Table Of ContentPOLYNOMIALITY FOR THE POISSON CENTRE OF
TRUNCATED MAXIMAL PARABOLIC SUBALGEBRAS.
FLORENCE FAUQUANT-MILLET AND POLYXENI LAMPROU
7
1
0
2
Abstract. WeshowthatthePoissoncentreoftruncatedmaximalparabolicsub-
b
e algebras of a simple Lie algebra of type B, D and E6 is a polynomial algebra.
F In roughlyhalf ofthe cases the polynomiality of the Poisson centre was already
known by a completely different method.
4
For the rest of the cases, our approach is to construct an algebraic slice in the
] sense of Kostant given by an adapted pair and the computation of an improved
T
upper bound for the Poisson centre.
R
.
h
t 1. Introduction
a
m
The base field k is assumed to be algebraically closed of characteristic zero.
[
InthispaperwecontinueourstudyonthePoissonsemicentre ofmaximalparabolic
2 subalgebras of a simple Lie algebra over k, that we initiated in [8] .
v
Let p be a parabolic subalgebra of a simple Lie algebra g over k. Recall that the
8
3 Poisson semicentre Sy(p) of p is the vector space generated by the semi-invariants of
2 the symmetric algebra S(p) of p and that the canonical truncation of p, denoted by
2
0 pΛ, is the largest subalgebra of p which vanishes on the weights of Sy(p). Hence one
1. has that Sy(pΛ) = Y(pΛ) := S(pΛ)pΛ. Observe that the latter, which is the algebra of
0 invariants of S(p ) under the adjoint action of p , is also the Poisson centre of S(p )
Λ Λ Λ
7
(of p for short), where S(p ) is equipped with its natural Poisson structure. Since
1 Λ Λ
v: p is algebraic, one has that Sy(p) = Y(pΛ) (see 2.3 for more details). Observe also
that the Poisson centre Y(p) of p is reduced to k, when p is not equal to g (see for
i
X
example [12, 7.9] or [4, Chap. I, Sec. B, 8.2 (iv)]), whereas the Poisson semicentre
r
a Sy(p) of p is never reduced to scalars by [2].
By [5] - see also [12], [13] in the more general case of biparabolic (seaweed) sub-
algebras - we know that Sy(p) is lower and upper bounded, up to gradations, by
polynomial algebras A and B respectively, having the same number of variables and
whose weights may either be equal or differ by one half. Moreover, it was shown
that the coincidence of these bounds is a sufficient condition for the polynomiality of
Key words and phrases. Poissoncentre,(bi)parabolicsubalgebras,polynomiality,adaptedpairs.
AMS Classification : 17B35, 17B22,16W22.
1
2 FLORENCE FAUQUANT-MILLETANDPOLYXENILAMPROU
Sy(p). The coincidence of these bounds occurs often, for instance when g is simple
of type A or C and p is any parabolic subalgebra of g.
However, the coincidence of bounds A and B is not a necessary condition for the
polynomiality of the Poisson semicentre and indeed there are examples where A and
B do not coincide but the Poisson semicentre is polynomial, for example in the Borel
case [11].
Since Sy(p ) = Y(p ), the field of invariant fractions C(p ) := (FractS(p ))pΛ
Λ Λ Λ Λ
is equal to the field of fractions Fract(Y(p )) of Y(p ), as each semi-invariant of
Λ Λ
FractS(a) is a quotient of two semi-invariants of S(a), for any finite dimensional
Lie algebra a by [2] or [4, Chap. I, Sec. B, 5.11, 5.12]. Hence the polynomiality of
Sy(p) = Y(p ) implies that thefieldof invariant fractionsC(p ) isapurely transcen-
Λ Λ
dental extension of the base field k and by [20, Thm. 66] so is the field of invariant
fractions C(p), since there exists a set of algebraically independent generators of
Sy(p) formed by weight vectors, that is by semi-invariants of S(p). This allows us to
answer positively Dixmier’s fourth problem for such parabolic subalgebras, namely
whether the field of invariant fractions is a purely transcendental extension of the
base field, for any finite dimensional Lie algebra. However the polynomiality of the
Poisson centre Y(p ) is a much stronger result.
Λ
Recently, several authors have been interested in the question of polynomiality of
the Poisson centre of non-reductive algebraic Lie algebras; parabolic and biparabolic
(seaweed) subalgebrasofasimpleLiealgebragoverk werestudiedin[5],[6],[12],[13]
and some particular semi-direct products were studied in [22], [23], [24], [27], [28],
where polynomiality of the Poisson centre was shown. In [20] the author gives nec-
essary and sufficient conditions for the Poisson centre or semicentre of certain finite
dimensional Lie algebra to be polynomial.
So far, only one counterexample to the polynomiality of the Poisson semicentre
of a biparabolic subalgebra p is known, namely when g is of type E and p is the
8
maximal parabolic subalgebra of g, whose canonical truncation coincides with the
centralizer of the highest root vector of g [26].
In [8] we studied Sy(p) for p a maximal parabolic subalgebra of a simple Lie
algebra g, when the lower and upper bounds A and B coincide (hence Sy(p) is
polynomial)andweconstructed slices forthecoadjointaction, extending theKostant
Slice Theorem [19, Thm. 0.10].
In this paper we study the remaining cases for g simple of type B, D and E and
6
we deduce the polynomiality of the Poisson semicentre Sy(p) by constructing slices
for the coadjoint action and computing an “improved upper bound” (see below).
The slices we constructed in [8] were given by adapted pairs (see 2.8) for the
canonical truncations p of the parabolic subalgebras p that we studied. In this
Λ
paper we construct adapted pairs for the remaining cases mentioned above.
POLYNOMIALITY FOR MAXIMAL PARABOLICS 3
Adapted pairs play the role of principal sl -triples in the non-reductive case and
2
were introduced in [14]. They give an improved upper bound B′ for the character of
Sy(p) = Y(p ) [16]. When this bound is attained, in particular when it coincides
Λ
with the character of the lower bound A mentioned above, polynomiality of Sy(p)
follows and the adapted pair gives an algebraic slice (in the sense of [17, 7.6]) also
called a Weierstrass section in [7], extending the Kostant Slice Theorem [19, Thm.
0.10] to non-reductive Lie algebras. By [7], this Weierstrass section is also an affine
slice for the coadjoint action (in the sense of [17, 7.3]).
Some particular cases had already been studied by other authors and different
methods. For example, it was shown in [21] that for all maximal parabolic subalge-
bras p whose canonical truncation is the centralizer of the highest root vector of the
simple Lie algebra (except in type E , where we have Yakimova’s counterexample),
8
the Poisson semicentre Sy(p) is a polynomial algebra over k.
Furthermore, Heckenberger [10] showed by computer calculations that in type B ,
n
2 ≤ n ≤ 4, the Poisson semicentre Sy(p) is polynomial for all parabolic subalgebras
p.
In [25] an affine slice for the coadjoint action of p was constructed for some non
truncated biparabolic subalgebras p of a simple Lie algebra, which gave a positive
answer to Dixmier’s fourth problem for C(p). These biparabolic subalgebras p do
not coincide with the maximal parabolic subalgebras we are interested in.
Below, labeling of simple roots follows Bourbaki [1, Planches I-IX].
Adapted pairs need not exist for all truncated parabolic subalgebras and are very
hard to construct in general. One may hope to construct such pairs when the trun-
cated Cartan subalgebra - that is, the subalgebra of the Cartan subalgebra, which
is contained in the canonical truncation of the parabolic subalgebra we consider - is
large enough, as it happens when g is of type A or when the parabolic subalgebra
p is maximal; however, we showed that even in these favourable cases adapted pairs
may not exist, as it happens for example when g is of type F and p is the maximal
4
parabolic subalgebra corresponding to π′ = {α , α , α } [8, Sect. 10]. In type A
1 2 4
adapted pairs were constructed for all truncated biparabolic subalgebras in [15].
When the parabolic subalgebra p is maximal associated to π′ = π \ {α } where
s
π is a set of simple roots α , 1 ≤ i ≤ n, in g and g is simple of type B or D , the
i n n
bounds A and B for Sy(p) coincide exactly when s is odd (in type D , n ≥ 4, under
n
the restriction s 6= n−1; additionally, when s = n−1 and s even, and finally in type
D for all s except for s = 2; in type B , n ≥ 2, also for n = s = 2 and n = s = 4).
4 n
In this paper we give an adapted pair for the rest of the truncated maximal para-
bolicsubalgebrasintypeBandD. Inparticular,weprovealemmaofnon-degeneracy
(3.8) which is a non-obvious generalization of [8, Lemma 5].
Using GAP [9], we also construct an adapted pair for the truncated maximal
parabolic subalgebras p in a simple Lie algebra of type E , when the bounds A and
Λ 6
4 FLORENCE FAUQUANT-MILLETANDPOLYXENILAMPROU
B do not coincide, that is when s = 1, 6 (for s = 2 an adapted pair was already
constructed in [16]).
FromthecaseD , s = 6,wealsodeduceanadaptedpairforthetruncatedmaximal
6
parabolic subalgebra of g of type E corresponding to π′ = π \{α } (6.7).
7 3
We compute the improved upper bound B′ (4.6.2, 5.6.2, 6.6.2, 6.7, 7) and we show
that it is attained and hence the Poisson centre Y(p ) of p is polynomial (4.6.3,
Λ Λ
5.6.3, 6.6.3, 6.7, 7). We deduce that for all such maximal parabolic subalgebras
p, Dixmier’s fourth problem is true for C(p). Furthermore, as in [8] we obtain an
algebraic and an affine slice for the dual of p .
Λ
Acknowledgements. This work was initiated when the second author was vis-
iting the University Jean Monnet, Saint-Etienne. We would like to thank A. Joseph
for many fruitful discussions on adapted pairs and for his interest in our work. We
are also grateful to A. Ooms for enlightening exchange of ideas on the polynomial-
ity of the Poisson semicentre. Part of these results were presented in the seminar
at the Weizmann Institute of Science in Israel in April 2016 and in the conference
“Representation Theory in Samos” in Greece in July 2016.
2. Preliminaries.
2.1. Let g be a finite dimensional simple Lie algebra over k and h a fixed Cartan
subalgebra of g.
Let ∆ be the root system of g with respect to h, π a chosen set of simple roots,
∆+ (resp. ∆−) the set of positive (resp. negative) roots. We adopt the labeling of
[1, Planches I-IX] for the simple roots in π.
For any α ∈ ∆, let g denote the corresponding root space of g and fix a nonzero
α
vector x in g . Then g = n⊕h⊕n−, where n = g and n− = g .
α α α∈∆+ α α∈∆− α
For all α ∈ π, denote by α∨ the corresponding coroot. For any subset A of ∆, set
L L
g = g .
A α∈A α
2.2. LFor any subset π′ of π, let ∆π′ be the subset of roots in ∆ generated by π′ and
∆+, ∆− the sets of positive and negative roots in ∆ respectively.
π′ π′ π′
Onedefinesthestandardparabolicsubalgebrap associatedtoπ′ tobethealgebra
π′
p = n⊕h⊕n− wheren− = g . Itsopposedalgebrathenisp− = n−⊕h⊕n ,
π′ π′ π′ α∈∆− α π′ π′
π′
with n defined similarly. The dual space p∗ identifies with p− via the Killing form
π′ L π′ π′
K on g.
We denote by W the Weyl group associated to π′ and by r , for γ ∈ ∆ the
π′ γ π′
reflection with respect to γ. Then W is the subgroup of the Weyl group of (g, h)
π′
generated by r , for all γ ∈ ∆ .
γ π′
2.3. Let a be a finite dimensional Lie algebra over k. The Poisson semicentre Sy(a)
ofitssymmetric algebraS(a)(ofaforshort)isdefinedtobethevector spacespanned
POLYNOMIALITY FOR MAXIMAL PARABOLICS 5
by the semi-invariants under the adjoint action of a that is, Sy(a) = S(a)
λ∈a∗ λ
where S(a) = {s ∈ S(a) | ∀x ∈ a, (adx)s = λ(x)s}. It is a subalgebra of S(a).
λ
L
When S(a) 6= {0}, λ is called a weight of the Poisson semicentre Sy(a). Let Λ(a)
λ
denote the set of weights of Sy(a). Then the canonical truncation a of a is :
Λ
a = ∩ kerλ. It is an ideal of a containing the derived subalgebra of a.
Λ λ∈Λ(a)
Equip S(a) with its natural Poisson structure coming from the Lie bracket on a.
The Poisson centre Y(a) of a is the centre of S(a) for this structure and it is also the
set of the invariants in S(a) under the adjoint action of a, that is Y(a) = S(a) . It
0
is an algebra contained in the Poisson semicentre Sy(a) of S(a).
If a is almost algebraic (for a definition see [6, B.2]), then we have that Sy(a) =
Sy(a ) = Y(a ) (see [6, B.2] or [20, Thm. 4]).
Λ Λ
2.4. The index of a, denoted by inda, is the minimal dimension of a stabilizer af for
f ∈ a∗. When a is algebraic the index of a is also equal to the minimal codimension
of a coadjoint orbit in a∗ [3, 1.11.3].
An element y ∈ a∗ is called regular in a∗ if its stabilizer ay is of minimal dimension
(equal to inda).
2.5. Let π′ ⊂ π and p be the canonical truncation of p which we recall is
π′,Λ π′
defined to be the largest subalgebra of p that vanishes on the weights of Sy(p ). It
π′ π′
has the property that the Poisson centre Y(p ) is equal to the Poisson semicentre
π′,Λ
Sy(p ) andalsoequaltoSy(p ), since p isalgebraicandsoalsoalmost algebraic.
π′,Λ π′ π′
The canonical truncation of p was given explicitly in [6]. It is of the form p =
π′ π′,Λ
n⊕ h ⊕ n− where h is a subalgebra of h called the truncated Cartan subalgebra
Λ π′ Λ
(this is the largest subalgebra of h which vanishes on the set of weights of all semi-
invariants in Sy(p )).
π′
The Gelfand-Kirillov dimension of Y(p ) is equal to the index of p . For more
π′,Λ π′,Λ
details, see [6, 2.4, 2.5, B.2].
Let h′ ⊂ h be the Cartan subalgebra of the Levi factor of p . When π′ = π\{α },
π′ s
then h = h′ that is, h is the vector space over k generated by all α∨ with α ∈ π′.
Λ Λ
Forconvenience, wereplacethetruncatedparabolicsubalgebrap byitsopposed
π′,Λ
algebra p− (that is, the canonical truncation of the opposed algebra p−). From
π′,Λ π′
now on, we denote it simply by p.
2.6. For any h-module M = M with finite dimensional weight spaces M :=
ν∈h∗ ν ν
{m ∈ M | ∀h ∈ h, h.m = ν(h)m}, we may define its formal character by
L
chM = dimM eν.
ν
ν∈h∗
X
Given two such h-modules M and M′ write chM ≤ chM′ if dim M ≤ dim M′ for
ν ν
all ν ∈ h∗ [13, 2.8].
6 FLORENCE FAUQUANT-MILLETANDPOLYXENILAMPROU
2.7. Here we recall the formal characters chA and chB of the lower and the upper
bounds mentioned in the introduction for chY(p) given in [13, Thm. 6.7].
Let E(π′) be the set of hiji-orbits of π, where i and j are the involutions of π
defined for example in [8, 2.2]. Denote by {̟ } (resp. {̟′ } ) the set of
α α∈π α α∈π′
fundamental weights associated to π (resp. to π′); the same sets sometimes are
denoted by {̟ } and {̟′} respectively. Let B (resp. B ) be the set of
i αi∈π i αi∈π′ π π′
weights of the Poisson semicentre of S(n⊕h) (resp. S(n ⊕h′)).
π′
For all Γ ∈ E(π′), set
δ = − ̟ − ̟ + ̟′ + ̟′
Γ γ γ γ γ
γ∈Γ γ∈j(Γ) γ∈Γ∩π′ γ∈i(Γ∩π′)
X X X X
and
1/2 if Γ = j(Γ), and ̟ ∈ B , and ̟′ ∈ B .
ε = γ∈Γ γ π γ∈Γ∩π′ γ π′
Γ
1 otherwise.
(
P P
It is shown in [13, Thm. 6.7] that
(1−eδΓ)−1 ≤ chY(p) ≤ (1−eεΓδΓ)−1.
Γ∈E(π′) Γ∈E(π′)
Y Y
In particular, if for all Γ ∈ E(π′), ε = 1, the above inequalities are equalities and
Γ
Y(p) is a polynomial algebra over k [5].
2.8. An adapted pair for p is a pair (h, y) ∈ h ×p∗ such that y is regular in p∗,
Λ
and (adh)y = −y where ad denotes the coadjoint action of p on p∗.
2.9. Assume that there exists an adapted pair (h, y) ∈ h × p∗ for p. One may
Λ
choose subsets S, T ⊂ ∆+ ⊔∆− such that y = a x , with a ∈ k \{0} for all
π′ γ∈S γ γ γ
γ ∈ S, and p∗ = (adp)y ⊕ g . Note that we may choose T such that |T| = indp.
T
P
Assume further that S is a basis for h∗. Then for each γ ∈ T there exists a unique
|hΛ Λ
t(γ) ∈ QS such that γ +t(γ) vanishes on h .
Λ
By [16, Lem. 6.11]
chY(p) ≤ (1−e−(γ+t(γ)))−1
γ∈T
Y
and we will call the right hand side an “improved upper bound” for chY(p); in this
work it is indeed always an improvement of the upper bound mentioned in 2.7.
Moreover by [16, Lem. 6.11] if the lower bound in 2.7 and this improved upper
bound coincide then the restriction map gives an isomorphism of algebras Y(p) ≃
R[y+g ], where R[y+g ] is the ring of polynomial functions on y+g , isomorphic
T T T
to S(g∗). Hence Y(p) is a polynomial algebra over k and y+g is an algebraic slice
T T
in the sense of [17, 7.6], also called a Weierstrass section in [7] and by [7] it is also
POLYNOMIALITY FOR MAXIMAL PARABOLICS 7
an affine slice in the sense of [17, 7.3] for the coadjoint action of the adjoint group
of p on p∗.
2.10. Assume that there exists an adapted pair (h, y) for p and denote by V an
h-stable complement of (adp)y in p∗. Assume further that Y(p) is a polynomial
algebra and let f ,..., f be homogeneous generators for Y(p) (l = indp). Then
1 l
by [18, Cor. 2.3] if m ,..., m are the eigenvalues of h on an h-stable basis of V,
1 l
one has that deg f = m +1 for all 1 ≤ i ≤ l, up to a permutation of indices.
i i
3. A lemma of regularity
Keep the notations of the previous section and let f ∈ g and Φ : g × g −→ k
f
be the skew-symmetric bilinear form on g defined by Φ (x, x′) = K(f, [x, x′]). Here
f
we recall ([8, Def. 2]) the definition of an Heisenberg set, of centre γ ∈ ∆. It is a
subset Γ of ∆ such that γ ∈ Γ and for all α ∈ Γ \ {γ}, there exists a (unique)
γ γ γ
α′ ∈ Γ \{γ} such that α+α′ = γ.
γ
Let p = n⊕h ⊕n− be the truncated parabolic subalgebra of g associated to
π′,Λ Λ π′
π′ ⊂ π. Let S be a subset of ∆+⊔∆− and for all γ ∈ S choose an Heisenberg set Γ
π′ γ
of centre γ in ∆+ ⊔∆−. Assume that the sets Γ are disjoint and set Γ = Γ
π′ γ γ∈S γ
and y = a x , where a ∈ k \ {0} for all γ ∈ S. Set O = Γ0, with
γ∈S γ γ γ γ∈S Fγ
Γ0 = Γ \{γ}, and o = g .
γ γ P −O F
3.1. The lemma below follows exactly like [8, Lem. 6].
Lemma. Retain the above notations and hypotheses and assume further that
(i) The restriction of Φ to o×o is non-degenerate.
y
(ii) S is a basis for h∗.
|hΛ Λ
(iii) |T| = indp , where T = (∆+ ⊔∆−)\Γ.
π′,Λ π′
Then p = (adp− )y⊕g , where ad denotes the coadjoint action. In particular, y
π′,Λ π′,Λ T
is regular in p . Moreover, if we uniquely define h ∈ h by the relations γ(h) = −1
π′,Λ Λ
for all γ ∈ S, then (h, y) is an adapted pair for p− .
π′,Λ
3.2. Given γ ∈ S, for all α ∈ Γ0 denote by α′ the unique root in Γ0 such that
γ γ
α +α′ = γ and let θ be the involution in Γ0 mapping α ∈ Γ0 to α′. Denote by θ
γ γ γ
the involution in O induced by all θ , γ ∈ S.
γ
Clearly, the non-degeneracy of the restriction of Φ to o×o is immediate if, for all
y
α ∈ O, the only root β in O such that α+β ∈ S is β = θ(α).
Unfortunately this will not be the case in general but the lemma in 3.8 will give
sufficient conditions for the non-degeneracy of the restriction of Φ to o×o. To state
y
this lemma, we need further notations. In particular for each root α ∈ O, we set
S = {β ∈ O | α+β ∈ S} and for all n ≥ 1, O = {α ∈ O | |S | = n}. Note that
α n α
O = {α ∈ O | ∀β ∈ O, α+β ∈ S =⇒ β = θ(α)}.
1
8 FLORENCE FAUQUANT-MILLETANDPOLYXENILAMPROU
3.3. Sequences of roots. Keep the notations of 3.2 above. Let α ∈ O. Set α0 = α
and for all i ∈ N define αi ∈ O inductively as follows. If θ(αi) ∈ O , set αi+1 = αi.
1
Otherwise, let αi+1 6= αi be a root in O such that αi+1 +θ(αi) ∈ S. For all i ∈ N,
set α(i) = θ(α)i.
We will say that (αi) is a sequence of roots in O constructed from α; such a
i∈N
sequence always exists but in general is not unique. If for all i ∈ N, θ(αi) ∈ O ⊔O ,
1 2
then (αi) will be called the sequence of roots in O constructed from α, since in
i∈N
this case, αi is uniquely defined, for all i ∈ N. Note that if θ(αi) ∈ O for some i ∈ N,
1
then αj = αi for all j ≥ i. Conversely, if αi = αi+1, then θ(αi) ∈ O and αj = αi,
1
for all j ≥ i. We call a minimal such i the rank of the sequence (αj) and we say
j∈N
that the sequence is stationary at rank i. Note that if θ(αi) 6∈ O then αi+1 6∈ O .
1 1
3.4. Stability condition. Keep the notations of 3.2 and let α ∈ O. We say that α
satisfies the “stability condition” if there exists α′ ∈ S \{θ(α)} such that α′ ∈ O
α 2
and θ(α′) ∈ O . We say then that α′ satisfies condition (St ).
1 α
In what follows (3.5), we will consider sequences (αi) of roots in O constructed
i∈N
from α, such that αi, θ(αi) ∈ O ⊔O ⊔O for all i ∈ N, and whenever one of them
1 2 3
belongs to O , then it satisfies the stability condition. First of all, for every β ∈ O
3 3
which satisfies the stability condition, we choose and fix a root β′ ∈ S \{θ(β)}which
β
satisfies condition (St ). Now let α ∈ O and assume that we have defined the roots
β
αi ∈ O inductively as in 3.3, for all 0 ≤ i ≤ t, t ∈ N. Then if θ(αt) ∈ O satisfies the
3
stability condition, the root αt+1 is defined to be the unique root in O distinct from
αt and distinct from the chosen root θ(αt)′ satisfying condition (St ) such that
θ(αt)
αt+1 +θ(αt) ∈ S; then (αt)1 = αt+1. We proceed similarly for the sequence (α(i)).
Note that if θ(αi) ∈ O ⊔O ⊔O , for all i ∈ N are such that whenever θ(αi) ∈ O ,
1 2 3 3
it satisfies the stability condition, then the sequence (αi) is uniquely defined.
i∈N
3.5. Stationary condition. Let α ∈ O and set A = {αi, θ(αi) | i ∈ N} for a
α
sequence (αi) of roots in O constructed from α. We say that α satisfies the
i∈N
“stationary condition” if the three conditions below are satisfied:
(i) A ⊂ O ⊔O ⊔O .
α 1 2 3
(ii) If β ∈ A ∩O then β satisfies the stability condition.
α 3
(iii) The sequence (αi) is stationary (and hence so is the sequence (θ(αi)) ).
i∈N i∈N
Remarks. (1) By 3.4, conditions (i) and (ii) for the set {θ(αi) | i ∈ N} imply that
the sequence (αi) is uniquely defined.
i∈N
(2) Suppose that (αi) is a sequence of roots in O constructed from α and that,
i∈N
for i ∈ N, conditions (i) and(ii) are satisfied for theset {θ(αi) | i ∈ N, i ≥ i }. Then
0 0
for all i, j ∈ N, i ≥ i , one has that (αi)j = αi+j (it follows from the conventions
0
in 3.4 and (1)).
POLYNOMIALITY FOR MAXIMAL PARABOLICS 9
(3) Assume that α satisfies the stationary condition. Then for all i ∈ N, αi
satisfies the stationary condition. Conversely, assume that there exists i ∈ N such
0
that A′ = {αi, θ(αi) | 0 ≤ i ≤ i − 1} satisfies (i) and (ii) and αi0 satisfies the
α 0
stationary condition. Then α satisfies the stationary condition. This easily follows
by (2) above.
3.6. Stationary roots. We say that α is “a stationary root” and write α ∈ O if
st
bothα andθ(α)satisfythestationarycondition. Clearly, ifα ∈ O , thenθ(α) ∈ O .
st st
Remarks. Let α ∈ O.
(1) Assume that conditions (i) and (ii) of Section 3.5 are satisfied for the sets A
α
and A and that for all i, s ∈ N, 0 ≤ s ≤ i − 1, θ(αs), αs+1 ∈ O . Then for all
θ(α) 2
j ≤ i, (αi)(j) = θ(αi−j) and for all j ≥ i, (αi)(j) = α(j−i).
(2) Assume that conditions (i) and (ii) of Section 3.5 are satisfied for the sets A
α
and A and that for all i, s ∈ N, 0 ≤ s ≤ i−1, θ(α(s)), α(s+1) ∈ O . Then for all
θ(α) 2
j ≤ i, (α(i))(j) = θ(α(i−j)) and for all j ≥ i, (α(i))(j) = αj−i.
(3) Assume that A := A ∪ A = {αi, α(i), θ(αi), θ(α(i)) | i ∈ N} ⊂ O ⊔ O .
α θ(α) 1 2
If α ∈ O then A ⊂ O and conversely if there exists i ∈ N such that αi0 or α(i0)
st st 0
belongs to O , then α ∈ O . This easily follows from (1) and (2) above and from
st st
remark (2) in 3.5.
(4) Assume that there exists i ∈ N such that αi0 ∈ O , A′ = {αi, θ(αi) | 0 ≤ i ≤
0 st α
i −1} ⊂ O ⊔O ⊔O and if β ∈ A′ ∩O , then β satisfies the stability condition,
0 1 2 3 α 3
and that θ(α) satisfies the stationary condition. From remark (3) in 3.5 we deduce
that α ∈ O . We have a similar statement if we interchange α and θ(α).
st
(5) Assume that there exists i , i ∈ N such that A′ = {αi, θ(αi) | 0 ≤ i ≤
0 1 α
i − 1} ⊂ O ⊔ O ⊔ O , A′ = {α(i), θ(α(i)) | 0 ≤ i ≤ i − 1} ⊂ O ⊔ O ⊔ O ,
0 1 2 3 θ(α) 1 1 2 3
αi0, α(i1) ∈ O and if β ∈ (A′ ∪A′ )∩O , then β satisfies the stability condition.
st α θ(α) 3
Then from (4) above it follows that α ∈ O .
st
(6) Assume that α ∈ O and satisfies the stationary condition. By (ii) of the
3
definition in 3.5, α also satisfies the stability condition. Let α′ ∈ S \{θ(α)} be the
α
chosen root satisfying condition (St ). Then α′ ∈ O : indeed, since α′ ∈ O ∩S ,
α st 2 α
one has that (α′)(1) = θ(α′)1 = α, then θ(α′) satisfies the stationary condition,
because α does, by remark (3) of 3.5. Moreover θ(α′) ∈ O implies that (α′)1 = α′,
1
then α′ also satisfies the stationary condition.
Lemma. Let α ∈ O and set A = {αi, θ(αi), α(i), θ(α(i)) | i ∈ N}. Let ϑ : O → O
st
be a permutation such that, for all γ ∈ O, γ +ϑ(γ) ∈ S. Then the restriction of ϑ
on A coincides with θ and if β ∈ A ∩ O , and β′ ∈ S \ {θ(β)} is the chosen root
3 β
satisfying condition (St ), then ϑ exchanges β′ and θ(β′).
β
Proof. Denote by n (resp. n ) the rank of the stationary sequence (αi) (resp.
0 1 i∈N
(α(i)) ).
i∈N
10 FLORENCE FAUQUANT-MILLETANDPOLYXENILAMPROU
Since θ(αn0) ∈ O (resp. θ(α(n1)) ∈ O ) the map ϑ necessarily sends θ(αn0) (resp.
1 1
θ(α(n1))) to αn0 (resp. α(n1)).
Now if θ(αn0−1) ∈ O , the permutation ϑ sends θ(αn0−1) to αn0−1 and for all
2
0 ≤ i ≤ n −1, as long as θ(αi) ∈ O we necessarily have ϑ(θ(αi)) = αi.
0 2
Suppose that there exists i ∈ N, 0 ≤ i ≤ n − 1, such that θ(αi) ∈ O ; then
0 3
it satisfies the stability condition. Let i be the largest such integer. Then the
0
image of θ(αi0) via the map ϑ could be either αi0 or θ(αi0)′ since αi0+1 has already
a preimage. Since θ(θ(αi0)′) ∈ O , the map ϑ sends θ(θ(αi0)′) to θ(αi0)′. Then the
1
permutation ϑ sends θ(αi0) to αi0. By decreasing induction on i, we then deduce
that, for all 0 ≤ i ≤ n , ϑ(θ(αi)) = αi. Similarly we obtain that, for all 0 ≤ i ≤ n ,
0 1
ϑ(θ(α(i))) = α(i).
It follows that if αi ∈ O then (αi)′ is sent to θ((αi)′) since αi has already a
3
preimage. Of course, since θ((αi)′) ∈ O , the latter is sent to (αi)′. An increasing
1
induction on i proves then that ϑ(αi) = θ(αi) for all 0 ≤ i ≤ n , and it follows that,
0
if θ(αi) ∈ O , then θ(αi)′ is sent to its image by θ.
3
Similarly one has that if α(i) ∈ O , then (α(i))′ is sent to its image by θ, ϑ(α(i)) =
3
θ(α(i)) for all 0 ≤ i ≤ n and if θ(α(i)) ∈ O then θ(α(i))′ is sent to its image by θ.
1 3
(cid:3)
3.7. Cyclic roots. Let α ∈ O. We say that α is “cyclic” and write α ∈ O if there
cyc
exist β, γ ∈ O such that the following conditions are satisfied :
(i) θ(α)+γ = β +θ(β)
(ii) θ(γ)+β = α+θ(α)
(iii) θ(β)+α = γ +θ(γ)
(iv) {α, β, γ, θ(α), θ(β), θ(γ)} ⊂ O ⊔O
2 3
(v) |{α, β, γ, θ(α), θ(β), θ(γ)}| = 6
˜ ˜
(vi) If δ ∈ {α, β, γ, θ(α), θ(β), θ(γ)}∩O , then there exists δ ∈ S such that δ
3 δ
satisfies the stationary condition.
For α ∈ O , set C = {α, β, γ, θ(α), θ(β), θ(γ)}. Note that for δ ∈ C ∩O then,
cyc α α 3
˜ ˜
with the above notations, δ is unique and S \S ∩C = {δ}.
δ δ α
Remarks. (1) If α ∈ O then all roots in C are cyclic roots.
cyc α
(2) Let α ∈ O . If δ ∈ C ∩O , then δ may satisfy the stability condition (5.3.1,
cyc α 3
first case, for instance) but we will see (6.3, paragraph (c) for instance) that it is not
always the case.
(3) If α ∈ O then it cannot satisfy the stationary condition nor it can be sta-
cyc
tionary, even if all roots in C ∩O satisfy the stability condition; in the latter case
α 3
the cyclic relations (i), (ii) and (iii) imply that α1 = γ, α2 = β and α3 = α, hence
the sequence (αi) is not stationary.
i∈N
