cid:101) w ) = e M ( T M , ∗ λn w ).Let M (cid:48) be the Levi part of the parabolic subgroup corresponding to ∆ M ∩ ∆ .Take µ ∈ Λ (cid:48) M (cid:48) (1) such that (cid:104) α, ν ( µ ) (cid:105) is sufficiently large for α ∈ Σ + M (cid:48) . Since (cid:101) w is M -negative, (cid:104) α, ν ( λ ) (cid:105) ≥ α ∈ Σ + \ Σ + M = Σ + M \ Σ + M (cid:48) . Hence we can take µ such that (cid:104) α, ν ( µλ ) (cid:105) ≥ α ∈ Σ + M = Σ + \ Σ + M , namely µλn w is M -negative. The element µ is in W M (cid:48) , aff (1). Hence using the same argument in the ODULO p PARABOLIC INDUCTION OF PRO- p -IWAHORI HECKE ALGEBRA 39 above, we have e M ( σ )( T M , ∗ λn w ) = e M ( σ )( T M , ∗ µλn w ). Since µ is also in W M , aff (1), wehave e ( σ )( T ∗ λn w ) = e ( σ )( T ∗ µλn w ). Finally, since µλn w ∈ W M (1) is M -negative,we have e M ( σ )( T M , ∗ µλn w ) = σ ( T M , ∗ µλn w ) = e ( σ )( T ∗ µλn w ). (cid:3) Statement of the classification theorem. We begin with the definition ofsupersingular representations. Definition 4.20 ([Oll12, Proposition-Definition 5.9], [Vigb, Definition 6.10]) . Foreach adjoint (cid:102) W (1)-orbit O in Λ(1), put z O = (cid:80) λ ∈O E ( λ ). Let J be the idealgenerated by { z O | (cid:96) ( λ ) > λ ∈ O ) } . An H -module π is called supersingular ifthere exists n ∈ Z > such that J n π = 0.Notice that (cid:96) ( λ ) is independent of λ ∈ O . By [Vig14, Theorem 1.2], z O is inthe center of H . We mainly use the characterization (3) and (4) in the followinglemma. Lemma 4.21. Let π be an irreducible H ⊗ C -module and J as in Definition 4.20.Then the following are equivalent. (1) The module π is supersingular. (2) We have J π = 0 . (3) For any irreducible A -submodule X of π , supp X = Λ ∆ (1) . (4) There exists an irreducible A -submodule X of π such that supp X = Λ ∆ (1) .Proof. Since J π is H -stable, the equivalence between (1) and (2) is clear. Obvi-ously, (3) implies (4).Notice that Λ ∆ (1) = Λ +∆ (1). Recall that supp X = Λ ∆ (1) if and only if theaction of A on X factors through χ ∆ . Let X be an irreducible A -submodule of π .We prove J X = 0 if and only if supp X = Λ ∆ (1). Let Θ ⊂ ∆ and w ∈ W such thatthe action of A on X factors through wχ Θ . By the length formula (2.1), (cid:96) ( λ ) = 0if and only if (cid:104) ν ( λ ) , α (cid:105) = 0 for any α ∈ Σ + , namely λ ∈ Λ ∆ (1). Hence if (cid:96) ( λ ) > χ ∆ ( E ( λ )) = 0. Therefore, if Θ = ∆, then J X = 0.Assume that Θ (cid:54) = ∆. We take λ ∈ Λ(1) as follows. Fix a uniformizer (cid:36) of F andconsider the embedding X ∗ ( S ) (cid:44) → S ( F ) defined by λ (cid:55)→ λ ( (cid:36) ) − . Then composition X ∗ ( S ) (cid:44) → S ( F ) (cid:44) → Z ( F ) → Λ(1) ν −→ V is equal to the embedding X ∗ ( S ) (cid:44) → V andin particular it is injective. Take λ ∈ X ∗ ( S ) such that its image λ in Λ(1) satisfies wχ Θ ( E ( λ )) (cid:54) = 0 and (cid:96) ( λ ) > 0. Since λ ∈ S ( F ) and S ( F ) is contained in the centerof Z ( F ), λ commutes with an element of Λ(1). Hence the orbit O through λ is { n w ( λ ) | w ∈ W } . Moreover, if w ∈ W satisfies ν ( n w ( λ )) = n w ( λ ), then we have w ( λ ) = λ . Hence n w ( λ ) = λ . Therefore O = { n w ( λ ) | w ∈ W/ Stab W ( ν ( λ )) } .Since wχ Θ ( E ( λ )) (cid:54) = 0, w − ( ν ( λ )) is dominant. Hence, if v ∈ W does not stabilize ν ( λ ), then w − v − ( ν ( λ )) is not dominant. Therefore we have wχ Θ ( E ( n v ( λ ))) =0. Hence wχ Θ ( z O ) = (cid:80) v ∈ W/ Stab( ν ( λ )) wχ Θ ( E ( n v ( λ ))) = wχ Θ ( E ( λ )) and it isinvertible. Therefore J X (cid:54) = 0.We assume that π is supersingular. Let X be an irreducible A -submodule. Then J X ⊂ J π = 0. Hence we have supp X = Λ ∆ (1). Thereofre (2) implies (3). Assumethat there exists an irreducible A -submodule X such that supp X = Λ ∆ (1). Then J X = 0. Hence { x ∈ π | J x = 0 } (cid:54) = 0. Since the left hand side is H -stable, { x ∈ π | J x = 0 } = π . Therefore J π = 0. Hence (4) implies (2). (cid:3) Let P = M N be a standard parabolic subgroup and σ an irreducible supersin-gular representation of H M . Define∆( σ ) = { α ∈ ∆ | (cid:104) ∆ M , ˇ α (cid:105) = 0 , σ ( E M ( λ )) = 1 for any λ ∈ Λ (cid:48) s α (1) } ∪ ∆ M . Notice that if (cid:104) ∆ M , ˇ α (cid:105) = 0 and λ ∈ Λ (cid:48) s α (1), then (cid:104) ∆ M , ν ( λ ) (cid:105) = 0 by Lemma 2.3.Hence (cid:96) M ( λ ) = 0. Therefore we have E M ( λ ) = T M λ . Let P ( σ ) = M ( σ ) N ( σ ) be the parabolic subgroup corresponding to ∆( σ ). Let Q be a parabolic subgroup such that P ( σ ) ⊃ Q ⊃ P . By Proposition 4.16, wehave the extension e Q ( σ ) of σ to H Q . By Proposition 4.19, if Q (cid:48) ⊃ Q , then wehave I Q (cid:48) ( e Q (cid:48) ( σ )) = Hom H − Q (cid:48) ( H , e Q (cid:48) ( σ ) | H − Q (cid:48) ) ⊂ Hom H − Q ( H , e Q (cid:48) ( σ ) | H − Q )= Hom H − Q ( H , e Q ( σ ) | H − Q ) = I Q ( e Q ( σ )) . Define an H -module I ( P , σ, Q ) by I ( P , σ, Q ) = I Q ( e Q ( σ )) (cid:44) (cid:88) P ( σ ) ⊃ Q (cid:48) (cid:41) Q I Q (cid:48) ( e Q (cid:48) ( σ )) The main theorem of this paper is the following. Theorem 4.22. For any triple ( P , σ, Q ) , I ( P , σ, Q ) is irreducible and any irre-ducible H -module have this form. Moreover, I ( P , σ, Q ) (cid:39) I ( P (cid:48) , σ (cid:48) , Q (cid:48) ) implies P = P (cid:48) , σ (cid:39) σ (cid:48) and Q = Q (cid:48) . Irreducibility. We prove that I ( P , σ, Q ) is irreducible. Lemma 4.23. The isomorphism I Q ( e Q ( σ )) (cid:39) (cid:76) w ∈ W Q wσ A induces I ( P , σ, Q ) (cid:39) (cid:76) ∆ w ∩ ∆( σ )=∆ Q wσ A .Proof. We remark that σ A = e Q ( σ ) A . Indeed, if λ ∈ Λ(1) is M -negative, then σ A ( E ( λ )) = σ ( E M ( λ )) = e Q ( σ )( E Q ( λ )) by the definition of σ A and the exten-sion. Since λ is also Q -negative, we have e Q ( σ )( E Q ( λ )) = e Q ( σ ) A ( E ( λ )). If λ is not M -negative, take λ − ∈ Λ(1) as in Remark 4.7. Then the above calcu-lation shows e Q ( σ ) A ( E ( λ − )) = σ ( E M ( λ − )) and it is invertible. On the otherhand, since λ is not M -negative, ν ( λ ) and ν ( λ − ) do not belong to the same closedchamber. Hence E ( λ ) E ( λ − ) = 0 by (3.1). Therefore we have e Q ( σ ) A ( E ( λ )) = e Q ( σ ) A ( E ( λ ) E ( λ − )) e Q ( σ ) A ( E ( λ − )) − = 0. By the definition of σ A , we have σ A ( E ( λ )) = 0. Hence we get σ A = e Q ( σ ) A .For P ( σ ) ⊃ Q (cid:48) ⊃ Q , we have I Q (cid:48) ( e Q (cid:48) ( σ )) I Q ( e Q ( σ )) (cid:77) w ∈ W Q (cid:48) wσ A (cid:77) w ∈ W Q wσ A . ∼ ∼ We write the low horizontal map explicitly. If ϕ ∈ I Q (cid:48) ( e Q (cid:48) ( σ )), then its imagein (cid:76) w ∈ W Q wσ A (cid:39) I Q ( e Q ( σ )) is ( ϕ ( T n w )) w ∈ W Q . Let w ∈ W and decompose w = w w such that w ∈ W Q (cid:48) and w ∈ W Q (cid:48) . Then we have ( ϕ ( T n w )) =( ϕ ( T n w T n w )) = ( ϕ ( T n w ) T n w ). Hence the low horizontal map of the above dia-gram is given by ( x w ) w ∈ W Q (cid:48) (cid:55)→ ( x w T n w ) w w . For α ∈ Σ + Q ⊂ Σ + Q (cid:48) , w ( α ) ∈ Σ Q (cid:48) .Hence w w ( α ) > w ( α ) > 0. Therefore, w w ∈ W Q if and only if w ∈ W Q .We have w ∈ W Q (cid:48) ∩ W Q ⊂ W Q (cid:48) ∩ W P . Since we have the orthogonal de-composition ∆ Q (cid:48) = ∆ P ∪ (∆ Q (cid:48) \ ∆ P ), the group W Q (cid:48) ∩ W P is generated by { s α | α ∈ ∆ Q (cid:48) \ ∆ P } . For such α , T n sα = 0 on e Q (cid:48) ( σ ) by the definition ofthe extension. Therefore, x w T n w = 0 if w (cid:54) = 1. Hence the low horizontal mapof the above diagram is the inclusion map induced by W Q (cid:48) (cid:44) → W Q . Therefore we ODULO p PARABOLIC INDUCTION OF PRO- p -IWAHORI HECKE ALGEBRA 41 have I ( P , σ, Q ) = (cid:77) w wσ A where w runs in W Q such that w / ∈ W Q (cid:48) for any P ( σ ) ⊃ Q (cid:48) (cid:41) Q . Notice that w ∈ W Q if and only if ∆ w ⊃ ∆ Q by the definition. Hence w ∈ W Q and w / ∈ W Q (cid:48) forany P ( σ ) ⊃ Q (cid:48) (cid:41) Q is equivalent to ∆ w ⊃ ∆ Q and α (cid:54)∈ ∆ w for any α ∈ ∆( σ ) \ ∆ Q which is equivalent to ∆ w ∩ ∆( σ ) = ∆ Q . (cid:3) We use the notation ∆( X ) from Corollary 3.17. Lemma 4.24. Let X be an irreducible A -submodule of σ A . Then supp X =Λ +∆ M (1) and ∆( X ) = ∆( σ ) .Proof. Take λ − ∈ Λ(1) as in Remark 4.7. Then A M = ( A M ∩ H − M ) E M ( λ − ) − by the argument in Remark 4.7. Put X (cid:48) = Xσ ( A M ) = (cid:83) n ∈ Z ≥ Xσ ( E M ( λ − ) − n ).We prove that X (cid:48) is an irreducible A M -module. Take x ∈ X (cid:48) \ { } . Then bymultiplying the power of E M ( λ − ), we have X ∩ x A M (cid:54) = 0. Since X is an irreducible A -module, we have X ⊂ ( X ∩ x A M ) σ A ( A ). The definition of σ A tells that σ A ( A ) = σ A ( A ∩ H − M ) = σ ( A M ∩ H − M ). Hence X ⊂ ( X ∩ x A M ) σ ( A M ∩ H − M ) ⊂ xσ ( A M ).Hence X (cid:48) = xσ ( A M ). We get the irreducibility of X (cid:48) .Since σ is supersingular, supp X (cid:48) = Λ ∆ M (1). Recall that if λ is M -negative,then the action of E ( λ ) on σ A is the action of E M ( λ ) on σ . Hence for M -negative λ , we have λ ∈ supp X if and only if λ ∈ Λ ∆ M (1). By the definition of σ A , if λ is not M -negative then λ / ∈ supp X . Hence λ is in supp X if and only if λ is M -negative and λ ∈ Λ ∆ M (1). Namely we have supp X = Λ +∆ M (1).Hence X is a C [Λ ∆ M (1)]-module. Assume that (cid:104) α, ∆ M (cid:105) = 0. Then for λ ∈ Λ (cid:48) s α (1), we have (cid:104) ν ( λ ) , ∆ M (cid:105) = 0. Namely, λ ∈ Λ ∆ M (1). Take λ , λ ∈ Λ +∆ M (1)such that λ = λ λ − . Then the action of τ λ is equal to that of E ( λ ) and, it isalso equal to the action of E M ( λ ) on X ⊂ σ . It is also true if we replace λ with λ . Since τ λ = τ λ τ − λ and E M ( λ ) = E M ( λ ) E M ( λ ) − , the action of τ λ on X and E M ( λ ) on X ⊂ σ coincide with each other. If α ∈ ∆( σ ), then E M ( λ ) is trivialon σ for any λ ∈ Λ (cid:48) s α (1). Hence τ λ is trivial on X , namely α ∈ ∆( X ). Conversely,assume that α ∈ ∆( X ). Then { x ∈ σ | xE M ( λ ) = x ( λ ∈ Λ (cid:48) s α (1)) } ⊃ X . Inparticular it is not zero. On the other hand, this subspace is stable under theaction of H M . Hence this space is σ . Therefore α ∈ ∆( σ ). (cid:3) Lemma 4.25. There exists λ ∈ n w ∆ w ∆ Q (Λ +∆ Q (1)) such that wσ A ( E ( λ )) = 0 forany w ∈ W Q \ { w ∆ w ∆ Q } . In particular, if X is an irreducible A -submodule of I ( P , σ, Q ) such that supp X = n w ∆ w ∆ Q (Λ +∆ Q (1)) , then we have X ⊂ w ∆ w ∆ Q σ .Proof. First we prove the following claim. Claim. If w ∈ W Q \ { w ∆ w ∆ Q } , then (Σ − \ Σ − Q ) ∩ ( w ∆ w ∆ Q ) − w (Σ + \ Σ + Q ) (cid:54) = ∅ .Set Q (cid:48) = n w ∆ w ∆ Q Q ( n w ∆ w ∆ Q ) − and v = w ( w ∆ w ∆ Q ) − . Then we have ∆ Q (cid:48) = − w ∆ (∆ Q ) = w ∆ w ∆ Q (∆ Q ). Hence v (∆ Q (cid:48) ) = w (∆ Q ) ⊂ Σ + . Therefore v ∈ W Q (cid:48) .We also have v (cid:54) = 1, hence v / ∈ W Q (cid:48) . Therefore v − / ∈ W Q (cid:48) . Hence there exists α ∈ Σ + \ Σ + Q (cid:48) such that v − ( α ) < 0. If v − ( α ) ∈ Σ − Q (cid:48) , then α = v ( v − ( α )) ∈ v (Σ − Q (cid:48) ) ⊂ Σ − , this is a contradiction. Hence v − ( α ) ∈ Σ − \ Σ − Q (cid:48) . Namely α ∈ v (Σ − \ Σ − Q (cid:48) ) ∩ (Σ + \ Σ + Q (cid:48) ). By the definition of v and Σ ∓ \ Σ ∓ Q (cid:48) = w ∆ (Σ ± \ Σ ± Q ) = w ∆ w ∆ Q (Σ ± \ Σ ± Q ),we get α ∈ w (Σ + \ Σ + Q ) ∩ w ∆ w ∆ Q (Σ − \ Σ − Q ). Hence ( w ∆ w ∆ Q ) − ( α ) gives an elementin the intersection of the claim.Take λ − ∈ Λ +∆ Q (1) as in Remark 4.7 and set λ = n w ∆ w ∆ Q ( λ − ). We have λ ∈ n w ∆ w ∆ Q (Λ +∆ Q (1)). Let w ∈ W Q \ { w ∆ w ∆ Q } . Take α from the intersection in the claim. Then since α ∈ Σ − \ Σ − Q , we have (cid:104) w − w ∆ w ∆ Q ( α ) , ν ( n − w ( λ )) (cid:105) = (cid:104) α, ν ( λ − ) (cid:105) < 0. Therefore n − w ( λ ) is not Q -negative. Hence ( wσ A )( E ( λ )) = 0.Assume that X is an irreducible A -submodule of I ( P , σ, Q ) such that supp X = n w ∆ w ∆ Q (Λ +∆ Q (1)). Take λ as in the above and x ∈ X \ { } . Let x w ∈ wσ suchthat x = (cid:80) x w . Since λ ∈ supp X , E ( λ ) is invertible on X . Hence 0 (cid:54) = xE ( λ ) = (cid:80) x w E ( λ ) = x w ∆ w ∆ Q E ( λ ). Therefore w ∆ w ∆ Q σ ∩ X is a non-zero A -submodule of X . Since X is irreducible, we have X ⊂ w ∆ w ∆ Q σ . (cid:3) We prove that I ( P , σ, Q ) is irreducible. Let π ⊂ I ( P , σ, Q ) be a non-zero H -submodule and take an irreducible A -submodule X (cid:48) of π . By Lemma 4.23, wehave the decomposition I ( P , σ, Q ) = (cid:76) ∆ w ∩ ∆( σ )=∆ Q wσ A . Take w such that thecomposition X (cid:48) (cid:44) → I ( P , σ, Q ) = (cid:76) ∆ w ∩ ∆( σ )=∆ Q wσ A (cid:16) wσ A is non-zero. Then wehave X (cid:48) (cid:44) → wσ A . Such irreducible representation has a form wX for an irreduciblesubmodule X of σ A . Hence we get a non-zero homomorphism wX ⊗ A H → π . ByLemma 4.24 and Corollary 3.17, if ∆ w (cid:48) ∩ ∆( σ ) = ∆ Q , then wX ⊗ A H (cid:39) w (cid:48) X ⊗ A H .Hence we have a non-zero homomorphism w (cid:48) X ⊗ A H → π , which gives a non-zero A -homomorphism w (cid:48) X → π . In particular we have w ∆ w ∆ Q X (cid:44) → π . By Lemma 4.25,the image of w ∆ w ∆ Q X (cid:44) → π (cid:44) → I ( P , σ, Q ) is contained in w ∆ w ∆ Q σ . Thereforewe have π ∩ w ∆ w ∆ Q σ (cid:54) = 0. By Proposition 4.14, we have w ∆ w ∆ Q σ ⊂ π . Put σ (cid:48) = w ∆ w ∆ Q σ . Let Q (cid:48) be a parabolic subgroup corresponding to w ∆ w ∆ Q (∆ Q ).By Proposition 4.14, σ (cid:48) is H + Q (cid:48) -stable and we have a homomorphism σ (cid:48) ⊗ H + Q (cid:48) H → π .By the construction, we have the following commutative diagram: σ (cid:48) ⊗ H + Q (cid:48) H I Q ( e Q ( σ )) π I ( P , σ, Q ) . Here σ (cid:48) ⊗ H + Q (cid:48) H → I Q ( e Q ( σ )) is a homomorphism in Proposition 4.15. Hence it isan isomorphism. Therefore π = I ( P , σ, Q ). Corollary 4.26. Let P = M N be a parabolic subgroup and σ an irreduciblesupersingular H M -module. Then the composition factors of I P ( σ ) is given by { I ( P , σ, Q ) | P ( σ ) ⊃ Q ⊃ P } .Remark . After finishing the proof of Theorem 4.22, we know that I P ( σ ) ismultiplicity free.4.7. Completion of the proof. We finish the proof of Theorem 4.22. First weprove that any irreducible H -module π is isomorphic to I ( P , σ, Q ) by induction ondim G . By Corollary 4.26, it is sufficient to prove that π is a subquotient of I P ( σ )for a parabolic subgroup P = M N and a supsersingular irreducible representation σ of H M . We use the following lemma. This is a generalization of a conjecture ofOllivier [Oll10, Conjecture 5.20]. Lemma 4.28. Let X be an irreducible A -module and Θ ⊂ ∆ such that supp X =Λ +Θ (1) . Let w, w (cid:48) ∈ W such that w (Θ) , w (cid:48) (Θ) ⊂ Σ + . Then wX ⊗ A H and w (cid:48) X ⊗ A H have the same composition factors with multiplicities.Proof. By Theorem 3.13, we may assume w = w ∆ w Θ and w (cid:48) = ws α for Θ ⊂ ∆and α ∈ Θ such that Θ ⊂ Θ. By Corollary 3.17, we may assume α ∈ ∆( X ).Take x ∈ V ∗ such that x ( ν (Λ(1))) ⊂ Z and (cid:104) x, ˇ α (cid:105) (cid:54) = 0. Consider an algebrahomomorphism C [Λ Θ (1)] → C [ t ± ] defined by τ λ (cid:55)→ t (cid:104) x,ν ( λ ) (cid:105) . Both X and C [ t ± ]are representations of Λ Θ (1). Hence we have a representation X [ t ± ] = X ⊗ C [ t ± ]of Λ Θ (1), equivalently C [Λ Θ (1)]-module. Via χ Θ , we get an A -module X [ t ± ]. ODULO p PARABOLIC INDUCTION OF PRO- p -IWAHORI HECKE ALGEBRA 43 Consider the H -module wX [ t ± ] ⊗ A H and w (cid:48) X [ t ± ] ⊗ A H . By Proposition 3.8,these modules are free C [ t ± ]-modules. From the proof of Corollary 3.17, we haveinjective homomorphisms wX [ t ± ] ⊗ A H → w (cid:48) X [ t ± ] ⊗ A H and w (cid:48) X [ t ± ] ⊗ A H → wX [ t ± ] ⊗ A H . The compositions are given by 1 − t (cid:104) x, − ˇ α (cid:105) . By [BL95, Lemma 31],we get the lemma. (cid:3) Let π be an irreducible representation of H and take an irreducible A -submodule X (cid:48) . Then there exists Θ ⊂ ∆ and w ∈ W such that supp X (cid:48) = n w (Λ +Θ (1)) and w (Θ) > 0. Take an irreducible A -module X such that X (cid:48) = wX . Then supp X =Λ +Θ (1). Since we have wX = X (cid:48) (cid:44) → π , we have a non-zero homomorphism wX ⊗ A H → π . Therefore π is a quotient of wX ⊗ A H . Taking w (cid:48) = 1 in the abovelemma, the composition factors of wX ⊗ A H and X ⊗ A H is the same. Hence π isa subquotient of X ⊗ A H .If Θ = ∆, then π is supersingular. So we have nothing to prove. We assumethat Θ (cid:54) = ∆. Let P (cid:48) = M (cid:48) N (cid:48) (resp. P = M N ) be the parabolic subgroupcorresponding to Θ (resp. − w ∆ (Θ)). Lemma 4.29. Put A (cid:48) = A∩ j + M (cid:48) ( H + M (cid:48) ) . Then X ⊗ A H (cid:39) X ⊗ A (cid:48) H and X ⊗ A (cid:48) H + M (cid:48) is the restriction of an H M (cid:48) -module.Proof. We have a natural homomorphism X ⊗ A (cid:48) H → X ⊗ A H . Take λ +0 ∈ Λ(1)as in Remark 4.7 for M (cid:48) . Then we have λ +0 ∈ Λ Θ (1). Hence E ( λ +0 ) is invertibleon X . If λ ∈ Λ(1) is not M (cid:48) -positive, then ν ( λ ) and ν ( λ +0 ) does not belongto the same closed Weyl chamber, hence E ( λ +0 ) E ( λ ) = 0. Hence x ⊗ E ( λ ) F = xE ( λ +0 ) − ⊗ E ( λ +0 ) E ( λ ) F = 0 in X ⊗ A (cid:48) H for x ∈ X and F ∈ H . Therefore x ⊗ E ( λ ) F = 0 = xE ( λ ) ⊗ F . Hence X ⊗ A (cid:48) H → X ⊗ A H is an isomorphism.The element E M (cid:48) ( λ +0 ) is in the center of H + M (cid:48) and j + M (cid:48) ( E M (cid:48) ( λ +0 )) = E ( λ +0 ) ∈ A (cid:48) is invertible on X . Hence E M (cid:48) ( λ +0 ) is invertible on X ⊗ A (cid:48) H + M (cid:48) . Since H M (cid:48) = H + M (cid:48) E M (cid:48) ( λ +0 ) − by Remark 4.7, we have X ⊗ A (cid:48) H + M (cid:48) (cid:39) ( X ⊗ A (cid:48) H + M (cid:48) ) E M ( λ +0 ) − (cid:39) X ⊗ A (cid:48) H M (cid:48) . (cid:3) Hence we have X ⊗ A H (cid:39) ( X ⊗ A (cid:48) H + M (cid:48) ) ⊗ H + M (cid:48) H . By Proposition 4.15, π is ansubquotient of I P ( σ ) for an H M -module σ . (Explicitly, σ is given by X ⊗ A (cid:48) H + M (cid:48) twisting by n w ∆ w ∆ M .) Hence for some irreducible subquotient σ (cid:48) of σ , π is asubquotient of I P ( σ (cid:48) ). By inductive hypothesis, σ (cid:48) is a subquotient of I P ∩ M ( σ )where P = M N ⊂ P is a parabolic subgroup of G and σ is a supersingular H M -module. Hence π is a subquotient of I P ( σ ).Finally, we prove that I ( P , σ, Q ) (cid:39) I ( P (cid:48) , σ, Q (cid:48) ) implies P = P (cid:48) , Q = Q (cid:48) and σ (cid:39) σ (cid:48) . Let M (resp. M (cid:48) ) be the Levi subgroup of P (resp. P (cid:48) ). Let X be anirreducible A -submodule of I ( P , σ, Q ). By Lemma 4.23 and Lemma 4.24, supp X = n w (Λ +∆ M (1)) for w ∈ W such that ∆ w ∩ ∆( σ ) = ∆ Q . Hence { n w (Λ +∆ M (1)) | ∆ w ∩ ∆( σ ) = ∆ Q } = { n w (Λ +∆ M (cid:48) (1)) | ∆ w ∩ ∆( σ (cid:48) ) = ∆ Q (cid:48) } . Notice that w Λ +Θ (1) = w (cid:48) Λ +Θ (cid:48) (1) implies Θ = Θ (cid:48) and w ∈ w (cid:48) W Θ (cid:48) . Hence ∆ M =∆ M (cid:48) . Therefore P = P (cid:48) . If w ∈ W satisfies ∆ w ∩ ∆( σ ) = ∆ Q , then ∆ w ⊃ ∆ Q ⊃ ∆ P . Hence w (∆ P ) ⊂ Σ + . Namely w ∈ W M . Therefore, if w, w (cid:48) ∈ W satisfies∆ w ∩ ∆( σ ) = ∆ Q , ∆ w (cid:48) ∩ ∆( σ (cid:48) ) = ∆ Q (cid:48) and w ∈ w (cid:48) W M , then w = w (cid:48) . Hence { w | ∆ w ∩ ∆( σ ) = ∆ Q } = { w | ∆ w ∩ ∆( σ (cid:48) ) = ∆ Q (cid:48) } . Therefore ∆( σ ) = ∆( σ (cid:48) ) and Q = Q (cid:48) .We have w ∆ w ∆ Q σ ⊂ I ( P , σ, Q ). Take λ ∈ Λ(1) as in Lemma 4.25. Then thesubspace w ∆ w ∆ Q σ ⊂ I ( P , σ, Q ) is characterized by { x ∈ I ( P , σ, Q ) | xE ( λ ) (cid:54) = 0 } .Hence an isomorphism I ( P , σ, Q ) → I ( P , σ (cid:48) , Q (cid:48) ) gives a morphism w ∆ w ∆ Q σ → w ∆ w ∆ Q σ (cid:48) . By Proposition 4.14, we have a non-zero homomorphism σ → σ (cid:48) as H − M -modules. By Remark 4.7, σ and σ (cid:48) are irreducible H − M -modules. Hence σ (cid:39) σ (cid:48) as H − M -modules and by Remark 4.7 again, σ (cid:39) σ (cid:48) as H M -modules.Finally we introduce the notion of supercuspidality and compare it with super-singularlity. Definition 4.30. An irreducible representation π of H is called supercuspidal if itis not isomorphic to a subquotient of I P ( σ ) where P = M N is a proper parabolicsubgroup of G and σ an irreducible representation of H M . Corollary 4.31. Let π = I ( P , σ, Q ) is an irreducible representation of H . Thenthe following is equivalent. (1) π is supersingular. (2) π is supercuspidal. (3) P = G .Proof. Obviously (3) implies (1). Since I ( P , σ, Q ) is the subquotient of I P ( σ ), (2)implies (3). Assume that π is not supercuspidal. Take ( P (cid:48) , σ (cid:48) ) such that π is asubquotient of I P (cid:48) ( σ (cid:48) ) and P (cid:48) is minimal subject to this condition. Then σ (cid:48) issupercuspidal. We have already proved that irreducible supercuspidal representa-tions are supersingular. Hence σ (cid:48) is supersingular. By Corollary 4.26, it is of aform I ( P (cid:48) , σ (cid:48) , Q (cid:48) ) for some Q (cid:48) . Hence P = P (cid:48) (cid:54) = G . 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Creative Research Institution (CRIS), Hokkaido University, N21, W10, Kita-ku, Sap-poro, Hokkaido 001-0021, Japan E-mail address ::