The Kottwitz conjecture for unitary PEL-type Rapoport–Zink spaces
-
Alexander Bertoloni Meli
and
Kieu Hieu Nguyen
Abstract
In this paper we study the cohomology of PEL-type Rapoport–Zink spaces associated to unramified unitary similitude groups over
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 ([45]) introduced p-adic analogues of Shimura varieties defined as moduli spaces of p-divisible groups with additional structures. The
We now describe our results in more detail. One considers triples
Attached to the group
as a representation of
We now further assume that we can give
given by
in the case when b is basic and ρ is an irreducible admissible representation of
The Kottwitz conjecture states the following:
Conjecture 1.1 (Kottwitz conjecture, [44, Conjecture 7.3]).
For irreducible admissible representations ρ of
where
We have not defined all the notation appearing in this conjecture, but this is described in detail in Section 5. One can extend the conjecture to general
The main goal of this paper is to prove Conjecture 1.1 when
Prior to this work, such a local Langlands correspondence was known for unitary groups by the works [40, Theorems 2.5.1 and 3.2.1] and [25, 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 2.8 we use Kaletha’s results in [23] 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
Theorem 1.2 (Theorems 2.8 and 2.12, Section 3.2).
There exists a local Langlands correspondence for odd unramified unitary similitude groups that satisfies the properties of [25, Theorem 1.6.1], in particular, the endoscopic character identities. By construction, the correspondence is compatible with that of [25, 40] via restriction of irreducible admissible representations to
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 [53] and crucially uses the endoscopic averaging formulas of [6]. We briefly describe these formulas for a connected reductive group
whose precise definition is given in Section 5.2. We remark that
where the first sum on the right-hand side is over irreducible factors of the representation
Note that equation (1.1) is an averaging formula in the sense that it gives a description of
The averaging formula is derived in [6] for PEL-type groups under a substantial list of assumptions. In this paper, we verify these assumptions for discrete parameters and hence prove:
Theorem 1.3.
For discrete L-parameters ϕ of
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
and the trace formulas for Shimura and Igusa varieties ([29, Theorem 7.2], [49, Theorem 13.1], [50, Theorem 7.2]). We denote respectively by
To carry out this approach, we need to define global A-parameters of
One important step in the proof of the averaging formula is the process of stabilization and destabilization of the trace formula for the cohomology of Shimura and Igusa varieties following [29] and [50]. The goal is to relate both sides of 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
Once we have done the destabilization step, we can put everything into equation (1.2) and derive the averaging formula. However, at this point 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
With the endoscopic averaging formula in hand, we prove the Kottwitz conjecture in Section 6. To do so, we observe that
2 Automorphic representations
2.1 The groups
Let F be a field of characteristic 0, E a quadratic extension of F and fix an algebraic closure
Then we give
We get an action of
We also need to define slightly more general groups
In this paper, we only consider the case where F is one of
The global groups we consider in this paper will be inner forms of
and
for any
In this paper we will assume that n is an odd number and that the localization
where
for
Recall that a reductive group
The set
Lemma 2.1.
The groups
Proof.
In the case that
so this is indeed the case. ∎
We also consider extended pure inner twists for connected reductive groups over
for F a number field and
for F a finite extension of
A maximal torus T defined over
The maximal split subtorus A of T is isomorphic to
where
where
A minimal parabolic subgroup of
From the description of unitary similitude groups, we see that there is an embedding
The tuple
Note that we can identify
and
respectively (see [41, p. 38] for details).
A maximal torus defined over
If we denote I, resp. J the set of indexes i such that
The relative Weyl group is
Lemma 2.2.
We have the equality
where
Proof.
For simplicity, we prove the equality when
We just need to show that
We see that
Moreover, by the above injection
Therefore
We now recall some facts from the theory of endoscopy.
Definition 2.3 (cf. [6, Definition 2.1]).
A refined endoscopic datum for
Suppose that
We record a set of representatives for the isomorphism classes of refined elliptic endoscopic data for
In the unitary case, 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
We further require that
In each case, we can extend η to get a map
2.2 The Langlands correspondence for unitary groups
In this subsection, we will review the Langlands correspondences for unitary groups in the local and global settings, largely following the works of [25, 40].
2.2.1 Local unitary groups
We start by considering a local field
and the group
Definition 2.4.
A local L-parameter for a connected reductive group
We denote
For global classifications, we will also need the notion of a local Arthur parameter.
Definition 2.5.
A local A-parameter for a connected reductive group
We denote by
We also have a “standard base change” morphism of L-groups ([40, p. 9]):
which allows us to identify the L-parameters of
given by projection of
For each A-parameter
Remark 2.6.
For
(although note that it is possible that
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
The following theorem gives the local Langlands correspondence for extended pure inner twists of
preserving the chosen
defines a character
Remark 2.7.
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
Theorem 2.8 ([25, Theorem 1.6.1], [40, Theorems 2.5.1 and 3.2.1]).
Fix a field
For each generic
where
The morphism
For each
Let
where
We have
and
Proof.
The contents of this theorem appear in the works of Mok ([40, Theorem 2.5.1]) and Kaletha–Minguez–Shin–White ([25, 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 results to define a geometrically normalized correspondence.
For
We now define
where if
We now verify the endoscopy character identity which is (4) in the theorem. To this end, fix
We now apply the endoscopic character identity proven in [25, Theorem 1.6.1] to get that the above equals
Now, since
as desired. ∎
Proposition 2.9.
Continue with the notation as in Theorem 2.8. Let
Proof.
Thanks to the results in [25, 40], the arguments in [23, Theorem 4.8] also work in our case. Indeed, the group
of the triple
Then by [40, Theorem 3.2.1] the linear form
is the unique distribution on
for all
Once we have this characterization, the proof of [23, Theorem 4.8] works without any change since [23, Proposition 4.4, Corollary 4.5 and Corollary 4.7] are valid for quasi-split unitary groups. ∎
Proposition 2.10.
Let
the contragredient representation
Proof.
These results are completely analogous to [23, Theorem 4.9]. The same arguments carry over to our case since an analogue of [23, Theorem 4.8] is still valid for unitary groups (Proposition 2.9). ∎
We also have the following basic fact.
Lemma 2.11.
For
Proof.
Pick some compact open subgroup
The operator
where the third equality uses the fact that
When we consider global parameters, we will also need a version of Theorem 2.8 for
Theorem 2.12.
Fix a field
For each generic
where
Let
where
Proof.
We sketch the proof following ideas in [25, 40]. The proof of (1) is in [25, Section 1.6.4]. They choose a standard parabolic subgroup
They then define
where
From the explicit description of
and that
as both parametrize irreducible representations of
for
It remains to verify the endoscopic character identity. Fix a refined endoscopic datum
Now,
under the natural inclusion
using the above procedure.
We need to verify that the resulting pairing satisfy the endoscopic character identity. Let
and similarly for
for regular
We now claim that
as desired. Note that in the above we use that
Suppose
and hence, since f and
as desired. Note that we use that the number of conjugacy classes of γ in the stable class is the same for
2.2.2 Global unitary groups
We now consider the global situation. Recall that we have fixed a quadratic imaginary extension
Due to the lack of a global L-group, we rely on the cuspidal automorphic representations of
where
We denote
We have a standard base change map
Definition 2.13.
The set of global L-parameters
is a map such that
We remark that
There is a localization morphism
see [40, pp. 18–19]. More precisely, if v is a place of
According to Theorem 2.8 and Theorem 2.12, for each
We denote
Since the localization maps
where
Definition 2.14.
Let
Let
Following [25, Section 3.3], for each global parameter
where the sum runs over the set of Hecke strings c which map to
Theorem 2.15 ([25, Theorem 1.7.1]).
There is an isomorphism of
If
In particular, if π is an automorphic representation of the unitary group
In order to show that
By the definition of
Remark 2.16.
We remark that the proof of the theorem as we have stated it here is completed in [25] up to assumptions in [40]. For instance, the careful reader will note that [25, Theorem 1.7.1] requires that
2.3 The Langlands correspondence for unitary similitude groups
In this subsection, we want to transfer the results about automorphic representations from unitary groups to unitary similitude groups (with an odd number of variables).
2.3.1 Local unitary similitude groups
Let v be a finite place of
We give
is trivial. In particular, this induced extended pure inner twist need not be isomorphic to
Corollary 2.17.
There is a natural bijection between the set
We use this corollary to define A-packets of representations for
and a parameter
such that
We can view
as a product of
Conversely, each
We now define the A-packet associated to
We now use the internal structure of
Lemma 2.18.
With
where
Proof.
For
where the second equality comes from the action of σ on
We now prove that
and
Thus,
We now construct a pairing
Let
We claim this character does not depend on our choice of
This map is induced by the determinant map on matrices and hence contains
Via
we get a character
Suppose that
is bijective onto
We have now proven:
Theorem 2.19.
Parts (1) and (2) of Theorem 2.8 and part (1) of Theorem 2.12 hold for
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 2.20 ([39, Section 8.4.4]).
Let
Corollary 2.21.
From the above description of local L-packets of
Remark 2.22.
Suppose that ϕ is as above and
2.3.2 Global unitary similitude groups
Fix a Hermitian form V and global group
Proposition 2.23 ([12, Section CHL.IV.C, Proposition 1.1.4]).
Fix
Let χ be an automorphic central character of
Now, by the proof of [12, Section CHL.IV.C, Proposition 1.1.4], we have
where
We now give a description of
Remark 2.24.
It would perhaps be possible to define these parameters and their centralizer groups in analogy with our definitions for
Definition 2.25.
Let
We now discuss localization. First, by the localization map for algebraic cocycles (see [32, Section 7]), the extended pure inner twists
Let
Similarly, we get a localization map
We now define
We associate to each
where
We showed previously that
Proposition 2.26.
For
We note that since we are assuming
Proof.
The left-hand side consists of all pairs
Hence we just need to show that
Remark 2.27.
For our purposes, we also need to generalize the above description of automorphic representations to the groups
3 Endoscopic character identities
Fix v a finite place of
if the endoscopic character identities hold for
if the endoscopic character identities hold for
We recall the statement of the endoscopic character identity for an extended pure inner twist
Let
where
According to a theorem of Harish-Chandra, the trace distribution
We remark that a Harish-Chandra character exists for parabolically induced representations
3.1 Endoscopic identities for
U
(
n
1
)
×
⋯
×
U
(
n
r
)
×
Res
E
/
ℚ
v
𝔾
m
In this subsection we use the notation
In fact, we will prove the following more general result. Fix quasi-split reductive groups
Fix
Then
We need to show that for two arbitrary
The packets
It is also a basic property of the Kottwitz sign that
Moreover, a function
and
Similarly, for every
and
In order to prove equation (3.2), it suffices to prove that for each
Lemma 3.1.
If
are
Proof.
Pick
where the sum is taken over the set of
By definition, for
Moreover, an element
and similarly
We will prove in Lemma 3.4 that
We can then rewrite the right-hand side of (3.4) as
and because
In other words, equation (3.3) is true. ∎
3.2 Endoscopic identities for
G
(
U
(
n
1
)
×
…
×
U
(
n
r
)
)
We now have the endoscopic character identities for
with kernel isomorphic to
We fix quasi-split groups
hence (after possibly modifying
A crucial input in the case we consider (where
given by pullback. The image of this map is the set of irreducible representations
Since the kernel of P is compact, it follows that any
where the middle equality holds by [43, equation (3.21)].
Analogously in the endoscopic case, we have a map
with kernel
We fix a refined endoscopic datum
giving a commutative diagram
We now fix an endoscopic datum of
as follows. We set
We now prove the following lemma.
Lemma 3.2.
Using the above normalizations, if
are
We begin by proving an auxiliary lemma.
Lemma 3.3.
For
Proof.
If
and clearly
We now aim to show that
Let
since the restriction of P to the first component is an injection. ∎
We now prove Lemma 3.2.
Proof.
We choose a strongly regular semisimple
Expanding this is equivalent to showing that
equals
Note that the kernels of
By Lemma 3.3, we can rewrite the equation above as
equals
In Lemma 3.8 we prove that there is an equality of transfer factors
Hence, the above equation reduces to
which is true by assumption. ∎
With this lemma in hand, we now prove the endoscopic character identities. Pick a parameter
Fix matching functions
Now, for ψ in the previous paragraph with packet
where on the right-hand side,
We showed above that there is a natural bijection
and we related the traces of corresponding representations. The pairing
is given as a product of the pairings for
Now, using that
Finally, we relate this to
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 Δ resp.
3.3.1 Transfer factors of a product
We temporarily return to the notation of Section 3.1. We denote by
We prove the following lemma
Lemma 3.4.
Let
Proof.
Each transfer factor is a product of terms
We state everything for
Now,
We denote by
We put
where the product is over all roots of S in
where the product is over all roots of
We define
By definition,
For the other terms in the definition of the transfer factors, we need to explain the notions of a-data and χ-data. A set of a-data for the set
which satisfies
and denote
satisfying the conditions
and if
Since
Now, we define
where the product is taken over the set
We have a similar formula for
In particular, we have
Next, we want to show that
To this end, for
Because
We recall the construction of λ for
Let
where the product runs over the subset
Now, we have
We see that
and we can take
We conclude that
We are now going to show that
The construction is as follows. First, we associate to the fixed χ-datum a
Next via the admissible isomorphism
gives another L-embedding
The term
In our case, we have
so it suffices to show that
Fix a Borel pair
To specify ξ, we have only to give a homomorphism
where
We then define
for
We recall briefly the construction of
This implies the desired product identity for
Finally, we show that
We have a natural isomorphism
that maps the class of
defining the above pairings. This implies the desired product formula. ∎
3.3.2 Transfer factors and changing the normalization
Lemma 3.5.
Let
Proof.
We consider first the ordinary endoscopic case. Suppose
Then we need to show that
Since
it suffices to show that the transfer factor
Recall that the transfer factor
which we need to use χ-data and a-data in order to define and moreover the transfer factors do not depend on the choices of χ-data and a-data.
By [33, Section 5.1], the transfer factor
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 χ-data and where the
Thus we have
Therefore
Since inverting the character φ leads to the inverse Whittaker datum
For the twisted endoscopic case, the same arguments still work. Indeed, in this case
Since
it suffices to show that the transfer factor
and the twisted transfer factor
Since
Moreover,
Thus we have
As in the standard endoscopy case, the second product is actually the twisted transfer factor
3.3.3 Endoscopy for
Res
E
/
ℚ
v
𝔾
m
We now study the endoscopy of
We must have
We now study transfer factors for the endoscopic datum
Lemma 3.6.
We have
Proof.
We will calculate each term in the definition of transfer factor. The virtual representation V in this case is 0 so that the factor
We now compute
The final term then contributes the factor
3.3.4 Transfer factors for
GU
and
U
×
Res
E
/
ℚ
v
𝔾
m
We use the notation of Section 3.2. We denote the Whittaker datum and extended pure inner twists of
Lemma 3.7.
Suppose
Proof.
This is [58, Lemma 3.6] adapted to the non-quasi-split 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 κ respect products, we get
By the functoriality of the Kottwitz map,
Hence we get
On the other hand, by [36, Lemma 4.4A], there is a character
Hence, it remains to show that
The characters λ and
We explain this diagram. The objects
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
Then the outer square fails to commute by the cocycle
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 v is a finite place of
Proposition 4.1.
Let
Proof.
Suppose
We mimic the proof of Lemma 4.1.1 in [54]. Denote by f the characteristic function of the standard special maximum compact subgroup of
In other words, the packet
Suppose now that
By the same argument we have
for every refined endoscopic datum
where
We now consider the case of general
4.2 On the hypothesis
S
T
ell
H
(
f
H
)
=
S
T
disc
H
(
f
H
)
In this subsection, we prove that for
We begin with some preparatory notation and lemmas. Let
Fix
Definition 4.2 (Stable cuspidal function at infinity).
We say that
vanishes outside
Definition 4.3 (Cuspidal function).
We say that
We record the following well-known lemma.
Lemma 4.4.
If
Proof sketch.
Due to [48], we can find a function
where ϕ is the L-parameter of π. Then F must be supported on finitely many discrete series packets since
We recall that
where the sum is over a set of representatives of the
Definition 4.5.
We define the term
where ψ is such that on (hence any) π in
Note that we have suppressed the term
Separately, we have for every Levi subgroup
for certain constants
We prove the following standard result.
Lemma 4.6.
Suppose
For any
If
Proof.
To prove the first part, we note that by definition, for
We now prove the second part. We first show that
where the sum is over discrete series L-packets of
Therefore we have
Now, by definition,
Since
where
Finally, we put everything together to get
The last equality follows from the fact that
Indeed,
To see this, first note that the set of conjugacy classes that are stably conjugate to
It then follows that
where the sum is over stable conjugacy classes in
Since
Suppose now that
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 [41, Theorem 5.4.1]
Let
where
Fix an odd positive integer n. By Proposition 2.23 and Remark 2.27 we have the following formula for each group
For a function
where
We will now prove by induction that for each group
We induct on
and hence it suffices to show that
We now settle the inductive step. Suppose we have shown
Then we can write Theorem 4.7 in the form
where for each non-trivial elliptic endoscopic group
To conclude, it suffices to show that we have an equality
We prove this by arguing as in [55, p. 30] (cf. [27, Section 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 [55, equation (11)] to get that the above equals
Now we use that
is 1 if
which equals
4.3 Some special global liftings
In this subsection, we work over
Denote by
Since
Moreover, the group
embeds diagonally into
and
Let
with the group
and
We say that a global parameter
4.3.1 First construction
(Cf. [25, Lemma 4.2.1].)
We choose auxiliary places
Note that such a Π will be cohomological by the first condition and the remark at the end of [30, Section 2].
By [17, Lemma 4.1.2], we can extend Π to an algebraic cuspidal automorphic representation
Consider the exact sequence
Since
There is an isomorphism of topological groups
Then there is a character
Therefore the global parameter
Moreover, since
Thus, we have proved the following lemma.
Lemma 4.8.
Let
We have
Any automorphic representation
4.3.2 Second construction
(We adapt [25, proof of Lemma 4.4.1].)
Consider an element
(where
Since all the
Taking the quotient by
Now take an automorphic representation Π of
Furthermore, if the element s belongs to
Thus we have proved the following lemma.
Lemma 4.9.
Let
We have
Any automorphic representation
4.4 Galois representations associated to global cohomological generic parameters
We have fixed a quadratic imaginary extension E of
Let
We first associate a
where
Similarly, if we denote
Denote by
The central character χ gives rise to a character of
From the local-global compatibility properties of ρ, we conclude that for every place
5 Rapoport–Zink spaces and an averaging formula
5.1 Rapoport–Zink spaces
We continue with our fixed prime number p as before. Let
be the completion of the maximal unramified extension of
We will be interested in the subset
Definition 5.1.
A Rapoport–Zink data of simple unramified unitary PEL type
an unramified extension
a
a symplectic Hermitian form
a conjugacy class of minuscule cocharacters
for all
a σ-conjugacy class
The cocharacter μ is determined by a pair of integers
To such a data, we associate the isocrystal
Theorem 5.2 ([45, Theorem 3.25]).
Let
X is a p - divisible group over S with a p - principle polarization
a
such that
We also require that
In order to introduce the usual level structures, we work with the rigid generic fiber
Definition 5.3.
Let
We have, in particular, that
(for
Let
We say that a simple unramified unitary Rapoport–Zink datum
Let
where V runs through the relatively compact open subsets of
The group
Let
For S a
By [38, Theorem 8], the
by
5.2 An averaging formula for the cohomology of Rapoport–Zink spaces
In this subsection we deduce an averaging formula for the cohomology of Rapoport–Zink spaces using the results of [6].
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. [6, Definition 2.18]).
Let
two embedded endoscopic data
We now fix a refined elliptic endoscopic datum
We define
Fix a maximal torus
By [6, comment before Proposition 2.27], we have that the set
Finally, for an inner form
We now fix
of η. We also get a natural map
It is clear that
We then make the following definition.
Definition 5.5.
We define
by
where
Our aim in this subsection is to establish the theorem below using the results of [6].
Theorem 5.6.
Let
be a discrete Langlands parameter such that there exists a Langlands parameter
where the first sum on the right-hand side is over irreducible factors of the representation
This theorem is [6, Theorem 6.4]. To verify this theorem, we essentially just need to check a number of hypotheses from [6].
First, we need a global group
We fix n an odd positive integer and define
We observe that, in accordance with [6, Sections 4.1 and 5.1], we have
We verify that
We now note an important difference between the exposition in [6, Section 4] and our current situation. This is that the group
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
Similarly, we define the element
Fix a place
Lemma 5.7.
Suppose that f is cuspidal at a finite place. Then we have
Proof.
Indeed, this follows from the fact we have a natural Γ-equivariant morphism for each i,
and the cuspidal part of
We remark on the definitions of the functions
Morel’s normalization of transfer factors away from p and
Since the transfer of a cuspidal function is cuspidal [3, Lemma 3.4] and
Proposition 5.8.
Suppose
Proof.
By Lemma 5.7 and [41, Theorem 7.1.7] (keeping in mind her remark that the result holds for general p) we have
Now, we apply Lemma 4.6 to the right-hand side to get the desired equality. ∎
At this point, we have finished using the work of Morel and have arrived at the formula [6, equation (4.17)]. We now need to show that we can perform the destabilization procedure as in [6, Section 4.7]. To do so we need to prove that we have a sufficiently good theory of the Langlands correspondence for
We need a construction of local Arthur packets of generic parameters at all localizations of
The local packets must satisfy the endoscopic character identities (Section 3).
A local generic A-packet contains a K-unramified representation if and only if the parameter is unramified. In the case that an A-parameter is unramified, this K-unramified representation is unique (Section 4.1).
We need a construction for global Arthur packets for generic “v-cuspidal” parameters. These consist of parameters that are supercuspidal at some fixed local place v. We need a description of the global A-packet in terms of the local packets (Section 2.3.2).
We need a decomposition of the generic v-cuspidal part of
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 (Section 4.4).
With these properties in hand, we can now apply the results of Section 4.2 (which is analogous to [6, Assumption 4.8]) to get
Following the argument of [6, Section 4.7], we derive the formula
where the first sum is over equivalence classes of v-cuspidal parameters and
We now define
in the Grothendieck group of
Lemma 5.9.
Let
we have
Proof.
The set
Following the argument at the end of [6, Section 4.7], we conclude that
in
So far we have discussed the computation of the cohomology of Shimura varieties and arrived at equation (5.1). We now want to carry out an analogous computation for the compactly supported cohomology of Igusa varieties as in [6, Section 5] starting from the stable trace formula as in [50, Theorem 1.1]. In this case the stabilization in [6, Section 5] does not require that
Lemma 5.10.
Let
be a compact open subgroup such that
Proof.
Consider the linear map from v-cuspidal functions to
where
Now, by the argument of [49, Lemma 6.4] and also [49, Lemma 6.3] it follows that every
By the argument of Lemma 5.9, we can find an
At this point, we have verified the assumptions of [6, Sections 4–5]. It remains to check those of [6, Section 6]. We first note that the Mantovan formula is known for the PEL-type Shimura varieties we consider. Indeed, this is [34, Theorem 6.32].
It remains to check [6, Assumptions 6.2 and 6.3]. We record some useful lemmas.
Lemma 5.11.
Suppose π is a discrete automorphic representation of
Proof.
Standard. For instance see [25, Lemma 4.3.1]. ∎
Lemma 5.12.
Suppose π is a discrete automorphic representation of
Proof.
Suppose π belongs to two A-packets with associated A-parameters
Before verifying [6, Assumptions 6.2 and 6.3], we need to understand the effect of an unramified twist on the
Lemma 5.13.
Let
Proof.
This lemma is an analogue of [53, 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 χ of
In
Then we prove that for each level
such that the resulting bijection of direct limits
is compatible with the action of
Note that there is a
and moreover there is a natural way to define an action of
where
We can now settle [6, Assumptions 6.2] in the cases we need. Let
We now tackle [6, Assumptions 6.2]. For a fixed discrete series representation
Suppose first that
By Lemma 4.8, we can choose
Now suppose we pick
and the packet
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. We remark that since
First of all, we show that
for b non-basic,
Indeed, the parameter
As b is non-basic, the group
as desired.
Now, for b basic, the main formula of Theorem 5.6 becomes
The endoscopic character identity (Equation (3.1)) and definition of
Substituting into the previous equation gives
Now, fix
We then average over
and
Now, for any irreducible representation χ of
We now isolate the term for a fixed
We would like to relate this to the term
which appears in the statement of the Kottwitz conjecture. Note that the dimension does not change if we tensor both
where we have the same abuse of notation as before that
This equals
and summing over ρ, we get
In conclusion we have proven
Theorem 6.1 (Kottwitz conjecture).
For irreducible admissible representations
in
Funding statement: 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
A Some computations with
GU
(
3
)
In this appendix we use the averaging formula (Theorem 5.6) to compute
Let
The torus T is a Levi subgroup of
Let
Let P be the parabolic subgroup with Levi factor equal to T and such that μ is dominant with respect to the positive root system determined by P. Let
The modulus character
We now describe the set
We also have
Hence
Let ϕ be an
We first consider the averaging formula for the trivial endoscopic group. This gives
On the other hand, by the Harris–Viehmann conjecture (known in our present situation since
for a certain
Now, we consider the non-trivial endoscopic group
with
We need to compute
It follows that
Let
where the
To figure out the right-hand side of the averaging formula, we need to understand which representations of the centralizer group of ϕ correspond to
In the unitary case, [46, Proposition 13.1.3 (d) ] indicates that the unitary group representations corresponding to
Hence, the endoscopic averaging formula becomes
Using Harris–Viehmann to compute
Hence,
To finish the computation, we use that
Proposition A.1.
We have
and
Additionally, we can consider the A-parameter
In this case, the trivial endoscopic group gives us the formula
and hence
In the non-trivial endoscopic case, we get
Hence,
Using these equations, we deduce
and
We briefly explain how these results relate to Ito and Mieda’s computation in [21]. Firstly, we note that our definition of
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. Finally, we thank the referees for their careful readings and helpful suggestions.
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Articles in the same Issue
- Frontmatter
- The Kottwitz conjecture for unitary PEL-type Rapoport–Zink spaces
- Milnor K-theory of p-adic rings
- Scrollar invariants, syzygies and representations of the symmetric group
- Residual categories of quadric surface bundles
- Uniqueness of entire graphs evolving by mean curvature flow
- Moduli spaces of complex affine and dilation surfaces
- Strominger connection and pluriclosed metrics
- Lattice cohomology and q-series invariants of 3-manifolds
Articles in the same Issue
- Frontmatter
- The Kottwitz conjecture for unitary PEL-type Rapoport–Zink spaces
- Milnor K-theory of p-adic rings
- Scrollar invariants, syzygies and representations of the symmetric group
- Residual categories of quadric surface bundles
- Uniqueness of entire graphs evolving by mean curvature flow
- Moduli spaces of complex affine and dilation surfaces
- Strominger connection and pluriclosed metrics
- Lattice cohomology and q-series invariants of 3-manifolds
