A simple character formula

In this paper we prove a character formula expressing the classes of simple representations in the principal block of a simply-connected semisimple algebraic group G in terms of baby Verma modules, under the assumption that the characteristic of the base field is bigger than 2h-1, where h is the Coxeter number of G. This provides a replacement for Lusztig's conjecture, valid under a reasonable assumption on the characteristic.

1. Introduction 1.1. Simple modules for reductive groups. Let G denote a connected reductive algebraic group over an algebraically closed field k of characteristic p ą 0, with simply connected derived subgroup. We fix a maximal torus and Borel subgroup T Ă B Ă G. Then for every dominant weight λ we have a Weyl module ∆pλq and its simple quotient Lpλq, both of highest weight λ. We obtain in this way a classification of the simple algebraic G-modules. A central problem in the field is to compute the characters of these simple modules.
Steinberg's tensor product theorem reduces this question to the case of p-restricted highest weights. For a p-restricted dominant weight λ it is known that Lpλq stays simple upon restriction to G 1 , the first Frobenius kernel of G. Moreover, all simple G 1 -modules occur in this way. Thus, understanding the simple G-modules is equivalent to understanding the simple G 1 -modules.
Instead of working with G 1 -modules, it is technically more convenient to work with G 1 T -modules. Simple G 1 T -modules stay simple upon restriction to G 1 , and thus we can instead try to answer our question in terms of G 1 T -modules. Via Brauer-Humphreys reciprocity, this question can be rephrased in terms of indecomposable projective G 1 T -modules (see §1.2 below for details).
In 1980 Lusztig [L2] proposed a conjecture for these characters if p is not too small (in the guise of "Jantzen's generic decomposition patterns"). His conjecture is in terms of the canonical basis in the periodic module for the affine Hecke algebra. This formula is known to hold for large p, see [KL,KT,L3,AJS,F3]. However it is also known to fail for "medium sized" p, see [W2]. In fact, at this point it is not known precisely when this formula holds.
Our goal in this paper is to define the p-canonical basis in the periodic module and prove that the p-analogue of Lusztig's conjecture is true, as long as p ě 2h´1 where h is the Coxeter number of G. Thus we obtain a character formula for simple G-modules in terms of p-Kazhdan-Lusztig polynomials. Our proof builds This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 677147). on a character formula for tilting G-modules, proved in a joint work with P. Achar and S. Makisumi [AMRW].
At present, p-canonical bases are very difficult to compute; for this reason our formula is certainly not the final answer to this problem. However it gives a good conceptual understanding of where the difficulties lie, and provides a way to compute characters or multiplicities which is much more efficient than classical techniques in representation theory. For instance, unpublished intensive efforts of Jantzen were not sufficient to completely answer the question of describing simple characters in types B 3 , C 3 and A 4 . Preliminary results of Jensen-Scheinmann indicate that our formula will allow one to solve these cases, and maybe the case of some bigger groups with the help of computer. (For example, Jensen and Scheinmann obtain the only missing multiplicity in Jantzen's work for A 4 in a few lines using our results.) We also believe our results will help answer the important question of when exactly Lusztig's character formula holds.
(1) A conjecture of Donkin [Do] would imply that our formula is valid for p ą h; see Remark 1.6 below for more details.
(2) It has been known for a long time that, in theory, knowing the characters of tilting modules is enough to determine the characters of simple modules (see e.g. [RW,§1.8] for details, and [Sob] for more recent advances on this question). However, obtaining a concrete character formula for simples out of a given character formula for tilting modules is a different story, which is the main topic of the present paper (taking as input the character formula from [AMRW]).
1.2. Representation theory of G 1 T . We continue with the notation of §1.1, and let X :" X˚pT q be the character lattice of T . Let also B`be the Borel subgroup of G opposite to B with respect to T . If as above G 1 T , resp. B1 T , denotes the preimage of the Frobenius twist 9 T of T under the Frobenius morphism of G, resp. B`, then for each λ P X we have a G 1 T -module p Zpλq (called the baby Verma module attached to λ) obtained by coinducing to G 1 T the 1-dimensional representation of B1 T defined by λ (see §5.2 below for details). The module p Zpλq has a unique simple quotient p Lpλq, and the assignment λ Þ Ñ p Lpλq induces a bijection between X and the set of isomorphism classes of simple G 1 T -modules. For any λ, µ P X we have T pµq (where we identify the weight lattice of 9 T with X in such a way that the pullback under the Frobenius morphism T Ñ 9 T corresponds to ν Þ Ñ pν, and k 9 T pµq is viewed as a G 1 T -module via the Frobenius morphism G 1 T Ñ 9 T ), and for λ dominant and p-restricted p Lpλq is the restriction to G 1 T of the simple G-module Lpλq considered in §1.1. In this way, understanding the simple G-modules, the simple G 1 -modules or the simple G 1 T are equivalent problems.
1.3. Characters of G 1 T -modules and alcoves. Assume now that p ě h.
Let ∆ Ă X be the root system of pG, T q. Let W f and W " W f˙Z ∆ denote the finite and affine Weyl groups, and denote their subsets of simple reflections (determined by B) by S f and S respectively. The affine Weyl group acts naturally on X b Z R, giving rise to the set of alcoves A . Let A fund denote the fundamental alcove; then the map x Þ Ñ xpA fund q gives a bijection We will also consider the "dot-action" of W on X, defined by pxt λ q¨p µ " xpµ`pλ`ρq´ρ for x P W f , λ P Z∆ and µ P X, where ρ is the halfsum of the positive roots. Let Rep 0 pG 1 T q denote the principal block of the category of algebraic finite-dimensional G 1 T -modules, i.e. the Serre subcategory generated by the simple modules p Lpλq with λ P W¨p 0. By construction the highest weights of simple and baby Verma modules belonging to this block are labelled by the affine Weyl group, and hence (via (1.1)) by alcoves. Given an alcove A, let us denote the corresponding simple and baby Verma modules by p L A and p Z A . We denote the projective cover (equivalently, injective hull) of p L A by p Q A . Each p Q A admits a baby Verma flag (that is, a finite filtration whose successive subquotients are isomorphic to baby Verma modues). We write p p Q A : p Z B q for the number of times the baby Verma module p Z B occurs in such a filtration. (This number is known to be independent of the chosen filtration.) For an alcove A P A , consider the element By Brauer-Humphrey's reciprocity [Hu], it is known that for any pair of alcoves pA, Bq. As the characters of the baby Verma modules are known (and easy!), knowledge of the elements q A for all alcoves A therefore implies knowledge of the characters of the simple G 1 T -modules in the principal block, and hence of all simple G 1 T -modules by Jantzen's translation principle.
1.4. Statement. Let H denote the Hecke algebra of the Coxeter system pW, Sq (an algebra over the ring Zrv˘1s, with standard basis pH w : w P W q), and let P denote its periodic (right) module. (We follow the notational conventions of [Soe].) As a Zrv˘1s-module, P is free with basis given by alcoves: In [L2], Lusztig has defined a canonical basis for P; following the conventions of [Soe,Theorem 4.3] we will denote this family of elements by tP A : A P A u. (Note that this terminology might be misleading: tP A : A P A u is not a Zrv˘1s-basis of P, but of a certain submodule.) Lusztig conjectured (see the last three paragraphs of the introduction to [L2]) that the canonical basis determines the characters of indecomposable projective modules, as follows: q A " pPÂq vÞ Ñ1 .
(Here, given R " ř p A A P P, R vÞ Ñ1 " ř p A p1qA P ZrA s denotes its specialisation at v " 1, and A Þ ÑÂ is a simple operation on alcoves, whose definition is recalled in Section 2.) See also [F1, §3] for the relation with Lusztig's original conjecture [L1] for characters of the modules Lpλq.
In this paper, we define the p-canonical basis t p P A : A P A u in the periodic module P, and we prove that it can be used to compute the elements q A , as follows.
Theorem 1.2. Assume p ě 2h´1. Then for any alcove A we have q A " p p PÂq vÞ Ñ1 .
1.5. Spherical and antispherical modules. Theorem 1.2 will be obtained as a consequence of a relation between the p-canonical bases in two other H-modules, namely the (twisted) spherical and the antispherical modules. We will denote by pH w : w P W q the Kazhdan-Lusztig basis of H, see [Soe,Theorem 2.1], and by p p H w : w P W q its p-canonical basis (see [JW, RW]).
The antispherical module M asph is defined as where H f is the Hecke algebra of pW f , S f q (which is a subalgebra in H in a natural way), and sgn is the right H f -module defined as Zrv˘1s with H s acting as multiplication by´v for s P S f . This module has a standard basis pN w : w P f W q parametrized by the subset f W Ă W consisting of elements w which are minimal in W f w, where N w :" 1 b H w . It also has a Kazhdan-Lusztig basis pN w : w P W q and a p-canonical basis p p N w : w P f W q, where The subset A`of A corresponding to f W under the bijection (1.1) consists of the alcoves contained in the dominant Weyl chamber C`; hence we will rather parametrize these bases by A`and denote them pN A : A P A`q, pN A : A P A`q and p p N A : A P A`q.
On the other hand, to any λ P X one can associate an automorphism τ λ of pW, Sq (which only depends on the image of λ in X{Z∆), see Remark 2.1 for details. We then set W λ :" τ λ pW f q, S λ :" τ λ pS f q. The pair pW λ , S λ q is a Coxeter system, and the associated Hecke algebra H λ is in a natural way a subalgebra of H. We will denote by triv λ the "trivial" right H λ -module (defined as Zrv˘1s, with H s acting as multiplication by v´1 for s P S λ ), and set M sph λ :" triv λ b H λ H.
Then M sph λ has a standard basis pM λ w : w P λ W q parametrized by the subset λ W Ă W consisting of elements w which are minimal in W λ w, where M λ w :" 1bH w . If w λ is the longest element in W λ , then the map 1 b h Þ Ñ H w λ¨h provides an embedding (1.2) ζ λ : M sph λ ãÑ H. For any w P λ W , the element H w λ w , resp. p H w λ w , belongs to the image of ζ λ , and if we denote by M λ w , resp. p M λ w , its preimage in M sph λ , then pM λ w : w P λ W q and p p M λ w : w P λ W q are bases of M sph λ , called the Kazhdan-Lusztig and the p-canonical basis. Recall that the set A also admits a natural right action of W ; through the identification (1.1) this action corresponds to the right multiplication of W on itself, and will be denoted pA, xq Þ Ñ A¨x. Then the assignment w Þ Ñ pλ`A fund q¨w identifies λ W with the subset Aλ Ă A consisting of alcoves contained in λ`C`. In this way the bases of M sph λ considered above can be labelled by Aλ , and will be denoted pM λ A : A P Aλ q and p p M λ A : A P Aλ q.
Remark 1.3. The bases pN w : w P W q and pM λ w : w P W q coincide with the bases considered in [Soe,Theorem 3.1] (for the parabolic subgroups W f and W λ respectively), see [Soe,Proof of Proposition 3.4]. In case λ " 0, we will write M sph for M sph 0 , M w for M 0 w , M w for M 0 w , and p M w for p M 0 w . 1.6. Outline of the proof of Theorem 1.2. For any λ P X, it is easily seen that the assignment 1 b h Þ Ñ N λ`A fund¨h induces an H-module morphism Now we assume (for simplicity) that G is semisimple, and specialize to the case λ " ρ. The key step in our approach to Theorem 1.2 is the following claim.
Theorem 1.4. Assume that p is good for G. Then for any A P Aρ we have (1) In the body of the paper, we find it more convenient to work with the extended affine Hecke algebra and its spherical/antispherical modules. This allows to get rid of the twist by τ ρ .
(2) Our proof of Theorem 1.4 also applies to the Kazhdan-Lusztig bases, and shows that for any A P Aρ we have ϕ ρ pM ρ A q " N A . This fact could have been stated many years ago (since it does not involve the p-canonical bases in any way), but seems to be new. Theorem 1.4 is obtained as a "combinatorial trace" of a statement of categorical nature. Namely, the modules M sph ρ and M asph can be "categorified" via some categories of parity complexes (in the sense of [JMW]) on the affine flag variety of the Langlands dual group G _ : M sph ρ corresponds to (twisted) spherical parity complexes, while M asph corresponds to Iwahori-Whittaker 1 parity complexes. The morphism ϕ ρ can also be categorified by a functor Φ ρ , given by convolution with a certain object. We then prove that this functor is full (but not faithful); in particular, it must send indecomposable objects to indecomposable objects, and Theorem 1.4 follows. This fullness result is deduced from a similar result in the case where the affine flag variety is replaced by the affine Grassmannian of G _ (in which case the corresponding functor is even an equivalence of categories) proved recently by the first author with R. Bezrukavnikov, D. Gaitsgory, I. Mirković and L. Rider, see [BGMRR].
Once Theorem 1.4 is proved, we deduce Theorem 1.2 from the fact that the p-canonical basis of M asph encodes the characters of indecomposable tilting Gmodules (as proved in joint work with P. Achar and S. Makisumi [AMRW]) and a result of Jantzen [J1] (in the interpretation of Donkin [Do]) saying that if the highest weight of an indecomposable tilting module is of the form pp´1qρ`µ with µ dominant and p-restricted, then this tilting module is indecomposable as a G 1 Tmodule. It is well known (and easy to check) that these modules are projective over G 1 T ; this therefore provides a useful relation between indecomposable tilting modules for G and indecomposable projective modules for G 1 T , which allows us to deduce Theorem 1.2 from Theorem 1.4 and the results of [AMRW].
Remark 1.6. A conjecture by Donkin states that Jantzen's result recalled above should be true in any characteristic, which would imply that Theorem 1.2 holds as soon as p ą h. Very recent work of Bendel-Nakano-Pillen-Sobaje [BNPS] shows that this conjecture is not true in full generality. It is currently not known whether the condition that p ą h is sufficient to ensure that this conjecture holds (which would be sufficient for our purposes).

The periodic module
2.1. Extended affine Weyl group. As in §1.2 we consider a connected reductive algebraic group G with simply-connected derived subgroup over an algebraically closed field k of characteristic p ą 0, with a fixed choice of maximal torus and Borel subgroup T Ă B Ă G. We set X :" X˚pT q, and denote by ∆ Ă X the root system of pG, T q, by ∆`Ă ∆ the positive system consisting of the opposites of the T -weights in the Lie algebra of B, by Σ Ă ∆ the corresponding subset of simple roots, by W f and W " W f˙Z ∆ the Weyl group and the affine Weyl group, and finally by S f and S their subsets of simple roots, see e.g. [J2,Chap. 6]. (Here, "f" stands for "finite.") The length function associated with the Coxeter system pW, Sq satisfies for w P W f and λ P Z∆. (Here, to avoid confusion, the image of λ P Z∆ in W is denoted t λ .) We will also consider the extended affine Weyl group The formula (2.1) defines a function ℓ : W ext Ñ Z, and we set Ω :" tw P W ext | ℓpwq " 0u. Then it is known that Ω is a subgroup of W ext , and that multiplication induces a group isomorphism More precisely, conjugation by any element of Ω defines a Coxeter group automorphism of W ; in other words it stabilizes S (hence preserves lengths). It is known also that the composition is then a group isomorphism. (In particular, Ω is abelian.) We consider the action of W and W ext on V :" R b Z X as in [Soe,Section 4], and denote by A the set of alcoves in V (i.e. the connected components of the complement of reflection hyperplanes associated with the action of W ) and by A fund P A the fundamental alcove, defined as The action of W ext on V preserves alcoves, hence induces an action on A . Moreover the assignment w Þ Ñ wA fund induces a bijection W " Ý Ñ A , see (1.1). If we denote by A`Ă A the subset of alcoves contained in the dominant chamber (denoted C in [Soe]), then this bijection restricts to a bijection Remark 2.1. The subset Ω Ă W ext can be characterized as consisting of the elements w P W ext such that wpA fund q " A fund . For any λ P X, the subset λ`A fund is an alcove, so that there exists x λ P W such that x λ pA fund q " λ`A fund , see (1.1).
Then ω λ " px λ q´1¨t λ belongs to Ω, and has image λ`Z∆ under (2.2). This procedure realizes the isomorphism inverse to (2.2). It also allows us to associate to any λ P X a Coxeter group automorphism τ λ of W , given by conjugation by ω λ in W ext .

2.2.
More about alcoves. Using again the identification (1.1) we may transport the Bruhat order on W to obtain a partial order on A , which we also denote by ď. We define the generic order on A by Equivalently, the generic order is uniquely determined by the fact that it agrees with the Bruhat order on A`and is invariant under translation by X.
For any µ P X we will consider the following subsets of V : Note that for µ, ν P X we have The setΠ 0 is often called the "fundamental box." (It is denoted Π in [Soe,§4].) The notation is intended to suggest thatΠ µ (resp.Π µ ) is "the fundamental box below (resp. above) µ." Given any alcove A P A there exists µ P X such that A ĂΠ µ . The stabiliser of µ is then t µ W f t´µ. We defineÂ where w f is the longest element in W f . From (2.4) we see thatÂ does not depend on the choice of µ, thatÂ ĂΠ µ , and that A Þ ÑÂ is a bijection A " Ý Ñ A . We denote its inverse by A Þ ÑǍ. (These operations agree with the maps denoted similarly in [Soe], see [Soe,Comments before Lemma 4.21].) 2.3. The periodic module and its canonical basis. The periodic module P is the right H-module (where, as in Section 1, H is the Hecke algebra of the Coxeter system pW, Sq) defined as follows. As a Zrv˘1s-module, P is free with basis the set of alcoves: The structure of a right H-module on P is characterized by the following formulas for s P S: The action of X on A by translations extends to an action of X on P: namely, given R " ř p A A P P, set R`µ :" ř p A pµ`Aq. This action does not commute with the H-action; it rather satisfies the following relation for all h P H: where we still denote by τ µ the automorphism of H defined by τ µ pH w q " H τµpwq . As in §1.4 we denote by tP A : A P A u the canonical basis of P. Then for any µ P X we have (see [Soe, Comments before Proposition 4.18]) and Lemma 2.2. Let A P A , let µ P X be such that A ĂΠ µ , and let w P W be the unique element such that pµ`A fund q¨w " A. Then we have Remark 2.3. The formula in Lemma 2.2 is compatible with (2.7) in view of (2.6).
2.4. The p-canonical basis of the periodic module. The usual procedure to define a p-canonical basis of an H-module (see [JW, RW, AR]) is to start with a categorification of this module in terms of a C-linear category (in practice, either via some diagrammatic category or some category of parity complexes) such that the classes of indecomposable objects correspond to the Kazhdan-Lusztig basis, and then to replace (in some appropriate way) the coefficients C by a field of characteristic p. In the case of the periodic module, the known categorifications involve semi-infinite geometry, and are beyond the authors' present understanding of the subject. So we will use a different strategy to define this basis: we will start with the formula of Lemma 2.2, and replace there the canonical basis of M sph µ by the p-canonical version. We expect that any reasonable categorification of P with characteristic-p coefficients should provide the same basis as the one constructed here.
Namely, for A P A , we choose µ P X such that A ĂΠ µ , and let w P W be the unique element such that pµ`A fund q¨w " A. Then we set The following properties are easy to check (using in particular (2.6) and the fact that τ µ " τ ν ifΠ µ "Π ν , as follows from (2.4)): (1) for any h P M sph µ the element P A fund`µ¨ζ µ phq belongs to π f¨P , so that p P A belongs to P; (2) the element p P A does not depend on the choice of µ; (3) for any ν P X we have It can also be shown (although this is less obvious, and will not be proved here) that for any alcove A we have 3. The extended affine Hecke algebra and its spherical and antispherical modules 3.1. The spherical and antispherical modules. We continue with the notation of Section 2, and fix a weight ς P X such that xς, α _ y " 1 for any α P Σ. (Such a weight exists thanks to our assumption on the derived subgroup of G. However, it might not be unique.) As mentioned in Remark 1.5, to avoid difficulties related to the twists τ λ , it will be more convenient to work with the Hecke algebra H ext associated with the "quasi-Coxeter" group W ext (see §2.1), i.e. the Zrv˘1s-algebra with a "standard" basis consisting of elements pH w : w P W ext q, with multiplication characterized by the following relations: (1) pH s`v q¨pH s´v´1 q " 0 for s P S; (2) H x¨Hy " H xy if x, y P W ext and ℓpxyq " ℓpxq`ℓpyq.
The algebra H ext contains H as a subalgebra (spanned by the elements H w with w P W ). Inducing from H f to H ext the modules considered in §1.5, we obtain the right H ext -modules M sph ext :" triv 0 b H f H ext and M asph ext :" sgn b H f H ext , which are called the spherical and antispherical module respectively.
We denote by f W ext Ă W ext the subset consisting of elements w which are of minimal length in the coset W f w (in other words, of the form wω with w P f W and ω P Ω). Then for w P f W ext we set M w :" 1 b H w P M sph ext , N w :" 1 b H w P M asph ext . The collections pM w : w P f W ext q and pN w : w P f W ext q are Zrv˘1s-bases of M sph and M asph respectively (called again the standard bases). Of course there are natural embeddings M sph ãÑ M sph ext and M asph ãÑ M asph ext , such that the elements denoted M w , resp. N w , in §1.5 (see Remark 1.3) correspond to the elements denoted similarly here. (It can be easily checked that we also have H ωw " H ω H w and p H ωw " H ω p H w for any ω P Ω and w P W .) Let us recall the following well-known property of the Kazhdan-Lusztig basis.
Lemma 3.2. Let w P W ext and s P S. If ℓpwsq ă ℓpwq, then H w¨H s " pv`v´1q¨H w .
As in the setting of §1.5 we have an H ext -module morphism ξ : H ext Ñ M asph ext defined by ξphq " N id¨h . This morphism is clearly surjective; moreover, for w P W ext we have (see [RW] for details). Now, let w f be the longest element in W f . Consider the endomorphism of H ext (as a right H ext -module) sending h to H w f¨h . It follows from Lemma 3.2 that this morphism factors through a morphism (3.1) ζ : M sph ext Ñ H ext . This morphism is injective, and satisfies for any w P f W ext . In particular, for ω P Ω we have (where the second equality uses [Soe, Proposition 2.9]).
Remark 3.3. Recall that for any w P W ext , there exists K w P Z ě0 such that p H w " H w for any prime number p such that p ě K w . (However, determining K w is a very difficult task.) A similar claim holds for the p-canonical bases in M asph ext and M sph ext . 3.3. Statement. We can now state a version of Theorem 1.4 in terms of the extended affine Hecke algebra (and for reductive groups which are not necessarily semisimple).
The statement will involve the element considered in the following lemma. (Here the second equality follows from [Soe,Lemma 5.7]. We will give a (geometric) proof of both equalities in §4.2 below.) Lemma 3.4. We have Moreover, for any s P S f we have N tς¨H s " pv`v´1q¨N tς .
Note for later use that, if ω P Ω, multiplying (3.4) on the right by H ω we obtain that Lemma 3.4 shows that the map H ext Ñ M asph ext defined by h Þ Ñ N tς¨h factors through a morphism of right H ext -modules (3.6) ϕ : M sph ext Ñ M asph ext . The main technical result of the paper is the following.
Theorem 3.5. Assume that p is good for G. Then for any w P f W ext we have Remark 3.6.
(1) Theorem 3.5 implies in particular that ϕ is injective. (Of course, this can also be seen more directly.) (2) To deduce Theorem 1.4 from Theorem 3.5, one simply observes that ϕ restricts to a morphism of right H-modules from the submodule of M sph ext generated by N ω´1 ς to M asph . Now the latter submodules identifies with M sph ς (see Remark 3.1), so that Theorem 1.4 becomes the special case of Theorem 3.6 when w P ω´1 ς W . 4. Proof of Theorem 3.5 4.1. Categorification and p-canonical bases. The proof of Theorem 3.5 will use the geometric description of the p-canonical bases in terms of parity complexes, which we now recall. For this we need to choose a field K of coefficients for the parity complexes, which should be of characteristic p but might differ from k. In fact, for technical reasons we will take for K a finite field. We also choose a prime number ℓ ‰ k, and assume that K contains a nontrivial ℓ-th root of unity.
We now fix an algebraically closed field F of characteristic ℓ. Let G _ be the connected reductive algebraic group over F which is Langlands dual to G. By definition, this group comes with a maximal torus T _ Ă G _ whose cocharacter lattice is X. We will denote by B _ Ă G _ the Borel subgroup containing T _ whose T _ -weights are the negative coroots of pG, T q. We set K :" Fppzqq, O :" Frrzss, and denote by G _ K , resp. G _ O , the ind-group scheme, resp. group scheme, over F which represents the functor R Þ Ñ G _`R ppzqq˘, resp. R Þ Ñ G _`R rrzss˘. We also denote by I _ Ă G _ K the Iwahori subgroup, i.e. the inverse image of B _ under the evaluation morphism G _ O Ñ G _ . We then consider the affine flag variety Fl :" G _ K {I _ . Following [JMW] we can consider the category Parity I _ pFl, Kq of I _ -equivariant parity (étale) K-complexes on Fl. The I _ -orbits on Fl are parametrized in a natural way by W ext ; we will denote by Fl w the orbit corresponding to w (so that dimpFl w q " ℓpwq). For any w P W ext , there exists a unique indecomposable parity complex E w on Fl which is supported on Fl w and whose restriction to Fl w is K Flw rℓpwqs. Then the assignment pw, nq Þ Ñ E w rns defines a bijection between W extˆZ and the set of isomorphism classes of indecomposable objects in Parity I _ pFl, Kq.
The usual convolution construction endows Parity I _ pFl, Kq with the structure of a monoidal category. (The fact that a convolution of parity complexes is parity is proved in [JMW,§4.1].) In particular, the split Grothendieck group rParity I _ pFl, Kqs has a natural product; we will in fact view this ring as a Zrv˘1s-algebra, where v acts via the automorphism induced by the cohomological shift r1s. It is well known (see [Sp, JMW, JW]) that there exists a unique Zrv˘1s-algebra isomorphism sending H s to rE s s for any s P S and H ω to rE ω s for any ω P Ω. Then for w P W ext , the element p H w is the inverse image of rE w s under (4.1), see e.g. [RW,Part 3].
Recall also that the p-Kazhdan-Lusztig polynomials are the elements p p h y,w q y,wPWext of Zrv˘1s such that p H w " ÿ yPWext p h y,z¨Hy .
In order to categorify the module M asph ext , we consider the category of "Iwahori-Whittaker" parity complexes Parity IW pFl, Kq on Fl. These objects are defined using the action of the unipotent radical I _,ù of the Iwahori subgroup I _,`a ssociated with the Borel subgroup B _,`o f G _ which is opposite to B _ with respect to T _ ; see [RW,§11] for details. (Here we use our assumption on ℓ-th roots of unity in K.) The I _,ù -orbits on Fl are parametrized in a natural way by W ext ; but only those corresponding to elements in f W ext support nonzero Iwahori-Whittaker local systems. Therefore the isomorphism classes of indecomposable objects in Parity IW pFl, Kq are naturally in bijection with f W extˆZ ; we will denote by E IW w the object associated with pw, 0q.
The convolution construction defines a right action of the monoidal category Parity I _ pFl, Kq on the category Parity IW pFl, Kq, and there exists a unique isomorphism of right H ext -modules (4.2) M asph ext " Ý Ñ rParity IW pFl, Kqs sending N id to E IW id . Then for w P f W ext , p N w is the inverse image of rE IW w s under this isomorphism, see [RW,§11].
Finally, we explain the categorification of M sph ext . We consider the "opposite affine Grassmannian" Gr op :" G _ O zG _ K . This variety admits an action of I _ induced by right multiplication on G _ K , and we can consider the corresponding category of parity complexes Parity I _ pGr op , Kq. The I _ -orbits on Gr op are parametrized by f W ext ; therefore the indecomposable objects in Parity I _ pGr op , Kq are parametrized in a natural way by f W extˆZ . The object associated with pw, 0q (for w P f W ext ) will be denoted F w .
Again, the convolution construction defines a right action of the monoidal category Parity I _ pFl, Kq on the category Parity I _ pGr op , Kq, and there exists a unique isomorphism of right H ext -modules M sph ext " Ý Ñ rParity I _ pGr op , Kqs sending M id to F id . Using [ACR,Lemma A.5] and the construction of the pcanonical basis in M sph ext , one can check that, for w P f W ext , p M w is the inverse image of rF w s under this isomorphism. 4.2. Parity complexes on affine Grassmannians. From now on we assume that p is good for G.
Consider the "usual" affine Grassmannian This category possesses a natural perverse t-structure, whose heart will be denoted Perv G _ O pGr, Kq. Under our assumptions that p is good for G (equivalently, for G _ ) and that G has a simply-connected derived subgroup (so that the quotient of X˚pT _ q by the coroot lattice of G is torsion-free), it is known that the equivariant cohomology Kq vanishes in odd degrees; see e.g. [JMW,§2.6] or [MR,§3.2] for references. Therefore, the theory developed in [JMW] applies in this context, and we will denote by Parity G _ O pGr, Kq the corresponding category of parity complexes.
For λ P X, we set L λ :" z λ¨G_ O P Gr. Then the assignment λ Þ Ñ G _ O¨L λ induces a bijection between X`and the set of G _ O -orbits on Gr. Therefore, the isomorphism classes of indecomposable objects in Parity G _ O pGr, Kq are parametrized in a natural way by X`ˆZ; for λ P X`we will denote by E sph λ the object associated with pλ, 0q. The following result is proved in [JMW2] under some technical assumptions, and in [MR,Corollary 1.6] in the present generality. (This claim is known to be false if we remove the assumption that p is good for G, see [JMW2].) Theorem 4.1. For any λ P X`, the object E sph λ is perverse.
Remark 4.2. From the combinatorial point of view, this theorem says that if w P W ext is maximal in W f wW f , then p H w belongs to À y Z¨H y . The other result which we will need is the main result of [BGMRR]. Here we consider the Iwahori-Whittaker derived category of sheaves on Gr, denoted D b IW pGr, Kq. This category is endowed with the perverse t-structure, whose heart will be denoted Perv IW pGr, Kq. This abelian category admits a natural structure of highest weight category (in the sense considered e.g. in [Ri,§7]), and moreover the realization functor provides an equivalence of triangulated categories IW pGr, Kq. The I _,ù -orbits on Gr are parametrized by X, and those which support a nonzero Iwahori-Whittaker local system are the ones parametrized by elements in ς`X( i.e. by strictly dominant weights). In particular, no orbit in the boundary of the orbit associated with ς supports such a local system; therefore the corresponding standard perverse sheaves is simple, and isomorphic to the associated costandard perverse sheaf (see [BGMRR,Equation (3.2)]). Hence this object is also parity, and will be denoted F IW ς . The following result is proved in [BGMRR]. (The first claim holds without any assumption on p; however for the second assertion we need the restriction that p is good.) Here we denote by ‹ G _ O the natural convolution bifunctor (see [BGMRR] for details).
Theorem 4.3. The functor O F is t-exact for the perverse t-structures, and restricts to an equivalence of categories Perv G _ O pGr, Kq " Ý Ñ Perv IW pGr, Kq. Moreover, for any λ P X, the object ΨpE sph λ q is a tilting perverse sheaf. The consequence of Theorems 4.1 and 4.3 that we will use below is the following.
Proof. Any object of Parity G _ O pGr, Kq is a direct sum of cohomological shifts of objects of the form E sph λ (with λ P X`); therefore to prove the corollary it suffices to prove that for any λ, µ P X`and n P Z the functor Ψ induces a surjection Now, by Theorem 4.3 the objects ΨpE sph λ q and ΨpE sph µ q are tilting perverse sheaves; therefore the right-hand side vanishes unless n " 0. And if n " 0, since E sph λ and E sph µ are perverse, and since Ψ restricts to an equivalence on perverse sheaves, the map (4.3) is an isomorphism in this case.
We can now give the proof of Lemma 3.4.
Proof of Lemma 3.4. One can easily check using (2.1) (and the fact that ℓpxq " ℓpx´1q for any x P W ext ) that t ς is of maximal length in t ς¨Wf . Therefore, the I _,ùorbit in Fl associated with t ς is the inverse image under the projection π : Fl Ñ Gr of the orbit of L ς . Using [ACR,Lemma A.5] we deduce that E IW tς " π˚pF IW ς qrℓpw f qs. It follows that p N tς " ř zPW f v ℓpzq N tς¨z , and that for any s P S f we have p N tς¨H s " pv`v´1q¨pN tς . The claims about N tς follow, taking p " 0 (see Remark 3.3).

4.3.
Fullness. To prove Theorem 3.5 we will consider a categorification of ϕ. For this, we work with the Kq. This category admits a right action of the I _ -equivariant derived category D b I _ pFl, Kq (by convolution, as usual), and it is clear that there exists a canonical equivalence of triangulated categories where Gr op is as in §4.1) and commuting with the right actions of D b I _ pFl, Kq on both sides. Moreover, the theory of parity complexes from [JMW] Kq are the objects ıpF w qrns for w P f W ext and n P Z, and we have a canonical isomorphism M sph ext " Ý Ñ rParity G _ O pFl, Kqs sending p M w to ıpF w q for any w P f W ext .
We now consider the functor Lemma 4.5. The functor Φ sends parity complexes to parity complexes. Moreover, the map on split Grothendieck groups induced by the restriction Proof. The proof of the first claim is similar to that of [BGMRR,Lemma 4.14]. For the second claim, we observe that the map induced by Φ Par is clearly a morphism of right H ext -modules. Since M sph ext is a cyclic module, this reduces the proof to checking that the image of rK G _ O {I _ rℓpw f qss corresponds to ϕpM e q " N tς under (4.2). However we have where π : Fl Ñ Gr is the projection. As observed in the proof of Lemma 3.4, the right-hand side is E IW tς , whose class in the Grothendieck group corresponds to p N tς by definition. The claim follows, using the first equality in Lemma 3.4.
The key step in our proof of Theorem 3.5 is the following claim.
Proposition 4.6. The functor Φ Par from Lemma 4.5 is full.
Proof. It is easily seen that any object in Parity G _ O pFl, Kq is a direct sum of direct summands of objects of the form Kq. Now any functor of the form p´q ‹ I _ E (with E in Parity I _ pFl, Kq) admits a left adjoint of the form p´q ‹ I _ E 1 with E 1 in Parity I _ pFl, Kq, hence this remark reduces the proof of fullness of Φ Par to proving that for any F in induced by Φ is surjective. If π is as in the proof of Lemma 3.4 (or of Lemma 4.5), then we have Since π˚-π ! r´2ℓpw f qs, using adjunction we deduce isomorphisms Now we have π ! ΦpGq -Ψpπ ! Gq, where Ψ is as in Theorem 4.3; hence we are reduced to proving that the morphism induced by Ψ is surjective. However, π ! G is parity, so that the claim follows from Corollary 4.4.
4.4. Proof of Theorem 3.5. Since the functor Φ is full, it must send indecomposable objects to indecomposable objects; in fact, using support considerations it is not difficult to check that for any w P f W ext we have ΦpıpF w qq -E IW tς w . Passing to classes in the split Grothendieck group we deduce the formula of Theorem 3.5.

Application: a character formula for simple G-modules
In this section we return to the setting of Sections 1-3; in particular, G is a connected reductive algebraic group with simply connected derived subgroup over an algebraically closed field k of characteristic p. We will assume that p ą h, where h is the Coxeter number of G. (In particular, this condition implies that p is good for G, so that Theorem 3.5 is applicable.) 5.1. The tilting character formula. We set M asph ext :" Z b Zrv˘1s M asph ext , where Z is considered as a Zrv˘1s-module via v Þ Ñ 1. This Z-module is a right module over Z b Zrv˘1s H ext " ZrW ext s. We will denote by ReppGq the abelian category of finite-dimensional algebraic G-modules. The simple objects in this category are labelled in a natural way by the subset X`Ă X of dominant weights; as in §1.1 we will denote by Lpλq the simple G-module of highest weight λ P X`.
We consider the dilated and shifted action of W on X defined by w¨p λ " wpλ`ςq´ς, t µ¨p λ " λ`pµ for w P W f and λ, µ P X. (It is a classical fact that this action does not depend on the choice of ς.) We then denote by Rep ∅ pGq the "extended principal block" of ReppGq, i.e. the Serre subcategory generated by the simple objects Lpw¨p 0q with w P f W ext . (Here, under our assumptions, for w P W ext we have w¨p 0 P X`iff w P f W ext .) For λ P X`we also denote by ∆pλq, ∇pλq and Tpλq the Weyl, induced, and indecomposable tilting G-modules of highest weight λ (see [RW,§3.1]). If λ " w¨p 0 for some w P f W ext , then these objects belong to Rep ∅ pGq.
In the following lemma, T ν µ is the translation functor from the µ-block to the ν-block of ReppGq, see [J2,Chapter II.7]. See also Remark 2.1 for the definition of ω ς .
Now by [J2,Remark in §II.3.19], the Steinberg module Lppp´1qςq is isomorphic to ∆ppp´1qςq and to ∇ppp´1qςq, hence is tilting. It is also clearly indecomposable, so that Tppp´1qςq " Lppp´1qςq. The first claim follows. The second claim follows from the first one (and the fact that Lppp´1qςq " ∇ppp´1qςq) in view of [J2,Proposition II.7.13].
For any s P S we choose a weight µ s as in [RW,§3.1] (i.e. a "generic" weight on the s-wall of the fundamental alcove for the dilated and shifted action) and consider the exact selfadjoint endofunctor of Rep ∅ pGq. (Here the sum might be infinite but, for each object V of Rep ∅ pGq, only finitely many of these functors applied to V do not vanish; so the functor Θ s is well defined.) If we denote by rRep ∅ pGqs the Grothendieck group of the abelian category Rep ∅ pGq, and by rM s the class of an object M , then it is well known (see e.g. [RW,§1.2]) that the assignment 1 b N w Þ Ñ r∆pw¨p 0qs " r∇pw¨p 0qs induces an isomorphism of abelian groups Using [J2,Propositions II.7.11 and II.7.12] one can check that, under this identification, the endomorphism of the right-hand side induced by Θ s corresponds to the action of p1`sq on the left-hand side.
The following statement was conjectured in [RW] (see in particular [RW,Corollary 1.4.1]) and proved in [AMRW].
Theorem 5.2. Under the isomorphism (5.1), 1 b p N w is sent to rTpw¨p 0qs for any w P f W ext .
Remark 5.3. In [RW, AMRW] we work with W instead of W ext ; but the extension is immediate (see e.g. [AR]).
Remark 5.5. The formula in Lemma 5.4 suggests that the Grothendieck group rRep ∅ pG 1 T qs is closely related with the right ZrW s-module Z b Zrv˘1s P. However, two important remarks are in order. First, since A is in bijection with W rather than W ext , to make this precise we would have to work with the "true" principal block Rep 0 pG 1 T q in ReppG 1 T q, i.e. the Serre subcategory generated by simple modules p Lpw¨p 0q with w P W . But even then a difficulty would remain, since the classes of baby Verma modules do not form a basis of the Grothendieck group rRep 0 pG 1 T qs. We will not try to address this problem here. 5.3. Injective/projective G 1 T -modules. For λ P X, we will denote by p Qpλq the injective hull of p Lpλq as a G 1 T -module, see [J2,§II.11.3]. As explained in [J2,Equation (3) in §II.11.5], p Qpλq is also the projective cover of p Lpλq in this category. As for simple and baby Verma modules, for λ, µ P X we have If λ P W ext¨p 0 then p Qpλq belongs to Rep ∅ pG 1 T q. By [J2,Proposition II.11.4], this module admits a filtration with subquotients of the form p Zpµq with µ P W ext¨p 0; moreover the number of occurrences of p Zpµq does not depend on the choice of such a filtration, and is equal to the multiplicity r p Zpµq : p Lpλqs. More generally, any projective object p Q in Rep ∅ pG 1 T q admits a filtration with subquotients of the form p Zpµq with µ P W ext¨p 0, and the number of occurrences of p Zpµq does not depend on the choice of filtration; this number will be denoted p p Q : p Zpµqq.
The first claim follows from the considerations above and Lemma 5.1.
We will say that an element w P W ext is restricted if w¨p0 is a restricted dominant weight. Note that this condition does not depend on p, and that restricted elements belong to f W ext .
In terms of the orbit W ext¨p 0, since w f pςq " ς´2ρ, Theorem 5.7 implies in particular that (if p ě 2h´2) for any w P W ext such that t ς w is restricted, we have (5.6) p Qpt ς w¨p 0q -Tpt ς w f w¨p 0q |G1T .
5.4. Characters of tilting modules as G 1 T -modules. Now we set M sph ext :" Z b Zrv˘1s M sph ext , and still denote by ϕ : M sph ext Ñ M asph ext and ζ : M sph ext Ñ ZrW ext s the (injective) morphisms induced by (3.6) and (3.1) respectively. We then consider the maps where the rightmost arrow is induced by the restriction functor (5.4).
Proposition 5.8. Let M be a tilting module in Rep ∅ pGq, all of whose direct summands are of the form Tpt ς w¨p 0q with w P f W ext . Then M |G1T is a projective G 1 T -module. Moreover, the inverse image a of rM s under (5.1) belongs to the image of ϕ, and the image under ζ of the preimage of a is equal to ÿ wPWext`M |G1T : p Zpw¨p 0`pςq˘¨w.
Proof. As explained in [J2,Lemma E.8], an indecomposable tilting module Tpλq (with λ P X`) is projective as a G 1 T -module iff λ´pp´1qς P X`. This implies the first claim in the proposition, and also that the G-modules M as in the statement are all isomorphic to direct sums of direct summands of modules of the form Θ s1 Θ s2¨¨¨Θsn pTppς`ω¨p 0qq with s 1 ,¨¨¨, s n in S and ω P Ω. This reduces the proof of the proposition to the case of modules of this form. We will prove this case by induction on n.
(Of course, this equality is also a special case of Theorem 5.2.) Using (3.3), we deduce that the image under ζ of the preimage of rTppς`ω¨p 0qs is ÿ On the other hand, by Lemma 5.6 we know that Tppς`ω¨p 0q |G1T " p Qpt ς w f ω¨p 0q, and we know the multiplicities of baby Verma modules in this projective module. Comparing with the formula above, we deduce the desired claim.
To prove the induction step, we will prove that if the claim is true for a module M , then it is true also for Θ s pM q for any s P S. As explained just after (5.1), if we denote by a the inverse image of rM s, then the inverse image of rΘ s pM qs is a¨pid`sq. Hence if a " ϕpbq, then this inverse image is ϕpa¨pid`sqq. Now by Lemma 5.4, for any w P W ext we have pΘ s pM |G1T q : p Zppς`w¨p 0qq " pM |G1T : p Zppς`w¨p 0qq`pM |G1T : p Zppς`ws¨p 0qq, and the desired claim follows.
compute the dominant weights appearing in p Lpw¨p 0q and their multiplicities. (See also [Sob,§4] for a different presentation of this procedure.) (3) Our assumptions on p in Theorem 5.9 are that p ě 2h´2 and p ą h. It is easily seen that these two conditions are equivalent to the condition that p ě 2h´1.
5.6. Proof of Theorem 1.2. We conclude the paper by explaining how Theorem 5.9 implies Theorem 1.2 from the introduction. As in §1.4, for m in P we denote by rms vÞ Ñ1 its image in Z b Zrv˘1s P -ZrA s. Recall that W acts on A on the right, see §1.5. This induces in the natural way a structure of right ZrW smodule on ZrA s, and it is clear from (2.5) that this action coincides with the one induced by the H-action on P (via the canonical isomorphism Zb Zrv˘1s H -ZrW s).