The Kottwitz conjecture for unitary PEL type Rapoport–Zink spaces
Abstract.
In this paper we study the cohomology of PEL-type Rapoport–Zink spaces associated to unramified unitary similitude groups over in an odd number of variables. We extend the results of Kaletha–Minguez–Shin–White to construct a local Langlands correspondence for these groups and prove an averaging formula relating the cohomology of Rapoport–Zink spaces to this correspondence. We use this formula to prove the Kottwitz conjecture for the groups we consider.
1. Introduction
Shimura varieties play an important role in the global Langlands program, which predicts a link between automorphic representations of linear algebraic groups and Galois representations. Rapoport and Zink ([RZ96]) introduced -adic analogues of Shimura varieties defined as moduli spaces of -divisible groups with additional structures. The -adic cohomology of these spaces should provide local incarnations of the Langlands correspondences and this is the subject of the Kottwitz conjecture ([RV14, Conjecture 7.3]). The goal of this paper is to prove the Kottwitz conjecture in the case of PEL type Rapoport–Zink spaces associated to unramified unitary similitude groups over in an odd number of variables. Prior to our work, the conjecture was proven for Lubin-Tate spaces by [Boy99], [HT01] and [Boy09]. By duality [Fal02], [FGL08], [SW20], the conjecture is also known in the Drinfeld case. The case of basic unramified EL type Rapoport–Zink spaces was proven by [Shi12a], [Far04] and the case of basic unramified PEL of unitary type of signature by [Ngu19]. Kaletha and Weinstein ([KHW17]) have proven, for all local Shimura varieties, a weakened form of the Kottwitz conjecture where, in particular, they do not consider the -action.
We now describe our results in more detail. One considers triples such that is a connected reductive group over and is a minuscule cocharacter of and is an element of the Kottwitz set . Then Rapoport–Zink attach to triples of PEL-type a tower of rigid spaces indexed by compact open subgroups .
Attached to the group and the element is a connected reductive group that is an inner form of a Levi subgroup of . The element is said to be basic when is in fact an inner form of . The tower carries an action of where is the field of definition of the conjugacy class of . For each one can take the compactly supported -adic cohomology of and hence consider the cohomology space
Then the Kottwitz conjecture describes the homomorphism of Grothendieck groups given by
in the case when is basic and is an irreducible admissible representation of with supercuspidal -parameter. This means that under the local Langlands correspondence, the -parameter is trivial when restricted to the factor and does not factor through a proper Levi subgroup of .
The Kottwitz conjecture states that
Conjecture 1.1.
For irreducible admissible representations of with supercuspidal -parameter, we have the following equality in :
where is the -packet of irreducible admissible representations of attached to .
We have not defined all the notation appearing in this conjecture, but this is described in detail in §5.
The main goal of this paper is to prove Conjecture 1.1 when is an unramified unitary similitude group over in an odd number of variables and the datum is basic and of PEL-type. Of course, to make sense of the Kottwitz conjecture for , one needs to establish the local Langlands correspondence for this group and show it satisfies an expected list of desiderata. In particular, one needs to check that the -packet has the expected structure determined by finite group related to the centralizer group of in and satisfies the endoscopic character identities.
Prior to this work, such a local Langlands correspondence was known for unitary groups by the works [Mok15, Theorem 2.5.1, Theorem 3.2.1] and [Kal+14, Theorem 1.6.1]. These authors work with the arithmetic normalization of the local Langlands correspondence whereby the Artin map is normalized so that uniformizers correspond to arithmetic Frobenius morphisms. However, it is more convenient for us to work with the opposite normalization. In Theorem 2.5 we use Kaletha’s results in [Kal13] on the compatibility of local Langlands correspondence and the contragredient to define a local Langlands correspondence for unitary groups under the geometric normalization whereby the Artin map takes uniformizers to geometric Frobenius morphisms.
We next construct a local Langlands correspondence for our groups by lifting the result for unitary groups to the group and then descending it to . We can carry out such an analysis because the map is a surjection on points for odd unitary groups. This property fails in the even case and is in fact the main reason we consider odd unitary similitude groups. We get
Theorem 1.2 (Theorem 2.5, Theorem 2.9, §3.2).
The local Langlands correspondence for odd unitary similitude groups is known and satisfies the properties of [Kal+14, Theorem 1.6.1], in particular, the endoscopic character identities.
With the local Langlands correspondence in hand, we can describe our proof of Conjecture 1.1 for the groups we consider. Our method of proof is similar to that of [Shi12a] and crucially uses the endoscopic averaging formulas of [BM21]. We briefly describe these formulas. Suppose that is an elliptic endoscopic datum for . Then there exists a complicated map
whose precise definition is given in §5.2. We remark that denotes the subgroup of with stable virtual character. Associated to each tempered -parameter of , we have a stable character denoted by . Suppose that is an -parameter of with parameter of such that . Then the endoscopic averaging formula is the following identity in :
| (1.1) |
where the first sum on the right-hand side is over irreducible factors of the representation and is the -isotypic part of . The averaging formula is derived in [BM21] under a substantial list of assumptions. In this paper, we verify these assumptions for supercuspidal parameters and hence prove:
Theorem 1.3.
For supercuspidal parameters of , the endoscopic averaging formulas hold.
For the sake of completeness, we briefly recall the strategy of the proof of this result as well as explain the important assumptions. The proof is via global methods. Thus we consider a global unitary similitude group defined over and a Shimura variety attached to which “globalizes” our Rapoport–Zink space. In particular, we have . We deduce the averaging formula by combining the Mantovan formula ([Man05, Theorem 22], [LS18, Theorem 6.26])
| (1.2) |
and the trace formulas for Shimura and Igusa varieties ([Kot90, Theorem 7.2], [Shi09, Theorem 13.1], [Shi10, Theorem 7.2]). We denote respectively by and the alternating sums of the compactly supported cohomology of Shimura and Igusa varieties evaluated at the -adic sheaf associated to some irreducible algebraic representation of .
To carry out this approach, we need to define global -parameters of without referring to the conjectural global Langlands group. We do so by adapting Arthur’s approach (also used in [Mok15] and [Kal+14]) where global parameters correspond to self-dual formal sums of cuspidal automorphic representations of . For us, a parameter of consists of a pair such that is a global parameter of in the sense of [Mok15] and is an automorphic character of . We attach global -packets to these parameters in the generic case and prove they satisfy the global multiplicity formula (Proposition 2.23).
One important step in the proof of the averaging formula is the process of stabilisation and destabilisation of the trace formula for the cohomology of Shimura and Igusa varieties following [Kot90] and [Shi10]. The goal is to relate both sides of the equality (1.2) to the global multiplicity formula. In order to achieve this, we need to prove a technical hypothesis concerning stable orbital integrals. More precisely, let be an endoscopic group of and a test function satisfying some local “cuspidality” conditions. We want to show that where is a sum of stable orbital integrals of with respect to and is, loosely speaking, the traces of all automorphic representations of evaluated against . This hypothesis is proven in Section §4.2.
Once we have done the destabilisation step, we can put everything into Equation (1.2) and derive the averaging formula. However, at this point the equality (1.2) is still quite complicated and we need to solve a lifting problem in order to extract the desired information. More precisely, for our choice of connected reductive group over such that and a cuspidal -parameter of , we need to construct global -parameters lifting and satisfying a number of conditions. For instance, we need to precisely control the centralizer group of in . These lifting problems are studied in [Art13a], [Kal+14] and we adapt their arguments to the unitary similitude case (Section §4.3).
With the endoscopic averaging formula in hand, we prove the Kottwitz conjecture in §6. To do so, we observe that whenever is non-basic and is supercuspidal. Hence, in this case, the only term on the left-hand side of the endoscopic averaging formula is the one for basic. We then combine the formulas for each elliptic to deduce the conjecture.
1.1. Acknowledgements
We would like to thank Tasho Kaletha for helpful discussions related to the arithmetic and geometric normalizations of the local Langlands correspondence. We are also grateful to Pascal Boyer, Laurent Fargues, and Sug Woo Shin for many helpful conversations regarding this paper. The first author was partially supported by NSF RTG grants DMS-1646385 and DMS-1840234. The second author was supported by ERC Consolidator Grant 770936: NewtonStrat.
2. Automorphic representations
2.1. The groups
Let be a field of characteristic , a quadratic extension of and fix an algebraic closure . Let be the anti-diagonal matrix defined by such that . We now define quasi-split groups and over as follows. We define and . Then we give an action of whereby acts by
We get an action of on by restriction.
We also need to define slightly more general groups defined by
In this paper, we only need to consider the case where is one of or . We now fix for once and for all a prime and a quadratic imaginary extension that is inert at . At each place of we get a rank two etale algebra over . Since we will not change , we can unambiguously use the notations and for the global groups we have defined and and for the local groups (for that do not split over ).
The global groups we consider in this paper will be inner forms of coming from hermitian forms. Namely, let be an -dimensional -vector space equipped with a hermitian form . Let , (resp. ) be the algebraic groups defined over by
for any -algebra .
In this paper we will assume that is an odd number and that the localization at every finite place is quasi-split. Such groups exist and the quasi-split condition we impose at the finite places does not constrain the isomorphism class of the group at the archimedean place. Indeed we can define
where is the identity matrix. Then for an -dimensional -vector space,
for odd and the nontrivial element, gives a unitary similitude group of type at the archimedean place that is quasi-split at the finite places.
We need to verify that the groups and arise as extended pure inner twists of and respectively. Recall that a group over arises as an extended pure inner twist of if there exists a tuple such that is an isomorphism over some finite extension and is such that for each and each projecting to , we have
The set is defined as in [Kot14]. In the case that has connected center, it is known by [Kot14, Proposition 10.4] that all inner twists of come from extended pure inner twists. In our case, we have and so this is indeed the case. We can also consider extended pure inner twists for connected reductive groups over . The definition is the same except for we have (where is the local gerb in [Kot14]). As in [Kot14], we define:
for a number field and
for a finite extension of .
A maximal torus defined over of and with maximal split rank is given by the diagonal subgroup. We have
The maximal split subtorus of is isomorphic to . The relative Weyl group is where is the permutation group of . The normalizer of inside is generated by and the following elements:
where and is the matrix with in the positions and for and elsewhere.
A minimal parabolic subgroup of is
From the description of unitary similitude groups, we see that there is an embedding given by . The tuple gives a -stable splitting of .
We can identify with and with . We fix the standard -splittings of and consisting of the where and are the diagonal subgroup and upper triangular subgroup respectively. The action of the Weil group on these dual groups factors through and the non-trivial element of acts via
and
respectively (see [Mor10, pg 38] for details).
A maximal torus defined over of with maximal split rank is given by
If we denote , resp. the set of indexes such that is odd, resp. even, then a maximal split sub-torus of is isomorphic to
And the relative Weyl group is
Lemma 2.1.
We have the equality where embeds into via the diagonal embedding.
Proof.
For simplicity, we prove the equality when . The general case follows by the same argument.
We just need to show that . Because is quasi-split, we have the Bruhat decomposition . We see that and by the above description of the normalizer of . Hence, and then since where is the unipotent radical of . By the assumption is odd and the description of , we have . Moreover, by the above injection , we also see that . Therefore . ∎
We now recall some facts from the theory of endoscopy.
Definition 2.2.
(cf. [BM21, Definition 2.1]) A refined endoscopic datum for a connected reductive group over is a triple such that
-
•
is a quasisplit reductive group over .
-
•
-
•
such that the conjugacy class of is -stable and .
Suppose that are refined endoscopic data. Then we say that an isomorphism is an isomorphism of endoscopic data if and and are conjugate in .
We say that a refined endoscopic datum is elliptic if .
We denote the set of isomorphism classes of refined endoscopic data of by .
We record a set of representatives for the isomorphism classes of refined elliptic endoscopic data for and . The description will be analogous in the local case. Compare with [Mor10, Proposition 2.3.1] but note that we have more isomorphism classes because we consider refined endoscopic data. For each , choose non-negative natural numbers and such that .
Then, in the unitary case, let be the group , let be the block diagonal embedding of dual groups and let
. These elliptic endoscopic data are all non-isomorphic and give a representative of each elliptic isomorphism class.
In the unitary similitude case we let be , let be the block diagonal embedding of dual groups, and let . We further require that is even.
2.2. Automorphic representations of unitary groups
We start by considering a local field for any place of . The local Langlands group is defined by if and by if is a prime. We also set as a topological group where is the Langlands dual group of . In our case we see that and the group acts trivially on .
Definition 2.3.
A local -parameter for a connected reductive group defined over is a continuous morphism which commutes with the canonical projections of and to and such that sends semisimple elements to semisimple elements.
We denote the set of -conjugacy classes of -parameters. An -parameter is called bounded, resp. discrete, if its image in projects to a relatively compact subset of , resp. if its image is not contained in any proper parabolic subgroup of . We denote by , resp. the subsets of consisting of bounded -parameters (resp. discrete parameters).
For global classifications, we will also need the notion of a local Arthur parameter.
Definition 2.4.
A local -parameter for a connected reductive group defined over is a continuous morphism such that the image of is a bounded -parameter.
We denote by the set of equivalence classes of -parameters. We also denote the set of the equivalence classes of continuous morphisms as above but where is not necessarily bounded . An -parameter (or ) is said to be generic if is trivial.
We also have a “base change” morphism of -groups ([Mok15, page 9]):
which allows us to identify the -parameters of with self-dual -parameters of . In particular, if is an -parameter then its image by the base change map is just .
For each -parameter we define centralizer groups as below, which play an important role in the local and global theory:
We also need to introduce some notation for representations. We denote the set of isomorphism classes of irreducible admissible representations of a connected reductive group by . We denote the set of tempered, essentially square integrable, and unitary representations by , , and respectively. We denote by .
The following theorem gives the local Langlands correspondence for extended pure inner twists of . We first fix some more notation.
Fix an odd natural number and let and be an extended pure inner twist of . Fix a invariant splitting of . Then induces a unique isomorphism preserving the chosen -splittings. The cocycle and the map
defines a character by . Fix a non-trivial character . Together with our chosen splitting of , this gives a Whittaker datum of . Attached to each refined endoscopic datum of we have a canonical local transfer factor normalized as in [BM21, §4.1]. These transfer factors correspond to the factors of [KS12, §5]. Since is simply connected, we can extend to a map of -groups.
We stress that in this paper, we are using the geometric normalization of the Langlands correspondence. This means that our Artin map is normalized to map a geometric Frobenius morphism to a uniformizer and explains why we normalize our transfer factors using the normalization. This normalization is consistent with [HT01] and [BM21] but is the inverse of the normalization in [Kal+14].
Theorem 2.5.
Fix an odd natural number and let and be an extended pure inner twist of . Fix a non-trivial character . Together with our chosen splitting of , this gives a Whittaker datum of . Then,
-
(1)
For each generic , there exists a finite set endowed with a morphism to . Our choice of defines a map
where denotes the set of irreducible representations of restricting on to .
-
(2)
The morphism is injective and its image is contained in . If is non-Archimedean then the map is a bijection.
-
(3)
For each in the image of , the central character has a Langlands parameter given by the composition
-
(4)
Let be a refined endoscopic datum and let be a generic parameter such that . If and are two -matching functions then we have
where is the Kottwitz sign.
-
(5)
We have
and
Proof.
The contents of this theorem appear in the works of Mok ([Mok15, Theorem 2.5.1]) and Kaletha–Minguez–Shin–White ([Kal+14, Theorem 1.6.1]) except using the arithmetic normalization of the Langlands correspondence. Hence our main task is to explain how we can use these arithmetically normalized correspondences to define a geometrically normalized correspondence.
For a generic parameter, we let denote the packet of representations assigned to by [Kal+14, Theorem 1.6.1] (the letter stands for arithmetic normalization) and define to consist of the contragredients of the representations in . By the compatibility of the local Langlands correspondence with contragredients (proven in our case in Proposition 2.7, cf. [Kal13, Equation (1.2)]), this is the same as saying that the packet of [Kal+14] is assigned to the parameter where is the extension to of the Chevalley involution, , of as described in [Kal13, p. 3-4].
We now define . For convenience, we will denote by the maps given by the arithmetic normalization. Then we define for that
where if is the Whittaker datum , then is the datum . Equivalently by taking the contragredient, we have
We now verify the endoscopy character identity which is (4) in the theorem. Fix and a -matching function.
By Lemma 3.5, we have that if is the inverse map, then and are matching for the transfer factors with respect to the endoscopic datum . We use the letter () resp. to denote the transfer factors that are compatible with the geometric normalization resp. arithmetic normalization of the local Artin reciprocity map. Then we will show in Proposition 2.6 that
Proposition 2.6.
Let be a tempered Langlands parameter. Then we have an equality of linear forms on
Proof.
Thanks to the results in [Mok15], [Kal+14], the arguments in [Kal13, Theorem 4.8] also work in our case. Indeed, the group can be extended to an endoscopic datum of the triple for a suitable outer automorphism of preserving the standard splitting. Then is a Langlands parameter of and denote by the representation of assigned to by the local Langlands correspondence. There is a unique isomorphism which preserves the -Whittaker functionals. Then we have the distribution
Then by [Mok15, Theorem 3.2.1] and [Kal+14, Theorem 1.6.1 (4)] the linear form is the unique distribution on having the properties that
for all and such that the -twisted orbital integrals of match the stable integrals of with respect to .
Once we have this characterization, the proof of [Kal13, Theorem 4.8] works without any change since Proposition 4.4, Corollary 4.5 and Corollary 4.7 in loc. cit. are valid for quasi-split unitary groups. ∎
Proposition 2.7.
Let be a tempered Langlands parameter and be a Whittaker datum. Let be a representation in and denote . Then
-
The contragredient representation belongs to the -packet ,
-
.
Proof.
We also have the following basic fact.
Lemma 2.8.
For an admissible representation of for a reductive group and , we have
Proof.
Pick some compact open such that is -bi-invariant. Let denote the contragredient of so that is the subspace of smooth vectors in the dual vector space of . Then we note that since each vector in is by definition smooth.
The operator acts on as the dual of the operator . Indeed for and
where the third equality uses that is unimodular. This implies the desired equality of traces. ∎
When we consider global parameters, we will also need a version of Theorem 2.5 for . The following theorem is essentially contained within the union of the remarks in [Mok15, pg33] and [Kal+14, §1.6.4].
Theorem 2.9.
Fix an odd natural number and let and be an extended pure inner twist of . Fix a non-trivial character . Together with our chosen splitting of , this gives a Whittaker datum of . Then,
-
(1)
For each generic , there exists a finite set of possibly reducible or non-unitary representations of . Our choice of defines a map
where denotes the set of irreducible representations of with central character . Each has a central character , these characters are the same for each element of .
-
(2)
Let be a refined endoscopic datum and let be a generic parameter such that . If and are two -matching functions then we have
where is the Kottwitz sign.
Proof.
We sketch the proof following ideas in [Kal+14] and [Mok15]. The proof of (1) is in [Kal+14, §1.6.4]. They choose a standard parabolic subgroup of that transfers to , a parameter and a character such that the induced central parameter satisfies that agrees with under the -embedding . Choose a representative in its equivalence class so that the restriction to is also an extended pure inner twist.
They then define by
where denotes normalized parabolic induction and is the character of corresponding to . Note that by definition of parabolic induction, if has central character , then will have central character . Since each element of has the same central character, this will also be true of .
From the explicit description of given in [Kal+14, pg 62], it follows that . In [Kal+14, §1.6.4] they show that and that and both extend uniquely to give the same character of that is trivial on . Now, we have an identification
since both parametrize irreducible representations of that restrict to on . One can now define
for .
It remains to verify the endoscopic character identity. Fix a refined endoscopic datum for such that for some . Then . By [BM21, Proposition 3.10], there exists a refined endoscopic datum and parameter corresponding to the pair . It is clear from construction that under the map of [BM21, §2.5], the image of the class of equals the class of . Now by [BM21, Proposition 2.20], we can choose a refined datum equivalent to fitting into an embedded datum . We observe that is a Levi subgroup of .
Now, induces a map and hence yields a central parameter of . It is easy to see that by definition under the natural inclusion . Hence, we can define a packet and pairing
using the above procedure.
We need to verify that the resulting pairing satisfy the endoscopic character identity. Let be -matching functions. Let and be the corresponding constant term functions. By the paragraph at the top of page and the remark on page of [Dij72] it follows that
and similarly for . We can restrict the splitting of to and together with the character , this gives a Whittaker datum . By [BM21, Proposition 5.3], the corresponding canonical transfer factor satisfies
for regular and where we recall that is defined to equal .
We now claim that and are -matching. If we can show this then we will have
as desired. Note that in the above we use that which is part of the proposition on page of [Kot83].
Suppose and are strongly regular elements that transfer to each other. Then by [Dij72, Lemma 9], we have the following equality of orbital integrals (and analogously for ):
and hence, since and are -matching:
as desired. Note that we use that the number of conjugacy classes of in the stable class is the same for and (this follows from the injection ).
∎
The global classification
We now consider the global situation. Recall that we have fixed a quadratic imaginary extension and are considering global unitary groups that are quasi-split at the finite places and with fixed quasi-split inner form . Due to the lack of global -group, we rely on the cuspidal automorphic representations of to define the notion of global parameters as in [Art13a] (cf. [Kal+14]). Let denote the set of all formal sums
where are positive integers, are cuspidal automorphic representations of and are algebraic representations of such that are pairwise disjoint.
We denote where the conjugate dual representation of . Now for , we say that is self-dual if there exists an involution of such that and . From a self-dual formal sum , we can construct a group and a map ([Mok15, p. 22, 23; definition 2.4.3])
Recall that we have a base change map .
Definition 2.10.
The set of global -parameters of is the set consisting of pairs where is a self-dual formal sum and is a map such that .
We remark that is determined by the base change map and , and as in the local case, from the map , we can define various centralizer groups , , , .
There is a localization morphism from to [Mok15, p. 18,19]. More precisely, if is a place of that splits in then and where , are the primes of above . Moreover, there is an identification and therefore we can define . If is a place of that does not split in then is a quadratic extension of . By [Mok15, Theorem. 2.4.10] the localization of factors through the base change map so that it defines a parameter in .
We denote
Since the localization maps induce the localization maps for centralizer groups ([Kal+14, pg 71]), we can associate to each a character of by the following formula
where is the image of by the localization morphism . The global pairing descends to a character of (see [Kal+14, pg 89]).
Definition 2.11.
Let where is the Arthur character of . If is a generic parameter then .
Theorem 2.12.
([Kal+14] theorem There is an isomorphism of -modules
If is generic then
-
if .
-
if .
In particular, if is an automorphic representation of the unitary group belonging to a generic global packet then .
Remark 2.13.
2.3. Automorphic representations of unitary similitude groups
In this section, we want to transfer the results about automorphic representations from unitary groups to unitary similitude groups (with an odd number of variables). We begin with the local case.
Let be a finite place of that does not split over , let be an odd positive integer, and let be an inner form of and denote the corresponding unitary group by . Fix a -invariant splitting of and restrict to get a -invariant splitting of . Fix also a character . This data gives us Whittaker data and of and respectively.
We give the structure of an extended pure inner twist . We can restrict to to get an inner twist . We give this the structure of an extended pure inner twist . Note that since we are assuming is odd, will automatically be quasi-split. By Lemma 2.1, we have and then the following result:
Corollary 2.14.
There is a natural bijection between the set and the set of pairs such that and is a character of such that where is the central character of .
We use this corollary to define -packets of representations for and the associated -parameters. Fix a character of corresponding to a morphism , and a parameter given by such that for each .
We can view as a product of and over where the first projection is given by and the second is given by . The above pair then defines a unique morphism . Conversely, each gives rise to a pair . We summarize these relationships in the following commutative diagram:
We define the associated -packet of representations for to be the set
We now use the internal structure of to describe that of . Let us first describe the relations between the various centralizer groups for and .
Lemma 2.15.
With and as above, we have
where .
Proof.
For and in and projecting to the nontrivial element of , we have
where the second equality comes from the action of on . In particular, if and only if and . In other words, .
We now prove that . By a direct calculation, we see that and (because odd). Hence . We also remark that the equality implies . Therefore, for every we have . Moreover, since , we have . Thus, as desired.
Finally, we have which implies the description of in the statement of the lemma. ∎
We now construct a pairing:
Let . Then and by Theorems 2.5 and 2.9 there is an associated character . Note that since , we can restrict this character to , and this restricted character factors to give a character of . Via and the map
we get a character of . Then in Lemma 2.15, we showed that . Hence we define
Suppose that is generic. We show that is bijective onto by constructing an inverse. Pick a character of which restricts on to the character . Since and generate and have trivial intersection, there is then a unique character of that restricts to on and on . By of Theorem 2.5, there then exists a that gets mapped to , and by construction, maps to . Hence is our desired inverse.
We have now proven
In the archimedean case, these results are known by work of Langlands and Shelstad.
In the next section, we will prove that this pairing also satisfies the endoscopic character identities.
We record the following proposition for later use.
Proposition 2.17.
[Mg07, section 8.4.4] Let be a discrete -parameter which is trivial over . Then the packet contains only supercuspidal representations. These -parameters are called supercuspidal.
Corollary 2.18.
From the above description of local -packets of , it follows that the -packet of a supercuspidal -parameter of will consist entirely of supercuspidal representations.
Remark 2.19.
Suppose that is as above and is an elliptic endoscopic datum and an -parameter such that . Then is also supercuspidal and hence the packet contains only supercuspidal representations.
2.4. The global classification for unitary similitude groups
Fix a Hermitian form and global group and . As in the local case, we give and the structure of extended pure inner twists and .
We begin by recalling the following result which relates automorphic representations of and of .
Proposition 2.20.
([Clo+11, Section. CHL.IV.C, Proposition 1.1.4]). Fix odd. Let be an irreducible automorphic representation of whose restriction to contains an irreducible automorphic representation . If has multiplicity in the discrete spectrum of then has multiplicity in the discrete spectrum of . Moreover, is the unique automorphic representation of with central character and containing in its restriction.
Let be an automorphic central character of and its restriction to the center of . Consider a generic -parameter for a global unitary group whose automorphic representations have as central character. The generic condition ensures the multiplicity one property of these automorphic representations by Theorem 2.12. As in the local case, a pair satisfying the above conditions determines a generic -parameter for . In the following, we will denote such an -parameter by if is clear from the context. We define the associated -packet to consist of the whose central character is and whose restriction to belongs to .
Now, by the proof of [Clo+11, Section. CHL.IV.C, Proposition 1.1.4], we have
where and . In particular, it follows from Theorem 2.12 that we can lift every representation to a representation of whose central character is . Combining with Proposition 2.20, we see that there is a bijection between and .
We now give a description of in the spirit of Definition 2.11. We have defined global generic -parameters of in terms of their counterpart for . We define the centralizer groups for such parameters of using the analogous groups for and using Lemma 2.15 as our guide.
Remark 2.21.
Definition 2.22.
Let be a generic parameter. We define:
We now discuss localization. First, by the localization map for algebraic cocycles (see [Kot14, §7]), the extended pure inner twists and give rise to local extended pure inner twists and for each place of .
Let be a generic -parameter. At each place of , we get a local parameter as well as a local character . We define the localization of at to be . The localization map restricts to give a map and hence we get a localization map
Similarly, we get a localization map .
We now define:
We associate to each a character of . Each corresponds to a pair where . We then define a global pairing by the formula
where is the image of under the localization map defined above. We claim that descends to a character on . Indeed, by definition we have
We showed previously that descends to and is trivial by [Kot14, Proposition 15.6].
Proposition 2.23.
For a generic -parameter of , we have the following equality of sets:
We note that since we are assuming is generic, we in fact have .
Proof.
The left-hand side consists of all pairs such that . By definition, we have
Hence we just need to show that is trivial if and only if is. But this is clear since these are the same character of . ∎
3. Endoscopic character identities
Let be a quadratic extension and an odd natural number. Our goal in this section is to prove the endoscopic character identities for elliptic endoscopic groups of with and is an inner form of . We prove this using the fact that these identities hold for as in [Mok15], [Kal+14].
-
•
We first show that if the endoscopic character identities hold for , then they also hold for where we note that is the center of .
-
•
We then show that if the endoscopic character identities hold for , then they hold for .
We recall the statement of the endoscopic character identity for an extended pure inner twist of a quasi-split reductive group over with refined endoscopic datum . Fix a local Whittaker datum of giving a Whittaker normalized transfer factor (as in [KT18, §4.3]) between and . Suppose that and are -matching functions.
Now, let and be such that . Let denote the respective -packets for the parameters. Then the endoscopic character identity states that
| (3.1) |
where is as defined in Theorem 2.5 and Theorem 2.9. The elements and are defined to be the image of under and respectively and is the Kottwitz sign.
According to a theorem of Harish-Chandra, the trace distribution is given by integrating against the Harish-Chandra character, which is a locally constant function of (where denotes the strongly regular semisimple elements of ). Then the above identity is equivalent to the equality
We remark that a Harish-Chandra character exists for representations by [Dij72, Theorem 3] and that this holds even in the case where the induction is not irreducible. Hence, have Harish-Chandra characters even in the case where .
3.1. Endoscopic Identities for
In this section we use the notation to denote the group . Our goal is to prove the endoscopic character identities for using the fact that these identities are known for by [Kal+14] (Theorems 2.5 and 2.9 in this paper).
In fact, we will prove the following stronger result. Fix quasi-split reductive groups for . Let be extended pure inner twists of . Let be refined endoscopic for . We denote by the corresponding endoscopic datum of . Fix a character and -splittings of . This induces a Whittaker datum of as well as the Whittaker datum of .
We will prove that if the endoscopic character identities are satisfied for and then they are also satisfied for and .
Fix and suppose is such that . Then factors as a product of parameters of and of . As a result, factors as a product of parameters of and of such that .
We need to show that for -matching functions and , the following identity holds:
| (3.2) |
The packets resp. consist of representations of the form resp. where resp. are representations in resp. . The pairings resp. are defined as resp. . It is not difficult to see that . It is also a basic property of the Kottwitz sign that .
Moreover, a function can be written as a sum of functions of the form where and . Hence, for every such we have an equality between the following quantities
and
Similarly, for every with a matching function of and a matching function of we have an equality between
and
Therefore, in order to prove the equation (3.2), it suffices to prove that for each , we may choose a -matching function such that and are -matching. This follows from the following lemma.
Lemma 3.1.
If and are -matching functions then and are -matching functions.
Proof.
Pick such that is strongly regular and transfers to a strongly regular . Then we need to show that
| (3.3) |
where the sum is taken over the set of that are stably conjugate to .
By definition, for and we have
Moreover, an element is stable conjugate to if and only if is stable conjugate to in and is stable conjugate in in . Therefore we have
and similarly
| (3.4) |
We will prove in Lemma 3.4 that
We can then rewrite the right hand side of (3.4) as
and because and are matching functions, this is exactly
In other words, Equation (3.3) is true. ∎
3.2. Endoscopic Identities for
We now have the endoscopic character identities for and need to show they also hold for where we use the letter to denote the group till the end of this section. Recall that we have a surjection of algebraic groups
| (3.5) |
with kernel .
We fix quasi-split groups and as well as extended pure inner twists and for and respectively. The extended pure inner twist restricts to give extended pure inner twists and of and respectively. We can choose these extended pure inner twists such that the projection takes to and such that and coincide under the map . We fix compatible -splittings of these groups as well as a character . Hence we get compatible Whittaker data which we denote by and respectively.
A crucial input in the case we consider (where is odd) is that the projection is also a surjection on -points. This follows from Lemma 2.1. Hence we get a map
given by pullback. The image of this map is the set of irreducible representations such . If this is satisfied by a single member of an -packet of , then it will be satisfied by the entire packet since elements of an -packet have the same central character ([Kal+14, Theorem 1.6.1] and Theorem 2.9). In light of Theorem 2.16, the -packets of are in a natural way a subset of the -packets of .
Since the kernel of is compact, any lifts to an element . Suppose is an admissible representation of and is the lift to . Then to prove the endoscopic character identities for it will be necessary to relate and . We have
where the middle equality holds by [PR94, (3.21)].
Analogously in the endoscopic case, we have a map
| (3.6) |
with kernel where such that is an endoscopic group of and is the associated similitude group. Suppose is odd. By Lemma 2.1, the map is a surjection on -points.
We fix a refined endoscopic datum for as in Section . The map induces a map of -groups . We get an analogous map for and since is the abelianization of , we get a commutative diagram
We now fix an endoscopic datum of which we denote by . We fix such that the restriction to induces an elliptic endoscopic datum for as in Section and compatible with our datum for and such that restricted to is just . In particular, we have a commutative diagram:
| (3.7) |
We now prove the following lemma.
Lemma 3.2.
Using the above normalizations, if and are -matching, then the pullbacks and are -matching.
We begin by proving an auxiliary lemma.
Lemma 3.3.
For , the map gives a bijection between conjugacy classes in that are stably conjugate to and conjugacy classes in that are stably conjugate to . The analogous result also holds for the map .
Proof.
Clearly, if is conjugate or stable conjugate to in , then and are conjugate or stably conjugate in . Now, suppose that are conjugate or stably conjugate. Then they must have the same similitude factor. In particular, this means that has trivial similitude factor and so and clearly .
We now aim to show that is conjugate or stably conjugate to . To simplify the notation, we just show that and are conjugate (although the argument to show stable conjugacy is similar).
Let be such that . We want to show that can be chosen to be an element of . Since the map is surjective on points, we can write such that and . Then lies in the center of and hence we have as desired. Finally, we finish the argument by observing that that since the restriction of to the first component is an injection. ∎
We now prove Lemma 3.2.
Proof.
Choose a strongly regular semisimple that transfers to a strongly regular . Then we need to show that
Expanding, this is equivalent to showing that
equals
Note that the kernels of are contained within and respectively. Hence we have and the analogous statement also holds for .
With this lemma in hand, we now prove the endoscopic character identities. Pick a parameter and let be the composition of with the map . We suppose factors through and pick so that . We can write where is the image of under the map . Diagram 3.7 implies there is a parameter such that .
For a packet , and matching functions , we have by definition
We showed above that there is a natural bijection between and and we related the traces of corresponding representations. The pairing is given as a product of the pairing for and . Hence we have the above equals
Now, using that (see [Kot83, pg. 292]) we can apply the previously established endoscopic character identity for to get that the above equals
Finally we relate this to using that and are both assumed to be trivial extended pure inner forms so that the pairings are especially simple. We get:
which is the desired formula.
3.3. Transfer factor identities
In this subsection, we prove a number of identities relating various transfer factors. These identities are used in the previous subsections. Remark that we use the letter resp. to denote the transfer factors that are compatible with the geometric normalization resp. arithmetic normalization of the local Artin reciprocity map.
3.3.1. Transfer factors of a product
We temporarily return to the notation of §3.1. We denote by the group and by the group .
We prove the following lemma
Lemma 3.4.
Let and be related elements. We have
Proof.
Each transfer factor is a product of terms
We state everything for but the definitions are analogous for . We now explain the terms in the above formula. Notably, all the terms except the last only depend on and (as opposed to ). Fix a such that is stably conjugate to . Recall that we have fixed -splittings for as well as the -splitting of .
Now, is the degree virtual Galois representation and is the additive character we fixed in order to define our Whittaker datum. The term is the local -factor of this representation normalized as in [Tat79, §3.6]. We also know that is additive for degree virtual representations (see [Tat79, Theorem. 3.4.1]), therefore .
We denote by the centralizer of and the centralizer of so that and are the centralizers of resp. .
We put where the product is over all roots of in . Similarly where the product is over all roots of in . In particular we have
We define and analogously and we also have the equality
By definition so that we have
For the other terms in the definition of the transfer factors, we need to explain the notions of -data and -data. A set of -data for the set of absolute roots of in is a function
which satisfies and for . We recall the notion of -data. For , we set and and denote , the fixed fields of resp. . A set of -data is then a set of characters
satisfying the conditions and if then is non-trivial but trivial on the subgroup of norms from .
Since acts on and preserves , we see that if and are -data resp -data of then and are -data resp -data of .
Now, we define
where the product is taken over the set . We have a similar formula for in which the product runs over the set . In particular we have
Next, we want to show that
First, for one constructs an element and then uses the Tate-Nakayama duality for tori in order to get a pairing between and . One can view as an element of , embed the latter into , and transport it to by the admissible isomorphism . We then define
Because and , to show the necessary product relation for this term, it is enough to show that .
We recall the construction of for and . Write for the absolute Weyl group and let be such that . For each there exists such that for all
Let be a reduced expression and let be the image of under the homomorphism attached to the simple root vector . Then is independent of the choice of the reduced expression. So is defined by the following -cocycle
where the product runs over the subset of where positivity is determined by the Borel subgroup . The construction is analogous for .
Now, we have
-
(1)
,
-
(2)
, ,
-
(3)
so that .
We see that and we can take so that . Therefore, . We conclude that .
We are now going to show that
The construction is as follows. First, we associate to the fixed -datum a -embedding ([LS87], 2.6). Next via the admissible isomorphism the -datum can be transferred to and gives an -embedding . The admissible isomorphism also provides dually an -isomorphism . The composition gives another -embedding . Via conjugation by an element of , we can arrange that and coincide on so that for some .
The term is given by where the paring is the Langlands duality for tori under the geometric normalization. More precisely, the element of is an -parameter of . By the local Langland correspondence for tori, gives rise to a character of .
In our case, we have and so it suffices to show that . In order to verify that, we need to review carefully the formation of the -embedding associated to a -datum [LS87, §2.6].
Fix a Borel pair of as well as a Borel subgroup (possibly not defined over ) of containing . The pair yields a set of positive coroots of and equivalently a set of elements of . Then is defined so that the restriction to maps to by the unique isomorphism mapping our chosen subset of to the set of positive roots of determined by .
Then, to specify we have only to give a homomorphism where . We require that if under then acts on as the transport by of the action of on .
We then define
for and under . The term is defined above, in the definition of and we have already seen that .
We recall briefly the construction of . We denote by the set and define to be the group of automorphisms of generated by and where acts on by (as in[LS87, Lemma 2.1A]). The group acts on and divides it into -orbits . For each -orbit , we define an element and then take the product over the orbits to obtain . Since and the group preserves , , we have that . This implies the desired product identity for .
Finally, we show that
We have a natural isomorphism that maps the class of to the product of the classes of and . Moreover, this product decomposition respects the Kottwitz maps defining the above pairings. This implies the desired product formula. ∎
3.3.2. Transfer factor and changing the normalization
Lemma 3.5.
Let and be -matching functions for an endoscopic datum of . If and are the inverse functions, then and are matching for the transfer factors with respect to the endoscopic datum .
Proof.
We consider first the ordinary endoscopic case. Suppose is strongly regular and transfers to a strongly regular element . By hypothesis, we have
Then we need to show that
Since and , it suffices to show that the transfer factor with respect to the endoscopic datum is the same as the transfer factor with respect to the endoscopic datum .
Recall that the transfer factor is a product of terms
which we need to use -data and -data in order to define and moreover the transfer factors do not depend on the choices of -data and -data.
By [KS12, Section 5.1], the transfer factor is defined by the same formula, except that one replaces the term by , inverts and inverts . If one keeps track of the dependence on -data and -data, then .
By using the definitions of the terms appearing in the transfer factors which we recalled in Lemma 3.4, we have
since these terms do not depend on -data and where the notation keeps track of whether we plug in or into the pairing defining . Moreover . Thus we have
Therefore with respect to the endoscopic datum is nearly the same as with respect to the endoscopic datum . The only difference is that in the above second product, the term is defined with respect to -data and the term is defined with respect to the -data. However, the and terms also depend on the Whittaker datum. According to [Kal13, page 16], we have .
Since inverting the character leads to the inverse Whittaker datum , the second product is actually the transfer factor with respect to the endoscopic datum .
For the twisted endoscopic case, the same arguments still work. Indeed, in this case and we need to show that
Since and , it suffices to show that the transfer factor with respect to the endoscopic datum is the same as the transfer factor with respect to the endoscopic datum . By the results in [KS12, Sections 5.3, 5.4], we know that the twisted transfer factor is a product of terms
and the twisted transfer factor is a product of terms
Since is the term computed for the inverse set of -data, we see that . Moreover . Thus we have
As in the standard endoscopy case, the second product is actually the twisted transfer factor with respect to the endoscopic datum . ∎
3.3.3. Endoscopy for
We now study the endoscopy of .
We must have and pick . We will be most interested in the case where is the identity map and so we assume this is the case. Then is determined up to conjugacy by an element of . By the Langlands correspondence for tori, this cocycle corresponds to a character of .
We now study transfer factors for the endoscopic datum of . Recall we have fixed an extended pure inner twist of such that . Consider which transfers to and . Our goal is to compute the transfer factor .
Lemma 3.6.
We have
Proof.
We will calculate each term in the definition of transfer factor. The virtual representation in this case is so that the factor . The terms , are trivial since has no absolute roots. The term is trivial since the group is trivial.
We now compute . The -maps , and are all the identity. Hence, by comparing with , we see that .
The final term then contributes the factor , completing the argument. ∎
3.3.4. Transfer factors for and
We use the notation of §3.2. We denote the Whittaker datum and extended pure inner twists of induced by restriction from by and . We record the following lemma:
Lemma 3.7.
Suppose that and are strongly regular and related. Then we have the following equality
Proof.
This is [Xu16, Lemma 3.6] adapted to the non-quasisplit setting. ∎
Finally, we prove the following lemma:
Lemma 3.8.
Suppose and are related. Then we have an equality of transfer factors
Proof.
First of all, by Lemma 3.4 we have
By Lemma 3.6, this equals
and by Lemma 3.7 we have
Since the Kottwitz set and the Kottwitz map respect products, we get that
By the functoriality of the Kottwitz map,
Hence we get
Hence, it remains to show that . We recall that is the character arising from the construction of the -term of the transfer factor for . From the description in [LS87, Lemma 4.4A], is the restriction to of the character arising from the -term of the transfer factor for .
The characters and are determined by the failure of the following diagram to commute:
We explain this diagram. The objects and are maximal tori in their respective groups that are isomorphic by an admissible embedding . The maps and are the -embeddings constructed in [LS87, §(2.6)] from a choice of -data. The lower two diagonal maps are induced by the embeddings and . Since the images of these embeddings lie in the image of the embeddings and respectively, we get induced maps and . These induce the upper diagonal maps in the above diagram. The outer vertical arrows are then defined so that the left and right trapezoids commute. Note that by definition of and the vertical maps are both the identity. The bottom trapezoid commutes by construction. Finally the top map is defined so that the top trapezoid commutes and will agree with on and map to .
Then the outer square fails to commute by the cocycle and the inner square fails to commute by . Since the trapezoids all commute, these cocycles agree under the natural map . This is the desired result. ∎
4. Properties of the local and global correspondences
In this section we prove a number of properties and compatibilities of the local and global Langlands correspondences. These properties are needed to derive our main theorem.
4.1. Unramified representations
In this subsection we suppose that is a finite place of and that is unramified. We let and be the trivial extended pure inner twists of and respectively. Let be the standard hyperspecial subgroup. Then we say that is -spherical if it has nontrivial -invariants.
Proposition 4.1.
Let be a generic parameter. Then contains a -spherical representation if and only if is unramified. In that case, contains a unique -spherical representation , which satisfies . The same results hold true for .
Proof.
We first consider the case where (or ). By Corollary 2.14, we see that a spherical representation of lifts to a spherical representation of and vice versa. Moreover, by the construction local packets for , we have that if and only if . Therefore it suffices to prove the proposition for unitary groups.
We mimic the proof of Lemma in [Tai17]. Denote the characteristic function of the standard special maximum compact subgroup of . If is unramified then by proposition [Mok15, Proposition 7.4.3] we have
In other words, the packet contains an unramified representation. The uniqueness comes from Theorem in [Mok15].
Suppose now that is ramified. Then the base change -parameter is also ramified. By the local Langlands correspondence for , one gets a representation of corresponding to . Then, as in [Mok15, §3.2], one lifts to a representation of , where is the automorphism of . Hence the corresponding representation of is ramified. We want to show that for every . If we denote the characteristic function of then . The twisted fundamental lemma implies that is the twisted transfer of and hence by [Mok15, Theorem 3.2.1 a] we have
By the same argument we have for every refined endoscopic datum of where is the characteristic function of a hyperspecial subgroup of . By the fundamental lemma, is the transfer of . Then, again by [Mok15, Theorem 3.2.1 ] we have
where corresponds to under [BM21, Proposition 3.10]. Hence we conclude that for every . Therefore the packet does not contain any unramified representations.
We now consider the case of general . This follows from the fact that is -spherical if and only if is -spherical for a standard Levi subgroup with parabolic subgroup . ∎
4.2. On the hypothesis
In this section, we prove that for a refined elliptic endoscopic datum of and that is stable cuspidal at infinity and cuspidal at a finite place , we have an equality of traces:
We begin with some preparatory notation and lemmas. Let be a connected reductive group defined over and let be a sufficiently regular (in the sense of Lemma 5.11) quasi-character of and be the set of functions smooth, with compact support modulo and such that for every . Fix a maximal compact subgroup of .
Definition 4.2.
(Stable cuspidal function at infinity) We say that is stable cuspidal if is left and right -finite and if the function
vanishes outside and is constant in the -packets of .
Definition 4.3.
(cuspidal function) We say that is cuspidal if if for each proper Levi subgroup we have that the constant term, , vanishes (as defined in [GKM97, (7.13.2)]).
We record the following well-known lemma
Lemma 4.4.
If is a stable cuspidal function and is an endoscopic triple of then there exists a stable cuspidal transfer function of .
Proof sketch.
By [She82] we can find a function that transfers to . Define the function on the set of unitary tempered representations of by setting
for . Then must be supported on finitely many discrete series packets since is stable cuspidal and is elliptic. Hence, by [CD90, Theorem 1] there exists a function that is stable cuspidal and . Thus, has the same stable orbital integrals as . This implies that is a stable cuspidal transfer of . ∎
We recall that is defined by the formula
| (4.1) |
where the sum is over a set of representatives of the -regular, semisimple, -elliptic, stable conjugacy classes in .
Definition 4.5.
We define the term to equal
Note that we have suppressed the term from this expression because our assumption on implies that all are generic by Lemma 5.11.
Separately, we have for every Levi subgroup of the term defined in [Mor10, pg 86] as well as the term defined by
for certain constants .
We prove the following standard result.
Lemma 4.6.
Suppose is stable cuspidal at infinity and cuspidal at a finite place. Then
-
•
For any we have
-
•
If then
Proof.
To prove the first part, we note that by definition, for a proper Levi subgroup, the “constant term” is (for instance see the definition before [Art88, Theorem 7.1]). This implies that .
We now prove the second part. We first show that . By [Art89, Theorem 5.1], we have that
where the sum is over discrete series -packets of with central character (the unique character of such that if a parameter has central character restricting to then has central character ). The representation is some representative of , and the value of does not depend on the choice of representative since is stable cuspidal. The in this formula that is seemingly at odds with the formula of Arthur is explained by [GKM97, 7.19].
Therefore we have
Now by definition,
Since is stable cuspidal, we have . Furthermore, it follows from the definitions and basic properties of the Kottwitz sign, that
where for an elliptic maximal torus of .
Finally, we put everything together to get
The last equality follows from the fact that only depends on the stable class of and
Indeed, is well known to equal for an elliptic maximal torus of . Hence, it suffices to show that
To see this, first note that the set of conjugacy classes that are stably conjugate to is in natural bijection with . For each such conjugacy class, we can choose a representative . This follows from the fact that since contains an elliptic maximal torus, any elliptic element of is contained in an elliptic maximal torus and all elliptic maximal tori are conjugate in . Then the set of classes in mapping to the class of in is in bijection with .
It then follows that
where the sum is over stable conjugacy classes in that are semisimple and elliptic in .
Since is stable cuspidal, its orbital integrals vanish on that are not elliptic at , so we may as well impose this condition. By [Mor10, Proposition 3.3.4, Remark 3.3.5] we may also restrict the sum to that are -regular. We then see that this is equal to . ∎
Suppose now that is stable cuspidal at infinity and cuspidal at a finite place. Then by the above Lemma 4.4 and [Art96, Lemma 3.4], for each elliptic endoscopic datum , we can find a function that is stable cuspidal at infinity, cuspidal at a finite place, and a transfer of .
Our proof of the main result of this section will be by induction. We now state the key formulas we will need.
First, we have the following theorem of Morel:
Theorem 4.7.
See [Mor10, Theorem 5.4.1] Let be a connected reductive group. Let where and . Assume that is stable cuspidal and that for every , there exists a transfer of . Then:
where is the set of isomorphism classes of elliptic endoscopic triples in the sense of Kottwitz and we recall that is defined to be the trace of on .
Now we fix an odd positive integer . By Proposition 2.20 and Remark 2.24 we have the following formula for each group of the form such that . We note that all such groups are quasisplit.
For a function :
where is the subset of containing those with trivial character .
We now prove by induction that for each group that we consider and for each stable cuspidal at infinity, we have
| (4.2) |
We induct on . Hence, the base case is when each . Such a group is a torus and hence has no non-trivial elliptic endoscopy. In particular, by Theorem 4.7 we have that
| (4.3) |
and hence it suffices to show that . By 5 since there is no non-trivial endoscopy, each and hence is the trivial character for all . The result follows.
We now settle the inductive step. Suppose we have shown for each satisfying . Suppose that satisfies . Pick a function that is stable cuspidal at infinity and for each elliptic endoscopic datum of we pick by Lemma 4.4 a transfer that is stable cuspidal at infinity.
Then we can write Theorem 4.7 in the form
where for each nontrivial elliptic endoscopic group appearing in the sum on right-hand side, we have verified by inductive assumption.
To conclude, it suffices to show that we have an equality
We prove this by arguing as in [Tai19, pg30] (cf [Kot84, §12]). Indeed, we have
Now, we apply at each place the endoscopic character identity we proved in Section 3 and argue as for the equation [Tai19, (11), pg30] to get that the above equals
Now we use that
is if and otherwise to get that the above equals
which equals as desired.
4.3. Some special global liftings
We remind that parameters with a dot above are global parameters. Now consider a supercuspidal -parameter. We denote the -parameter of obtained from by the projection . There is a (standard) base change morphism :
| (4.4) |
Denote by the image of by this morphism. Then is just the restriction of to . Since acts trivially on , if we denote the projection of to , then it is an dimensional representation of and moreover is the the image of by the (standard) base change morphism.
Since is a supercuspidal -parameter (in particular, a discrete -parameter), the group is finite (see [Kot84, Lemma 10.3.1]) and we can write where the are irreducible and pairwise distinct. By the computation in [Kal+14, page 62, 63], all the are conjugate-orthogonal and we have
Moreover, the group embeds diagonally into . Furthermore and . By Lemma 2.15 we have
and
Let be a discrete global -parameter of . The corresponding -packet consists of automorphic representations of whose central character is and whose restriction to is an automorphic representation in the -packet of . Again, we denote by the isobaric sum of automorphic representations of corresponding to . As in the local case, we see that is finite and by [Kal+14, page 69] we have then
with the group embedded diagonally into and an isomorphism . Thus
We say that a global parameter is a global lifting of if we have = where is the localization at . In this case, there exist morphisms , and . Since the the local and global parameters and are discrete, these maps are injective (see [Mok15, page 28-31] for more details). In this section, we construct some global liftings such that the above maps , and have some special properties.
4.3.1. First construction
(c.f. Lemma 4.2.1 in [Kal+14])
We choose an auxiliary place of which splits over as . Therefore is isomorphic to . By [Shi12, Theorem 5.7], there exists a cuspidal automorphic representation of satisfying the following properties
-
is discrete series corresponding to a regular highest weight and with sufficiently regular infinitesimal character in the sense of [Ngu19, Def. 2.2.10],
-
belongs to the packet ,
-
is a supercuspidal representation of .
Note that such a will be cohomological by the first condition and the remark at the end of § of [Kot92].
By Lemma 4.1.2 of [GL16], we can extend to an algebraic cuspidal automorphic representation of . Furthermore, we can assume that is cohomological since is.
Consider the exact sequence
Since belongs to the packet , the central character and the central character of any representation in must agree on . The map restricted to has kernel equal to so that factors to give a character of which (since is odd) is the norm subgroup . We can choose a lift of this character to and hence we conclude that there is some character such that belongs to the packet .
There is an isomorphism of topological groups
Then there is a character of such that is trivial on , and . This character descends to a Hecke character of such that where is an unramified character and is trivial. In particular if we denote then it is still cohomological (since is) and the local representation belongs to the packet up to an unramified character twist.
Therefore the global parameter is a globalisation of , up to an unramified twist (where is the global parameter of and corresponds to the central character of ). Since has sufficiently regular infinitesimal character, is generic (Lemma 5.11). The last condition implies that is a cuspidal automorphic representation of which is self-dual and conjugate orthogonal. Therefore we have ([Kal+14, page 69]) so . The above second condition implies that is a global lift of . Since the map is injective, we see that is the diagonal embedding of into .
Moreover since and , the map is the trivial map. The group is also trivial and the map is given by
4.3.2. Second construction
. (We adapt the proof of Lemma 4.4.1 in [Kal+14])
Consider an element whose image in is denoted by . We can suppose that for and for . Denote and (where and ). Since all the are conjugate orthogonal, by [Mok15, Lemma 2.2.1], the -parameters resp. come from -parameters resp. of unitary groups resp. by the base change map (see (4.4)). Now as in the first construction, for these -parameters we can construct cuspidal automorphic representations resp. , of resp. . Since these automorphic representations are self-dual and conjugate-orthogonal, the isobaric sum factors through the base change map ([Kal+14, Proposition 1.3.1], [Mok15, page 27]). Denote this global -parameter of by . Again by [Kal+14, page 69] we know that . As in the first construction, the -parameter is generic (Lemma 5.11) and is a global lift of . Moreover the localization map is defined as follows
Now, taking the quotient by we see that and the map is given by :
Now take an automorphic representation of in the packet . By the same argument as in the first construction, we can extend it to an automorphic representation of such that belongs to the packet up to an un-ramified twist. Thus the global parameter of is a globalisation of . We have then and .
Furthermore, if the element belongs to then belongs to since the map is injective and restricts to a map from to . Therefore, we have the following description of the map
4.4. Galois representations associated to global cohomological generic parameters
We have fixed a quadratic imaginary extension of . In this subsection, we associate representations of to certain global parameters.
Let be a global parameter of a global unitary similitude group . In particular is a global parameter for the corresponding unitary group . We suppose further that the localization at infinity is regular and sufficiently regular so that will be generic.
We first associate a representation to . Associated to , we have the quadratic base change, , which is an automorphic representation of . Since the global parameter is generic, the representation is of the form where are self dual cuspidal generic and cohomological automorphic. Now, fix a place of and an isomorphism . Then by [Shi11, Theorem 1.2], for each representation there is a unique -adic -representation such that for each place of not dividing , we have the following isomorphism of Weil-Deligne representations
where is the local parameter associated to under the local Langlands correspondence.
Similarly, if we denote , then for each place dividing and not dividing , we have that
Denote by the localization of at . By the definition of localization map of global parameters ([Mok15, p. 18, 19]), we see that the local -parameter (not necessarily bounded) corresponding to is if is split in . If is inert in then and is a quadratic extension of . In this case corresponds to the image of via the base change map and equals .
The central character gives rise to a character of and hence an -adic character . The pair then gives us a morphism
From the local-global compatibility properties of , we conclude that for every place dividing a prime , the restriction equals where is the localization of the global parameter at the prime .
5. Rapoport–Zink spaces and an averaging formula
5.1. Rapoport–Zink spaces
We continue with our fixed prime number as before. Let the completion of the maximal unramified extension of and the geometric Frobenius automorphism of .
We will be interested in the subset of associated with a minuscule cocharacter as defined in [Kot97, §6.2]. The Bruhat ordering on the image of the Newton map induces a partial order on .
Definition 5.1.
A Rapoport–Zink data of simple unramified unitary PEL type consists of the following
-
an unramifed extension of degree of with a non trivial involution ,
-
a -vector space of dimension ,
-
a symplectic Hermitian form for which there is a self-dual lattice ,
-
a conjugacy class of minuscule cocharacters where is the similitude unitary group defined over by
for all -algebra and .
-
a -conjugacy class . We also suppose that where is the similitude factor of .
Denote then the cocharacter is determined by the integral couples such that and .
To such a data, we associate the isocrystal with an action and an alternating non degenerate form where . By Dieudonne’s theory, the isocrystal corresponds to a -divisible group defined over provided with an action of and a polarization .
Theorem 5.2.
[RZ96, Theorem 3.25] Let be the functor associating to each scheme on which is locally nilpotent the set of pairs where:
-
-
is a -divisible group over with a -principle polarization and an action such as the Rosati involution inducing by induces on .
-
-
A -linear quasi-isogeny such that is a -multiple of in . (here, is the modulo reduction of ).
We also require that satisfies the Kottwitz determinant condition. More precisely, under the action of , we have a decomposition: then is locally free of rank . This functor is then represented by a formal scheme defined over .
In order to introduce the usual level structures, we work with the rigid generic fiber of over . We set , a maximal compact subgroup of .
Definition 5.3.
Let be the local system defined by the -adic Tate module of the universal -divisible group on . For we define as the etale covering of which classifies the - trivializations modulo of by . We also require that the trivialization preserves the alternating form up to .
We have, in particular, that . We then get a tower of analytic spaces on provided with finite étale transition maps (for )
which forget the level structure. The map is Galois of Galois group if is normal in .
Let be the group of -linear quasi-isogenies of such that is a - multiple of . The group acts on the left on by the formula
We say that a simple unramified unitary Rapoport–Zink datum is basic if the associated group is an inner form of . The above datum is basic if and only if is the unique minimal element in . In this case, we also say that is basic.
Let be a prime number. Let be a level. As in [Far04, remark 2.6.3] we denote:
where runs through the relatively compact open subsets of .
The group acts on and this action extends to so that acts on . Since is odd, the reflex field of the conjugacy class of is . We can also define an action of the Weil group on these cohomology groups thanks to the Rapoport–Zink descent data defined as below.
Let the relative Frobenius automorphism with respect to . We denote by the Frobenius morphism induced on . For a -divisible group defined over , we note the relative Frobenius morphism. We construct a functor isomorphism as follows.
For a scheme on which is nilpotent as well as a point , the point associated in is defined as follows: with the action of , with the polarization and . The isomorphism of functors is the Rapoport–Zink descent data associated with . Since the descent data commute with the action of , the groups has an action of . In addition, when varies, the system has an action of . Thus, this system has an action of . Let be an admissible -adic representation of , we define
By [Man08, Theorem 8], the are admissible and are zero for almost all . Finally, we define the homomorphism of Grothendieck groups by
5.2. An averaging formula for the cohomology of Rapoport–Zink spaces
In this section we deduce an averaging formula for the cohomology of Rapoport–Zink spaces using the results of [BM21].
We begin with some endoscopic preliminaries. To state the formula, we need the following notion of endoscopic data for Levi subgroups.
Definition 5.4.
(cf. [BM21, Definition 2.18]) Let be a Levi subgroup. We say that is an embedded endoscopic datum of relative to if is a refined endoscopic datum of and the restriction gives a refined endoscopic datum of .
We say that two embedded endoscopic data and are isomorphic if there exists an isomorphism of refined endoscopic data and whose restriction to gives an isomorphism of and . We denote the set of isomorphism classes of embedded endoscopic data of relative to by .
We now fix a refined elliptic endoscopic datum of . Note that for each standard Levi subgroup , there is a natural forgetful map
We define to be the set of embedded endoscopic data such that and whose class lies in the fiber modulo the relation that two data and are equivalent if there exists an inner automorphism of inducing an isomorphism of the embedded endoscopic data.
Fix a maximal torus and define . By the comment before [BM21, Proposition 2.27], we have that the set is parametrized by the set of double cosets where and are the Weyl groups of and respectively and is defined in [BM21, Definition 2.23].
Finally, for an inner form of , we define the subset to consist of those equivalence class of endoscopic data such that there exists a maximal torus that transfers to .
We now fix and let be a decent lift. We get a standard Levi subgroup of and an extended pure inner twist of . Let (where is the maximal split torus in the center of ) denote the image of the Newton map applied to . Fix an elliptic endoscopic group of and a set, , of representatives of . Furthermore, for each we may choose an extension of . We also get a natural map . Then we define to be the composition of with this map.
We then make the following definition.
Definition 5.5.
We define
| (5.1) |
by
| (5.2) |
where denotes the transfer of distributions from to and denotes the Grothendieck group of admissible representations of and is the subgroup of consisting of those elements with stable distribution character.
Our aim in this subsection is to establish the theorem below using the results of [BM21].
Theorem 5.6.
Let be a refined elliptic endoscopic datum of . Let be a supercuspidal Langlands parameter such that there exists a Langlands parameter of with . Then we have the following equality in :
where the first sum on the right-hand side is over irreducible factors of the representation and is the -isotypic part of .
This theorem is [BM21, Theorem 6.4]. To verify this theorem we essentially just need to check a number of hypotheses from [BM21].
First, we need a global group such that and such that there exists a Shimura variety of PEL type such that the global conjugacy class of cocharacters of associated to localizes to the conjugacy class of . Since is assumed minuscule, its weights are equal to and . In particular, is determined by a pair such that and denotes the number of weights and denotes the number of weights.
We fix an odd positive integer and define to be the group coming from the hermitian form as in Section 2. Following [Mor10, §2.1], we have a PEL Shimura for this group (in Morel’s notation, this is the datum ). As we observed in Section 2, the group can be equipped with the structure of an extended pure inner twist . As in [BM21a], this twist gives us for each refined endoscopic datum of a normalized transfer factor at each place .
We observe that, in accordance with [BM21, §4.1, §5.1 ], we have is simply connected and is unramified. The center is isomorphic to which has split rank equal to . Since is an imaginary quadratic extension, the split rank of also equals .
We verify that satisfies the Hasse principle. By [Kot84, Lemma 4.3.1] it suffices to show that vanishes but this latter group is trivial.
We now note an important difference between the exposition in [BM21, §4] and our current situation. This is that the group will not in general be anisotropic modulo center. For this reason, the stabilization of the trace formula carried out in that paper does not carry over exactly to our case. Instead, we use Morel’s work on the cohomology of these Shimura varieties to establish the desired stabilization. However, Morel’s work is on the intersection cohomology of Shimura varieties whereas we need to study compactly supported cohomology. We introduce some necessary notation.
Let be a compact open subgroup that factors as where is a hyperspecial subgroup of . Following the notation of [Mor10], we let be the Baily-Borel-Satake compactification of the Shimura variety . Fix primes and and an algebraic representation of . Choose the highest weight of to be ‘sufficiently regular’ in the sense of [Ngu19, Def. 2.2.10]. Let be a number field containing the field of definition of and let be a place of over . Then let denote the intersection complex on with coefficients in . Then we define an element in the Grothendieck group of representations by
Similarly, we let be the local system on associated to and define the element in the Grothendieck group of representations by
Let and assume that factors as . Fix a place of above and let be a lift of of the geometric Frobenius at . We will often consider functions such that where is stable cuspidal and at some finite place , we have and is cuspidal. For instance, could be a coefficient for a supercuspidal representation. Recall that these terms were defined in Section 4.2.
Lemma 5.7.
Suppose that is cuspidal at a finite place. Then we have .
Proof.
Indeed, this follows from the fact we have a natural -equivariant morphism for each
| (5.3) |
and the cuspidal part of lies in the image of this map (see, for instance, [Ngu19, Proposition 3.2]). ∎
We now remark on the definitions of the functions defined in [BM21, §4] and [Mor10, §6.2] respectively. Morel’s normalization of transfer factors away from and is arbitrary up to the global constaint given by [Kot86, 6.10b]. At the definitions of and coincide up to differences in transfer factor normalization. At , Morel normalizes her transfer factors as in [Kot90, pg180]. If one chooses a different normalization at , then Kottwitz explains ([Kot90, pg180-181]) how to modify the function by a constant such that it satisfies and analogous fundamental lemma formula. At , Morel uses the normalization given on [Kot90, pg184]. We can again modify the function by a constant so that it satisfies the same formulas. Hence, so long as one modifies the normalizations of the transfer factors at each place in such as way that the global constraint is still satisfied, one gets an analogous modification of the function such that it satisfies the same transfer formulas. By examining the constructions at each place, it is clear that if is modified to be compatible with our chosen normalization of transfer factors, then the functions and can be chosen to be equal.
Since the transfer of a cuspidal function is cuspidal [Art96, Lemma 3.4] and is stable cuspidal by definition, we have that satisfies the hypotheses of Lemma 5.7 and Lemma 4.6. In particular, we have the following proposition.
Proposition 5.8.
Suppose is a cuspidal at a finite place and factors as . Then
Proof.
At this point, we have finished using the work of Morel and have arrived at the formula [BM21, Equation (4.17)]. We now need to show that we can perform the destabilization procedure as in [BM21, §4.7]. To do so we need to prove that we have a sufficiently good theory of the Langlands correspondence for and its localizations. Globally, we will work with “automorphic parameters” in the style of [Kal+14] and [Art13a] and as we defined in 2.4. Since our ultimate goal is to prove a local formula, these parameters are sufficient for our purpose. We list the following properties we need and where these facts have been proven.
-
(1)
We need a construction of local Arthur packets of generic parameters at all localizations of and descriptions of the elements in each local -packet in terms of representations of the various centralizer groups (Theorem 2.16).
-
(2)
The local packets must satisfy the endoscopic character identities (Section 3).
-
(3)
A local generic -packet contains a -unramified representation if and only if the parameter is unramified. In the case that an -parameter is unramified, this -unramified representation is unique (Subsection 4.1).
-
(4)
We need a construction for global Arthur packets for generic “-cuspidal” parameters. These consist of parameters that are supercuspidal at some fixed local place . We need a description of the global -packet in terms of the local packets (Section 2.4).
- (5)
-
(6)
We need a decomposition of the generic -cuspidal part of in terms of global Arthur packets and this decomposition should satisfy the global multiplicity formula (Section 2.4).
-
(7)
We need to attach to a global generic parameter a global Galois representation whose localizations at each place are compatible with the corresponding localization of the global parameter (Subsection 4.4).
With these properties in hand, we can now apply the results of Section 4.2 (which is analogous to [BM21, Assumption 4.8]) to get
Following the argument of [BM21, §4.7], we derive the formula
| (5.4) |
in the Grothendieck group of -modules where the first sum is over equivalence classes of -cuspidal parameters.
Suppose is a representation of appearing in the cohomology of Shimura varieties whose associated automorphic -parameter is -cuspidal. We need to compute the -isotypic part, , of . To do so, we apply the argument at the end of [BM21, §4.7] along with the following lemma.
Lemma 5.9.
Suppose is an admissible representation of such that the -parameter at is supercuspidal. There exists a compact open such that and factors as and there exists a -cuspidal function such that and for any with nontrivial -invariants and appearing in either or
we have
Proof.
The set of isomorphism classes of satisfying the above conditions is finite. Hence we can find a function such that for all unless in which case the trace is nonzero. Now, at we have that is supercuspidal and so we choose to be a coefficient for . Then has the desired properties. Indeed any not isomorphic to will differ from either at or away from it, and hence . ∎
We now need to show that a similar result holds for the compactly supported cohomology of Igusa varieties. In this case the stabilization in [BM21, §5] does not require that is anisotropic modulo center and so that argument goes through essentially unchanged. The only difference is that we only prove the equality of and in the case that is cuspidal at a finite place. In particular, this means that when we compute the -isotypic part of the cohomology of Igusa varieties, we need the following lemma.
Lemma 5.10.
Suppose is an irreducible admissible representation of such that the corresponding local -parameter at is supercuspidal. Let be a compact open subgroup such that and factors as . Let be a finite set of isomorphism classes of irreducible admissible representations such that . Then there exists a -cuspidal function that is acceptable in the sense of [Shi09, Definition 6.2] such that factors as and for if and only if .
Proof.
Consider the linear map from -cuspidal functions to given by where . It suffices to show this map is surjective. If the map is not surjective, then its image is a proper subspace and hence lies in a hyperplane of . Hence we can find some element such that for all -cuspidal , we have .
At this point, we have verified the assumptions of § and § of [BM21]. It remains to check those of §. We first note that the Mantovan formula is known for the PEL type Shimura varieties we consider. Indeed this is [LS18, Theorem 6.32].
It remains to check Assumptions and of [BM21]. We record some useful lemmas.
Lemma 5.11.
Suppose is a discrete automorphic representation of contained in an -packet . Suppose further that the infinitesimal character of is sufficiently regular in the sense of [Ngu19, Def. 2.2.10]. Then the -parameter associated to is generic.
Proof.
Standard. For instance see [Kal+14, Lemma 4.3.1]. ∎
Lemma 5.12.
Suppose is a discrete automorphic representation of contained in an -packet and such that has sufficiently regular infinitesimal character. Then this is the unique -packet containing . Moreover, if is another discrete automorphic representation of such that has sufficiently regular infinitesinal character and such that then and are in the same -packet.
Proof.
Suppose that belongs to two -packets with associated -parameters and . Since and correspond to the central character of , they are equal. We need to show that , are also equal. At almost all finite unramified places where is unramified, the localizations and are equal. Indeed, our sufficiently regular assumption implies that these parameters are generic. Following [Mok15, pg 189], these local parameters factor through where is the minimal Levi subgroup of and correspond to the same spherical parameter of (for more details, see [Mok15], page ). This implies that and give rise to the same Hecke string. Then, by [JS81], [Art13, Theorem 4.3] we see that and are equal. It is clear that the second statement also follows from exactly the same argument. ∎
Before verifying Assumptions on and of [BM21], we need to understand the effect of an unramified twist on the map. Let be the similitude factor character. For non-basic, the group is an inner form of a Levi subgroup of . Then the similitude character restricted to can be transferred to . Hence by abuse of language, we also denote the corresponding character on .
Lemma 5.13.
Let be an unramified unitary Rapoport–Zink PEL datum and suppose is an unramified character. Then the following holds in
Proof.
This lemma is an analogue of [Shi12a, Lemma 4.9] and the same proof applies in our situation. Thus we just briefly give an idea of how to proceed.
Define a character of by
Then we prove that there is an isomorphism of -vector spaces
compatible with the action of .
Notice there is a -equivariant map ([RZ96, sec. 3.52]) and moreover there is a natural way to define an action of on such that the map is equivariant with respect to ([Far04, remark 2.6.11]).
We can prove the lemma by using the fact that acts trivially on and
where is the inverse image of by and is the subgroup of that acts trivially on .
∎
We can now settle Assumption in the cases we need. Let be a representation of and a discrete automorphic representation of such that . Suppose further that appears in either the formula for the cohomology of Igusa varieties or . Then since has sufficiently regular infinitesimal character, it follows that the same is true of . Now suppose is a discrete automorphic representation of appearing in either of the above formulas and such that . We then have by Lemma 5.12 that and are in the same packet.
We now tackle Assumption 6.3 of [BM21]. For a fixed supercuspidal representation with local parameter , we have the local centralizer group . For any global -parameter such that , we have a natural embedding . The formula immediately before Assumption 6.3 of [BM21] includes a sum indexed over a set of representatives of . We must show that we can pick different globalizations, , of to derive the formula below assumption 6.2 for each element of .
Suppose first that projects to the identity element of . Then by Construction 4.3.1 we can choose so that the image of in is and the packet differs from the packet by an unramified twist of the form . Then we simply pick to contain the unique element of mapping to . This establishes the formula for projecting to the identity of . By Lemma 5.13, we obtain the formula for projecting to the identity of .
Now suppose pick that projects to a non-identity element. By Construction 4.3.2, we may choose such that the image of in is precisely and the packet differs from the packet by an unramified twist of the form . Choose to contain the unique elements mapping to and denote these and respectively. Then each side of the formula before Assumption 6.2 for the parameter has two terms indexed by and respectively. Again, by Lemma 5.13, we can draw the same formula for . The terms are already known to be equal by the previous paragraph. It therefore follows that the terms are equal as well.
This completes the verification of Theorem 5.6.
6. Proof of the main theorem
To prove the Kottwitz conjecture for the groups we consider, we use Theorem 5.6.
First of all, we show that
for non basic, an elliptic endoscopic datum of and a supercuspidal parameter.
Indeed, the parameter is again a supercuspidal -parameter. In particular, the representations are supercuspidal. Now by definition we have
As is non-basic, the group is an inner form of a proper Levi subgroup of . Suppose that = . In this case equals and is isomorphic to an endoscopic group of . This is a contradiction because by the classification of the endoscopic groups of and its Levi subgroups, we know that the elliptic endoscopic groups of are not endoscopic groups of any proper Levi subgroup of . We conclude that is a proper parabolic subgroup of so that
as desired.
Now, for basic, the main formula of Theorem 5.6 becomes
Simplifying the left-hand side using the endoscopic character identities gives
Now, fix and multiply the above equation by . Then one can check that both sides only depend on the projection . We then average over . This gives equality between
and
Now, for any irreducible representation of , we have is if is trivial and otherwise. Hence we get the equality
We now isolate the term for a fixed and representation . It is
which equals
This equals
Hence summing over , we get
In conclusion we have proven
Theorem 6.1 (Kottwitz Conjecture).
For irreducible admissible representations of with supercuspidal -parameter , we have the following equality in :
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