Iwahori-Hecke model for mod p representations of GL(2,F)
aa r X i v : . [ m a t h . R T ] J a n IWAHORI-HECKE MODEL FOR MOD p REPRESENTATIONS OF GL ( F ) U. K. ANANDAVARDHANAN AND ARINDAM JANAA bstract . For a p -adic field F , the space of pro- p -Iwahori invariants of a universalsupersingular mod p representation τ of GL ( F ) is determined in the works of Breuil,Schein, and Hendel. The representation τ is introduced by Barthel and Livn´e andthis is defined in terms of the spherical Hecke operator. In [AB13, AB15], an Iwahori-Hecke approach was introduced to study these universal supersingular representationsin which they can be characterized via the Iwahori-Hecke operators. In this paper, weconstruct a certain quotient π of τ , making use of the Iwahori-Hecke operators. When F is not totally ramified over Q p , the representation π is a non-trivial quotient of τ . Wedetermine a basis for the space of invariants of π under the pro-p Iwahori subgroup.A pleasant feature of this ”new” representation π is that its space of pro- p -Iwahoriinvariants admits a more uniform description vis-`a-vis the description of the space ofpro- p -Iwahori invariants of τ .
1. I ntroduction
For a p -adic field F , the study of irreducible smooth mod p representations ofGL ( F ) started with the famous work of Barthel and Livn´e [BL94]. They showed thatthere exist irreducible smooth representations, called supersingular representations,which cannot be obtained as a subquotient of a parabolically induced representation.It is shown in [BL94] that a supersingular representation can be realized as thequotient of a universal module constructed as follows. Let G = GL ( F ) and let K be its standard maximal compact subgroup. Let Z denote the center of G . For anirreducible representation σ of KZ , let ind GKZ σ be the representation of G compactlyinduced from σ . Its endomorphism algebra is a polynomial algebra in one variable:End G (cid:16) ind GKZ σ (cid:17) ≃ F p [ T ] ,where T is the standard spherical Hecke operator and F p denotes an algebraic closureof the finite field F p of p elements [BL94, Proposition 8]. The universal module inconsideration is τ = ind GKZ σ ( T ) and a supersingular representation of G is an irreducible quotient of the universalmodule for some σ of KZ up to a twist by a character [BL94].Explicitly constructing a supersingular representation of GL ( F ) is a challengingproblem when F = Q p [BP12]. When F = Q p , Breuil proved that the universal repre-sentation τ itself is irreducible [Bre03, Theorem 1.1]. The key step in Breuil’s proof ofthe irreducibility of τ is the explicit computation of its I ( ) -invariant space, which isof dimension 2, where I ( ) is the pro- p -Iwahori subgroup of K [Bre03, Theorem 3.2.4]. Mathematics Subject Classification.
Primary 20G05; Secondary 22E50, 11F70.
The space of I ( ) -invariants of τ is infinite dimensional when F = Q p . An explicitbasis for this infinite dimensional space is computed by Schein when F is totally ram-ified over Q p [Sch11, §2] and by Hendel more generally for any p -adic field F [Hen19,Theorem 1.2].One can also construct a universal module from the perspective of the Iwahori-Hecke operators instead of the spherical Hecke operator T [AB13, AB15]. For this,instead of doing compact induction from an irreducible representation of KZ , we startwith a regular character χ of IZ , where I is the Iwahori subgroup K , and considerthe compactly induced representation ind GIZ χ . Its endomorphism algebra is [BL94,Proposition 13]: End G ( ind GIZ χ ) ≃ F p [ T − , T ]( T − T , T T − ) ,where T − and T are the Iwahori-Hecke operators. When F is a totally ramifiedextension of Q p , it is proved in [AB15, Proposition 3.1 & Remark 1] that the image ofone of these operators is equal to the kernel of the other; i.e.,(1) Im T − = Ker T & Im T = Ker T − .Let F q be the residue field of F where q = p f . Assume 0 < r < q − r = r + r p + · · · + r f − p f − with 0 ≤ r i ≤ p − ≤ i ≤ f −
1. Let σ r = Sym r F p ⊗ Sym r F p ◦ Frob ⊗ · · · ⊗
Sym r f − F p ◦ Frob f − be an irreducible representation of GL ( F q ) , where Frob is the Frobenius morphism.We continue to denote the corresponding irreducible representation of K , obtainedvia inflation, by σ r . Similarly, let χ r be the character of I , valued in F × p , obtained viathe character of the Borel subgroup of GL ( F q ) defined by (cid:18) a b d (cid:19) d r .We fix a uniformizing element ̟ of the ring of integers O of F . The representation σ r is treated as a representation of KZ by making diag ( ̟ , ̟ ) acting trivially andsimilarly the character χ r is treated as a character of IZ .For g ∈ G and v ∈ σ r , let g ⊗ v be the function in ind GKZ σ r supported on KZg − thatsends g − to σ r ( k ) v . Similarly, for g ∈ G , by [ g , 1 ] we define the function in ind GIZ χ r which is supported on IZg − and sending g − to 1. It can be seen that every elementof ind GIZ χ r (resp. ind GKZ σ r ) is a finite sum of these type of functions [ g , 1 ] (resp. g ⊗ v ).Now [AB15, Theorem 1.1] takes the form: Theorem 1.1.
Let F be a finite extension of Q p with residue field F q and residue degree f .Let < r < q − and r = r + r p + · · · + r f − p f − with ≤ r i ≤ p − . Then τ r = ind GKZ σ r ( T ) ≃ ind GIZ χ r ( Im T , Ker T ) . WAHORI-HECKE MODEL FOR MOD p REPRESENTATIONS OF GL ( F ) Moreover, this isomorphism is determined by Id ⊗ f − O j = x r j j mod T [ β , 1 ] mod ( Im T , Ker T ) . Remark . Theorem 1.1 is stated and proved in [AB15, Theorem 4.1] when F is atotally ramified extension of Q p (see [AB15, Remark 3]) and exactly the same proofgoes through in the general case as well. Remark . As mentioned earlier, the space of I ( ) -invariants of τ r is computed byHendel [Hen19, Theorem 1.2]. Stating an explicit basis for this space involves fourcases; (i) e = f =
1, (ii) e > f =
1, (iii) e = f >
1, and (iv) e > f > π r = ind GIZ χ r ( Ker T − , Ker T ) which is a further quotient of τ r . Note that this representation equals τ r when F istotally ramified over Q p by (1). We show that when F is not totally ramified over Q p ,we have strict containments(2) Im T − ( Ker T & Im T ( Ker T − .and thus we have a new representation to investigate for its properties (cf. Remark 4).At this stage, we also note that the representation π r is indeed non-trivial (cf. Lemma3.4).The main result of this paper gives an explicit basis for the space of I ( ) -invariantsof π r . This space turns out to be infinite dimensional as well as in the case of [Hen19,Theorem 1.2]. However, in this case the basis can be written in a uniform mannerwhenever F = Q p . Thus, the statement involves only two cases; (i) F = Q p and (ii) F = Q p . It is interesting to compare our result with that of Hendel in this aspect (cf.Remark 2).In order to state the theorem, we introduce a few more notations. Set I = { } , andfor n ∈ N , let I n = n [ µ ] + [ µ ] ̟ + · · · + [ µ n − ] ̟ n − | µ i ∈ F q o ⊂ O ,where, for x ∈ F q , we denote its multiplicative representative in O by [ x ] . If 0 ≤ m ≤ n , let [ · ] m : I n → I m be the truncation map defined by n − ∑ i = [ λ i ] ̟ i m − ∑ i = [ λ i ] ̟ i .Let us denote α = (cid:18) ̟ (cid:19) , β = (cid:18) ̟ (cid:19) , w = (cid:18) (cid:19) , U. K. ANANDAVARDHANAN AND ARINDAM JANA and observe that β = α w normalizes I ( ) . For any n ∈ N , we denote s kn = ∑ µ ∈ I n µ kn − (cid:20)(cid:18) ̟ n µ (cid:19) , 1 (cid:21) , t sn = ∑ µ ∈ I n µ sn − (cid:20)(cid:18) ̟ n − [ µ ] n − (cid:19) (cid:18) [ µ n − ] (cid:19) w , 1 (cid:21) ,where 0 ≤ k , s ≤ q −
1. For 0 ≤ l ≤ f − m ≥
1, we define the following sets S lm = { s q − − r + p l n } n ≥ m ∪ { β s q − − r + p l n } n ≥ m S m = f − [ l = S lm , T lm = { t r + p l n } n ≥ m ∪ { β t r + p l n } n ≥ m , T m = f − [ l = T lm .Now we state the main theorem of this paper. Theorem 1.2.
Let F be a finite extension of Q p with ramification index e. Let F q be theresidue field of F with q = p f . Let < r < q − and r = r + r p + · · · + r f − p f − with < r j < p − for all ≤ j ≤ f − . When f = , we assume < r < p − . Then a basisof the space of I ( ) -invariants of the representation π r = ind GIZ χ r ( Ker T − , Ker T ) as an F p -vector space is given by the images of the following sets in π r : (1) { [ Id, 1 ] , [ β , 1 ] } when F = Q p (2) S S { [ Id, 1 ] , [ β , 1 ] } S T when F = Q p .Remark . The representation π r that we construct and investigate in this paper is aquotient of the representation τ r considered in [BL94, Bre03, Sch11, Hen19];0 → Ker T − Im T → τ r → π r → F is totally ramified over Q p , the representations τ r and π r are isomorphicby Theorem 1.1 together with the equality of spaces in (1). However, π r is a “new”representation when F is not totally ramified over Q p . That there is no isomorphismbetween τ r and π r can be checked, for instance, from the characterization of the spaceof I ( ) -invariants of π r in Theorem 1.2 and that of τ r in [Hen19, Theorem 1.2]. Wegive more details in §4.5.Following the argument in [Hen19, Conclusion 3.10] word to word, we get thefollowing corollary to Theorem 1.2. Corollary 1.3.
The representation π r is indecomposable; i.e., End G ( π r ) ≃ F p . WAHORI-HECKE MODEL FOR MOD p REPRESENTATIONS OF GL ( F ) The plan of the paper is as follows. We collect many results about the Iwahori-Hecke operators in Section 3. Several of these results are contained in some form in[AB13, AB15]. Theorem 1.2 and the key ideas in its proof are inspired by the workof Hendel [Hen19], though the Iwahori-Hecke approach which is employed in thispaper as in [AB13, AB15] seems to be more amenable to carrying out the necessarycalculations. We take up the proof in Section 4.2. T wo basic results
As in the work of Hendel [Hen19], we will need to frequently make use of thefollowing two results in our computations.The first one is the classical result in modular combinatorics due to Lucas whichgives a condition for a binomial coefficient ( nr ) to be zero modulo p . Theorem 2.1 (Lucas) . Let n , r ∈ N be such that n = k ∑ i = n i p i and r = k ∑ i = r i p i , where ≤ n i ≤ p − and ≤ r i ≤ p − Then (cid:18) nr (cid:19) ≡ k ∏ i = (cid:18) n i r i (cid:19) mod p . Corollary 2.2.
Let n , r ∈ N . Then p divides ( nr ) if and only if n i < r i for some ≤ i ≤ k .The next result gives a formula for adding multiplicative representatives in O [Hen19, Lemma 1.7]. As in [Hen19], this formula will play a crucial role in the calcu-lations to follow. Lemma 2.3.
Let x , y ∈ F q with q = p f . Then [ x ] + [ y ] ≡ [ x + y ] + ̟ e [ P ( x , y )] mod ̟ e + , where P ( x , y ) = x qe + y qe − ( x + y ) qe ̟ e .3. P reliminaries on the I wahori -H ecke operators For n ∈ N ∪ { } and λ ∈ I n , define g n , λ = (cid:18) ̟ n λ (cid:19) & g n , λ = (cid:18) ̟λ ̟ n + (cid:19) .We have the relations g = Id, g = α , β g n , λ = g n , λ w .Now G acts transitively on the Bruhat-Tits tree of SL ( F ) , whose vertices are in a G -equivariant bijection with the cosets G / KZ and whose oriented edges are in a G -equivariant bijection with the cosets G / IZ . We have the explicit Cartan decompositiongiven by G = ∐ i ∈{ } n ≥ λ ∈ I n g in , λ KZ U. K. ANANDAVARDHANAN AND ARINDAM JANA and an explicit set of coset representatives of G / IZ is given by(3) (cid:26) g n , λ , g n , λ (cid:18) µ (cid:19) w , g n , λ w , g n , λ w (cid:18) µ (cid:19) w (cid:27) n ≥ λ ∈ I n ,where µ ∈ I .Now we recall a few details about the Iwahori-Hecke algebra [BL94, §3.2]. Bydefinition, this algebra, denoted by H ( IZ , χ r ) , is the endomorphism algebra of thecompactly induced representation ind GIZ χ r . For n ∈ Z , let φ n , n + denote the convo-lution map supported on IZ α − n I such that φ n , n + ( α − n ) = T n , n + the corresponding element in H ( IZ , χ r ) . By [BL94, Proposition 13],for 0 < r < q −
1, we have: H ( IZ , χ r ) ≃ F p [ T − , T ]( T − T , T T − ) .Substituting n = T − and T :(4) T − ([ g , 1 ]) = ∑ λ ∈ I h gg λ , 1 i ,(5) T ([ g , 1 ]) = ∑ λ ∈ I (cid:20) g β (cid:18) λ (cid:19) w , 1 (cid:21) .The following proposition characterizes the kernel of the Iwahori-Hecke operators T − and T [AB13, AB15]. Proposition 3.1.
We have: (1) Ker T − is generated as a G-module by the vectors (a) ( − ) q − − r s + t r , (b) t s where ≤ s ≤ r − , (c) t s where s > r and ( q − − rq − − s ) ≡ p. (2) Ker T is generated as a G-module by the vectors (a) t + s q − − r , (b) s k where ≤ k ≤ q − − r, (c) s k where k > q − − r and ( rq − − k ) ≡ p.Proof. We indicate the proof for Ker T , with the other case being similar. An arbi-trary vector in ind GIZ χ r is an F p -linear combination of vectors [ g , 1 ] , where g is in theset of coset representatives (3) of G / IZ . Arguing as in the proof of [AB15, Proposition3.1], we can restrict our attention to the vectors (cid:26) [ Id, 1 ] , [ β , 1 ] , [ g µ , 1 ] , (cid:20)(cid:18) µ (cid:19) w , 1 (cid:21)(cid:27) for µ ∈ I . Now the proof boils down to elementary linear algebra as in [AB15,Lemma 3.2], where one is led to analyse the indices i for which ∑ µ ∈ F q µ i ( µ − λ ) r = WAHORI-HECKE MODEL FOR MOD p REPRESENTATIONS OF GL ( F ) for λ ∈ F q . Alternatively, this last step can be deduced directly from the explicitformulas for the Iwahori-Hecke operators in [AB13, p. 63-64]. (cid:3) Remark . We remarked in (2) in Section 1 that we have strict containments(6) Im T − ( Ker T & Im T ( Ker T − .when F is not a totally ramified extension of Q p . The reason for this is that the thirdtype of vectors in both (1) and (2) in Proposition 3.1 do not belong to the images ofthe Iwahori-Hecke operators. Note that such vectors do not exist when f =
1; i.e.,when q = p . By the argument in [AB15, Lemma 3.2], it can be shown that the firsttwo types of vectors are indeed in the image of the relevant Iwahori-Hecke operator. Corollary 3.2.
A basis of the space of I ( ) -invariants of Ker T − is given by { t n , β t n } n ≥ and that of Ker T is given by { s n , β s n } n ≥ . Moreover, the action of I is given by (cid:18) a b ̟ c d (cid:19) · v = ( a r v v = t n or β s n ,d r v v = s n or β t n .Proof. The first part of Proposition 3.1 together with the observation that the space of I ( ) -invariants of the full induced representation is given by (cid:16) ind GIZ χ r (cid:17) I ( ) = h s n , t n , β s n , β t n i n ≥ .For the second part, observe that since I / I ( ) = (cid:26)(cid:18) a d (cid:19) | a , d ∈ F q × (cid:27) ,it follows that (cid:18) a b ̟ c d (cid:19) & (cid:18) a d (cid:19) have the same action on any I ( ) -invariant vector. Now, for any k ≥
0, we have (cid:18) a d (cid:19) s kn = (cid:18) a d (cid:19) ∑ µ ∈ I n µ kn − (cid:20)(cid:18) ̟ n µ (cid:19) , 1 (cid:21) = ∑ µ ∈ I n µ kn − (cid:20)(cid:18) ̟ n ad − µ (cid:19) (cid:18) a d (cid:19) , 1 (cid:21) = d r ( da − ) k s kn .A similar computation gives (cid:18) a d (cid:19) t sn = a r ( da − ) s t sn .Similarly, we can check the action on β s kn and β t kn . (cid:3) Next, we recall [AB15, Proposition 3.3], whose proof in [loc. cit.] is valid for any q . Proposition 3.3.
We have
Ker T − ∩ Ker T = { } .As a corollary to Proposition 3.3, we have the following lemma. U. K. ANANDAVARDHANAN AND ARINDAM JANA
Lemma 3.4.
For the iwahori-Hecke operators T − and T , we have ind GIZ χ r = Ker T − ⊕ Ker T . Proof.
If possible, let ind
GIZ χ r = Ker T − ⊕ Ker T .Then we get [ Id, 1 ] = v + v for some v ∈ Ker T − and v ∈ Ker T . Then, for anelement g ∈ I , g ( v + v ) = (cid:18) a b ̟ c d (cid:19) ( v + v ) = d r [ Id, 1 ] = d r ( v + v ) and this implies gv − d r v = − gv + d r v = v and v are I ( ) -invariant. By Corollary 3.2, v is a linear combination of vectors of the form { β t n } n ≥ and v is a linear combinationof vectors of the form { s n } n ≥ . But [ Id, 1 ] cannot be written as a linear combination ofthese types of vectors. (cid:3) We end this section with two more results which immediately follow from consid-erations similar to Proposition 3.1. We state these in a ready to use format here (seealso [AB13, p. 63-64]).
Lemma 3.5.
Let ≤ i j ≤ q − for ≤ j ≤ n − and µ = [ µ ] + [ µ ] ̟ + · · · +[ µ n − ] ̟ n − ∈ I n . Write i n − = i n − + i n − p + · · · + i n − f − p f − . Then (1) ∑ µ · · · ∑ µ n − µ i . . . µ i n − n − h g n , µ , 1 i ∈ Ker T if and only if ≤ i n − ≤ q − − r ori n − > q − − r such that i n − j < p − − r j for some ≤ j ≤ f − . (2) ∑ µ · · · ∑ µ n − µ i . . . µ i n − n − (cid:20) g n − [ µ ] n − (cid:18) [ µ n − ] (cid:19) w , 1 (cid:21) ∈ Ker T − if and only if ≤ i n − ≤ r − or i n − > r such that i n − j < r j for some ≤ j ≤ f − .Remark . Note that in Lemma 3.5, the range for j is 0 ≤ j ≤ f − i n − > q − − r = ⇒ i n − f − ≥ p − − r f − . Remark . Note that the condition i n − > q − − r & i n − j < p − − r j for some 0 ≤ j ≤ f − (cid:18) rq − − i n − (cid:19) ≡ p which is related to the condition in (c) of Proposition 3.1 (2). Similarly, the condition i n − > r & i n − j < r j for some 0 ≤ j ≤ f − q . WAHORI-HECKE MODEL FOR MOD p REPRESENTATIONS OF GL ( F ) Lemma 3.6.
Let µ = [ µ ] + · · · + [ µ n − ] ̟ n − ∈ I n . Then modulo ( Ker T − , Ker T ) , wehave the identities (1) ∑ µ n − ∈ I µ q − − rn − h g n , µ , 1 i = − (cid:20) g n − [ µ ] n − (cid:18) [ µ n − ] (cid:19) w , 1 (cid:21) ,(2) ∑ µ n − ∈ I µ rn − (cid:20) g n − [ µ ] n − (cid:18) [ µ n − ] (cid:19) w , 1 (cid:21) = ( − ) r − h g n − [ µ ] n − , 1 i . Remark . In fact, (1) is true modulo Ker T and (2) is true modulo Ker T − (cf.[AB13, (4) & (5) on p. 62]). 4. P roof of T heorem A set of I ( ) -invariants. First we make the following observation [Hen19, §2.1].For a , b , c ∈ O , any matrix in I ( ) can be written as (cid:18) + ̟ a b ̟ c + ̟ d (cid:19) = (cid:18) ( + ̟ d ) − b (cid:19) (cid:18) ̟ ct − (cid:19) (cid:18) t
00 1 + ̟ d (cid:19) ,where t = + ̟ ( a − bc ( + ̟ d ) − ) . Hence to prove that a certain vector is I ( ) -invariant modulo ( Ker T − , Ker T ) , it is enough to check for invariance under (cid:18) b (cid:19) , (cid:18) ̟ c (cid:19) , (cid:18) + ̟ a
00 1 (cid:19) ,where a , b , c ∈ O .We first prove that the set of vectors S and T are I ( ) -invariants when consideredas vectors in π r ; i.e., when we consider the images of these vectors modulo Ker T − ⊕ Ker T . The first step in achieving this is an inductive argument which reduces thegeneral case to the case n = Lemma 4.1.
If s kn − (resp. t sn − ) is I ( ) -invariant modulo ( Ker T − , Ker T ) , then, for alln ≥ , the vector s kn (resp. t sn ) is also I ( ) -invariant modulo ( Ker T − , Ker T ) .Proof. We prove the case of s kn and the case of t sn is similar. Assume that s kn − is I ( ) -invariant modulo ( Ker T − , Ker T ) . Now, (cid:18) b (cid:19) s kn = ∑ µ ∈ I n µ kn − (cid:20)(cid:18) b (cid:19) (cid:18) ̟ n µ (cid:19) , 1 (cid:21) = ∑ µ ∈ I n µ kn − (cid:18) b (cid:19) (cid:18) ̟ [ µ ] (cid:19) ̟ n − n − ∑ i = [ µ i ] ̟ i − , 1 = ∑ µ ∈ I n µ kn − (cid:18) ̟ [ µ + b ] (cid:19) (cid:18) B ( µ , b ) (cid:19) ̟ n − n − ∑ i = [ µ i ] ̟ i − ,0 1 , 1 where(7) B ( µ , b ) = ̟ e − [ P ( µ , b )] + [ b ] + [ b ] ̟ + . . .and(8) P ( µ , b ) = µ q e + b q e − ( µ + b ) q e ̟ e is obtained from the formula in Lemma 2.3. Let µ ′ = [ µ ] + [ µ ] ̟ + · · · + [ µ n − ] ̟ n − .We continue by making the substitution µ → µ − b . Thus, (cid:18) b (cid:19) s kn = ∑ µ ∈ I (cid:18) ̟ [ µ ] (cid:19) (cid:18) B ( µ − b , b ) (cid:19) ∑ µ ′ ∈ I n − µ ′ kn − (cid:20)(cid:18) ̟ n − µ ′ (cid:19) , 1 (cid:21) = ∑ µ ∈ I (cid:18) ̟ [ µ ] (cid:19) ∑ µ ′ ∈ I n − µ ′ kn − (cid:20)(cid:18) ̟ n − µ ′ (cid:19) , 1 (cid:21) + x µ ,by our assumption, where x µ ∈ ( Ker T − , Ker T ) . Thus, we get (cid:18) b (cid:19) s kn = s kn + ∑ µ ∈ I (cid:18) ̟ [ µ ] (cid:19) x µ ,and hence (cid:18) b (cid:19) s kn − s kn ∈ ( Ker T − , Ker T ) .Checking for invariance under (cid:18) ̟ c (cid:19) & (cid:18) + ̟ a
00 1 (cid:19) is even easier which we skip. (cid:3)
WAHORI-HECKE MODEL FOR MOD p REPRESENTATIONS OF GL ( F ) Now we take the case n =
2. Recall that e (resp. f ) is the ramification index (resp.residue degree) of F over Q p . We write r = r + r p + · · · + r f − p f − where 0 ≤ r j ≤ p − ≤ j ≤ f − a , b , c ∈ O , we have(9) (cid:18) + ̟ a bc ̟ + d ̟ (cid:19) (cid:18) ̟ [ µ ] (cid:19) = (cid:18) ̟ [ µ + b ] (cid:19) k ,for k ∈ I ( ) . Indeed,LHS = (cid:18) ̟ ( + ̟ a ) [ µ + b ] + ̟ ( ∗ ) c ̟ + ̟ ( ∆ ) (cid:19) = (cid:18) ̟ [ µ + b ] (cid:19) (cid:18) + a ̟ − [ µ + b ] c ̟ ( ∗ ) − ( µ + b ) ∆ c ̟ + ̟ ∆ (cid:19) ,where ∗ , ∆ ∈ O . Similarly, one can show that(10) (cid:18) + ̟ a bc ̟ + d ̟ (cid:19) (cid:18) [ µ ] (cid:19) w = (cid:18) [ µ + b ] (cid:19) wk ′ for some k ′ ∈ I ( ) . Lemma 4.2.
Assume < r j < p − , and if f = , assume further that < r < p − .Then when ( e , f ) = (
1, 1 ) , we havegs q − − r + p l − s q − − r + p l ∈ ( Ker T − , Ker T ) and gt r + p l − t r + p l ∈ ( Ker T − , Ker T ) for all g ∈ I ( ) and ≤ l ≤ f − Proof.
We have (cid:18) b (cid:19) s q − − r + p l = ∑ µ ∈ I µ q − − r + p l (cid:20)(cid:18) b (cid:19) (cid:18) ̟ [ µ ] + [ µ ] ̟ (cid:19) , 1 (cid:21) = ∑ µ ∈ I µ q − − r + p l (cid:20)(cid:18) b (cid:19) (cid:18) ̟ [ µ ] (cid:19) (cid:18) ̟ [ µ ] (cid:19) , 1 (cid:21) = ∑ µ ∈ I µ q − − r + p l (cid:20)(cid:18) ̟ [ µ + b ] (cid:19) (cid:18) B ( µ , b ) (cid:19) (cid:18) ̟ [ µ ] (cid:19) , 1 (cid:21) ,where B ( µ , b ) is given by (7) in the proof of Lemma 4.1. Now write B ( µ , b ) = [ b + Z ] + ( ∗ ) ̟ where Z = e > Z = P ( µ , b ) for e = To continue, the above expression equals ∑ µ ∈ I µ q − − r + p l (cid:20)(cid:18) ̟ [ µ + b ] (cid:19) (cid:18) [ b + Z ] + ( ∗ ) ̟ (cid:19) (cid:18) ̟ [ µ ] (cid:19) , 1 (cid:21) which equals ∑ µ ∈ I µ q − − r + p l (cid:20)(cid:18) ̟ [ µ + b ] (cid:19) (cid:18) ̟ [ µ + b + Z ] (cid:19) k , 1 (cid:21) for k ∈ I ( ) , by (9). We continue by making the change of variables µ → µ − b − Z & µ → µ − b ,and we get ∑ µ ∈ I ( µ − b − Z ) q − − r + p l (cid:20)(cid:18) ̟ µ (cid:19) , 1 (cid:21) = s q − − r + p l + ∑ µ ∈ I q − − r + p l − ∑ i = (cid:18) q − − r + p l i (cid:19) ( − b − Z ) q − − r + p l − i µ i (cid:20)(cid:18) ̟ µ (cid:19) , 1 (cid:21) .Now we read the above expression modulo Ker T . We claim that only the termcorresponding to i = q − − r remains amongst the q − − r + p l terms in the innersummation in the above expression. By Lemma 3.6 (1), we know that ∑ µ ∈ I µ i (cid:20)(cid:18) ̟ µ (cid:19) , 1 (cid:21) ∈ Ker T precisely when 0 ≤ i ≤ q − − r or i > q − − r such that i j < p − − r j for some0 ≤ j ≤ f −
2. Note that if i > q − − r and i j ≥ p − − r j for all 0 ≤ j ≤ f − i j > p − − r j for some 0 ≤ j ≤ l − i ≤ q − − r + p l − (cid:18) q − − r + p l i (cid:19) ≡ p by Corollary 2.2. Thus, modulo Ker T , we get (cid:18) b (cid:19) s q − − r + p l = s q − − r + p l + ∑ µ ∈ I (cid:18) q − − r + p l q − − r (cid:19) ( − b − Z ) p l µ q − − r (cid:20)(cid:18) ̟ µ (cid:19) , 1 (cid:21) = s q − − r + p l + ∑ µ ∈ I ( p − r l )( − b − Z ) p l (cid:20)(cid:18) [ µ ] (cid:19) w , 1 (cid:21) by Lemma 3.6 (1) and the binomial coefficient here is computed via Theorem 2.1.Now if e > Z =
0. Therefore, it follows, by Lemma 3.5 (2), that (cid:18) b (cid:19) s q − − r + p l − s q − − r + p l ∈ Ker T − , WAHORI-HECKE MODEL FOR MOD p REPRESENTATIONS OF GL ( F ) and thus we have proved (cid:18) b (cid:19) s q − − r + p l ≡ s q − − r + p l mod ( Ker T − , Ker T ) .If e = Z = P ( µ , b ) . As F is unramified over Q p , we have ̟ = p . Now byCorollary 2.2, it follows that Z = µ q e + b q e − ( µ + b ) q e ̟ e ≡ − p − ∑ i = p (cid:18) p f ip f − (cid:19) b p f − ip f − µ ip f − mod p .In this case, if further f = T , (cid:18) b (cid:19) s q − − r + p l − s q − − r + p l = ∑ µ ∈ I ( p − r l )( − b − Z ) p l (cid:20)(cid:18) [ µ ] (cid:19) w , 1 (cid:21) = ∑ µ ∈ I r l ( b p l + Z p l ) (cid:20)(cid:18) [ µ ] (cid:19) w , 1 (cid:21) .Note that both ∑ µ ∈ I (cid:20)(cid:18) [ µ ] (cid:19) w , 1 (cid:21) & ∑ µ ∈ I µ ip l − (cid:20)(cid:18) [ µ ] (cid:19) w , 1 (cid:21) are in Ker T − , by Lemma 3.5 (2). Thus, once again we have proved (cid:18) b (cid:19) s q − − r + p l ≡ s q − − r + p l mod ( Ker T − , Ker T ) .Now we analyze invariance for the lower unipotent representative of I ( ) . We have (cid:18) ̟ c (cid:19) s q − − r + p l = ∑ µ ∈ I µ q − − r + p l (cid:20)(cid:18) ̟ c (cid:19) (cid:18) ̟ [ µ ] + [ µ ] ̟ (cid:19) , 1 (cid:21) = ∑ µ ∈ I µ q − − r + p l (cid:20)(cid:18) ̟ c (cid:19) (cid:18) ̟ [ µ ] (cid:19) (cid:18) ̟ [ µ ] (cid:19) , 1 (cid:21) which we express as ∑ µ ∈ I µ q − − r + p l (cid:20)(cid:18) ̟ [ µ ] (cid:19) (cid:18) − ̟ c [ µ ] − [ µ ] c ̟ c + ̟ c [ µ ] (cid:19) (cid:18) ̟ [ µ ] (cid:19) , 1 (cid:21) and this equals ∑ µ ∈ I µ q − − r + p l (cid:20)(cid:18) ̟ [ µ ] (cid:19) (cid:18) ̟ [ µ − c µ ] (cid:19) k , 1 (cid:21) for k ∈ I ( ) by (9). Changing µ → µ + c µ , we get (cid:18) ̟ c (cid:19) s q − − r + p l = ∑ µ ∈ I ( µ + c µ ) q − − r + p l (cid:20)(cid:18) ̟ µ (cid:19) , 1 (cid:21) which we read modulo Ker T and get s q − − r + p l + ∑ µ ∈ I (cid:18) q − − r + p l q − − r (cid:19) ( c µ ) p l µ q − − r (cid:20)(cid:18) ̟ µ (cid:19) , 1 (cid:21) by Corollary 2.2 together with Lemma 3.5 (1), exactly as we have argued before. Nowthis equals, modulo Ker T , s q − − r + p l + ∑ µ ∈ I ( p − r l ) c p l µ p l (cid:20)(cid:18) [ µ ] (cid:19) w , 1 (cid:21) by Theorem 2.1 and Lemma 3.6 (1). By Lemma 3.5 (2), this vector belongs to Ker T − (with the extra assumption that 3 ≤ r when f = (cid:18) ̟ c (cid:19) s q − − r + p l ≡ s q − − r + p l mod ( Ker T − , Ker T ) .The proof for showing that (cid:18) + ̟ a
00 1 (cid:19) s q − − r + p l − s q − − r + p l ∈ ( Ker T − , Ker T ) is similar and therefore we skip it.The argument for gt r + p l − t r + p l ∈ ( Ker T − , Ker T ) for all g ∈ I ( ) is similar to the one for s q − − r + p l . Note that corresponding to thecase 3 ≤ r in the totally ramified case for s q − − r + p l , in the case of t r + p l we will get r ≤ p − (cid:3) Linear independence.
The following lemma gives the action of the Iwahori sub-group I on the I ( ) -invariant vectors (cf. [Hen19, Lemma 3.6]). Lemma 4.3.
Let (cid:18) a bc d (cid:19) ∈ I . Let s kn and t sn be I ( ) -invariants modulo ( Ker T − , Ker T ) . Then they are I-eigenvectors and those actions are given by (1) (cid:18) a bc d (cid:19) · s kn = d r ( da − ) k s kn ,(2) (cid:18) a bc d (cid:19) · t sn = a r ( da − ) s t sn . Proof.
The proof is straightforward and we have already done it in the proof of thesecond part of Corollary 3.2. (cid:3)
Remark . Lemmas 4.1, 4.2 and 4.3 remain true for β s kn and β t sn . Proposition 4.4.
The set of vectors in S ∪ T of Theorem 1.2 are linearly independent.Proof. Note that the vectors in S ∪ T consist of vectors of the form s q − − r + p l n , β s q − − r + p l n , t r + p l n , β t r + p l n WAHORI-HECKE MODEL FOR MOD p REPRESENTATIONS OF GL ( F ) for n ≥ ≤ l ≤ f −
1. These are invariant under I ( ) modulo ( Ker T − , Ker T ) except for the case when both e = f = v ∈ ind GIZ χ r , note that v and β v cannot cancel each other (pictori-ally they are on two different sides of the tree of SL ( F ) ). Therefore, it is enough toshow that the set { s q − − r + p l n , t r + p l n } , for n ≥ ≤ l ≤ f −
1, is linearly indepen-dent. Since s q − − r + p l n and t r + p l n have different I -eigenvalues, it is enough to show that { s q − − r + p l n } and { t r + p l n } , for n ≥ ≤ l ≤ f −
1, are linearly independent.We show that the vectors in { s q − − r + p l n } n ≥ ≤ l ≤ f − are linearly independent, and the proof for { t r + p l n } is similar. Suppose that n ∑ i = c i s q − − r + p l i ∈ ( Ker T − , Ker T ) where c i ∈ F p and n ∈ N . Since no reduction is possible in the above expression andalso these vectors obviously cannot be in Ker T − , it follows that n ∑ i = c i s q − − r + p l i ∈ Ker T .For i = j with 2 ≤ i , j ≤ n , once again from the formula for T , there cannot be anycancellation between T ( c i s q − − r + p l i ) and T ( c j s q − − r + p l j ) , so we get c i s q − − r + p l i ∈ Ker T for all 2 ≤ i ≤ n . By Lemma 3.5 (1), it follows that c i = ≤ i ≤ n . (cid:3) Remark . It follows by eigenvalue considerations as in the proof of Proposition 4.4that the set S ∪ { [ Id, 1 ] , [ β , 1 ] } ∪ T is linearly independent.4.3. Auxiliary lemmas.
We will have to make use of the following elementary lemma[Hen19, Lemma 2.8].
Lemma 4.5.
Let n ≥ and φ : I n → F p be any set map. Then there exists a uniquepolynomial Q ( x , . . . , x n − ) ∈ F p [ x , x , . . . , x n − ] in which degree of each variable is atmost q − and φ ( µ ) = Q ( µ , µ , . . . , µ n − ) for all µ ∈ I n .The next two lemmas are the first steps towards the proof of Theorem 1.2. Lemma 4.6.
Let µ = [ µ ] + [ µ ] ̟ + · · · + [ µ n − ] ̟ n − ∈ I n and r = r + r p + · · · + r f − p f − with < r j < p − for all ≤ j ≤ f − Letf n = f ′ n + f ′′ n be such that f ′ n = ∑ µ ∈ I n a ( µ , µ , . . . , µ n − ) (cid:20)(cid:18) ̟ n µ (cid:19) , 1 (cid:21) and f ′′ n = ∑ µ ∈ I n b ( µ , µ , . . . , µ n − ) (cid:20)(cid:18) ̟ n − [ µ ] n − (cid:19) (cid:18) [ µ n − ] (cid:19) w , 1 (cid:21) , where a ( µ , . . . , µ n − ) and b ( µ , . . . , µ n − ) are polynomials in µ , . . . , µ n − . Suppose (cid:18) − ̟ n − (cid:19) f n − f n ∈ ( Ker T − , Ker T ) . Then (1) the possible powers of µ n − , say k = k + k p + · · · + k f − p f − , in a ( µ , . . . , µ n − ) will satisfy one of the following three conditions: (a) there exists some ≤ j ′ ≤ f − such that k j ′ < p − − r j ′ ,(b) k j = p − − r j for all ≤ j ≤ f − k j = p − − r j for j = l and k l = p − r l for some ≤ l ≤ f − the possible powers of µ n − , say k = k + k p + · · · + k f − p f − , in b ( µ , . . . , µ n − ) will satisfy one of the following three conditions: (a) there exists some ≤ j ′ ≤ f − such that such that k j ′ < r j ′ ,(b) k j = r j for all ≤ j ≤ f − k j = r j for j = l and k l = r l + for some ≤ l ≤ f − Proof of Lemma 4.6.
We will prove ( ) and the proof of ( ) is similar. Suppose ( ) doesnot hold. Then there exists k such that k j ≥ p − − r j for all 0 ≤ j ≤ f − k j > p − − r j for some 0 ≤ j ≤ f − k = ( p − r j ) p j + f − ∑ j = j = ( p − − r j ) p j .Then either there exists j with j = j such that k j > p − − r j or k = k j p j + f − ∑ j = j = ( p − − r j ) p j with k j > p − r j . Choose k with the above property such that there is no othermonomial µ k ′ n − in a ( µ , . . . , µ n − ) with k j ≤ k ′ j for all 0 ≤ j ≤ f −
1. Since a polynomialis of finite degree, such a k exists. Let g = (cid:18) − ̟ n − (cid:19) .We have g f n − f n = ( g f ′ n − f ′ n ) + ( g f ′′ n − f ′′ n ) ∈ ( Ker T − , Ker T ) .Note that, g f ′ n − f ′ n = ∑ µ ∈ I n [ a ([ µ ] n − , µ n − + ) − a ([ µ ] n − , µ n − )] (cid:20)(cid:18) ̟ n µ (cid:19) , 1 (cid:21) and g f ′′ n − f ′′ n = ∑ µ ∈ I n [ b ([ µ ] n − , µ n − + ) − b ([ µ ] n − , µ n − )] (cid:20)(cid:18) ̟ n − [ µ ] n − (cid:19) (cid:18) [ µ n − ] (cid:19) w , 1 (cid:21) . WAHORI-HECKE MODEL FOR MOD p REPRESENTATIONS OF GL ( F ) Let ∆ a = a ([ µ ] n − , µ n − + ) − a ([ µ ] n − , µ n − ) considered as a polynomial in µ n − with coefficients in F p [ µ , . . . , µ n − ] . By Theorem2.1, we have ( µ n − + ) k − µ kn − ≡ k − ∑ i = f − ∏ j = (cid:18) k j i j (cid:19) µ in − mod p .Now if there exists j with j = j such that k j > p − − r j , take k ′ = ( k j − ) p j + f − ∑ j = j = k j p j .The coefficient of µ k ′ n − in ∆ a is (cid:18) kk ′ (cid:19) = (cid:18) k j k j − (cid:19) p by Theorem 2.1 and Corollary 2.2. Note that the term involving µ k ′ n − in g f ′ n − f ′ n cannot get cancelled by any other term in g f n − f n . Indeed, it cannot get cancelledwith any other term in g f ′ n − f ′ n because of the choice of k and anyway no term in g f ′ n − f ′ n can get cancelled with a term in g f ′′ n − f ′′ n (pictorially they represent edges ofopposite orientation on the tree of SL ( F ) ). So this term involving µ k ′ n − must be therein ( Ker T − , Ker T ) , but then Lemma 3.5 (1) would imply that there exists some0 ≤ l ≤ f − k ′ l < p − − r l , which contradicts our assumption. So k mustbe of the form k = k j p j + f − ∑ j = j = ( p − − r j ) p j with k j > p − r j . Taking k ′ = ( k j − ) p j + f − ∑ j = j = ( p − − r j ) p j ,and using the same argument as in the previous case, we arrive at a contradiction. (cid:3) Remark . The idea of choosing k as in Lemma 4.6 is already employed by Hendel in[Hen19, Lemma 3.13].Now we state one more lemma whose main idea of proof also comes from [Hen19,Lemma 3.13]. In what follows, B ( t ) denotes the ball of radius m on the tree of SL ( F ) with center at the vertex representing the trivial coset G / KZ . Explicitly it consists oflinear combinations of vectors of the form B ( t ) = (cid:26) [ g n , µ , 1 ] , (cid:20) g n − [ µ ] n − (cid:18) [ µ n − ] (cid:19) w , 1 (cid:21)(cid:27) n ≤ t ,and B ( t ) = (cid:26) [ g n − [ µ ] n − w , 1 ] , (cid:20) g n − [ µ ] n − w (cid:18) [ µ n − ] (cid:19) w , 1 (cid:21)(cid:27) n ≤ t ,where µ = [ µ ] + [ µ ] ̟ + · · · + [ µ n − ] ̟ n − ∈ I n . Lemma 4.7.
Let f ′ n = ∑ µ ∈ I n f − ∑ l = P l ([ µ ] n − ) µ q − − r + p l n − h g n , µ , 1 i and f ′′ n = ∑ µ ∈ I n f − ∑ l = Q l ([ µ ] n − ) µ r + p l n − (cid:20) g n − [ µ ] n − (cid:18) [ µ n − ] (cid:19) w , 1 (cid:21) , where P l ([ µ ] n − ) and Q l ([ µ ] n − ) are polynomials in µ , . . . , µ n − . Let f n = f ′ n + f ′′ n . Letf = f n + f ′ be such that f ′ ∈ B ( n − ) and (cid:18) − ̟ n − m (cid:19) f − f ∈ ( Ker T − , Ker T ) , for all ≤ m ≤ n − Then we havef ′ n = ∑ µ ∈ I n f − ∑ l = a l µ q − − r + p l n − h g n , µ , 1 i and f ′′ n = ∑ µ ∈ I n f − ∑ l = b l µ r + p l n − (cid:20) g n − [ µ ] n − (cid:18) [ µ n − ] (cid:19) w , 1 (cid:21) , where a l and b l are constants.Proof of Lemma 4.7. We do the proof only for f ′ n , as the case of f ′′ n is similar. Theproof is by induction on n . Note that P l ([ µ ] n − ) is independent of µ n − . Suppose it isindependent of µ n − , . . . , µ n − m + . Then f ′ n = ∑ µ ∈ I n f − ∑ l = P l ([ µ ] n − m , µ n − m ) µ q − − r + p l n − h g n , µ , 1 i We show that it is independent of µ n − m . It is given to us that (cid:18) − ̟ n − m (cid:19) f − f = (cid:20)(cid:18) − ̟ n − m (cid:19) f n − f n (cid:21) + (cid:20)(cid:18) − ̟ n − m (cid:19) f ′ − f ′ (cid:21) ∈ ( Ker T − , Ker T ) .Now, (cid:18) − ̟ n − m (cid:19) ̟ n n − ∑ i = [ µ i ] ̟ i = (cid:18) ̟ n [ µ ] + · · · + [ µ n − ] ̟ n − − ̟ n − m (cid:19) and this equals ̟ n n − m − ∑ i = [ µ i ] ̟ i + [ µ n − m − ] ̟ n − m + [ µ ′ n − m + ] ̟ n − m + + · · · + [ µ ′ n − ] ̟ n − where µ ′ k = µ k + c k ( µ n − m , . . . , µ n − ) for n − m + ≤ k ≤ n − WAHORI-HECKE MODEL FOR MOD p REPRESENTATIONS OF GL ( F ) Note that the transformation µ ′ k µ k − c k ( µ n − m , . . . , µ n − ) does not affect the vari-ables µ k for n − m + ≤ k ≤ n − P l ([ µ ] n − ) , as it is independent of these variables.This transformation together with µ n − m µ n − m + (cid:18) − ̟ n − m (cid:19) f ′ n = ∑ µ ∈ I n f − ∑ l = P l ([ µ ] n − m , µ n − m + )( µ n − − c n − ) q − − r + p l [ g n , µ , 1 ] .In the above expression, by c n − we mean c n − ( µ n − m , . . . , µ n − ) . Now, (cid:18) − ̟ n − m (cid:19) f ′ n − f ′ n = ∑ µ ∈ I n f − ∑ l = α ( µ , l )[ g n , µ , 1 ] ,where α ( µ , l ) = h P l ([ µ ] n − m , µ n − m + )( µ n − − c n − ) q − − r + p l − P l ([ µ ] n − m , µ n − m ) µ q − − r + p l n − i .Thus, (cid:18) − ̟ n − m (cid:19) f ′ n − f ′ n = ∑ µ ∈ I n f − ∑ l = [ P l ([ µ ] n − m , µ n − m + ) − P l ([ µ ] n − m , µ n − m )] µ q − − r + p l n − [ g n , µ , 1 ]+ ∑ µ ∈ I n f − ∑ l = q − − r + p l − ∑ i = β ( µ , l , i )[ g n , µ , 1 ] ,where β ( µ , l , i ) = P l ([ µ ] n − m , µ n − m + )( − ) i (cid:18) q − − r + p l i (cid:19) ( − c n − ) q − − r + p l − i µ in − .Now we read this modulo Ker T . Thus, we get (cid:18) − ̟ n − m (cid:19) f ′ n − f ′ n = ∑ µ ∈ I n f − ∑ l = [ P l ([ µ ] n − m , µ n − m + ) − P l ([ µ ] n − m , µ n − m )] µ q − − r + p l n − [ g n , µ , 1 ]+ ∑ µ ∈ I n f − ∑ l = P l ([ µ ] n − m , µ n − m + ) (cid:18) q − − r + p l q − − r (cid:19) ( − c n − ) p l µ q − − rn − [ g n , µ , 1 ] ,by Corollary 2.2 and Lemma 3.5 (1), exactly as we have argued before in the proof ofLemma 4.2. Now by Lemma 3.6 (1), it follows that, modulo ( Ker T − , Ker T ) , wehave (cid:18) − ̟ n − m (cid:19) f ′ n − f ′ n = ∑ µ ∈ I n f − ∑ l = [ P l ([ µ ] n − m , µ n − m + ) − P l ([ µ ] n − m , µ n − m )] µ q − − r + p l n − [ g n , µ , 1 ] + g n −
10 U. K. ANANDAVARDHANAN AND ARINDAM JANA where g n − ∈ B ( n − ) . As r l =
0, by Lemmas 3.5 (1) and 3.6 (1) we have ∑ µ ∈ I n µ q − − r + p l n − h g n , µ , 1 i / ∈ ( Ker T − , Ker T ) .Also the term involving µ q − − r + p l n − cannot get cancelled by any other term in theexpression (cid:18) − ̟ n − m (cid:19) f − f .So it follows that P l ([ µ ] n − m , µ n − m + ) − P l ([ µ ] n − m , µ n − m ) = P l ([ µ ] n − ) is independent of µ n − m . Therefore, by induction P l ([ µ ] n − ) is a con-stant. (cid:3) Proof of Theorem 1.2.
Clearly the vectors [ Id, 1 ] and [ β , 1 ] are fixed by I ( ) . ByLemmas 4.1 and 4.2 and Remark 8 the vectors in S and T are I ( ) -invariant modulo ( Ker T − , Ker T ) except for the case when both e = f =
1. By Remark 9, theset S ∪ { [ Id, 1 ] , [ β , 1 ] } ∪ T is linearly independent.Now let f ∈ ind GIZ χ r be an I ( ) -invariant of π r = ind GIZ χ r ( Ker T − , Ker T ) .We write f = f + f where f (resp. f ) is a linear combination of vectors on the zero side (resp. one side)of the tree of SL ( F ) . By this, we mean f is a linear combination of vectors of theform [ g n , µ , 1 ] , (cid:20) g n − [ µ ] n − (cid:18) [ µ n − ] (cid:19) w , 1 (cid:21) and f is a linear combination of vectors of the form [ g n − [ µ ] n − w , 1 ] , (cid:20) g n − [ µ ] n − w (cid:18) [ µ n − ] (cid:19) w , 1 (cid:21) .Then, g f i − f i ∈ ( Ker T − , Ker T ) ,for all i ∈ {
0, 1 } and g ∈ I ( ) . Since β f is a linear combination of vectors on the zeroside and β normalizes I ( ) , without loss of generality, we may assume f = f . Write f = f n + f ′ with f n = f ′ ∈ B ( n − ) , for n maximal. Now, f n = ∑ µ ∈ I n a µ h g n , µ , 1 i + ∑ µ ∈ I n b µ (cid:20) g n − [ µ ] n − (cid:18) [ µ n − ] (cid:19) w , 1 (cid:21) , WAHORI-HECKE MODEL FOR MOD p REPRESENTATIONS OF GL ( F ) where µ = [ µ ] + [ µ ] ̟ + · · · + [ µ n − ] ̟ n − and a µ , b µ ∈ F p . By Lemma 4.5, the coeffi-cients a µ and b µ can be replaced by the polynomials a ( µ , . . . , µ n − ) and b ( µ , . . . , µ n − ) respectively, where each µ i has maximum degree q −
1. Write f n = f ′ n + f ′′ n ,where f ′ n = ∑ µ ∈ I n ∑ i a ( i , i , . . . , i n − ) µ i . . . µ i n − n − h g n , µ , 1 i ,and f ′′ n = ∑ µ ∈ I n ∑ j b ( j , j , . . . , j n − ) µ j . . . µ j n − n − (cid:20) g n − [ µ ] n − (cid:18) [ µ n − ] (cid:19) w , 1 (cid:21) .Let g ′ = (cid:18) − ̟ n − (cid:19) ∈ I ( ) .Since f ′ belongs in B ( n − ) , it is easy to check that g ′ fixes f ′ . This gives g ′ f n − f n ∈ ( Ker T − , Ker T ) .Now Lemma 3.5 (1) together with Lemma 4.6 (1) gives f ′ n = ∑ µ ∈ I n ∑ i a ( i , . . . , i n − , q − − r ) µ i . . . µ q − − rn − h g n , µ , 1 i + ∑ µ ∈ I n f − ∑ l = a l ([ µ ] n − ) µ q − − r + p l n − h g n , µ , 1 i ,which in turn implies that f ′ n − ∑ µ ∈ I n f − ∑ l = a l ([ µ ] n − ) µ q − − r + p l n − h g n , µ , 1 i = ∑ µ ,..., µ n − ∑ i ,..., i n − a ( i , . . . , q − − r ) µ i . . . µ q − − rn − h g n , µ , 1 i which modulo Ker T equals ∑ µ ,..., µ n − ∑ i ,..., i n − a ( i , . . . , q − − r ) µ i . . . µ i n − n − (cid:20) g n − [ µ ] n − (cid:18) [ µ n − ] (cid:19) w , 1 (cid:21) ,by Lemma 3.6 (1). This vector belongs to B ( n − ) which we call g ′ n − . We get f ′ n = ∑ µ ∈ I n f − ∑ l = a l ([ µ ] n − ) µ q − − r + p l n − h g n , µ , 1 i + g ′ n − .Similarly, working with f ′′ n , we get f ′′ n = ∑ µ ∈ I n f − ∑ l = b l ([ µ ] n − ) µ r + p l n − (cid:20) g n − [ µ ] n − (cid:18) [ µ n − ] (cid:19) w , 1 (cid:21) + g ′′ n − for some g ′′ n − ∈ B ( n − ) , by Lemmas 3.5 (2), 4.6 (2) and 3.6 (2). For 1 ≤ m ≤ n −
1, we note that (cid:18) − ̟ n − m (cid:19) ∈ I ( ) .Using the condition (cid:18) − ̟ n − m (cid:19) f − f ∈ ( Ker T − , Ker T ) ,by Lemma 4.7, we have f ′ n = ∑ µ ∈ I n f − ∑ l = a l , n µ q − − r + p l n − h g n , µ , 1 i + g ′ n − and f ′′ n = ∑ µ ∈ I n f − ∑ l = b l , n µ r + p l n − (cid:20) g n − [ µ ] n − (cid:18) [ µ n − ] (cid:19) w , 1 (cid:21) + g ′′ n − ,where a l and b l are constants.Hence f n takes the form f n = f − ∑ l = a l , n s q − − r + p l n + f − ∑ l = b l , n t r + p l n + g n − ,where g n − = g ′ n − + g ′′ n − ∈ B ( n − ) .Thus it follows that f − f − ∑ l = a l , n s q − − r + p l n − f − ∑ l = b l , n t r + p l n = g n − + f ′ is an I ( ) -invariant vector modulo ( Ker T − , Ker T ) in B ( n − ) .Applying this argument on vectors in B ( n − ) and repeating this process, we get f = f − ∑ l = a l , n s q − − r + p l n + f − ∑ l = b l , n t r + p l n + · · · + f − ∑ l = a l ,2 s q − − r + p l + f − ∑ l = b l ,2 t r + p l + f ,where f is an I ( ) -invariant in B ( ) . Write f = f ′ + f ′′ ,where f ′ = ∑ µ ∈ I ∑ i a i µ i h g µ , 1 i ,and f ′′ = ∑ µ ∈ I ∑ j b j µ j (cid:20)(cid:18) µ (cid:19) w , 1 (cid:21) .Using the action of u = (cid:18) (cid:19) WAHORI-HECKE MODEL FOR MOD p REPRESENTATIONS OF GL ( F ) on f , by Lemma 4.6 (1), the possible powers i of µ in f ′ will satisfy either 0 ≤ i ≤ q − − r or i = q − − r + p l for some 0 ≤ l ≤ f −
1. If i = q − − r + p l , then (cid:18) (cid:19) f ′ − f ′ = ∑ µ ∈ I µ q − − r h g µ , 1 i ∈ ( Ker T − , Ker T ) .This, by Lemma 3.6 (1), gives [ β , 1 ] ∈ ( Ker T − , Ker T ) , which is not possible. Sowe must have 0 ≤ i ≤ q − − r . Then, by Lemma 3.5(1) and Lemma 3.6 (1), we have f ′ = [ β , 1 ] mod ( Ker T − , Ker T ) .Similarly, by Lemmas 3.5 (2) and 4.6 (2) and Lemma 3.6 (2) , we can show that f ′′ = [ Id, 1 ] mod ( Ker T − , Ker T ) .Thus, we have f = f − ∑ l = a l , n s q − − r + p l n + f − ∑ l = b l , n t r + p l n + . . . + f − ∑ l = a l ,2 s q − − r + p l + f − ∑ l = b l ,2 t r + p l + c [ β , 1 ] + d [ I d , 1 ] .Now assume e = f =
1. Let f ∈ ind GIZ χ r be an I ( ) -invariant vector modulo ( Ker T − , Ker T ) . As in the previous case, we concentrate only on the zero side ofthe tree and assume that f = f . We write f = f n + f ′ where f n = f ′ ∈ B ( n − ) .We further write f n = f ′ n + f ′′ n where f ′ n and f ′′ n are same as in the previous case.Following the steps in the previous case, we have f ′ n = ∑ µ ∈ I n a µ p − rn − h g n , µ , 1 i + g ′ n − ,and f ′′ n = ∑ µ ∈ I n b µ r + n − (cid:20) g n − [ µ ] n − (cid:18) [ µ n − ] (cid:19) w , 1 (cid:21) + g ′′ n − ,where a and b are constants and g ′ n − , g ′′ n − ∈ B ( n − ) . Thus, f n = ∑ µ ∈ I n a µ p − rn − h g n , µ , 1 i + ∑ µ ∈ I n b µ r + n − (cid:20) g n − [ µ ] n − (cid:18) [ µ n − ] (cid:19) w , 1 (cid:21) + g n − ,where g n − = g ′ n − + g ′′ n − ∈ B ( n − ) . Write f = f n + f n − + f ′ . We get (cid:18) p n − (cid:19) f − f = (cid:20)(cid:18) p n − (cid:19) f n − f n (cid:21) + (cid:20)(cid:18) p n − (cid:19) f n − − f n − (cid:21) ∈ ( Ker T − , Ker T ) .For e =
1, we have (cid:18) p n − (cid:19) f ′ n − f ′ n = ∑ µ ∈ I n a h ( µ n − − ( ∗ )) p − r − µ p − rn − i h g n , µ , 1 i , where ( ∗ ) = p − ∑ s = ( − ) p − s ( ps ) p µ sn − .Then, by Lemmas 3.6 (1) and 3.5 (1), modulo ( Ker T − , Ker T ) , the above expressionbecomes(11) − ∑ µ ∈ I n − a (cid:18) p − rp − − r (cid:19) ( ∗ ) (cid:20) g n − [ µ ] n − (cid:18) [ µ n − ] (cid:19) w , 1 (cid:21) .Writing f n − = f ′ n − + f ′′ n − , we have, (cid:18) p n − (cid:19) f n − − f n − = (cid:20)(cid:18) p n − (cid:19) f ′ n − − f ′ n − (cid:21) + (cid:20)(cid:18) p n − (cid:19) f ′′ n − − f ′′ n − (cid:21) .No term in the first summand of the above equation can cancel a term in (11). Also, byLemma 4.6 (2), the possible powers, say k , of µ n − in f ′′ n − must satisfy either 0 ≤ k ≤ r or k = r +
1. As r < p −
1, we have max ( r + ) = p −
1. So the maximum power of µ n − in the second summand of the above equation is p −
2. In both the cases, theterm involving µ p − n − in (11) will not get cancelled. Since there is no reduction, thisterm must be in Ker T − , which is not possible by Lemma 3.5 (2). Thus we arrive at acontradiction. So i n − can not be p − r . Thus one can always modify f ′ n by a vector g ′ n − in B ( n − ) . Similarly, working with f ′′ n , we can modify it by a vector g ′′ n − in B ( n − ) .Thus f n is congruent to a vector f n − in B ( n − ) modulo ( Ker T − , Ker T ) andhence by induction, f is congruent to a vector f in B ( ) modulo ( Ker T − , Ker T ) .Write f = f ′ + f ′′ , where f ′ = ∑ i ∑ µ ∈ I a i µ i h g µ , 1 i ,and f ′′ = ∑ j ∑ µ ∈ I b j µ j (cid:20)(cid:18) µ (cid:19) w , 1 (cid:21) .Considering the action of (cid:18) (cid:19) on f as in the previous case, we have 0 ≤ i ≤ p − − r and 0 ≤ j ≤ r , by Lemma 3.6 and Lemma 4.6. Then, by Lemma 3.6 andLemma 3.5, modulo ( Ker T − , Ker T ) , we get f ′ = [ β , 1 ] and f ′′ = [ Id, 1 ] . Thus wecan conclude that f = c [ Id, 1 ] + d [ β , 1 ] .This finishes the proof of Theorem 1.2.4.5. A remark on π r . We show that there is no isomorphism between τ r = ind GKZ σ r ( T ) and π r = ind GIZ χ r ( Ker T − , Ker T ) when f =
1; i.e., F is not a totally ramified extension of Q p (cf. Remark 3). WAHORI-HECKE MODEL FOR MOD p REPRESENTATIONS OF GL ( F ) Note that any G -linear isomorphism ϕ : π r → τ r must preserve I ( ) -invariants and the corresponding I -eigenvalues.Suppose e = f =
1; i.e., F / Q p is unramified. In this case, s q − − r + p l n , for n ≥
2, isan I ( ) -invariant in π r such that (cid:18) a b ̟ c d (cid:19) · s q − − r + p l n = a r − p l d p l · s q − − r + p l n by Lemma 4.3. By [Hen19, Theorem 1.2], a basis of the I ( ) -invariants in τ r consistsof the vectors Id ⊗ f − O j = x r j j , α ⊗ f − O j = y r j j , c p l ( r l + ) n , β c p l ( r l + ) n for n ≥
1, where c kn = ∑ µ ∈ I n (cid:18) ̟ n µ (cid:19) ⊗ µ kn − f − O j = x r j j .By [Hen19, Lemma 3.6], (cid:18) a b ̟ c d (cid:19) · c kn = a r − k ( ad ) k · c kn ,and it follows that there is no I ( ) -invariant vector in τ r with I -eigenvalue a r − p l d p l .Thus there is no vector in τ r where s q − − r + p l n can be mapped under ϕ . This gives acontradiction.Now, suppose e > f >
1. In this case t r + p l n , n ≥
2, is an I ( ) -invariant vector in π r with I -eigenvalue a q − − p l d r + p l , by Lemma 4.3. A basis of the I ( ) -invariants in τ r consists of the vectorsId ⊗ f − O j = x r j j , α ⊗ f − O j = y r j j , c p l ( r l + ) n , β c p l ( r l + ) n , d ln , β d ln ,for n ≥
1, where d ln = ∑ µ ∈ I n (cid:18) ̟ n µ (cid:19) ⊗ f − O l = j = x r j j ⊗ x r l − l y l ,by [Hen19, Theorem 1.2]. By [Hen19, Lemma 3.6], (cid:18) a b ̟ c d (cid:19) · d ln = a r − p l ( ad ) p l · d ln ,and once again it can be checked that there is no I ( ) -invariant vector in τ r with I -eigenvalue a q − − p l d r + p l , where t r + p l n can be mapped under φ , giving a contradiction. A cknowledgements The second author would like to thank Council of Scientific and Industrial Research,Government of India (CSIR) and Industrial Research and Consultancy Centre, IITBombay (IRCC) for financial support.R eferences [AB13] U. K. Anandavardhanan and Gautam H. Borisagar,
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