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Stability estimates for solving Stokes problem with nonconforming finite elements

Paolo Sentinelli

Comptes Rendus
Mathématique
Paolo Sentinelli
Reflections and quotients in Coxeter groups
Volume 363 (2025), p. 235-239
Online since: 8 April 2025
https://doi.org/10.5802/crmath.714
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2025, Vol. 363, p. 235-239
https://doi.org/10.5802/crmath.714
Research article / Article de recherche
Algebra, Combinatorics / Algèbre, Combinatoire
Reflections and quotients in Coxeter groups
Réflexions et quotients dans les groupes de Coxeter
Paolo Sentinelli a
a Dipartimento di Matematica, Politecnico di Milano, Milan, Italy
E-mail: paolosentinelli@gmail.com
Abstract. We present a formula relating the set of left descents of an element of a Coxeter group with the sets
of left descents of its projections on maximal quotients indexed by simple right descents. This formula is an
instance of a general result involving quotients and descents.
Résumé. Nous présentons une formule reliant l’ensemble des descentes gauches d’un élément d’un groupe
de Coxeter aux ensembles des descentes gauches de ses projections sur les quotients maximaux indexés par
les descentes droites simples. Cette formule est un exemple d’un résultat général impliquant les quotients et
les descentes.
Manuscript received 15 May 2024, revised and accepted 19 December 2024.
1. Introduction
A reflection t in a Coxeter system (W,S) is a conjugate of a generator, and t is a left descent of
an element w ∈ W if ℓ(t w) < ℓ(w), where ℓ is the length function. Denoting by TL(w) the set of
left descents of w , it is well known that the right weak order ⩽R on W is isomorphic to the set{
TL(w) : w ∈W
}
ordered by inclusion. In this note we prove the following formula:
TL(v) = TL(u) ∪
⋃
s∈DR (u−1v)
TL
(
vS\{s}
)
,
for all u, v ∈ W such that u ⩽R v . Here DR (w) is the set of right simple descent (right descents
of length one) of w , and w J is the element of minimal length in the coset wWJ , for any J ⊆ S.
This formula is a special case of a general result we prove in Theorem 2. We then deduce some
corollaries; for example, our result implies that the set
{
wS\{s} : s ∈ DR (w)
}
admits a join and it is
precisely w (Corollary 5), i.e.
w = ∨
s∈DR (w)
wS\{s},
for all w ∈ W . We are regarding the poset (W,⩽R ) as a complete meet-semilattice, for which the
joins may not exist.
We now fix notation and recall some definitions useful for the rest of the paper. We refer to the
book of Björner and Brenti [1] for Coxeter groups.
Let (W,S) be a Coxeter system, i.e. a presentation of the group W given by a set S of involutive
generators and relations encoded by a Coxeter matrix m : S ×S → {1,2, . . . ,∞}. A Coxeter matrix
over S is a symmetric matrix which satisfies the following conditions for all s, t ∈ S:
(1) m(s, t ) = 1 if and only if s = t ;
(2) m(s, t ) ∈ {2,3, . . . ,∞} if s ̸= t .
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
236 Paolo Sentinelli
The presentation (W,S) of the group W is then the following:{
generators: S,
relations: (st )m(s,t ) = e,
for all s, t ∈ S, where e denotes the identity in W . The Coxeter matrix m attains the value ∞ at (s, t )
to indicate that there is no relation between the generators s and t . The class of words expressing
an element of W contains words of minimal length; the length function ℓ : W →N assigns to an
element w ∈ W such minimal length. The identity e is represented by the empty word and then
ℓ(e) = 0. A reduced word or reduced expression for an element w ∈W is a word of minimal length
representing w . The set of reflections of (W,S) is defined by T := {w sw−1 : w ∈W, s ∈ S}. For J ⊆ S
and v ∈W , we let
W J := {w ∈W : ℓ(w) < ℓ(w s) ∀ s ∈ J},
DR (v) :=
{
s ∈ S : ℓ(v s) < ℓ(v)}.
With WJ we denote the subgroup of W generated by J ⊆ S; in particular, WS = W and W∅ = {e}.
We let ⩽R and ⩽ be the right weak order and the Bruhat order on W , respectively. The covering
relations of the right weak order are characterized as follows: u◁R v if and only if ℓ(u) < ℓ(v) and
u−1v ∈ S. For w ∈W , define
TL(w) :=
{
t ∈ T : ℓ(t w) < ℓ(w)},
TR (w) :=
{
t ∈ T : ℓ(w t ) < ℓ(w)}.
The right weak order has the following property (see [1, Proposition 3.1.3]):
u ⩽R v ⇐⇒ TL(u) ⊆ TL(v).
The covering relations of the Bruhat order are characterized as follows: u◁v if and only if
ℓ(u) = ℓ(v) − 1 and u−1v ∈ T . The posets (W,⩽R ) and (W,⩽) are graded with rank function
ℓ and (W,⩽R ) ,→ (W,⩽). For J ⊆ S, each element w ∈ W factorizes uniquely as w = w J w J ,
where w J ∈ W J , w J ∈ WJ and ℓ(w) = ℓ(w J )+ℓ(w J ); see [1, Proposition 2.4.4]. We consider the
idempotent function P J : W →W defined by
P J (w) = w J ,
for all w ∈ W . This function is order-preserving for the Bruhat order (see [1, Proposition 2.5.1]).
Moreover I ⊆ J ⊆ S implies P J ◦P I = P J (see [2, Lemma 5.4]). We let
P (s) := P S\{s},
for all s ∈ S.
For the following proposition see [1, Corollary 2.6.2].
Proposition 1. Let u, v ∈W . Then
u ⩽ v ⇐⇒ P (s)(u)⩽ P (s)(v) ∀ s ∈ DR (u).
Now we recall some results about the standard geometric representation of a finitely generated
Coxeter group. Let (W,S) be a Coxeter system with |S| < ∞ and Coxeter matrix m. Let V be the
free R-vector space on the set
{
αs : s ∈ S
}
and σs ∈ End(V ) defined by σs (v) = v − 2(αs |v)αs ,
where the bilinear form ( · | · ) is defined by setting (αs |αt ) := −cos
(
π
m(s,t )
)
, for all s, t ∈ S; then
the assignment w 7→ σw := σs1 · · ·σsk , where s1 · · · sk is any reduced word of w ∈ W , provides a
faithful representation W → GL(V ) (see, e.g. [1, Chapter 4]). Then we let w(v) := σw (v), for all
v ∈ V , w ∈ W . Let Φ := {w(αs ) : w ∈ W, s ∈ S}, V + := {∑s∈S asαs : as ⩾ 0 ∀ s ∈ S} \ {0} and V − :={∑
s∈S asαs : as ⩽ 0 ∀ s ∈ S
}
\{0}; then it is well known thatΦ=Φ+⊎Φ− = {αt : t ∈ T }⊎{−αt : t ∈ T },
Φ+ ⊆V + and Φ− ⊆V −. It holds that t (v) = v −2(αt |v)αt and w(αt ) ∈Φ− if and only if w t < w , for
all w ∈W , t ∈ T (see, e.g. [1, Section 4.4]). Moreover, if s ∈ S and t ∈ T \ {s}, then s(αt ) =αst s .
Paolo Sentinelli 237
2. Reflections and quotients
In this section we prove the main result of this paper, i.e. the formula mentioned in the introduc-
tion, as a corollary of the following theorem.
Theorem 2. Let (W,S) be a Coxeter system and u, v ∈W such that u ⩽R v. If E ⊆P (S) is such that⋂
J∈E J = S \ DR (u−1v), then
TL(v) = TL(u) ∪
⋃
J∈E
TL(v
J ).
Proof. It is known that
TL(x y) = TL(x)+xTL(y)x−1, (1)
for all x, y ∈ W , where + is the symmetric difference. Hence, for any v ∈ W , we have that
∅ = TL(v−1v) = TL(v−1)+ v−1TL(v)v ; this implies TR (v) = TL(v−1) = v−1TL(v)v , for all v ∈ W .
Consider J ⊆ S. Then
(v J )−1TL(v J )v J ∩WJ = TR (v J )∩WJ =∅. (2)
Notice that, if ℓ(x y) = ℓ(x)+ℓ(y), the symmetric difference in (1) is a disjoint union. Therefore
we have that
TL(v) = TL(v J )⊎ v J TL(v J )(v J )−1, (3)
for all v ∈W , J ⊆ S. Equalities (2) and (3) imply
TL(v
J ) = TL(v) \ v J TL(v J )(v J )−1
= TL(v) \ v J WJ (v J )−1
= TL(v) \ vWJ v−1.
(4)
If u ⩽R v and w := u−1v , we have that TL(v) = TL(u)⊎uTL(w)u−1. Moreover, by (4),
uTL(w
J )u−1 = u(TL(w) \ wWJ w−1)u−1
= uTL(w)u−1 \ vWJ v−1
⊆ TL(v) \ vWJ v−1 = TL(v J ).
(5)
Therefore, by (4) and (5), if we let K :=⋂J∈E J , we obtain⋃
J∈E
TL(v
J ) ⊇ ⋃
J∈E
uTL(w
J )u−1
= u
(⋃
J∈E
(
TL(w) \ wWJ w
−1))u−1
= u(TL(w) \ wWK w−1)u−1
= u(TL(w) \ wWS\DR (w)w−1)u−1
= uTL
(
wS\DR (w)
)
u−1
= uTL(w)u−1.
This implies that
TL(u) ∪
⋃
J∈E
TL(v
J ) ⊇ TL(u)∪uTL(w)u−1 = TL(v).
On the other hand, since u ⩽R v and v J ⩽R v for all J ∈ E , we also have that TL(v) ⊇ TL(u) ∪⋃
J∈E TL(v J ) and this concludes the proof. □
By the fact that
⋂
s∈DR (u−1v)
(
S \ {s}
) = S \ DR (u−1v), we obtain the following corollary, which is
the finest instance of the statement of Theorem 2.
238 Paolo Sentinelli
Corollary 3. Let (W,S) be a Coxeter system and u, v ∈W such that u ⩽R v. Then
TL(v) = TL(u) ∪
⋃
s∈DR (u−1v)
TL
(
vS\{s}
)
.
In the following example we consider a Coxeter system of type B4. The set of generators is
S = {s0, s1, s2, s3}; the Coxeter matrix takes the values m(s0, s1) = 4 and m(s1, s2) = m(s2, s3) = 3.
Example 4. In type B4, let u = s2s3s2 and v = s2s3s2s1s0s2s3. Then u−1v = s1s0s2s3, DR (v) =
{s0, s2, s3} and DR (u−1v) = {s0, s3}. We have that
• TL(u) =
{
s2, s3, s2s3s2
}
,
• TL
(
vS\{s3}
)= TL(s1s2s3) = {s1, s1s2s1, s1s2s3s2s1},
• TL
(
vS\{s0}
)= TL(s3s2s1s0) = {s3, s2s3s2, s1s2s3s2s1, s3s2s1s0s1s2s3},
• TL(v) =
{
s1, s2, s3, s1s2s1, s2s3s2, s1s2s3s2s1, s3s2s1s0s1s2s3
}
.
An element w ∈W is uniquely determined by its descent set; hence, by Theorem 2 for u = e, we
have that w is uniquely determined by the set
{
wS\{s} : s ∈ DR (w)
}
. This fact can be also deduced
from Proposition 1. An additional information provided by Theorem 2 is the following one, which
concerns the join operation in the right weak order.
Corollary 5. Let w ∈W . Then w = ∨
s∈DR (w)
wS\{s}.
Proof. Clearly TL(w J ) ⊆ TL(w), for all J ⊆ S, or, equivalently, w J ⩽R w , for all J ⊆ S. Hence the
set
{
wS\{s} : s ∈ DR (w)
}
is bounded and then it admits a join w ′ (see [3, Theorem 1.5]). Moreover,
by [4, Proposition 2.14] we have that TL(w ′) = TL(w) and this concludes the proof. □
By repeated use of Corollary 3, we obtain the following result.
Corollary 6. Let s1s2 · · · sk be a reduced expression for w ∈W . Then
TL(w) =
k−1⋃
i=0
TL
(
(s1 · · · sk−i )S\{sk−i }
)
.
We now deduce from Theorem 2 the statement of [5, Lemma 4.2]. Such result has inspired our
main formula and it is related to the shellability of Coxeter complexes.
Corollary 7. Let u, v, w ∈ W be such that u ⩽R v and u ⩽R w. If P (s)(v) = P (s)(w) for all
s ∈ DR (u−1v), then v ⩽R w.
Proof. Since u ⩽R w , we have that TL(u) ⊆ TL(w). Moreover TL
(
vS\{s}
) = TL(wS\{s}) ⊆ TL(w)
for all s ∈ DR (u−1v). Hence by Corollary 3 we obtain TL(v) ⊆ TL(w), which is equivalent to
v ⩽R w . □
For n ∈ N, let Bn be the power set P
(
{1, . . . ,n}
)
ordered by inclusion and, for w ∈ W and
K ⊆ S \ DR (w),
T Kw :=
{
TL
(
wK∪J
)
: J ⊆ DR (w)
}
⊆P (T ),
ordered by inclusion.
Proposition 8. Let w ∈W , K ⊆ S \ DR (w) and n := |DR (w)|. Then, as posets, T Kw ≃ Bn .
Proof. If w = e the result is trivial. So let w ̸= e and H := {s ∈ S : s ⩽ w}. We consider the
standard geometric representation of the finitely generated Coxeter group WH . Then w ∈WH and
assume TL
(
wK∪I
)= TL(wK∪J ), for some I , J ⊆ DR (w). Consider the factorization w = wK∪J wK∪J .
We have that s ∈ J ⊆ DR (w) if and only if s ∈ DR
(
wK∪J
)
, i.e. if and only if wK∪J (αs ) ∈ Φ−. Let
ts := w sw−1; then αts =−w(αs ). We have that(
wK∪J
)−1(αts ) =−(wK∪J )−1w(αs ) =−wK∪J (αs ).
Paolo Sentinelli 239
Hence
s ∈ J ⇐⇒ s ∈ DR
(
wK∪J
)
⇐⇒ (wK∪J )−1(αts ) ∈Φ+
⇐⇒ ts ̸∈ TL
(
wK∪J
)
⇐⇒ ts ̸∈ TL
(
wK∪I
)
⇐⇒ s ∈ DR
(
wK∪I
)
⇐⇒ s ∈ I .
Therefore I = J . We have proved that |Tw | = |Bn | = 2n . This also proves that TL
(
wK∪I
) ⊆
TL
(
wK∪J
)
implies J ⊆ I . On the other hand, if I ⊆ J , then K ∪ I ⊆ K ∪ J and we have that
wK∪J = (P K∪J ◦P K∪I )(w) ⩽R wK∪I , i.e. TL(wK∪J )⊆ TL(wK∪I ). Then the poset Tw is isomorphic
to the dual of Bn and the result follows since Bn is self-dual. □
Proposition 8 implies that the atoms of the poset T S\DR (w)w are the sets TL
(
wS\{s}
)
, with
s ∈ DR (w). The proof of this proposition also shows that if s ∈ DR (w) then
w sw−1 ∈ TL
(
wS\{s}
)
\
⋃
r∈DR (w)\{s}
TL
(
wS\{r }
)
,
i.e. the union in Corollary 7 is minimal with respect to its indexing set.
Acknowledgments
I thank the anonymous referee for having generalized the original main result (now Corollary 3)
and proposing a more elegant proof. I also thank Christophe Hohlweg for some useful conversa-
tions and for having pointed out the result of Corollary 5.
Declaration of interests
The author does not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and has declared no affiliations other than their research
organizations.
References
[1] A. Björner and F. Brenti, Combinatorics of Coxeter groups, Graduate Texts in Mathematics,
Springer, 2005, pp. xiv+363.
[2] V. V. Deodhar, “On some geometric aspects of Bruhat orderings. II. The parabolic analogue
of Kazhdan–Lusztig polynomials”, J. Algebra 111 (1987), no. 2, pp. 483–506.
[3] M. Dyer, “On the weak order of Coxeter groups”, Can. J. Math. 71 (2019), no. 2, pp. 299–336.
[4] C. Hohlweg and J.-P. Labbé, “On inversion sets and the weak order in Coxeter groups”, Eur.
J. Comb. 55 (2016), pp. 1–19.
[5] P. Sentinelli, “Linearly shellable complexes”. Online at https://link.springer.com/article/10.
1007/s11083-024-09688-4. To appear in Order.


Reflections and quotients in Coxeter groups

Pierre Cardaliaguet

A note on contractive semi-groups on a 1:1 junction for scalar conservation laws and Hamilton–Jacobi equations

Kęstutis Česnavičius

The affine Grassmannian as a presheaf quotient

Rodolphe Richard

Andrei Yafaev

Comptes Rendus
Mathématique
Rodolphe Richard and Andrei Yafaev
Generalised André–Pink–Zannier conjecture for Shimura varieties of abelian type
Volume 363 (2025), p. 873-878
Online since: 21 July 2025
https://doi.org/10.5802/crmath.751
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2025, Vol. 363, p. 873-878
https://doi.org/10.5802/crmath.751
Research article / Article de recherche
Algebraic geometry, Number theory / Géométrie algébrique, Théorie des nombres
Generalised André–Pink–Zannier conjecture
for Shimura varieties of abelian type
Conjecture d’André–Pink–Zannier généralisée pour les
variétés de Shimura de type abélien
Rodolphe Richard a and Andrei Yafaev ∗,a
a UCL Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK
E-mails: r.richard@ucl.ac.uk, yafaev@ucl.ac.uk
Abstract. This note describes the results of [6]. The main result is the proof of the Generalised André–
Pink–Zannier conjecture in Shimura varieties of abelian type. The core result is a lower bound, in terms
of height functions defined in [7], for the sizes of Galois orbits of points in generalised Hecke orbits, which is
unconditional for Shimura varieties of abelian type.
Résumé. Dans cette note nous décrivons les résultats de [6]. Le résultat principal est la preuve de la Conjec-
ture d’André–Pink–Zannier genéralisée pour les variétés de Shimura de type abélien. Le résultat central
énonce des bornes inférieures, en termes des fontions hauteurs de [7], pour la taille des orbites galoisiennes
dans une orbite de Hecke généralisée, qui sont inconditionnelles pour les variétés de Shimura de type abelien.
Keywords. Shimura varieties, Hecke orbits, Zilber–Pink, heights, Siegel sets, Mumford–Tate conjecture, adelic
linear groups.
Mots-clés. Variétés de Shimura, orbites de Hecke, Zilber–Pink, hauteurs, ensembles de Siegel, conjecture de
Mumford–Tate, groupes linéaires adéliques.
2020 Mathematics Subject Classification. 03C64, 11G18, 11G50, 11F80, 14L30, 20G35, 15A16, 14G35.
Funding. This project was funded by Leverhulme Trust grant RPG-2019-180.
Manuscript received 21 December 2022, revised 17 November 2023 and 1 May 2025, accepted 16 April 2025.
1. Statements of results
In this note we describe the results of [6], which is the sequel to [7]. A previous note [8] described
the results of [7]. We refer to [2] for the notions related to Shimura varieties, and to [9, Section 3]
for the notion of weakly special subvarieties. We study the following conjecture, introduced in
the following form in [7]. We refer to the previous note [8] for the conjecture and for the notions
involved.
Conjecture 1 (Generalised André–Pink–Zannier Conjecture [7, Conjecture 1.1]). Let S be a
Shimura variety and Σ a subset of a generalised Hecke orbit in S. Then, irreducible components
of the Zariski closure of Σ are weakly special subvarieties.
∗Corresponding author
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
874 Rodolphe Richard and Andrei Yafaev
Our most notable result is the following. We refer to [2] for the notion of Shimura of abelian
type. This is the most important class of Shimura varieties, and it contains in particular the
moduli spaces of abelian varieties Ag ,d .
Theorem 2. If S is of abelian type, and Σ is a subset of a generalised Hecke orbit in S, then the
Zariski closure of Σ is a finite union of weakly special subvarieties.
The main strategy underlying the proof of Theorem 2 is the Pila–Zannier strategy in the form
laid out in [7]. This relies on a variant [5] of Pila–Wilkie theorem, some functional transcendence
results [4] and the so-called “geometric part of André–Oort conjecture” [9]. The article [7]
introduces some height functions with good properties which are required for this strategy to
work. We refer to [8] for more details.
A main difficulty in this approach is to obtain lower bounds for the sizes of some Galois orbits.
This is Theorem 7. The use of Theorem 5, which is deduced from Faltings’ theorems, allows us
to reduce Theorem 2 to Theorem 6. The proof of Theorem 6 follows from [7] and Theorem 7.
Theorem 7 is reduced to Theorem 8, whose proof is discussed in Section 2. This proof relies on
the tool presented in Section 3.
1.1. Main results
Theorem 2 follows from two statements. The first one, Theorem 5, is a refined version of
Faltings theorems’ on Tate conjecture for abelian varieties. We encode this refined form into
the Definition 3 of “uniform integral Tate property”. The second statement, Theorem 6, proves
Conjecture 1 assuming that a point s ∈Σ satisfies this uniform integral Tate property.
In order to state Definition 3 we let (G , X ) be the Shimura datum (G , X ) and we let K be the
open compact subgroup of G(A f ) such that S = ShK (G , X ). We write s = [x,1] ∈ S with x ∈ X . We
denote by M ≤G the Mumford–Tate group of x, choose a field E ≤ C such that s ∈ S(E) and such
that we have a Galois representation, as in [7, Section 3],
ρx : Gal(E/E) → M(A f )∩K . (1)
The following definition makes sense for two Q-algebraic groups M ≤ G and a faithful repre-
sentation ρG : G → GL(d). It does not depend on the choice of ρG .
Definition 3 (Uniform integral Tate property). Let U ≤ M(A f ) be a compact subgroup. Let Up :=
U ∩M(Qp ) and, for p ≫ 0, let U (p) ≤ GL(d ,Fp ) be the image of Up .
We say that U satisfies the uniform integral Tate property with respect to M , G and ρG if the
following is satisfied.
(1) For every p,
ZGQp (Up ) = ZGQp (Up 0) = ZG (M)Qp (2a)
and
the action of Up onQ
d
p is semisimple.
1 (2b)
(2) For every D, there exists an integer M(D) such that, for every p ≥ M(D) and every U ′ ≤U (p)
of index
[
U (p) : U ′
]≤ D, we have
ZGFp (U
′) = ZGFp (MFp ) (3a)
and
the action of U ′ on Fp
d
is semisimple. (3b)
Definition 4. We say that x0 satisfies the uniform integral Tate conjecture if U = ρx0
(
Gal(E/E)
)
satisfies the uniform integral Tate property with respect to M, G in the sense of Definition 3.
1Also known as completely reducible.
Rodolphe Richard and Andrei Yafaev 875
The following is deduced (see [6, Section 6.2]) from Faltings’ theorems, a specialisation argu-
ment of Noot, and a result of Serre.
Theorem 5. In a Shimura variety of abelian type, every point satisfies the uniform integral Tate
conjecture in the sense of Definition 4.
Our main result towards Conjecture 1 is the following. We recall that a generalised Hecke orbit
is a finite union of “geometric Hecke orbits” ([7, Definition 2.3 and Theorem 2.4]).
Theorem 6. Let s0 = [x0,1] be a point in a Shimura variety ShK (G , X ), and assume the uniform
integral Tate conjecture for x0 in X in the sense of Definition 4. Let Z be a subvariety whose
intersection with the geometric Hecke orbit of s0 is Zariski dense in Z . Then, Z is a finite union
of weakly special subvarieties of S.
We observe that the “uniform integral Tate property” follows from the “weakly adelic
Mumford–Tate property” of [7]. As a result, Theorem 6 supersedes the main result of [7].
2. Polynomial Galois bounds
The proof of Theorem 6 is found in [6, Section 3]. It follows [7], which uses Pila–Zannier strategy,
as summarised in [8]. The main new difficulty which is dealt with in [6] is to prove lower bounds
on Galois orbits assuming the uniform integral Tate property.
Theorem 7 (Galois bounds in Shimura varieties). Let S = ShK (G , X ) be a Shimura variety,
let M ≤ G be the Mumford–Tate group of an element x ∈ X , and denote by W the conjugacy
class G ·φ0 ≃ G/ZG (M) of the inclusion φ0 : M → G (viewed as a Q-algebraic variety). We choose
integral structures mZ of m and gZ of g, and consider the “finite height function” H f : W (Q) →Z≥1
defined by
H f (φ) = min
{
k ∈Z≥1
∣∣ dφ(mZ) ≤ 1k ·gZ},
where dφ : m→ g denotes the differential of φ : M →G.
We define W + :=G(R)+ ·φ0 ⊆W (R) and W (Q)+ :=W +∩W (Q).
Let E ≤C be a subfield of finite type overQ and such that x0 satisfies Definition 4.
Then, as functions of φ ∈W (Q)+,
#Gal(E/E) · [φ◦x0,1] ≈ H f (φ).
We recall that
{
[φ◦x0,1]
∣∣φ ∈W (Q)+} is the geometric Hecke orbit of x0.
Using [7, Section 3.3], we deduce Theorem 7 from Theorem 8 applied to U as in Definition 4.
The assumption (1) in Theorem 8 follows from the properties of canonical models of Shimura
varieties. The assumption (2) in Theorem 8 follows from the assumption that x0 satisfies Defini-
tion 4.
We believe Theorem 8 to be of independent interest.
Theorem 8 ([6, Theorem 5.1]). Let M ≤G be connected reductive Q-groups. Let U ≤ M(A f ) be a
subgroup satisfying the following.
(1) The image of U in M ab is MT in M ab in the sense of [7, Defintion 6.1].
(2) The group U satisfies the uniform integral Tate property as in Definition 3.
Denote by φ0 : M → G the identity homomorphism and W = G ·φ0 its conjugacy class. Then as φ
varies in W (A f ), for any compact open subgroup K ≤G(A f ), where KM = K ∩M(A f ), we have[
φ(U ) :φ(U )∩K ]≈ [φ(KM ) :φ(KM )∩K ]≽ H f (φ).
as functions W (A f ) →Z≥1.
876 Rodolphe Richard and Andrei Yafaev
Theorem 8 can be studied prime by prime. Without loss of generality, we assume K = ∏p Kp
where Kp := GL(d ,Zp )∩G(Qp ). We write M(Zp ) := GL(d ,Zp )∩M(Qp ). It is enough to prove the
following.
• For every prime p, we have
∃ a(p), c(p) ∈R>0, ∀φ ∈W (Qp ),
[
φ(U ) :φ(U )∩K ]≥ a(p) ·[φ(M(Zp )) :φ(M(Zp ))∩Kp]c(p). (4)
• There exists a,c ∈R>0 such that, for all but finitely many primes,
∀φ ∈W (Qp ),
[
φ(Up ) :φ(Up )∩Kp
]≥ a · [φ(M(Zp )) :φ(M(Zp ))∩Kp]c . (5)
At a given prime, property (2a) ensures that the conjugacy classes of M(Zp ) → G(Qp ) and Up ,→
G(Qp ) are the same variety W . Then (4) follows from the functoriality of height.
The main difficulty is to ensure uniform and sharper estimates (5) for p ≫ 0, and properties (2)
in Definition 3 are crucial. This is the most technical part of [6].
The approach is as follows (see [6, Section 5] for details).
We construct from Up a vector, say v , in a representation of G . This representation is a product
of copies of the adjoint representation G → GL(g) and of a fixed representation G → GL(d). In
particular the weights are bounded. Moreover, the orbit (G ·v)(Qp ) can be identified with W (Qp ).
Using the tools of [7], we can bound from below
[
φ(Up ) : φ(Up )∩Kp
]
in terms of the p-adic
height Hp (g · v) = max
{
1;∥g · v∥}, where φ ∈W (Qp ) is identified with g · v ∈ (G · v)(Qp ).
Let v ′ be the vector constructed from the case Up = M(Zp ). The local height Hp (g · v ′) is the
local factor of a global height function W (A f ) →Z≥1. Theorem 10 is used, for p ≫ 0, to compare
the local height Hp (g · v) = max
{
1;∥g · v∥} with the local height Hp (g · v ′) = max{1;∥g · v ′∥}.
Because the constant C in (13) does not depend on p, but only on the weights of the represen-
tation, we obtain the uniformity in (5).
3. p-adic Kempf–Ness theorem
Theorem 9 is key in order to make use of part (2) of Definition 3. Theorem 9 can be seen as an
analogue of Kempf–Ness theorem [3, Theorem 0.1(b)]. Kempf–Ness theorem deals with closed
orbits in linear representations equipped with an archimedean metric. Theorem 9 is a p-adic
analogue. While Kempf–Ness theorem is easy to prove, our p-adic analogue requires a fine study
of the “good reduction” of the orbit under consideration and the use of group schemes.
We believe this is a result of independent interest.
Theorem 9 (p-adic Kempf–Ness Theorem). Let FZp ≤GZp ≤ GL(n)Zp be smooth reductive group
schemes such that FZp →GZp → GL(n)Zp are closed immersions and GZp is connected.
Let v ∈Znp , denote by v ∈ Fnp its reduction and assume that
StabGQp (v) = FQp and dim
(
StabGFp (v)
)= dim(FFp ), (6)
(using Krull dimensions) and assume that the orbits
GQp · v ⊆AnQp and GFp · v ⊆A
n
Fp
(7)
are closed.
Then, denoting by Zp [G/F ] := Zp [G] ∩Qp [G]F the algebra of F -invariant functions G → A1
defined over Zp , for every g ∈G(Qp ), we have,
g · v ∈Zp n if and only if ∀ f ∈Zp [G/F ], f (g ) ∈Zp . (8)
Moreover, Spec
(
Zp [G/F ]
)
is smooth over Zp , and we have(
G(Qp ) · v
)∩Zp n =G(Zp ) · v. (9)
Rodolphe Richard and Andrei Yafaev 877
Remarks. Some of the hypotheses can be rephrased as follows.
The Qp -algebraic groups F and G are reductive, the compact subgroups FZp (Zp ) ≤ F (Qp )
and GZp (Zp ) ≤ G(Qp ) are hyperspecial subgroups, and we have FZp (Zp ) = F (Qp ) ∩ GL(n,Zp )
and GZp (Zp ) =G(Qp )∩GL(n,Zp ).
The property (7) is related to semi-stability and residual semi-stability of the vector v in the
sense of [1].
In (6), the hypothesis on dimensions means that StabGFp (v)
0,red (the reduced subgroup of the
neutral component) is equal to (FFp )
0. Equivalently, StabGFp (v)
0(Fp ) = F 0(Fp ). This is implied by
the stronger condition
StabG(Fp )(v) = F (Fp ) (10)
and the stronger one
StabGFp (v) = FFp as group schemes. (11)
3.1. Comparison of local height functions
Theorem 9 is a key result in proving the following, which can be seen as a refined, more precise,
version of the functoriality of local heights.
Theorem 10 (Local relative stability estimates). Let F ≤G ρ→ GL(d) be as in Theorem 9. Let v, v ′ ∈
Zdp be non zero vectors, denote by v , v
′ ∈ Fdp their reduction, and assume that
(1) the orbits GQp · v and GQp · v ′ ⊆AdQp are closed subvarieties;
(2) the stabiliser groups Fv := StabG (v),Fv ′ := StabG (v ′) satisfy
Fv = Fv ′ = F ;
and that
(3) the orbits GFp · v and GFp · v ′ ⊆AdFp are closed subvarieties;
(4) the stabiliser groups Fv := StabG (v), Fv ′ := StabG (v ′) satisfy, as group schemes2,
Fv = Fv ′ = FFp . (12)
We denote by ∥·∥ : Cp d →R the standard ultrametric norm, and we define two functions G(Cp ) →R
given by
Hv : g 7→ max
{
1;∥g · v∥} and Hv ′ : g 7→ max{1;∥g · v ′∥}.
Then, the functions hv = log Hv and hv ′ = log Hv ′ satisfy
hv ≤C ·hv ′ and hv ′ ≤C ·hv , (13)
in which C =C(Σ(ρ)) depends only on the set of weights of ρ (cf. [6, 6.1.1.2]).
Declaration of interests
The authors do not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and have declared no affiliations other than their research
organizations.
2It amounts to the property that Fv and Fv are smooth.
878 Rodolphe Richard and Andrei Yafaev
References
[1] J.-F. Burnol, “Remarques sur la stabilité en arithmétique”, Int. Math. Res. Not. (1992), no. 6,
pp. 117–127.
[2] P. Deligne, “Variétés de Shimura: interprétation modulaire, et techniques de construction de
modèles canoniques”, in Automorphic forms, representations and L-functions (Proc. Sym-
pos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proceedings of Symposia
in Pure Mathematics, vol. XXXIII, American Mathematical Society, 1979, pp. 247–289.
[3] G. Kempf and L. Ness, “The length of vectors in representation spaces”, in Algebraic geome-
try (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), Lecture Notes in Mathe-
matics, vol. 732, Springer, 1979, pp. 233–243.
[4] B. Klingler, E. Ullmo and A. Yafaev, “The hyperbolic Ax–Lindemann–Weierstrass conjecture”,
Publ. Math., Inst. Hautes Étud. Sci. 123 (2016), pp. 333–360.
[5] J. Pila, “On the algebraic points of a definable set”, Sel. Math., New Ser. 15 (2009), no. 1,
pp. 151–170.
[6] R. Richard and A. Yafaev, “Generalised André–Pink–Zannier Conjecture for Shimura vari-
eties of abelian type”, 2021. Online at https://arxiv.org/abs/2111.11216v2.
[7] R. Richard and A. Yafaev, “Height functions on Hecke orbits and the generalised André–
Pink–Zannier conjecture”, 2021. Online at https://arxiv.org/abs/2109.13718v1.
[8] R. Richard and A. Yafaev, “On the Generalised André–Pink–Zannier conjecture”. To appear
in C. R. Math.
[9] E. Ullmo, “Applications du théorème d’Ax–Lindemann hyperbolique”, Compos. Math. 150
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