The Cohomology of Unramified Rapoport-Zink Spaces of EL-type and Harris's Conjecture

We study the $l$-adic cohomology of unramified Rapoport-Zink spaces of EL-type. These spaces were used in Harris and Taylor's proof of the local Langlands correspondence for $\mathrm{GL_n}$ and to show local-global compatibilities of the Langlands correspondence. In this paper we consider certain morphisms, $\mathrm{Mant}_{b, \mu}$, of Grothendieck groups of representations constructed from the cohomology of the above spaces, as studied by Harris and Taylor, Mantovan, Fargues, Shin, and others. Due to earlier work of Fargues and Shin we have a description of $\mathrm{Mant}_{b, \mu}(\rho)$ for $\rho$ a supercuspidal representation. In this paper, we give a conjectural formula for $\mathrm{Mant}_{b, \mu}(\rho)$ for all $\rho$ and prove it when $\rho$ is essentially square integrable. Our proof works for general $\rho$ conditionally on a conjecture appearing in Shin's work. We show that our description agrees with a conjecture of Harris in the case of parabolic inductions of supercuspidal representations of a Levi subgroup.


Introduction
Our goal in this paper is to give a description of the l-adic cohomology of unramified Rapoport-Zink spaces of EL-type. These spaces are moduli spaces of p-divisible groups associated to unramified Weil-restrictions of general linear groups and can be thought of as generalizations of Lubin-Tate spaces.
This work generalizes, for these particular spaces, the Kottwitz conjecture stated in [RV14,Conj 7.3]. The Kottwitz conjecture describes the supercuspidal part of the l-adic cohomology of Rapoport-Zink spaces, and is known in the cases we consider by work of Shin [Shi12, Cor 1.3]. We prove our description of this cohomology is compatible with a conjecture of Harris [Har01,Conj 5.4] generalizing the Kottwitz conjecture to parabolic inductions of supercuspidal representations.
Our result describes the cohomology of these Rapoport-Zink spaces as a formal alternating sum (indexed by certain root theoretic data) of representation-theoretic constructions including the local Langlands correspondence, parabolic inductions, and Jacquet modules.
We prove our result inductively using two formulas from the literature. The first of these is Shin's averaging formula [Shi12,Thm 7.5] which is proven using Mantovan's formula [Man05,Thm 22] connecting the cohomology of Rapoport-Zink spaces, Igusa varieties and Shimura Varieties. The second formula is the Harris-Viehmann conjecture of [RV14,Conj 8.4] which relates the cohomology of so-called non-basic Rapoport-Zink spaces to a product of Rapoport-Zink spaces of lower dimension. A proof of this conjecture is expected to appear in a forthcoming paper of Scholze.
To carry out our induction, we prove combinatorial analogues of the above formulas phrased purely in terms of root-theoretic data. Interestingly, we are able to prove these analogues for general quasisplit reductive groups, though at present we can only connect them to the cohomology of Rapoport-Zink spaces of unramified EL-type.
We now describe our main results more precisely. We study Rapoport-Zink spaces of unramified EL-type which we denote M b,µ . These are moduli spaces of p-divisible groups coming from an unramified EL-datum consisting of (1) a finite unramified extension F of Q p , (2) a finite dimensional F vector space V which defines the group G " Res F {Qp GLpV q, (3) a conjugacy class of cocharacters rµs with µ : G m Ñ G Qp , (4) an element b of a finite set BpG, µq which defines a group J b which is an inner twist of a Levi subgroup M b of G. Roughly one can think of b, µ as specifying the Newton and Hodge polygons of a p-divisible group and J b as the automorphism group of the isocrystal b.
The spaces M b,µ are formal schemes over y Q ur p . One constructs a tower of rigid spaces M rig U,b,µ over the generic fiber M rig b,µ of M b,µ , where the index U runs over compact open subgroups of GpQ p q. Associated to such a tower we have a cohomology space rH ‚ pG, b, µqs which is an element of the Grothendieck group GrothpGpQ p qˆJ b pQ p qˆW E q of admissible representations of GpQ p q, J b pQ p q and W E , where the latter group is the Weil group of the reflex field, E, of rµs. This construction can be thought of as an alternating sum of a direct limit over U Ă G of l-adic cohomology groups with the actions of GpQ p q and J b pQ p q arising from Hecke correspondences and isogenies of p-divisible groups, respectively.
The cohomology object rH ‚ pG, b, µqs gives rise to a map of Grothendieck groups Mant G,b,µ : GrothpJ b pQ p qq Ñ GrothpGpQ p qˆW E q which maps a representation ρ to the alternating sum of the J b pQ p q-linear Ext groups of rH ‚ pG, b, µqs and ρ. We refer to section 3.1 for a precise definition. The map Mant G,b,µ has been studied by many authors. Harris and Taylor [HT01] used this construction to prove the local Langlands correspondence for general linear groups. It also appears naturally in Mantovan's work relating the cohomology of Shimura varieties, Igusa varieties, and Rapoport-Zink spaces [Man05]. Fargues studied Mant G,b,µ for basic b in some EL and P EL-cases in [Far04]. Shin combined Mantovan's formula with his trace formula description of the cohomology of Igusa varieties to prove instances of local-global Langlands compatibilities [Shi11].
In [Shi12], Shin proved an averaging formula for Mant G,b,µ which is key to our work. He defined a map Red b : GrothpGpQ p qq Ñ GrothpJ b pQ p qq which up to a character twist is given by composing the un-normalized Jacquet module Jac G P b : GrothpGpQ p qq Ñ GrothpM b pQ p qq with the Jacquet-Langlands map of Badulescu [Bad1] LJ : GrothpM b pQ p qq Ñ GrothpJ b pQ p qq.
Shin uses global methods and so necessarily works with a large but inexplicit class of representations which he denotes accessible. This set loosely consists of those representations appearing as the p-component of an automorphic representation of a certain global group. In particular, the essentially square integrable representations in GrothpGpQ p qq are accessible.
In what follows r´µ is a finite dimensional representation of p G¸W E which restricts to the representation of highest weight´µ on p G, and LL is the semisimplifed local Langlands correspondence. Shin shows the following result.
Theorem 1.0.1 (Shin). Assume π is an accessible representation of GpQ p q. Then ÿ bPBpG,µq where the above formula is correct up to a Tate twist which we omit for clarity.
Additionally we have the conjecture of Harris and Viehmann which allows us to write Mant G,b,µ for non-basic b (b is basic when it corresponds to an isocrystal with a single slope) in terms of Mant G 1 ,b 1 ,µ 1 such that G 1 is a general linear group of smaller rank than G. This conjecture was formulated in work of [Har01] and [RV14] and is expected to be proven in forthcoming work of Scholze. In what follows, Ind is the un-normalized parabolic induction functor.
where we omit a Tate twist which we discuss at length in section 3.2. The finite set I G,µ M b ,b 1 is described in proposition 2.5.5.
Shin's averaging formula and the Harris Viehmann conjecture allow one to compute Mant G,b,µ˝R ed b recursively. The latter lets us compute Mant G,b,µ for nonbasic b given that we know Mant G 1 ,b 1 ,µ 1 for G 1 of lower rank and the former lets us compute Mant G,b,µ for the unique basic b P BpG, µq if we know it for all nonbasic b P BpG, µq. One of our main results is to give a non-recursive description of Mant G,b,µ˝R ed b which we now describe.
Let G " Res F {Qp GLpV q as before and choose a rational Borel subgroup B of G, and a rational maximal torus T Ă B Ă G. Then we consider pairs pM S , µ S q where M S Ą T is a Levi subgroup of a parabolic subgroup P S containing B, and µ S P X˚pT q is dominant as a cocharacter of M S . We call a pair of the above form a cocharacter pair for G.
We associate to a cocharacter pair pM S , µ S q the map of representations rM S , µ S s : GrothpGpQ p qq Ñ GrothpGpQ p qˆW E tµ S u M S q, which up to a Tate twist is given by π Þ Ñ rpInd G PS˝r µ S s˝Jac G PS qpπqs and rµ S s : GrothpM S pQ p qq Ñ GrothpM S pQ p qˆW E tµ S u M S q given by π Þ Ñ rπsrr´µ S˝L Lpπqs Then our main result, which follows from theorem 3.3.7 in this paper is Theorem 1.0.3. Suppose Mant G,b,µ corresponds to a tower of unramified Rapoport-Zink spaces of EL-type. We assume that the Harris-Viehmann conjecture is true. Then if ρ P GrothpGpQ p qq is essentially square-integrable, we have where R G,b,µ is a collection of cocharacter pairs with a combinatorial definition and p´1q LM S ,M b is an easily determined sign.
Based on a conjecture of Taylor, Shin conjectures ([Shi12, Conj 8.1]) that the averaging formula holds for all admissible representations of GpQ p q. If this were indeed the case, then our result would also immediately hold for all admissible representations of GpQ p q.
A crucial part of the proof of the above theorem is the following unconditional result, which is perhaps interesting in its own right.
Theorem 1.0.4 (Imprecise version of theorem 2.5.4 and corollary 2.5.8 of our paper). For general quasisplit G and a cochcaracter µ (not necessarily minuscule), combinatorial analogues of Shin's formula and the Harris-Viehmann conjecture hold true.
This result suggests that perhaps the combinatorics of cocharacter pairs is related to Mant G,b,µ in cases more general than Rapoport-Zink spaces of unramified ELtype. However, we caution the reader that the existence of nontrivial L-packets and nontrivial endoscopy in more general groups will likely complicate the situation.
In section 4 of the paper, we use our combinatorial formula to prove the EL-type cases of a conjecture of Harris ([Har01, Conj 5.4]) describing Mant G,b,µ pI G M pρqq for ρ a supercuspidal representation of M pQ p q for M a Levi subgoup of G. In this case, I G M denotes normalized parabolic induction. In particular, we show the following result, which is conjecture 4.0.4 in our paper.
Theorem 1.0.5 (Harris conjecture). We assume that Shin's averaging formula holds for all admissible representations of GpQ p q and that the Harris-Viehmann conjecture is true. Let ρ be a supercuspidal representation of M pQ p q. Then up to a precise Tate twist, for an explicit set of cocharacter pairs Rel G,µ M,b . We prove our result for I G M pρq not necessarily irreducible and b not necessarily basic, which is a generalization of what Harris conjectured for the G we consider.
Finally, in Appendix A we give an example to show that for general representations, one cannot hope for an expression as simple as that in Harris's conjecture.
Acknowledgements. I would like to thank Sug Woo Shin for suggesting I study the cohomology of Rapoport-Zink spaces and for countless helpful discussions on this topic. I thank Michael Harris for a fruitful conversation and for suggesting that my work might allow the verification of some cases of his conjecture. I thank Peter Scholze for a discussion in which he explained that the extra Tate twist in the Harris-Viehmann conjecture arises naturally in his work. This work is partially supported by NSF grant DMS-1646385 (RTG grant).

Cocharacter Formalism
In this section we define and study the notion of a cocharacter pair. This notation will be used in the third and fourth sections of this paper, where we describe the cohomology of certain Rapoport-Zink spaces in terms of cocharacter pairs. We endeavor to use a similar notation to [Kot97].
This section is divided into five subsections. These are structured so that the first contains the basic definitions and the fourth and fifth subsections contain the most important results. The second and third subsections prove a number of technical lemmas that the reader may want to skip at first and refer to as necessary.

Notation and Preliminary Definitions.
For the remainder of this section, we fix G a connected quasisplit reductive group defined over Q p . This is a significantly more general setting than we will need for applications in this paper. However, we choose to work in this generality because doing so is both conceptually clearer and potentially useful for future applications. The ideas in §5 of [Kot97] might allow one to remove the quasisplit assumption, but we do not attempt this here as it is unnecessary for the applications. Moreover, Kottwitz's study of the set BpGq in that section relies on understanding the quasisplit case first.
Remark 2.1.1. The reader will notice that most of this section makes sense over an arbitrary field. The assumption that we work over Q p is used in section 2.4 when we connect cocharacter pairs to the set BpGq defined by Kottwitz. However, in §5.1 of [Kot97], Kottwitz shows that over Q p , the set BpGq is parametrized by a disjoint union of sets of the form X˚pZp y M S q Γ q`for M S a standard Levi subgroup of G. These latter sets make sense over general fields and one could make sense generally of all the results of this section by replacing BpGq with the sets parametrizing it.
Since G is quasisplit, we can pick a Borel subgroup B Ă G defined over Q p and a maximal split torus A Ă B of G. We choose T to be a maximal torus defined over Q p satisfying A Ă T Ă B. We define X˚pAq and X˚pAq respectively to be the character and cocharacter groups of A Qp .
The group G has a relative root datum pX˚pAq, Φ˚pG, Aq, X˚pAq, Φ˚pG, Aqq, where Φ˚pG, Aq and Φ˚pG, Aq respectively denote the set of relative roots and relative coroots of G and the torus A. Our choice of Borel subgroup B determines a decomposition Φ˚pG, Aq " Φ˚pG, Aq`š Φ˚pG, Aq´of positive and negative roots and a subset ∆ Ă Φ˚pG, Aq`of simple roots. Analogous statements are also true for the coroots. The set of parabolic subgroups P Ą B defined over Q p are called standard parabolic subgroups. We define P S to be the unique standard parabolic subgroup such that Φ˚pP S , Aq " Φ˚pG, Aq`Y pΦ˚pG, Aq´X Span Z pSqq. There is an inclusion preserving bijection between the set of standard parabolic subgroups and subsets of ∆ given by S Þ Ñ P S .
We let N S be the unipotent radical of the standard parabolic subgroup P S . It is a standard result that there exists a connected reductive subgroup M Ă P S so that the natural map M Ñ P S {N S is an isomorphism. In particular, this gives us a Levi decomposition P S " M N S and the subgroup M is called a Levi subgroup of P S . The subgroup M is not unique but any two Levi subgroups of P S are conjugate by an element of N S . However, we have fixed a maximal torus T and there is a unique Levi subgroup M S containing T . The subgroup M S is constructed explicitly as the centralizer C G pZq, where Z Ă T is the connected component of the intersection of the kernels of the roots in S. We refer to the Levi subgroups M S that we produce in this way as standard Levi subgroups.
Define A :" X˚pAq. We have the closed rational Weyl chamber C Q " tx P A Q : xx, αy ě 0, α P ∆u.
We define for each standard Levi subgroup, A MS,Q :" tx P A Q : xx, αy " 0, α P S, xx, αy ‰ 0, α P ∆zSu, and denote the dominant elements of A MS ,Q by

AM
S ,Q " tx P A Q : xx, αy " 0, α P S, xx, αy ą 0, α P ∆zSu, and we have ž There is a partial ordering of A Q given by µ ĺ µ 1 if µ 1´µ is a non-negative rational combination of simple roots.
Definition 2.1.2. We define a cocharacter pair for a group G (relative to some fixed choice of T and B defined over Q p ) to be a pair pM S , µ S q such that M S Ă G is a standard Levi subgroup and µ S P X˚pT q satisfies xµ S , αy ě 0 for each positive absolute root α of T in the Lie algebra of M S,Qp . Positivity for absolute roots is determined by the Borel subgroup B which we have fixed. We denote the set of cocharacter pairs for G by C G .
Remark 2.1.3. We caution the reader that the cocharacter µ S need not be an element of X˚pAq, even though M S is defined over Q p . We could define cocharacter pairs more canonically as conjugacy classes of cocharacters of Levi subgroups. We choose not to as in practice we will often need to work with the unique dominant cocharacter in a conjugacy class relative to a fixed based root datum.
Let Γ " GalpQ p {Q p q. Since we have assumed T and B are defined over Q p , Γ acts on T Qp and B Qp . This gives us a natural left action of Γ on X˚pT q given explicitly by pγ¨µqpgq " γpµpγ´1pgqq for µ P X˚pT q and γ P Γ. We get an analogous left action on X˚pT q and one can easily check that the pairing X˚pT qˆX˚pT q Ñ Z is Γ invariant under these actions.
We have X˚pT q Γ " A.
Indeed, a Γ-invariant cocharacter µ factors through the identity component of T Γ , where T Γ is the subscheme defined by T Γ pQ p q " T pQ p q Γ . But the identity component of T Γ is the torus A. Conversely any cocharacter of A induces a Γ-invariant cocharacter via the natural inclusion A ãÑ T .
Given µ P X˚pT q, we construct an element µ Γ of A Q as follows: where Γ µ is the stabilizer of µ in Γ. Then µ Γ P X˚pT q Γ Q " A Q . Given a standard Levi subgroup M S , we let W rel MS be the subgroup of the relative Weyl group, W rel , which is generated by the transpositions corresponding to simple roots in S.
Definition 2.1.4. We define a map given by We are now ready to describe a formalism that will prove useful in studying the cohomology of certain Rapoport-Zink spaces. Crucial to everything that follows is a partial ordering on the set C G of cocharacter pairs for G.
Definition 2.1.5. We define a partial ordering on C G which we denote by the symbol ď. Unfortunately, our definition is somewhat indirect: we first define when pM S2 , µ S2 q ď pM S1 , µ S1 q for M S2 Ă M S1 and S 1 zS 2 contains a single element (in other words, M S2 is a maximal proper Levi subgroup of M S1 ). We then extend the relation to all cocharacter pairs by taking the transitive closure.
Let M S2 , M S1 be standard Levi subgroups of G such that M S2 Ă M S1 and S 1 zS 2 is a singleton. For cocharacter pairs pM S2 , µ S2 q, pM S1 , µ S1 q P C G , we write pM S2 , µ S2 q ď pM S1 , µ S1 q if µ S2 is conjugate to µ S1 in M S1 Qp and θ MS 2 pµ S2 q ą θ MS 1 pµ S1 q. We then take the transitive closure to extend to a partial ordering on C G .
Example 2.1.6. Consider G " GL 4 with T the diagonal torus and B the upper triangular matrices. We can pick a basis for X˚pT q of cocharacters p e i defined so that p e i pgq is the diagonal matrix with 1 in every position except for the ith, which equals g. Then we can identify an element of X˚pT q with its coordinate vector in this basis. Finally, we use additional parenthesis to indicate the product structure of the standard Levi subgroup M S . Using this notation, the set of cocharacter pairs that are less than or equal to pGL 4 , p1 2 , 0 2 qq is given in the diagram at the start of appendix A.
Finally, we remark that the fact that all the related cocharacter pairs in the above example have equal (as opposed to just conjugate) cocharacters is very much a result of us choosing a fairly small group G. Even for G " GL 5 , this is not the case.
Definition 2.1.7. We define a cocharacter pair pM S , µ S q for G to be strictly decreasing if θ MS pµ S q P AM S ,Q . We denote by SD Ă C G the strictly decreasing elements of C G and by SD µ (for dominant µ P X˚pT q) the strictly decreasing elements pM S , µ S q P C G such that pM S , µ S q ď pG, µq.
Remark 2.1.8. The θ MS map can be thought of as associating a tuple of slopes to a cocharacter pair. Then the strictly decreasing cocharacter pairs with Levi subgroup M S are the ones whose slope tuple lies in the image of the Newton map ν : BpGq MS Ñ A MS ,Q . The above statement is made precise by proposition 2.4.3.

2.2.
An Alternate Characterization of the Averaging Map. The following two subsections consist of a collection of lemmas developing the theory of the map θ MS and the set of strictly decreasing elements SD of C G .
In this section, we give an alternate description of the map θ MS . To do so, we will need several properties of cocharacters and root data which we record in the following lemma. For this lemma only, we consider T and G defined over a more general class of fields so that these results also apply to the complex dual groups p T and p G.
Lemma 2.2.1. Let F Ą Q be a field and G a connected quasisplit reductive group defined over F . Suppose that T Ă G is a maximal torus defined over F and that the group scheme T F admits an action defined over F by a finite group Λ. Let X˚pT Λ q denote the characters of the subgroup scheme of Λ-fixed points of T F . The anti-equivalence of categories between tori and finitely generated free Abelian groups given by T F Þ Ñ X˚pT q induces an action of Λ on X˚pT q. We then have the following.
(1) There is a unique isomorphim X˚pT Λ q -X˚pT q Λ such that the following diagram commutes.
X˚pT q X˚pT Λ q X˚pT q Λ res proj (2) Let M S Ă G be a standard Levi subgroup. Let W abs MS , W rel MS denote the absolute and relative Weyl groups of M S and let Γ " GalpF {F q. Then There is an anti-equivalence of categories between diagonalizable groups over F and finitely generated Abelian groups. The diagram for the universal property for Λ-invariants is that of Λ-coinvariants but with all the arrows reversed. Thus, there must exist a unique isomorphism between X˚pT Λ q and X˚pT q Λ that makes the diagram

X˚pT q
X˚pT Λ q X˚pT q Λ res proj commute. This proves p1q.
In [Kot84, Lem 1.1.3], Kottwitz proves that pX˚pT q Γ q{W rel MS -pX˚pT q{W abs MS q Γ . Thus, to prove p2q, we need only show that this isomorphism gives a bijection of the singleton orbits. Kottwitz's isomorphism maps the W rel MS -orbit of µ P X˚pT q Γ to its W abs MS orbit in X˚pT q. Thus, it suffices to show that if µ P X˚pT q Γ is invariant by W rel MS then it is also invariant by W abs MS . If µ is invariant by W rel MS , then the pairing of µ with each relative root of M S is 0. Thus the image of µ lies in the intersection of the kernels of the relative roots of M S which is ZpM S q X A. Therefore, µ is invariant under the action of W abs MS . For p3q, we need to construct an inverse to the map is independent of the choice of lift of rµs to X˚pT q Q and gives an inverse to the map above.
Let A MS be the maximal split torus in the center of M S . Then We now prove a lemma that we will need to use to describe the alternate characterization of θ MS .
Proof. By 2.2.1, we have the following isomorphisms.
We explicate the isomorphism X˚pT q This follows from the isomorphism X˚pAq W rel M S -X˚pA MS q which we now describe. Suppose we have µ P X˚pAq W rel M S . Equivalently, for each relative root α of M S , we have σ α pµq " µ (where σ α is the reflection in the Weyl group corresponding to α). Since σ α pµq " µ´xµ, αyα, this is equivalent to xµ, αy " 0 for all relative roots α of M S , which in turn is equivalent to the statement that impµq Ă Ş α kerα. Finally, this is equivalent to impµq Ă ZpM S q X A. Since the image of a cocharacter is connected, we in fact have that µ P X˚pA MS q.
To finish the argument, we need to construct an isomorphism Note that it is necessary to take the tensor product with Q here as Zp y M S q and p T W abs M S need not be isomorphic.

It suffices to show that
X˚pZp y where both maps are averages over the relevant group. As we now show, this is the same as the composition where the first two maps are averages and the third is as in 2.2.1 (2). Indeed, The commutativity essentially follows from the definition of the averaging maps. The benefit of this is that now we can write θ MS as the composition of Thus, the previous expression equals comparing with 2.2.2, we can rewrite θ MS as We record the following useful corollary of the ideas discussed in the above argument.
Proof. By the observation at the start of 2.2.4, θ MS is equivalently defined as the composition In particular, µ and µ 1 are mapped to the same element under the first map in the above composition.
2.3. Strictly Decreasing Cocharacter Pairs. In this section, we prove a number of properties of strictly decreasing cocharacter pairs and their relation to the partial order we defined in 2.1.5. As always, we let σ α denote the reflection in the relative Weyl group corresponding to the relative root α.
Proof. For the first part of the lemma, we claim that if we can show that xσpxq, αy ě 0 for each σ P W rel MS and α P ∆zS, then we are done. This follows because if a collection of cocharacters pair non-negatively with α, then so will their average. Thus for α P ∆zS, we get xy, αy ě 0. For α P S, we automatically have xy, αy " 0 since 0 " y´σ α pyq " xy, αyα.
Pick α P ∆zS. Then the root group of α is contained in the unipotent radical N S of P S . The group N S is normalized by M S and therefore by W rel MS . In particular, for any σ P W rel MS , the root group of σ´1pαq is contained in N S and hence is also positive. Thus xσpxq, αy " xx, σ´1pαqy ě 0 as desired.
To prove the second part, we notice since xx, αy ą 0, the term in y corresponding to σ " 1 has positive pairing with α. Since all the other terms have non-negative pairing with α, we must have that xy, αy ą 0.

Lemma 2.3.2. If x as in the previous lemma is dominant, then
Proof. It suffices to show that for any σ P W rel MS , we have σpxq ĺ x. This is a standard fact ([Bou68, Ch6 1.6.18, p. 158]).
Corollary 2.3.3. Let pM S , µ S q P SD be a strictly decreasing cocharacter pair and let pM S 1 , µ S 1 q P C G and suppose that pM S , µ S q ď pM S 1 , µ S 1 q. Then pM S 1 , µ S 1 q P SD.
Proof. We need to show that for each β P ∆zS 1 , that xθ M S 1 pµ S 1 q, βy ą 0. By 2.2.5, θ M S 1 pµ S 1 q " θ M S 1 pµ S q. Further, we observe that Since θ MS pµ S q is dominant by assumption and satisfies xθ MS pµ S q, βy ą 0, we can apply 2.3.1 to get the desired result.
The following easy uniqueness result is quite useful.
We now define the notion of a cocharacter pair being strictly decreasing relative to a Levi subgroup.
Definition 2.3.5. Let M S Ĺ M S 1 be standard Levi subgroups of G. We say pM S , µ S q is strictly decreasing relative to M S 1 if xθ MS pµ S q, αy ą 0 for α P S 1 zS.
Remark 2.3.6. Recall that by construction, xθ MS pµ S q, αy " 0 for α P S. Thus, pM S , µ S q P SD exactly when it is strictly decreasing relative to G. Lemma 2.3.7. Let pM S1 , µ S1 q, pM S 1 1 , µ S 1 1 q P C G be cocharacter pairs such that pM S1 , µ S1 q ď pM S 1 1 , µ S 1 1 q. Let M S2 Ą M S1 be a standard Levi subgroup of G and suppose pM S1 , µ S1 q is strictly decreasing relative to M S2 . Then pM S 1 1 , µ S 1 1 q is strictly decreasing relative to M S 1 1 YS2 . Proof. We first reduce to the case where M S1 is a maximal Levi subgroup of M S 1 1 (i.e. S 1 1 " S 1 Y tαu for some α P ∆zS 1 ). To do so, we recognize that the relation pM S1 , µ S1 q ď pM S 1 1 , µ S 1 1 q definitionally implies that there is a finite sequence of cocharacter pairs pM S1 , µ S1 q " Thus, if we prove the lemma in the maximal Levi subgroup case, we can inductively prove it in the general case.
We now assume that M S1 Ă M S 1 1 is a maximal Levi subgroup so that S 1 1 " S 1 Y tαu for some α P ∆zS 1 . We need to show that xθ M S 1 1 pµ S 1 1 q, βy ą 0 for each β P S 1 1 Y S 2 zS 1 1 . First note that any such β is an element of S 2 zS 1 . By 2.2.5, since µ S1 and µ S 1 1 are conjugate in M S 1 1 , we have θ M S 1 1 pµ S1 q " θ M S 1 1 pµ S 1 1 q. Thus we are reduced to showing xθ M S 1 1 pµ S1 q, βy ą 0 for β P S 2 zS 1 . Note that since pM S1 , µ S1 q is strictly decreasing relative to M S2 , we have θ MS 1 pµ S1 q is dominant relative to the root datum of M S2 and xθ MS 1 pµ S1 q, βy ą 0. Therefore, by equation 1 and lemma 2.3.1, xθ M S 1 1 pµ S1 q, βy ą 0 as desired.
Proposition 2.3.8. Let pM S , µ S q P C G and suppose it is strictly decreasing relative to some standard Levi subgroup In the case where S 1 " S Y tαu for α P ∆zS, the converse is true. Specifically, if pM S , µ S q P C G and there exists pM S 1 , µ S 1 q P C G satisfying pM S 1 , µ S 1 q ě pM S , µ S q with S 1 " S Y tαu, then pM S , µ S q is strictly decreasing relative to M S 1 .
Proof. We begin by proving the first statement. Uniqueness follows from 2.3.4. For existence, we first reduce to the case where M S is a maximal Levi subgroup of M S 1 . Suppose we have proven the proposition in this reduced case. We might then try to prove the general case by iteratively applying the reduced case of the proposition to a chain of standard Levi subgroups M S " M S0 Ă ... Ă M S k " M S 1 such that each is maximal in the next. Such a chain clearly exists, but to apply the reduced case of the proposition we need to show that if we have constructed a cocharacter pair pM Si , µ Si q ě pM S , µ S q then pM Si , µ Si q is strictly decreasing relative to M S 1 . This follows from 2.3.7. Now, we let µ S 1 be the unique conjugate of µ S which is dominant in M S 1 . If we can show that θ M S 1 pµ S 1 q ă θ MS pµ S q, then pM S 1 , µ S 1 q will satisfy the conditions of the proposition. By 2.2.5 and equation 1, for any y satisfying xy, αy ą 0 for α P S 1 zS and xy, αy " 0 for α P S. Any such y is dominant in the root datum of M S 1 and so by lemma 2.3.2, Further, the above equation cannot be an equality because y has positive pairing with each root of S 1 zS while σpyq has 0 pairing with these roots.
To prove the converse, suppose that pM S , µ S q ď pM S 1 , µ S 1 q and S 1 " S Y tαu for some α P ∆zS. Then by 2.2.5 and so Since by assumption θ M S 1 pµ S 1 q ă θ MS pµ S q, it follows that xθ MS pµ S q, αy ą 0.
Remark 2.3.9. Note that the converse of the above proposition is not true in the general case.
Corollary 2.3.10. Fix a standard Levi subgroup M S and roots α 1 , α 2 P ∆zS. Suppose we have cocharacter pairs pM S , µ S q, pM SYtα1u , µ SYtα1u q, pM SYtα1,α2u , µ SYtα1,α2u q P C G satisfying pM S , µ S q ď pM SYtα1u , µ SYtα1u q ď pM SYtα1,α2u , µ SYtα1,α2u q and that pM S , µ S q is strictly decreasing relative to M SYtα2u . Then the extension of pM S , µ S q to M SYtα2u , which we denote pM SYtα2u , µ SYtα2u q, satisfies Proof. By the second statement of 2.3.8, we have that pM S , µ S q is strictly decreasing relative to M SYtα1u . Then by 2.3.7, pM SYtα2u , µ SYtα2u q is strictly decreasing relative to M SYtα1,α2u . Thus by 2.3.8, we have pM SYtα2u , µ SYtα2u q ď pM SYtα1,α2u , µ SYtα1,α2u q as desired.
Proposition 2.3.11. Let S Ă S 1 Ă S 2 be subsets of ∆ and suppose pM S , µ S q, pM S2 , µ S2 q P C G with pM S , µ S q ď pM S2 , µ S2 q and pM S , µ S q is strictly decreasing relative to M S1 . Then the unique extension pM S1 , µ S1 q of pM S , µ S q to M S1 satisfies pM S1 , µ S1 q ď pM S2 , µ S2 q.
Proof. Since pM S , µ S q ď pM S2 , µ S2 q, there is an increasing chain of cocharacter pairs pM S , µ S q " pM S 0 , µ S 0 q ď ... ď pM S k , µ S k q " pM S2 , µ S2 q such that each standard Levi subgroup is maximal in the next. The content of this proposition is that we can pick a chain such that pM S1 , µ S1 q appears. By 2.3.7, we can assume that M S is maximal in M S1 . Let α be the unique element of S 1 zS.
Pick a chain of cocharacter pairs pM S , µ S q " pM S 0 , µ S 0 q ď ... ď pM S k , µ S k q " pM S2 , µ S2 q as above. Chains of cocharacter pairs are determined by an ordering on the roots in S 2 zS " tα 1 , ..., α k u, such that the S i " S Y tα 1 , ..., α i u. The root α appears in this chain so α " α i for some i. If i " 1 we are done. Otherwise, we consider q is strictly decreasing relative to M S i´2 Ytαu and so by 2.3.10 (applied so that pM S i´2 , µ S i´2 q takes the place of pM S , µ S q in 2.3.10), we get a new chain of cocharacter pairs between pM S , µ S q and pM S2 , µ S2 q where we switch the positions of α, α i´1 in the corresponding ordering of S 2 zS. By repeating this argument, we can construct a chain where α " α 1 , which is what we need.
The preceding propositions give us the following picture. Given a cocharacter pair pM S , µ S q we check for which simple roots α satisfy xθ MS pµ S q, αy ą 0. Suppose there are n such simple roots. Then we get 2 n standard Levi subgroups containing M S corresponding to adding different subsets of these simple roots. The cocharacter pair pM S , µ S q has a unique extension to each of the Levi subgroups and the poset lattice of these co-character pairs can be thought of as a directed n dimensional cube (such that the vertices are cocharacter pairs and the directed edges correspond to the ă relation).

Connection With Isocrystals.
We now investigate the relation between strictly decreasing cocharacter pairs and Kottwitz's theory of isocrystals with additional structure. See [Kot97] for omitted details on the theory of isocrystals.
An isocrystal is a pair pV, Φq where V is a finite dimensional y Q ur p vector space and Φ : V Ñ V is an additive transformation satisfying Φpavq " σpaqΦpvq for a P y Q ur p , v P V and σ a lift of Frobenius. As before, let G be a connected quasisplit reductive group defined over Q p and consider the set of isomorphism classes of exact b-functors from ReppGq to Isoc, the category of isocrystals. Such isomorphism classes are classified by H 1 pW Qp , Gp y Q ur p qq which we denote BpGq (where W Qp is the Weil group of Q p ).
In §4.2 of [Kot97], Kottwitz constructs the Newton map ν : BpGq Ñ C Q and the Kottwitz map κ : BpGq Ñ X˚pZp p Gq Γ q. An element of BpGq is uniquely determined by its image under these maps.
We say that the standard Levi subgroup M S is associated to b P BpGq if νpbq P A MS,Q . Henceforth, we will often denote the standard Levi subgroup associated to b by M b . Notice that many elements of BpGq could be associated to the same Levi

Kottwitz uses the Kottwitz map for M S to construct canonical bijections
where Kottwitz constructs a canonical isomorphism and X˚pZp y M S q Γ q`denotes the subset of X˚pZp y M S q Γ q mapping to AM S ,Q . In fact, Kottwitz shows that the composition of the above isomorphisms gives the Newton map BpGq MS Ñ AM S ,Q ãÑ C Q . For a further discussion of equation 3, we refer the reader to 2.2.2 and the subsequent remark.
We now prove an important lemma that will be used to relate the set BpGq to the strictly decreasing elements of C G .
Proof. We first describe the set νpBpGq MS q. By equations 2 and 3, the set νpBpGq MS q is equal to the image of X˚pZp y M S q Γ q`in A MS,Q . Thus, to prove this proposition, it suffices to show that θ MS factors through the map X˚pZp y BpG, µq :" tb P BpGq : νpbq ĺ θ T pµq, κpbq " µ| Zp p Gq Γ u. Now we prove the key result of this section, which permits us to associate an element of BpGq to each strictly decreasing cocharacter pair.
Proposition 2.4.3. We have a natural map T : SD Ñ BpGq defined as follows. Let pM S , µ S q P SD. Then there exists a b P BpGq so that κpbq " µ S | ZpĜq Γ and νpbq " θ MS pµ S q. We note that by construction, b is unique. Then we define T ppM S , µ S qq " b. In particular we show that Recall that the composition of this isomorphism with equation 3 induces the Newton map restricted to BpGq MS . Thus, we have θ MS pµ S q " νpbq. Equation (4.9.2) of [Kot97] implies that κpbq " µ S | ZpĜq Γ .
It remains to show that if pM S , µ S q P SD µ then the element b P BpGq that we have constructed lies in the set BpG, µq. For this, we need to show that νpbq " θ MS pµ S q ĺ θ T pµq.
We claim that θ T pµq ľ θ T pµ S q. After all, by ([Bou68, Ch6 1.6.18, p. 158]), we have µ ľ µ S . Then the claim follows from B.0.4. Now we claim that θ T pµ S q is dominant in the relative root system of M S . To prove the claim, we first observe that µ S is dominant relative to the absolute root system of M S . As above, the Galois group Γ preserves the Weyl chamber corresponding to the positive absolute roots given by B. Thus, γpµ S q is dominant for each γ P Γ, and so θ T pµ S q is dominant relative to the absolute roots of M S . The intersection of the closed positive Weyl chamber for the absolute root datum of M S with A Q is the Weyl chamber for relative root datum of M S . Thus, θ T pµ S q is dominant with respect to the relative roots as desired.
Finally, we apply lemma 2.3.2 and equation 1 to get which finishes the proof.
Question 2.4.4. Is it true that T pSD µ q " BpG, µq?
This would be an intrinsically interesting result. However, the author does not know a proof of this statement.
2.5. The Induction and Sum Formulas. We are now ready to prove our main theorems on cocharacter pairs. We begin by defining some key subsets of C G , the set of cocharacter pairs for G.
Definition 2.5.2. Let ZxC G y denote the free Abelian group generated by the set of cocharacter pairs for G.
We define M G,b,µ P ZxC G y by We will show in theorem 3.3.7 that at least in the case where G is a restriction of scalars of a general linear group, M G,b,µ is related to the cohomology of Rapoport-Zink spaces for G. Thus one expects there to be an analogue of the Harris-Viehmann conjecture (which in this setting we call the induction formula). Perhaps the more surprising result is that there is also an analogue of Shin's averaging formula (which we call the sum formula) [Shi12, Thm 7.5]. We first prove the sum formula.
Theorem 2.5.4 (Sum Formula). The following holds in ZxC G y: ÿ bPBpG,µq Proof. We need to show that ÿ bPBpG,µq We prove this equality by counting how many times a given cocharacter pair shows up on the left-hand side. The pair pG, µq shows up exactly once in the left-hand sum as an element of R G,b,µ for b the unique basic element of BpG, µq. SupposepM S , µ S q P C G is some other cocharacter pair. Then define We are reduced to showing ÿ Our general strategy will be to show that the left-hand side of equation 4 vanishes for each pM S , µ S q ă pG, µq by inducting on the size of ∆zS. However, in the case that pM S , µ S q P SD µ , we can prove the vanishing without an inductive argument. We show this first before discussing the induction.
Suppose now that pM S , µ S q P SD µ . By 2.3.3, every pair pM S 1 , µ S 1 q P C G satisfying pM S , µ S q ď pM S 1 , µ S 1 q ď pG, µq is strictly decreasing and thus by 2.4.3, we have T ppM S 1 , µ S 1 qq P BpG, µq. These are precisely the elements b P BpG, µq so that pM S , µ S q P R G,b,µ . By the discussion after 2.3.11, this set forms a cube and it is a standard fact that if we associate one of t1,´1u to each vertex of a cube so that adjacent vertices have opposite signs, then the sum of all the signs is 0. This implies that the left-hand side of 4 vanishes in the strictly decreasing case. Now we discuss the inductive argument. The base case will be for pairs pM S , µ S q ă pG, µq satisfying |∆zS| " 1. The second statement of proposition 2.3.8 implies that in this case pM S , µ S q is strictly decreasing relative to G, which means that pM S , µ S q P SD µ . Thus, the base case is proven by the previous paragraph.
We now discuss the inductive step. Suppose pM S , µ S q ă pG, µq. If pM S , µ S q is strictly decreasing, then we are done by the above. Suppose now that pM S , µ S q is not strictly decreasing. We claim that pM S , µ S q must be strictly decreasing with respect to at least some standard Levi subgroup of G that properly contains M S . After all, since pM S , µ S q ă pG, µq, there must exist at least some α P ∆zS and pM SYtαu , µ SYtαu q P C G so that pM S , µ S q ď pM SYtαu , µ SYtαu q. Then by 2.3.8, this implies that pM S , µ S q is strictly decreasing relative to M SYtαu .
Thus, let M S 1 be the maximal standard Levi subgroup of G such that pM S , µ S q is strictly decreasing relative to M S 1 . We can write S 1 " S Y tα 1 , ..., α n u where α i ‰ α j for i ‰ j and each α i P ∆zS. We denote by X the n-cube of cocharacter pairs above pM S , µ S q as in the discussion after 2.3.11.
We claim that ÿ Given this claim, we see that to finish the proof, it suffices to show that the righthand side is identically 0. However, the right-hand side consists of a sum of a number of terms similar to the left-hand side but for pairs pM S 1 , µ S 1 q in place of pM S , µ S q. Note that each S 1 is strictly larger than S and thus we are done by induction.
We now prove the claim. Moving all the terms to one side, we need only show that ÿ Then it suffices to show the contribution from b in the above formula vanishes. Thus, we must show ÿ We examine the structure of X X R G,b,µ when it is nonempty. If we can show that the cocharacter pairs in this set form a sub-cube of X of positive dimension, then we will be done by the standard fact that if we place alternating signs on the vertices of a cube and add up all the signs we get 0.
Clearly, any pM S 1 , µ S 1 q P X X R G,b,µ must satisfy M S Ă M S 1 Ă M b . The subset of X satisfying this latter property forms a sub-cube of X since its elements are indexed by subsets of S b zS, where S b is the subset of ∆ corresponding to M b in the standard way (note that by 2.3.4, there is at most one element of X X R G,b,µ for each standard Levi M S 1 ). Moreover, this latter set cannot form a cube of dimension 0 for then we would have M S " M b and so X X R G,b,µ " tpM S , µ S qu which would imply that pM S , µ S q is strictly decreasing contrary to assumption.
Thus to finish the proof, we need only show that every pM S 1 , µ S 1 q such that Then the desired result follows from 2.3.11.
We now turn to the induction formula. The following definition will be important in relating cocharacter pairs of a group G to those of a standard Levi. Compare with [RV14, eqn 8.1].
Definition 2.5.5. Let M S be a standard Levi subgroup of G, let µ P X˚pT q be a dominant cocharacter and choose b P BpG, µq. We take b S P BpM S q as constructed in the previous paragraph and define the set We first check the following transitivity property of I G,µ MS ,bS . Proposition 2.5.6. Fix pG, µq P C G and b P BpG, µq. Suppose M S2 and M S1 are standard Levi subgroups of G such that M b Ă M S2 Ă M S1 . Then Proof. We claim that for distinct pM S1 , µ S1 q, pM S1 , µ 1 S1 q P I G,µ MS are disjoint. Indeed suppose pM S2 , µ S2 q is an element of both sets. Then µ S2 is conjugate to both µ S1 and µ 1 S1 in M S1 . Since µ S1 , µ 1 S1 are dominant in M S1 , this implies they are equal. Thus to prove the proposition, it suffices to show each set is a subset of the other. Take pM S2 , µ S2 q P I G,µ MS 2 ,bS 2 . Let µ S1 be the unique dominant cocharacter conjugate to µ S2 in M S1 . Then we consider pM S1 , µ S1 q as an element of C MS 1 and just need to show that b S1 P BpM S1 , µ S1 q since we already know that b S2 P BpM S2 , µ S2 q by assumption. Thus, we need only show that νpb S1 q ď θ T pµ S1 q and κpb S1 q " µ S1 | Zp z MS 1 q Γ . We prove the inequality first. By assumption, νpb S2 q ĺ θ T pµ S2 q and by equations 2 and 3, νpb S1 q " νpbq " νpb S2 q. Since µ S1 and µ S2 are conjugate in M S1 and µ S1 is dominant, it follows from [Bou68, Ch6 1.6.18, p. 158] that µ S2 ĺ µ S1 . Then, by B.0.4 it follows that θ T pµ S2 q ĺ θ T pµ S1 q in the relative root system. Combining all this data, we get νpb S1 q " νpb S2 q ĺ θ T pµ S2 q ĺ θ T pµ S1 q, as desired.
To prove κpb S1 q " µ S1 | Zp z MS 1 q Γ , we note that by equation (4.9.2) of [Kot97] and the fact that b S2 P BpM S2 , µ S2 q, we have κpb S1 q " µ S2 | Zp z MS 1 q Γ . Then µ S1 and µ S2 are conjugate in M S1 so there exists a w P W abs MS 1 so that wpµ 1 q " µ 2 . This implies that µ 1 and µ 2 are conjugate in z M S1 and in particular equal when restricted to Zp z M S1 q. This implies the desired equality. To show the converse inclusion, we start with pM S2 , µ S2 q P I MS 1 ,µS 1 MS 2 ,bS 2 for some pM S1 , µ S1 q P I G,µ MS 1 ,bS 1 and need to show that b S2 P BpM S2 , µ S2 q and that µ S2 is conjugate to µ in G. But pM S2 , µ S2 q P I MS 1 ,µS 1 MS 2 ,bS 2 implies that b S2 P BpM S2 , µ S2 q and also that µ S2 is conjugate to µ S1 in M S1 . Further, pM S1 , µ S1 q P I G,µ MS 1 ,bS 1 implies that µ S1 is conjugate to µ in G. Thus, µ S2 is conjugate to µ in G as desired.
The set I G,µ MS ,bS will primarily be useful because it allows us to relate the set T G,b,µ to analogous constructions in M S . This is encapsulated in the following proposition. Since M b Ă M S , it follows from the discussion after equation 6 that Thus, pick an arbitrary element of T G,b,µ of the form i G MS pM b , µ b q for pM b , µ b q P C MS . The cocharacter pair i G MS pM b , µ b q is strictly decreasing, and therefore so is pM b , µ b q P C MS . By 2.3.8 we can find pM S , µ S q P C MS such that pM b , µ b q ď pM S , µ S q. Observe that since i G MS pM b , µ b q ď pG, µq, the cocharacter µ b is conjugate to µ in G and therefore µ S must be as well by construction. If we can show that T ppM b , µ b qq " b S , then we will be done because by 2.4.3, this implies that b S P BpM S , µ S q and so therefore that pM S , µ S q P I G,µ MS ,bS and pM b , µ b q P T MS ,bS,µS . By assumption, T pi G MS pM b , µ b qq " b P BpG, µq. Recall that the map T is defined so that a strictly decreasing pM b , µ b q P C G which satisfies pM b , µ b q ď pG, µq is mapped first to the element Then, this element is identified with an element of BpGq via the isomorphisms of equation 2: where the second isomorphism above is induced by the inclusion M b ãÑ G. We have the commutative diagram where each map is induced from the inclusion of groups. By definition, the element b 1 P BpM b q`maps to b P BpGq and b S P BpM S q respectively. Thus, we see that by This implies that i G MS pM b , µ b q P SD. By 2.3.8, we have an extension of i G MS pM b , µ b q to G, and since µ b and µ are conjugate in G by assumption, it follows that this extension is pG, µq so that i G MS pM b , µ b q ď pG, µq. It follows from these facts that i G MS pM b , µ b q P T G,b,µ . Finally, we remark that for distinct pM S , µ S q, pM S , µ 1 S q P I G,µ MS ,bS the sets T MS ,bS ,µS and T MS ,bS ,µ 1 S are indeed disjoint by 2.3.4. As a corollary of this result, we have the induction formula.
Corollary 2.5.8 (Induction Formula). We continue using the notation of the previous proposition. The natural map We remark that for distinct pM S , µ S q, pM S , µ 1 S q P I G,µ MS ,bS we have R MS ,bS ,µS X R MS ,bS ,µ This result can be thought of as an analogue of the Harris-Viehmann conjecture which we discuss in the next section.
In the cases we are interested in, we will also need a description of how cocharacter pairs behave with respect to products.
Suppose G " G 1ˆ. ..ˆG k and T " T 1ˆ. ..ˆT k such that T i is a maximal torus for G i . Then X˚pT q -X˚pT 1 q ' ... ' X˚pT k q, and any standard Levi subgroup admits a product decomposition Then any cocharacter pair pM S , µ S q of G corresponds to a tuple of cocharacter pairs ppM S1 , µ S1 q, ..., pM S k , µ S k qq P C G1ˆ. ..ˆC G k , in the obvious way. The pair pM S , µ S q P C G is strictly decreasing if and only if each pair pM Si , µ Si q P C Gi is, and if T ppM S , µ S qq " b P BpG, µq, then we also have We record the following proposition Proposition 2.5.9. We use the notation of the previous two paragraphs.
The natural bijection

Cohomology of Rapoport-Zink spaces and the Harris-Viehmann Conjecture
In this section, we define the Rapoport-Zink spaces we will work with and show how we can describe their cohomology using the language developed in the previous section. We also give a statement of the Harris-Viehmann conjecture, and explain the necessity of a small correction to the conjecture. We follow [Far04], [Shi12], and [RV14].
3.1. Rapoport-Zink Spaces of EL-Type. We fix the following notation. Suppose G is a reductive group defined over a field k and µ P X˚pGq. Then if H is a subgroup of G, we denote by tµu H the Hpkq conjugacy class of µ and by E tµuH the field of definition of tµu H (i.e the smallest extension of k so that each element of Galpk{E tµuH q stabilizes tµu H ). Now we define the Rapoport-Zink data we consider.
Definition 3.1.1. An unramified Rapoport-Zink datum of EL type is a tuple Let X be a p-divisible group defined over F p with an action of O F and covariant isocrystal isomorphic to pV F , bσq. We consider the moduli functor M b,µ such that for S a scheme over O z Q ur p with p locally nilpotent, M b,µ pSq " tpX, i, ρqu{ ". Where X is a p-divisible group defined over S, i : O F Ñ End F pXq, and ρ : XˆF p S Ñ X is a quasi-isogeny (S, X are the reductions modulo p).
By work of Rapoport and Zink [RZ96, Thm 3.25], the above moduli problem is represented by a formal scheme over O z Q ur p which we also denote by M b,µ . We have the generic fiber M rig b,µ which is a rigid analytic space over y Q ur p . Further, we get a tower of coverings M rig b,µ,U of M rig b,µ for each compact open subgroup U Ă GpQ p q. For a fixed prime l ‰ p, we denote by H j c pM rig b,µ,Uˆy Q ur p , Q l q the etale cohomology with compact supports. This is a Q l vector space which is a smooth representation of J b pQ p qˆW E tµu G , where J b is the inner form of M b associated to b (as constructed in §3.3 of [Kot97]) and W E tµu G is the Weil group of E tµuG (for example see [RV14, Prop 6.1]).
We use the notation Grothp¨q for the Grothendieck group of admissible representations of topological groups. See §I.2 of [HT01] for the precise definition of these Grothendieck groups.
We let P b be the standard parabolic subgroup with Levi factor M b and denote the opposite parabolic by P op b . We define J G P , Jac G P to be the normalized and unnormalized Jacquet module functors, and we define I G P , Ind G P to be the normalized and un-normalized parabolic induction functors. Often, if M Ă P is the standard Levi subgroup of P and we are taking I G P , J G P to be maps of Grothendieck groups, we will write I G M , J G M to remind the reader that these maps do not depend on choice of P, P op when considered as maps of Grothedieck groups.
We follow the construction given in [Shi11] 1 . We define Red b by LJ : GrothpM b pQ p qq Ñ GrothpJ b pQ p qq, is the map defined by Badulescu extending the inverse Jacquet-Langlands correspondence (see [Bad07, Prop 3.2]) and epJ b q is the Kottwitz sign as defined in [Kot83]. Since the maps induce the same maps on Grothendieck groups, we can rewrite the above as Red b : π Þ Ñ epbqpLJ˝Jac G P b pπqq. We now describe the main result of [Shi12]. The cocharacter µ of G is a map µ : G m,Qp Ñ ś τ PHompF,Qpq GL n,Qp such that the weights in each GL n factor are 0s or 1s. Thus we let p τ , q τ denote the number of 1 and 0 weights respectively in the factor corresponding to τ . The following formula is the main theorem in [Shi12, Thm 7.5].
Theorem 3.1.2 (Shin). We have the following equality for accessible representa- Loosely speaking, accessible representations in Shin's paper are character twists of the local components of global representations that can be found within the cohomology of Shimura varieties. Shin shows that all essentially square-integrable representations are accessible.
In this case, LL is the semisimplified local Langlands correspondence (known by the work of [HT01] for instance). The map r´µ is the algebraic representation of p G¸W E tµu G Ă L G defined by Kottwitz ([Kot84, Lem 2.1.2]). It is characterized by the fact that r´µ| p G is the irreducible representation of extreme weight´µ and if we take a Γ-invariant splitting of p G, then the subgroup W E tµu G of L G acts trivially on the highest weight vector of r´µ associated with this splitting.
Remark 3.1.3. The Tate twist appearing on the right-hand side of the above formula comes from the dimension formula for Shimura varieties and is equal to´xρ G , µy where ρ G is the half sum of the positive roots in G.
The above theorem is analogous to the sum formula for cocharacter pairs. The induction formula is related to the Harris-Viehmann conjecture. A proof of this conjecture is expected to appear in forthcoming work of Scholze. 1 We believe the construction given before lemma 6.2 of [Shi12] has a slight typo. There, Red b is defined by π Þ Ñ epJ b qpLJ˝Jac G P op b pπqq. However as maps of Grothendieck groups, Jac G . But this is not equal to J G P op b pπq b δ 1 2 P , which is the construction given in [Shi11] 3.2. Harris-Viehmann Conjecture. We now state the Harris-Viehmann conjecture following Rappoport and Viehmann in [RV14]. We return to the notation of section 2 so that in particular, G is a connected, quasisplit reductive group defined over Q p . Choose a dominant µ P X˚pT q with weights as in 3.1.1 (where we can consider µ as a cocharacter of G since T Ă G) and a b P BpG, µq . Associated to b, we have the standard Levi subgroup M b . Suppose we have a standard rational Levi subgroup M S so that M b Ă M S Ă G. We define b 1 , b S as we did before 2.5.5.
In [RV14,(6.2)], the authors associate a cohomological construction to the triple pG, b, µq which they denote H ‚ ppG, rbs, tµuqq. This construction agrees with Mant G,b,µ in the case above. We will denote this construction H ‚ pG, b, µq since we deal with dominant cocharacters instead of conjugacy classes. Then they have the following conjecture.

Conjecture 3.2.1 (Harris-Viehmann). We have the equality
The parabolic induction only modifies the GrothpGpQ p qq parts of these representations.
Remark 3.2.2. We need to explain several things in the above conjecture. First we explain why the right-hand side is a representation of W E tµu G , second we check that the conjecture satisfies a transitivity property, and third we give an example justifying the extra character twist appearing in our formulation. This twist is not present in the original formulation of the conjecture.
We first explain why the right-hand side is a representation of W E tµu G . We start with a general lemma.
Lemma 3.2.3. Suppose a group Λ acts on a finite set S. Suppose further that for each s P V , we attach a vector space V s and for each λ P Λ and s P S we have an isomorphism ips, λq : V s Ñ V λpsq . We suppose further that ips, 1q is the identity map and that ipλ 1 psq, λ 2 q˝ips, λ 1 q " ips, λ 2 λ 1 q. Then À sPS V s is naturally a representation of Λ.
Let ts 1 , ..., s k u Ă Λ be a set containing one representative from each Λ-orbit in where Ind refers to the induced representation ( not parabolic induction).
Proof. The proof is more or less clear from the definition of induced representation.
Moreover, we record the following transitivity property for later use.  . To prove the first part of the claim, we pick γ P W E tµu G and observe that since M S and P S are defined over Q p , we have γpM S q " M S and γpµ S q is dominant in M S . Thus pM S , γpµ S qq P C MS so we need only check that b S P BpM S , γpµ S qq and γpµ S q " G µ. The first check follows from the fact that θ T pµ S q " θ T pγpµ S qq, and µ S | Zp y MS q Γ " γpµ S q| Zp y MS q Γ . The second check follows because γ stabilizes tµu G .
To prove the second part of the claim, we note that if µ S " γpµ S q then γ stabilizes tµ S u MS . Conversely, if γ stabilizes tµ S u MS then since it maps dominant elements relative to M S to dominant elements, we must have γpµ S q " µ S .
We observe that we have now shown that W E tµu G acts on the collection of Rapoport-Zink data pM S , b S , µ S q for pM S , µ S q P I G,µ MS ,bS . By [RV14,prop 5.3.iv], these actions induce morphisms of the corresponding towers of rigid spaces and therefore the spaces H ‚ pM S , b S , µ S q. Thus by 3.2.3 we get an action of W E tµu G on the sum of vector spaces ÿ pMS ,µS qPI G,µ and therefore on ÿ We remark that the character twist by´dimM rig b,µ,U in the definition of H ‚ pM S , b S , µ S q is not an obstacle to defining the W E tµu G -action as the dimensions of the spaces associated to pM S , b S , µ s q and pM S , b S , γpµ S qq are the same (for γ P W E tµu G ). Also we observe that the twist by r1sr|¨| xρG,µS y´xρG,µy s is harmless as it is constant over orbits of W E tµ S u G . This concludes our discussion of the W E tµu G action.
We now check that the Harris-Viehmann conjecture is transitive. By this, we mean that if we have standard Levi subgroups M S1 and M S2 of G such that M b Ă M S2 Ă M S1 Ă G, then first applying the conjecture to pG, b, µq and the inclusion M S1 Ă G and then applying the conjecture to each resulting pM S1 , b S1 , µ S1 q for pM S1 , µ S1 q P I G,µ MS 1 ,bS 1 and the inclusion M S2 Ă M S1 should be the same as applying the conjecture to pG, b, µq and the inclusion M S2 Ă G.
We need to check that the character twists match, that and that the W E tµu G actions are the same.
This reduces to showing the equality xρ GzMS 1 , µ S1 y " xρ GzMS 1 , µ S2 y, where ρ GzMS 1 is the half-sum of the absolute roots of G that are not roots of M S1 . Since µ S2 and µ S1 are conjugate in M S1 , there exists a w P W abs MS 1 so that wpµ 1 q " µ 2 . Then the desired equality follows from the fact that the pairing x¨,¨y is W abs MS 1 -invariant and that W abs MS 1 stabilizes the set of positive absolute roots in G but not M S1 . To prove this second fact, note that M S1 normalizes the unipotent radical U S1 of P S1 and that the roots of LiepU S1 q are precisely the positive absolute roots of G that are not contained in M S1 .
The second check is precisely 2.5.6, and the third check follows from 2.5.6 and 3.2.4. Now we compute an example to illustrate the necessity of the extra Tate twist in our statement of 3.2.1.
Example 3.2.5. Let n 1 ă n 2 be coprime positive integers and let G " GL n1`n2 . Fix T the standard maximal torus of diagonal matrices and B the Borel subgoup of upper triangular matrices. Let µ be the minuscule cocharacter with weight vector p1 2 , 0 n1`n2´2 q and b P BpG, µq satisfying ν b " pp1{n 1 q n1 , p1{n 2 q n2 q. Let ρ 1 , ρ 2 be supercuspidal representations of GL n1 pQ p q, GL n2 pQ p q respectively. Define the standard Levi subgroup M b " GL n1ˆG L n2 , and consider the representation π " I G M b pρ 1 b ρ 2 q. We will be interested in computing Mant G,b,µ pRed b pπqq. The key point is that we can use Shin's formula (Theorem 3.1.2 of this paper) and known cases of the Harris-Viehmann conjecture due to Mantovan ([Man08]) to do this computation, even though the Harris-Viehmann conjecture is not known to be true in the case of M b since b is not of Hodge-Newton type.
We observe that there are only 3 elements b 1 P BpG, µq that satisfy After all, the fact that ρ 1 , ρ 2 are supercuspidal and the geometric lemma of Bernstein-Zelevinski ( §2.11 of [BZ77]) forces M b 1 to be one of G, GL n1ˆG L n2 , GL n2ˆG L n1 . In the case where M b 1 " G, we also get 0 since LJpπq " 0. Thus, if we write out Shin's formula for our π, the only elements of BpG, µq whose terms contribute to the left-hand side are b, b 1 , b 2 where b is as before and b 1 , b 2 are defined by Thus, we have M b1 " M b " GL n1ˆG L n2 and M b2 " GL n2ˆG L n1 . Note that b 1 , b 2 are both of Hodge-Newton type so that we can apply the results of Mantovan.

Now
LLpπq " LLpρ 1 q ' LLpρ 2 q. Thus, we need to restrict r´µ to x M b Ă p G (we have been ignoring the Galois part of L G in this example since G is a split group). Using the theory of weights, we get r´µ| x M " rr p´1 2 ,0 n 1´2q b r p0 n 2 q s ' rr p´1,0 n 1´1q b r p´1,0 n 2´1q s ' rr p0 n 1 q b r p´1 2 ,0 n 2´2q s, and so we see that the contributions for b 1 , b 2 which we computed above will cancel terms on the righthand side of Shin's formula leaving us with Mant G,b,µ pRed b pπqq " rπsrpr p´1,0 n 1´1 q br p´1,0 n 2´1q q˝pLLpρ 1 q`LLpρ 2 qqb|¨| 2´n1´n2 s.
However, if the Harris-Viehmann conjecture without the extra Tate twist were to hold for b, we would get " rπsrr p´1,0 n 1´1q b r p´1,0 n 2´1q˝p LLpρ 1 q`LLpρ 2 qq|¨| 1´n2 s. Thus, we see the Tate twists do not agree.
In general, the righthand side of Shin's formula has a twist of´xρ G , µy where ρ G is the half sum of the positive roots of G. If we have a term on the lefthand side of Shin's formula corresponding to some b P BpG, µq, then we would expect the Galois part of Mant M b ,b,µ b terms to come with a twist of´xρ M b , µ b y and that we would get an extra twist of´x , µ b y corresponding to the twist on r´µ b˝L Lpρq that we get by twisting ρ by δ 1 2 P b . We note that Thus, we see that the difference between these Tate twists is which is the twist in conjecture 3.2.1 Remark 3.2.6. We note that in the Hodge-Newton case studied by Mantovan, µ " µ b so that this extra twist vanishes, agreeing with Mantovan's results.
We now give an alternate version of the Harris-Viehmann conjecture that we will use in numerous arguments in this paper. Suppose that G, b, µ are as in theorem 3.1.2. The standard Levi subgroup M b has a natural product decomposition Then the local Shimura variety datum pM b , b 1 , µ b q decomposes into a collection pM 1 , b 1 1 , µ b,1 q, ..., pM k , b 1 k , µ b,k q. In §5.2.piiq of [RV14], the authors show that the local Shimura variety associated to pM b , b 1 , µ b q is the product of those associated to pM i , b 1 i , µ b,i q. Furthermore using the Kunneth formula (as in [Man08,15]), we get that Thus, we have the following alternate form of the Harris-Viehmann conjecture for the Rapoport-Zink spaces we consider.
Conjecture 3.2.7 (Alternate Form of Harris-Viehmann Conjecture). We use the notation of the previous paragraphs so that in particular, pG, b, µq comes from an unramified Rapoport-Zink space of EL-type as in 3.1.1. Then we have the following equality in GrothpGpQ p qˆW E tµu G q: 3.3. Proof of Theorem 1.0.3. The combination of the Harris-Viehmann conjecture and sum formula allows us to relate the cohomology of Rapoport-Zink spaces to the cocharacter pairs studied in section 2. To do so, we attach a map of Grothendieck groups to each cocharacter pair. Fix a cocharacter pair pG, µq P C G . Suppose pM S , µ S q P C G and satisfies µ S " G µ. We associate pM S , µ S q to a map of representations given by π Þ Ñ pInd G PS˝r µ S s˝Jac G PS qpπq b r1sr|¨| xρG,µS y´xρG,µy s, with rµ S s : GrothpM S pQ p qq Ñ GrothpM S pQ p qˆW E tµu M S q, given by π Þ Ñ rπsrr´µ S˝L Lpπq| WE tµ S u M S b |¨|´x ρM S ,µS y s.
Remark 3.3.1. We note that the map rM S , µ S s is only defined relative to a cocharacter pair pG, µq. Also, in the case where G " G 1ˆ. ..ˆG k where each G i is a general linear group or Weil restriction of a general linear group, LLpπ 1 b ... b π k q is defined to be LLpπ 1 q ' ... ' LLpπ k q for π 1 b ... b π k a smooth irreducible representation of GpQ p q.
Remark 3.3.2. We observe an interesting property of the maps rM S , µ S s. Fix pG, µq and consider pM S , µ S q such that µ S " G µ. Since the normalized Jacquet module and parabolic induction functors behave better with respect to the local Langlands correspondence, it makes sense to rewrite rM S , µ S s in terms of these maps. We get xρG,µS´µy s. Note that the twists by the modular character cancel in the admissible part but do not cancel in the Galois part. Thus, the total Tate twist of the Galois part is This twist does not depend on pM S , µ S q but rather only on pG, µq. Thus, as we will see in the computations of the next section, it is possible for large cancellations to occur in computations of Mant G,b,µ pρq for various ρ.
We now prove some lemmas relating to these maps before tackling the main theorem.
Lemma 3.3.3. Let M S1 , M S2 be standard Levi subgroups of G satisfying M S2 Ă M S1 . Consider the natural map i G MS 1 : C MS 1 Ñ C G , as defined in equation 6. Let pM S2 , µ S2 q P C MS 1 . Suppose further that we have fixed pairs pM S1 , µ S1 q P C MS 1 and pG, µq P C G so that µ S2 " MS 1 µ S1 and µ S2 " G µ.
Then for π P GrothpG Qp q, to denote the map associated to i G MS 1 ppM S2 , µ S2 qq in the manner above.
This is the same as equation 7.
Lemma 3.3.4. Suppose we are in the situation of 2.5.9 so that G " G 1ˆ. ..ˆG k is a connected reductive group with standard Levi subgroup M S " M S1ˆ. ..M S k . Fix cocharacter pairs pM S , µ S q, pG, µq P C G with µ S " G µ. The bijection C G -C G1ˆ. ..C G k takes pM S , µ S q to ppM S1 , µ S1 q, ..., pM S k , µ S k qq and pG, µq to ppG 1 , µ 1 q, ..., pG k , µ k qq and we have µ Si " Gi µ i . For each i, we have Then we have the following equality of homomorphisms of Grothendieck groups: b k i"1 rM Si , µ Si s " rM S , µ S s. Proof. We claim that b k i"1 rµ Si s " rµ S s as maps GrothpM S pQ p qq Ñ GrothpM S pQ p qˆW E tµ S u M S q.
Indeed, for π " π 1 b ... b π k a smooth irreducible representation of GpQ p q, we have For some finite subset C Ă C G , such that each pM S , µ S q P C satisfies µ S " G µ, we would like to make sense of a sum ÿ pMS ,µS qPC rM S , µ S s.

This makes sense as a map GrothpGpQ
However, for our purposes, we would like to understand when we can extend the image of this map to a representation in GrothpGpQ p qˆW E tµu G q.
Lemma 3.3.5. Fix a pair pG, µq P C G . Consider a finite subset C Ă C G such that if pM S , µ S q P C then µ S " G µ. Furthermore, suppose that for each γ P W E tµu G and element pM S , µ S q P C, we have pM S , γpµ S qq P C. Then ÿ pMS ,µS qPC rM S , µ S s, is a map GrothpGpQ p qq Ñ GrothpGpQ p qˆW E tµu G q in a natural way.
Proof. Our construction is analogous to that of 3.2.3. We fix ρ P GrothpGpQ p qq and give the structure of a GpQ p qˆW E tµu G -representation and then define the GpQ p qŴ E tµu G -structure on V C to be the direct sum of the V Ci . Suppose now that C contains a single W E tµu G orbit. In this case, we will show that à pMS ,µS qPC rM S , µ S spρq, can be given the structure of a GrothpGpQ p qˆW E tµu G q representation equal to where r is the induced representation (not parabolic induction) given by for a fixed choice of pM S , µ S q P C. The isomorphism class of r will not depend on this choice. We study the representation r. Fix representatives γ 1 , ..., γ k P W E tµu G {W E tµ S u M S so that γ 1 " 1. Then r is defined to be the sum of k copies of r´µ S indexed by the γ i and acted on by W E tµu G in the standard way. We check that the ith copy of r´µ S is a representation of y M S¸WE tγ i pµ S qu M S and isomorphic to r´γ ipµS q . Let V i be the underlying vector space of the ith copy of r´µ S . Then V i is naturally a representation of y Now suppose v P V 1 is a weight vector of p T Ă y M S of weight µ 1 . Then we show that p1, γ i qv P V i has weight γ i pµ 1 q. After all, for t P p T , we have " p1, γ i qµ 1 pγ´1 i ptqqv " γ i pµ 1 qptqp1, γ i qv.
In particular, we have shown that V i is irreducible of extreme weight´γ i pµ S q as an y M S -representation (since r´µ S is irreducible of extreme weight´µ S as an y M Srepresentation). It is a simple check similar to the above that W E tγ i pµ S qu M S acts trivially on the highest weight space of V i . This proves that V i is isomorphic to r´γ ipµS q .
In particular, this shows that we can give à To conclude the proof, we just need to check that the |¨| twists on each rM S , γ i pµ S qsterm are the same. This follows because ρ G and ρ MS are both invariant by W E tµu G .
We write rM G,b,µ s for the map of representations given by replacing each term in the formal sum for M G,b,µ by the corresponding map of representations. We would like to check the following: Lemma 3.3.6. The sum M G,b,µ gives a map rM G,b,µ s : GrothpGpQ p qq Ñ GrothpGpQ p qˆW E tµu G q.
Proof. By 3.3.5, it suffices to show that M G,b,µ is invariant under the natural action of W E tµu G on ZxC G y. Pick γ P W E tµu G . Since the action of γ on a cocharacter pair fixes the standard Levi subgroup in the first factor, signs will not be an issue and we will be done if we can check that R G,b,µ is γ-invariant. But if pM b , µ b q P T G,b,µ then it is a consequence of the definition of T that so is pM b , γpµ b qq. Furthermore if pM S , µ S q ď pM b , µ b q then pM S , γpµ S qq ď pM b , γpµ b qq by definition of the partial order relation (remarking that θ MS pµ S q " θ MS pγpµ S qq). This shows that R G,b,µ is γ-invariant as desired.
If we combine the previous lemma with 2.5.9, and 3.3.4 we get b k i"1 rM Gi,bi,µi s " rM G,b,µ s.
We now prove the key result of this section which provides the connection between Mant and cocharacter pairs. Theorem 3.3.7. Assume that theorem 3.1.2 holds for all admissible representations of GrothpGpQ p qq and that the Harris-Viehmann conjecture is true for the general linear groups we consider. Then we have the following equality of morphisms GrothpGpQ p qq Ñ GrothpGpQ p qˆW E tµu G q: See the remark after the proof for a discussion of which of the conditions we can remove in the essentially square integrable case.
Proof. We prove this result by induction on the rank of X˚pT q.
If the rank of X˚pT q is 1, then BpG, µq is a singleton and so the result follows from theorem 3.1.2.
Suppose the result holds for all non-basic b P BpG, µq with RkpX˚pT qq ď r. Then by theorem 3.1.2 and 2.5.4, the result holds for all b P BpG, µq with RkpX˚pT qq ď r.
Finally, suppose the result holds for all b P BpG, µq with RkpX˚pT qq ď r. Then suppose X˚pT q has rank r`1 and choose b P BpG, µq such that b is not basic. By the Harris-Viehmann formula, By inductive assumption we get and now by equation 9 and so by 2.5.8 and 3.3.3 " rM G,b,µ s.
We must check that the W E tµu G structure coming from 3.2 is compatible with that of 3.3.5. Pick ρ P GrothpGpQ p qq. By inductive assumption and 3.3.3, for each are the same. Thus by 3.2.3, the W E tµu G -structure on Mant G,b,µ pRed b pρqq is a direct sum over the W E tµu G -orbits of I G,µ M b ,b 1 of induced representations of the form This W E tµu G -structure matches the one on rM G,b,µ s (coming from 3.3.5) by the transitivity of the induced representation construction (see 3.2.4 for instance).
Remark 3.3.8. To conclude the proof of theorem 1.0.3, we need to show that if we restrict ourselves to the essentially square integrable representations Irr 2 pGpQ p qq Ă GrothpGpQ p qq, then we can remove the first assumption. In particular, these representations are accessible, so we have 3.1.2 unconditionally. In the above proof we need only observe that the Jacquet module J G M pρq is a sum of essentially square integrable representations for ρ P Irr 2 pGpQ p qq. Thus, to get the result for Irr 2 pGpQ p qq by induction, our inductive assumption need only hold for all Irr 2 pG 1 pQ p qq for rkG 1 ă rkG. This shows that under the condition that the Harris-Viehmann conjecture is true in the cases we consider, the theorem is true for essentially square integrable representations without any other assumptions.

Harris's Generalization of the Kottwitz Conjecture (proof of Theorem 1.5)
In this section, we discuss an explicit computation using the above results. In particular, we prove that Shin's formula for all admissible representations combined with the Harris-Viehmann conjecture proves Harris's conjecture for the general linear groups considered in section 3. This conjecture is distinct from the Harris-Viehmann conjecture and is [Har01,Conj 5.4].
We begin by discussing the Kottwitz conjecture, which appears as [Shi12,Cor 7.7] in the cases we consider, and more generally as [RV14,Conj 7.3].
Fix G as in section 3 of this paper and a cocharacter pair pG, µq such that µ is minuscule. Let b P BpG, µq be the unique basic element. Now, consider ρ a representation of J b pQ p q such that JLpρq is a supercuspidal representation of GpQ p q. Then Mant G,b,µ pRed b pJLpρqqq " Mant G,b,µ pρq, but by theorem 3.3.7, the lefthand side equals rM G,b,µ spJLpρqq.
Now we see that since JLpρq is supercuspidal, each term of the form rM S , µ S spJLpρqq is 0 when M S is a proper Levi subgroup of G. Thus, This result is the Kottwitz conjecture for G. Alternatively, if b P BpG, µq is not basic, then no cocharacter pairs with G as the Levi subgroup will appear in M G,b,µ and so Mant G,b,µ pρq " 0. Of course, these results are already known by [Shi12], but we review them as motivation for Harris's conjecture.
We begin with the following useful definition.
Definition 4.0.1. Fix pG, µq P C G and b P BpG, µq. Let M S be a standard Levi subgroup such that M S Ă M b . We define the subset Rel G,µ MS ,b Ă C G as the set tpM S , µ S q P C G : The notation µ S " M b µ b is defined to mean that µ S and µ b are conjugate in M b . Note that we do not require pM S , µ S q ď pG, µq or pM S , µ S q ď pM b , µ b q.
We record the following useful properties of Rel G,µ MS ,b . Lemma 4.0.2. We use the same notation as in the previous definition. Then Proof. If pM S , µ S q P Rel G,µ MS ,b , then there is an pM b , µ b q P T G,b,µ such that θ M b pµ b q " θ M pµ S q and µ S " M b µ b . Then by 2.5.7, there is a unique pM b , µ 1 q P I G,µ M b ,b 1 such that pM b , µ b q P T M b ,b 1 ,µ 1 and so pM S , µ S q P Rel M b ,µ b MS ,b 1 . The reverse inclusion is analogous. Recall that by assumption, G is quasi-split over Q p and A is a split torus of G of maximal rank. Pick g P N G pAqpQ p q so that g projects to w P W rel " N G pAqpQ p q{Z G pAqpQ p q. Then the equation wpxq " x implies that g P Z G pxqpQ p q. This centralizer is a Levi subgroup, and since x P AQ ,MS , we have Z G pxq " M S . In particular, g P N MS pAqpQ p q and so w P W rel MS . We remark that strictly speaking, x is not a cocharacter, but that Z G pxq still makes sense as there is an induced action of G on X˚pAq Q .
We can now prove the following key proposition.
Proposition 4.0.10. Fix pG, µq P C G and suppose pM S , µ S q P C G satisfies µ S " G µ. Then there exists a unique b P BpG, µq and a unique w P W MS ,M b so that pwpM S q, wpµ S qq P Rel G,µ wpMS q,b .
Proof. We note that W MS ,M b Ă W rel acts on X˚pT q by B.0.2. We prove uniqueness first. By assumption, w must map M S to a standard Levi subgroup wpM S q. This induces an equality wW rel MS w´1 " W rel wpMS q . In particular, it follows that wpθ MS pµ S qq " θ wpMS q pwpµ S qq.
Since pwpM S q, wpµ S qq P Rel G,µ wpMSq,b , it follows that θ wpMSq pwpµ S qq is dominant in the relative roots system. In particular, θ wpMSq pwpµ S qq must be equal to the unique element x in the W rel orbit of θ MS pµ S q which is dominant in A Q . Now x P AM S 1 ,Q for a unique M S 1 . Since any pM b , µ b q P T G,b,µ is definitionally strictly decreasing, it follows that even though we can't yet conclude the uniqueness of b, we have shown that any other b 1 must satisfy M b1 " M b " M S 1 . Now, suppose we had w, w 1 P W MS ,M b such that Then in particular, w 1 w´1 stabilizes x and so by 4.0.9, w 1 w´1 P W rel M b . So w and w 1 are in the same right coset W rel M b w. However, W MS ,M b Ă pW M b q´1. By lemma 4.0.8, pW M b q´1 contains a unique representative of each right coset of pW M b q´1 and so there is a unique w P pW M b q´1 satisfying wpθ MS pµ S qq " x. In particular, this implies that w " w 1 . Thus, we have shown that w is unique, if it exists. There is exactly one µ 1 P X˚pT q such that µ 1 " M b wpµq and µ 1 is dominant in M b . Then pM b , µ 1 q P T G,b,µ for at most one b P BpG, µq. This shows uniqueness.
To prove existence, we again define x to be the unique dominant element in the W rel -orbit of θ MS pµ S q. Define M S 1 " Z G pxq and take the unique w P pW M S 1 q´1 such that wpθ MS pµ MS qq " x. We would like to show that w P W MS ,M S 1 .
By definition, Suppose it is not the case that wpM S X Bq Ă B. In particular, w maps a positive root r of M S to a root wprq of M S 1 which is not positive. In particular,´wprq is positive and so w´1p´wprqq "´r is positive (since w P pW M S 1 q´1). But this is clearly a contradiction. Thus, in fact w P W MS ,M S 1 .
By 4.0.7, wpM S q X M S 1 " wpM S q is a standard Levi. It remains to show that pwpM S q, wpµ S qq is a cocharacter pair and an element of Rel G,µ wpMS q,b . Now if r is a positive root in the absolute root system of wpM S q, then xr, wpµ S qy " xw´1prq, µ S y ě 0 (since pM S , µ S q is a cocharacter pair and w´1prq is a positive root of M S ). Thus, pwpM S q, wpµ S qq is a cocharacter pair. By construction, x " θ wpMSq pwpµ S qq " θ M S 1 pwpµ S qq. Suppose µ 1 P X˚pT q is the unique cocharacter conjugate to wpµ S q in M S 1 and dominant in M S 1 . Then by 2.2.5, pM S 1 , µ 1 q is strictly decreasing and therefore pM S 1 , µ 1 q P T G,b,µ for some b and so pwpM S q, wpµ S qq P Rel G,µ wpMS q,b .
Corollary 4.0.11. Fix a cocharacter pair pG, µq P C G and a standard Levi subgroup Then the previous lemma gives a bijection Proof. By the construction in the previous proposition, it is clear that given an pM S , µ S q P C G we get an element of the right-hand side of the above equation.
Conversely, an element pwpM S q, µ 1 q of the right-hand side comes with a fixed w P W b and so we can recover pM S , w´1pµ 1 qq on the left-hand side.
We are now ready to finish the proof of conjecture 4.0.4. By inductive assumption we assume we've shown 4.0.4 for G with maximal torus of rank less than n. Then proposition 4.0.5 implies that 4.0.4 holds for G with maximal torus of rank n in the case where b is not basic. It remains to prove the basic case, for which it suffices to show that theorem 3.1.2 is compatible with 4.0.4. We have ÿ bPBpG,µq By the geometric lemma of [BZ77], we have By the assumption that ρ is supercuspidal we must have M 1 S " M S and M 1 b " wpM S q. In this case, we have from the geometric lemma that wpM S q is a standard Levi subgroup. Thus we get that the previous expression is equal to ÿ bPBpG,µq This can now be computed by cocharacter pairs using the results of section 3. If I G MS pρq is assumed to be irreducible, then for each cocharacter pair pM S 1 , µ S 1 q of G, we have where W ρ is the subset of w P W MS ,M S 1 such that wpM S q Ă M S 1 . Then the above equals Thus we see that applying various rM S 1 , µ S 1 s to I G MS pρq in the irreducible case will always yield the same term of GrothpGpQ p qq (namely rI G MS pρqs) and so when evaluating Mant G,b,µ pRed b pI G MS pρqq as a sum of cocharacter pairs, the different Galois terms must cancel to give conjecture 4.0.4. Thus, if we can show that in the reducible case, the GrothpGpQ p qq part of each rM S 1 , µ S 1 spI G MS pρqq is fixed and the Galois part is identical to the irreducible case, then conjecture 4.0.4 must hold for this case as well.
The first part of our previous computation did not depend on the irreducibility of I G MS pρq so we still have as desired.

Appendix A. Examples
In this section, we give an example to show that even in the unramified EL-type case, we do not get an expression as simple as Harris's conjecture for Mant G,b,µ pρq for general ρ. We generally use the same notation as in the computation in example 3.2.5.
Let G " GL 4 , suppose µ has weights p1 2 , 0 2 q, and take b basic. Let T be the diagonal maximal torus and B be the Borel subgroup of upper triangular matrices. Then the set of cocharacter pairs less than or equal to pG, µq is as follows.
pGL 4 , p1 2 , 0 2 qq, , pGL 3ˆG L 1 , p1 2 , 0qp0qq pGL 2 2 , p1 2 qp0 2 qq pGL 1ˆG L 3 , p1qp1, 0 2 qq Let ρ P GrothpGL 1 pQ p qq and consider π the unique essentially square integrable quotient of I G We want to compute Mant G,b,µ pRed b pπqq. We introduce some notation which will allow us to describe the answer to this question. The results of §2 of [Zel80] show that I G so that the irreducible subquotients correspond to a partition of Ω. We use the following shorthand: we define the notation p0123q to refer to the representation ρp0q b ρp1q b ρp2q b ρp3q. Following Zelevinsky, our 8 irreducible subquotients naturally correspond to vertices of a 3-dimensional cube, and so we denote them by binary strings of length 3. Then if we denote the subset of Ω corresponding to some subquotient π 1 by Ωpπ 1 q,we have Ωpr111sq " tp3210qu Ωpr011sq " tp2310q, p2130q, p2103qu Ωpr101sq " tp3120q, p1320q, p1302q, p3102q, p1032qu Ωpr110sq " tp3201q, p3021q, p0321qu Ωpr001sq " tp1203q, p1023q, p1230qu Ωpr010sq " tp2013q, p2031q, p0213q, p0231q, p2301qu Ωpr100sq " tp3012q, p0312q, p0132qu Ωpr000sq " tp0123qu In particular, our representation π corresponds to r111s under the above notation. A tedious computation using theorem 3.3.7 yields the following We finish by remarking that the set of cocharacter pairs less than or equal to pG, µq has some special properties in the above case that make the general case more complicated.
For instance, each T G,b,µ has at most a single element. However, if G has a nontrivial action by Γ, this need not be the case.
Further, in the above example, each cocharacter pair pM S , µ S q had the property that µ S was dominant as a cocharacter of G relative to B. In general this need not be the case. In fact, pGL 5 1 , p1qp1qp0qp1qp0qq ď pGL 5 , p1 3 , 0 2 qq.

Appendix B. Relative Root Systems and Weyl Chambers
In this section we prove a fact about root systems that is needed in the text (for instance in the proof of 2.4.3). We assume that G is a quasisplit group over a field k of characteristic 0 and pick a separable closure k sep . We fix a split k-torus A of maximal rank in G and choose a maximal torus T and Borel subgroup B both defined over k and such that A Ă T Ă B. Associated to this data, we have an absolute root datum pX˚pT q, Φ˚pG, T q, X˚pT q, Φ˚pG, T qq, and a relative root datum pX˚pAq, Φ˚pG, Aq, X˚pAq, Φ˚pG, Aqq.
Our choice of B also gives sets ∆ of absolute simple roots and k ∆ of relative simple roots. Note that we also have a natural restriction map res : X˚pT q Ñ X˚pAq, and that by definition an absolute root in Φ˚pG, T q restricts to an element of Φ˚pG, Aq Y t0u.
We record two standard consequences of our assumption that G is quasisplit.
The key point being that no absolute simple root restricts to the trivial character.
We have the following easy consequence on the structure of the Weyl group of the relative root system. Recall that the absolute Weyl group W equals N G pT qpk sep q{Z G pT qpk sep q, and the relative Weyl group W rel is N G pAqpkq{Z G pAqpkq.
Proof. It suffices to show that Z G pAq " Z G pT q and that N G pAqpkq " N G pT qpkq. For the first equality, we note that by the quasisplit assumption, Z G pAq " T " Z G pT q. For the second equality, we note that any g P N G pAqpkq must also normalize the centralizer of A which is T . Conversely, if g P N G pT qpkq then g normalizes the unique maximal k-split sub-torus of T which is A.
Define the absolute Weyl chamber CQ Ă X˚pT q Q by tx P X˚pT q Q : xα, xy ě 0, α P ∆u and define the relative Weyl chamber k CQ Ă X˚pAq Q analogously. The key result of this section is that respCQq " k CQ.
Despite its simple statement, the author has been unable to locate a convenient reference of this fact. For x P X˚pT q Q and α P ∆, we need to relate xq α, xy and x respαq, respxqy. If we let σ α P W be the transposition corresponding to the root α, then we have x´σ α pxq " xq α, xyα.
and analogously for respαq. Thus it will suffice to relate σ α and σ respαq . Note that since B is defined over k, we have γp∆q " ∆ for every γ P Γ. Moreover, for each α P ∆, we have respγpαqq " respαq. After all, Γ acts trivially on X˚pAq Q and the restriction map is Γ-equivariant. Now fix α P ∆ and let W α be the subgroup of W generated by the elements σ γpαq for each γ P Γ. We claim that if we can find a nontrivial Γ-invariant element of W α , then it must equal σ respαq . To prove this, we first recall the construction of σ α and σ respαq (see [Bor91,pg 230]) for instance). Given a root α P Φ˚pG, T q we can define a group G α " Z G pT α q where T α " kerpαq 0 Ă T . Then N Gα pT qpk sep q{Z Gα pT qpk sep q embeds into W and has a unique nontrivial element which is σ α . Analogously, we define A respαq and G respαq " Z G pA respαq q. Then N G respαq pAqpkq{Z G respαq pAqpkq embeds into W rel and has a unique nontrivial element that is identified with σ respαq . Now, by B.0.2 we have N G respαq pAqpkq{Z G respαq pAqpkq " N G respαq pT qpkq{Z G respαq pT qpkq.
Thus to complete the proof of the claim, we need to show that N Gα pT qpk sep q{Z Gα pT qpk sep q ãÑ N G respαq pT qpk sep q{Z G respαq pT qpk sep q.
After all, the unique nontrivial Γ-invariant element of the group on the right is σ respαq and the group on the left contains σ α . Since we get the same equation if we replace α everywhere with γpαq, this will imply that W α Ă N G respαq pT qpk sep q{Z Gres pT qpk sep q. Now, equation 12 follows from the fact that Z Gα pT q " Z G respαq pT q " T, and N Gα pT q Ă N G respαq pT q.
We are now interested in finding a nontrivial Γ-invariant element of the group W α defined above. In fact, W α will be a finite Coxeter group and the element we seek is the unique element of longest length. We need to compute this element explicitly, which we now do. We treat two cases. Suppose first that the elements of the Γ-orbit of σ α commute pairwise. Then clearly the product ś γPΓ{stabpσαq σ γpαq is Γ-invariant. In the second case, suppose that the Γ-orbit of σ α has precisely two elements which we denote X and Y . Then we have pXY q k " 1 for some k ě 2 which we assume to be minimal. If k is even, then pXY q k{2 is invariant and nontrivial and if k is odd, then Y pXY q pk´1q{2 is invariant and nontrivial.
We now prove that any Γ action on the simple roots ∆ of G is a combination of these cases. The action of Γ on ∆ induces an action on the associated (not necessarily connected) Dynkin diagram D. Each γ P Γ maps connected components of D to connected components and so there is an induced action of Γ on the set of connected components π 0 pDq. Now fix an α P ∆ and consider the Γ-orbit Γα of α. Suppose D i is a connected component of D such that D i X Γα ‰ H. Then via the classification of connected Dynkin diagrams, we see that Γα X D i contains either a single node, 2 non-adjacent nodes, 2 adjacent nodes, or 3 nodes where no two are adjacent. In particular, these are all covered by the cases we considered above, so we can find an element w i of W α that is invariant by the action of stabpD i q Ă Γ. Then Γα consists of finitely many disjoint copies of one of the above possibilities and so we see that ś i w i is Γ-invariant and an element of W α and therefore equal to σ respαq . Equipped with this description, we now give a proof of the main result of this section.
Proposition B.0.3. We continue to observe the assumptions made above. In particular, G is a quasisplit group over k. Then the map res : X˚pT q ։ X˚pAq induces an equality respCQq " k CQ.
Proof. We first show that respCQq Ă k CQ. Pick x P CQ and α P ∆. Then we need to show that x respαq, respxqy ě 0 or equivalently, that respxq´σ respαq prespxqq is a non-negative multiple of respαq. Note that res is W Γ -equivariant (where W Γ acts as W res on X˚pAq). Thus, it suffices to show that respx´σ respαq pxqq is a non-negative multiple of respαq. Thus, we need to compute x´σ respαq pxq. We do so using our description of σ respαq . We first consider the case where the Γ-orbit of σ α consists of pairwise commuting elements. Equivalently, the elements of Γα are pairwise orthogonal. Then σ respαq " σ αn˝. ..˝σ α1 for tα 1 , ..., α n u " Γα. Since x is dominant in the absolute root system, we have x´σ αi pxq " a i α i for some a i ě 0. Then since α i is orthogonal to α j for i ‰ j, we have σ αi pα j q " α j . Thus, x´σ respαq pxq " Thus in this case, respx´σ respαq pxqq " pa 1`. ..`a n qrespαq and a 1`. ..`a n ě 0 as desired.
Finally, we must consider the case where Γα equals tα 1 , β 1 , ..., α n , β n u such that α i and β i are connected by a single edge in D but for i ‰ j, neither α i nor β i are connected to either α j or β j . We compute x´pσ βi˝σαi˝σβi qpxq as in the previous paragraph. Then if we let w i " σ βi˝σαi˝σβi , we have σ respαq " w 1˝. .˝w n . Now we can compute x´σ respαq pxq as in the commuting case, substituting w i for σ αi . We see in this case that respx´σ respαq pxqq " 2pa 1`b1`. ..`a n`bn qrespαq.
This concludes the proof that respCQq Ă k CQ.
It remains to show that we actually have equality. We claim it suffices to show that the fundamental weight δ respαq is an element of respCQq. Recall that δ respαq is the element in the Q-span of the relative roots defined so that the pairing with respαq is 1 and the pairing is 0 with all the other relative simple coroots. To show the claim proves our result, we note there is a natural isomorphism X˚pAq Q -X˚pA 0 q QˆX˚p A 1 q Q where A 0 is the maximal k-split central torus and A 1 is the identity component of the intersection of A with the derived subgroup of G. Then k CQ corresponds under this identification to the product of X˚pA 0 q Q with the projection of k CQ to X˚pA 1 q. Then we have a natural map X˚pZpGq 0 q Q ։ X˚pA 0 q Q where ZpGq 0 is the identity component of the center of G and X˚pZpGq 0 q Q Ă CQ. Thus it suffices to show that respCQq surjects onto the projection of k CQ to X˚pA 1 q. This latter space is identified with the set of non-negative linear combinations of the fundamental relative weights, thus proving the claim.
To prove that δ respαq is an element of respCQq, we make use of an equivalent description of δ respαq . It is the unique element in the Q-span of the relative roots so that σ respβq pδ respαq q " δ respαq for respαq and respβq distinct simple roots and σ respβq pδ respαq q " δ respαq´r espβq when respαq " respβq.
In the case where the elements of Γα are mutually orthogonal, we have by the above characterization of fundamental weights that the absolute fundamental weight δ α restricts to δ respαq . In the case where Γα has two elements that are connected in D, then δ α restricts to 2δ respαq . In the final case, δ α restricts to 2δ respαq . Thus, in all cases, we can find an element of X˚pT q Q that restricts to δ respαq . This completes the proof.
We record an important corollary of this proposition.
Corollary B.0.4. Suppose µ, µ 1 P X˚pT q Q and µ ľ µ 1 . Let µ Γ be the average of µ over its Γ orbit. Then µ Γ ľ µ 1Γ in X˚pAq Q . We caution that the first inequality means that µ´µ 1 is a non-negative combination of absolute simple coroots, while the second means that µ Γ´µ1Γ is a non-negative combination of relative simple coroots.
Proof. Recall that the action of Γ stabilizes q ∆. Thus for each γ P Γ, we have γpµq ľ γpµ 1 q and so also µ Γ ľ µ 1Γ in the absolute root system. Thus, we are reduced to showing that if x P X˚pT q Γ Q is a non-negative combination of simple absolute coroots, then it is also a non-negative combination of simple relative coroots (under the identification X˚pAq Q " X˚pT q Γ Q ).
Equivalently, we need to show that if x has non-negative pairing with every element of CQ, then x has non-negative pairing with every element of k CQ. This is indeed equivalent because x has non-negative pairing with each element of CQ if and only if it has non-negative pairing with each fundamental weight δ α and this is the case if and only if x is a non-negative combination of simple roots.
Finally, this equivalent statement is an immediate consequence of the proposition.