Littelmann paths and brownian paths

0.36 MB·English

by  Philippe Biane

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0
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2 Littelmann paths and Brownian paths

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a
Philippe Biane, Philippe Bougerol, and Neil O’Connell
J

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1 Abstract. WestudysomepathtransformationsrelatedtoPitman’stheorem
onBrownianmotionandthethreedimensionalBesselprocess. Werelatethese

] toLittelmannpathmodel,andgiveapplicationstorepresentationtheoryand
T
toBrownianmotioninaWeylchamber.
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.
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t
a 1. Introduction
m
Some transformations defined on continuous paths with values in a vector
[
space have appeared in recent years, in two separate parts of mathematics. On

2
the one hand Littelmann [22] developed his path model in order to give a unified
v
combinatorial setup for representation theory, generalizing the theory of Young
1
7 tableaux to semi-simple or Kac-Moody Lie algebras of type other than A. On the
1 other hand, in probability theory, several path transformations have been intro-
3 duced that yield a construction of Brownian motion in a Weyl chamber starting
0
from a Brownian motion in the corresponding Cartan Lie algebra. The oldest and
4
simplest of these transformations comes from Pitman’s theorem [28] which states
0
that if (B ) is a one-dimensional Brownian motion, then the stochastic process
/ t t 0
h ≥
R := B −2inf B is a three dimension Bessel process, i.e. is distributed
t t t 0≤s≤t s
a as the euclidean norm of a three dimensional Brownian motion (actually Pitman
m stated his theorem with the transformation 2sup B −B , but thanks to the
0 s t s t
: symmetryofBrownianmotionthisisclearlyequiva≤le≤nttotheabovestatement). It
v
turns out that the fact that, here, the dimension of the Brownian motion is equal
i
X to 1, the rank of the group SU(2), while 3, the dimension of the Bessel process,
r is the dimension of the group SU(2) is not a mere coincidence but a fundamental
a
fact which we will clarify in the following. Pitman’s theorem has been extended in
severalways. Thefirststephasbeenthe resultofGravner,TracyandWidom,[15]
and of Baryshnikov [1] which states that the largest eigenvalue of a random n×n
Hermitian matrix in the GUE is distributed as the random variable
n
sup (B (t )−B (t ))
i i i i 1
1=tn≥tn−1≥...≥t1≥t0=0 Xi=1 −
where (B ,...,B ) is a standard n-dimensional Brownian motion. This result in
1 n
turn was generalized in [7] and [27]. These extensions involve path transforma-
tions which generalize Pitman’s and are closely related to the Littelmann path
model. One of the purposes of this paper is to clarify these connections as well as
1991 Mathematics Subject Classification. Primary;Secondary.
1
2 PHILIPPEBIANE, PHILIPPEBOUGEROL,ANDNEILO’CONNELL
to settle a number of questions raised in these works. In the course of these inves-
tigations we will derive several applications to representation theory. These path
transformations occur in quite different contexts, since the one in [7] is expressed
byrepresentationtheoreticmeans, whereasthe one in[27] is purelycombinatorial,
and arises from queuing theory considerations.
Let us describe more precisely the content of the paper. We start by defining
the Pitman transforms which are the main object of study in this paper. These
transforms operate on the set of continuous functions π : [0,T] → V, with values
in some real vector space V, such that π(0)=0. They are given by the formula
P π(t)=π(t)− inf α (π(s))α, t∈[0,T].
α ∨
t s 0
≥ ≥
Here α ∈ V and α ∈ V (where V is the dual space of V) satisfy α (α) = 2.
∨ ∨ ∨ ∨
These are multidimensional generalizations of the transform occuring in Pitman’s
theorem. They are related to Littelmann’s operators as shown in section 2.2. We
showthatthesetransformssatisfybraidrelations,i.e. ifα,β ∈V andα ,β ∈V
∨ ∨ ∨
are such that α (α) = β (β) = 2, and α (β) < 0,β (α) < 0 and α (β)β (α) =
∨ ∨ ∨ ∨ ∨ ∨
4cos2 π, where n≥2 is some integer, then one has
n
P P P ...=P P P ...
α β α β α β
wheretherearenfactorsineachproduct. ConsidernowaCoxetersystem(W,S)(cf
[8],[18]). To each fundamental reflection s we associate a Pitman transform P .
i αi
Thebraidrelationsimplythatifw ∈W hasareduceddecompositionw =s ...s ,
1 n
then the operator P = P ...P is well defined, i.e. it depends only on w and
w α1 αn
not on the reduced decomposition. We show that if W is a Weyl group, w ∈ W
0
is the longest element, and π is a dominant path ending in the weight lattice, then
for any path η in the Littelmann module generated by π, one has
(1.1) π =P η.
w0
The path transformation introduced in [27] can be expressed as P where w is
w0 0
the longest element in the Coxeter group of type A.
WederivearepresentationtheoreticformulaforP ,inthecaseofaWeylgroup,
w
expressed in terms of representations of the Langlands dual group, see Theorem
3.12. This formula is canonical, in the sense that it is independent of any choice
of a reduced decomposition of w in the Weyl group. It is obtained by lifting the
path to a path g(t) with values in the Borel subgroup of the simply connected
complex Lie group associated with the root system. Then one obtains integral
transformations which relate the diagonal parts in the Gauss decompositions of
the elements wg(t). The Pitman transforms are obtained by going down to the
Cartan algebra by applying Laplace’s method. By (1.1) we obtain in this way a
new formula for the dominant path in some Littelmann module, in terms of any
of the paths of the module, which is a generalization to arbitrary root systems of
Greene’sformula (see [14]). As a byproductof this formula wealso obtaina direct
proof of the symmetry of the Littlewood-Richardsoncoefficients.
This formulaappearedin [7]whereit wasconjecturedthatthe associatedmap
transforms a Brownian motion in the Cartan Lie algebra into a Brownian motion
in the Weyl chamber. This conjecture was proved in [7] for some classical groups.
Here we give a completely different proof, valid for all root systems.
Thispaperisorganizedasfollows. Insection2wedefinetheelementaryPitman
transformationsoperatingoncontinuouspathswithvaluesinsomerealvectorspace
LITTELMANN PATHS AND BROWNIAN PATHS 3
V,takingthevalue0at0. Thefirstresultisaformulafortherepeatedcompositions
oftwoPitmantransforms,whichimpliesthattheysatisfythebraidrelations. Then
we define Pitman transformations P associated to a Coxeter system (W,S). In
w
section 3 we prove our main result which is a representation theoretic formula for
these operators P in the case where W is a Weyl group. This formula unifies
w
the results of [27] and of [7]. Results of Berenstein and Zelevinsky [2] and of
Fomin and Zelevinsky [12] on totally positive matrices play a crucial role in the
proof. In section 4 we make some comments on a duality transformationnaturally
defined on paths, which generalizes the Schu¨tzenberger involution, and give an
application to the symmetry of the Littlewood-Richardson rule. In section 5 we
givetwo proofs ofthe generalizationofthe representationof Brownianmotionin a
Weyl chamber obtained in [27] and [7]. One of the proofs relies essentially on the
duality properties, while the other uses Littelmann paths in the context of Weyl
groups. Finallysection6isanappendixwherewehavepostponedatechnicalproof.
Acknowledgements. We would like to thank P. Littelmann for a useful con-
versation at an early stage of this work, and P. Diaconis and S. Evans for helpful
discussions. We also thank the referee for useful comments.
2. Braid relations for the Pitman transforms
2.1. Pitman transforms. LetV be a realvectorspace,withdualspace V .
∨
Let α∈V and α ∈V be such that α (α)=2.
∨ ∨ ∨
Definition 2.1. The Pitman transform P is defined on the set of continuous
α
paths π :[0,T]→V, satisfying π(0)=0, by the formula:
P π(t)=π(t)− inf α (π(s))α, T ≥t≥0.
α ∨
t s 0
≥ ≥
This transformation seems to have appeared for the first time in [28] in the
one-dimensional case. Note that P actually depends on the pair (α,α ). For
α ∨
simplicity we shall use the notation P , it will be always clear from the context
α
which α is involved.
∨
When, for some v ∈ V, π is the linear path π(t) = tv then P π = π when
α
α (v)≥0 and P π =s π when α (v)≤0 where s is the reflection on V
∨ α α ∨ α
(2.1) sαv =v−α∨(v)α
for v ∈V.
We list a number of elementary properties of the Pitman transform below.
Proposition 2.2. (i) For any λ > 0 the Pitman transformation associated
with the pair (λα,α /λ) is the same as the one associated with the pair (α,α ).
∨ ∨
(ii) One has α (P π(t)) ≥ 0 for all t ∈ [0,T]. Furthermore P π = π if and
∨ α α
only if α (π(t))≥0 for all t∈[0,T].
∨
(iii ) The transformation P is an idempotent, i.e. P P π =P π for all π.
α α α α
(iv) Let π : [0,∞[→V be a path, then −inf α (π(t)) ∈ [0,α (P π(T))].
0 t T ∨ ∨ α
≤ ≤
Conversely, given a path η satisfying η(0) = 0, α (η(t)) ≥ 0 for all t ∈ [0,T]
∨
and x ∈ [0,α (η(T))], there exists a unique path π such that P π = η and x =
∨ α
−inf α (π(t)). Actually π is given by the formula
T t 0 ∨
≥ ≥
(2.2) π(t)=η(t)−min x, inf α∨(η(s)) α.
T s t
(cid:18) ≥ ≥ (cid:19)
4 PHILIPPEBIANE, PHILIPPEBOUGEROL,ANDNEILO’CONNELL
Proof. Items (i) and (ii) are trivial, and (iii) follows immediately from (ii).
Hopefully the reader can give a formal proof of (iv), see section 6 for such a proof,
but it is perhaps more illuminating to stare for a few minutes at Fig. 1, which
shows, in the one dimensional case, with α = 1,α = 2, the graph of a function
∨
g :[0,1]→Ras wellas thoseofI,−I andf =P g whereI(s)=inf g(u). ♦.
α 0 u s
≤ ≤
f
−I
x
g
I
Fig. 1
2.2. Relation with Littelmann path operators. Using Proposition 2.2
(iv) we can define generalizedLittelmann transformations. Recall that Littelmann
operators are defined on paths with values in the dual space a of some real Lie
∗
algebra a. The image of a path is either another path or the symbol 0 (actually
the zero element in the Z-module generated by all paths). We define continuous
versions of these operators.
Definition 2.3. Let π : [0,T] → V be a continuous path satisfying π(0) = 0,
and x∈R, then Exπ is the unique path such that
α
P Exπ =P π and α (Exπ(T))=α (π(T))+x
α α α ∨ α ∨
if −2α (π(T))+2inf α (π(t)) ≤ x ≤ −2inf α (π(t)) and Exπ = 0
∨ 0 t T ∨ 0 t T ∨ α
≤ ≤ ≤ ≤
otherwise.
One checks easily that E0π = π and ExEyπ = Ex+yπ as long as Eyπ 6= 0.
α α α α α
Whenαisarootandα itscoroot,insomerootsystem,thenE2 andE 2 coincide
∨ α α−
with the Littelmann operators e and f , defined in [22]. Recall that a path π is
α α
calledintegralif its endpoint π(T)is inthe weightlattice and, for eachsimple root
α, the minimum of the function α (π(t)) is an integer. The class of integral paths
∨
isinvariantundertheLittelmannoperators. Forsuchpaths,theactionofaPitman
LITTELMANN PATHS AND BROWNIAN PATHS 5
transform can be expressed through Littelmann operators by
(2.3) Pαπ =enαα(π)
where n is the largest integer n such that en(π)6=0.
α α
2.3. Braid relations. An important property of the Pitman transforms is
the following result.
Theorem2.4. Letα,β ∈V andα ,β ∈V besuchthatα (α)=β (β)=2,
∨ ∨ ∨ ∨ ∨
andα (β)<0,β (α)<0andα (β)β (α)=4cos2 π,wheren≥2issomeinteger,
∨ ∨ ∨ ∨ n
then one has
P P P ...=P P P ...
α β α β α β
where there are n factors in each product.
We shall prove Theorem 2.4 as a corollary to the result of section 2.4. Note
that if α (β) = β (α) = 0 then P P = P P by a simple computation. For
∨ ∨ α β β α
crystallographic angles (i.e. n = 2,3,4,6) a proof of Theorem 2.4 could also be
deduced fromLittelmann’s theory (see [23] or [19]). We shall providestill another
(hopefully more conceptual) proof for these angles in section 3, see Remark 3.10.
The general case seems to be new.
2.4. A formula for P P P P .... Let α,β ∈ V and α ,β ∈ V be such
α β α β ∨ ∨ ∨
thatα (β)<0andβ (α)<0. ByProposition2.2(i)wecan-andwill-assumeby
∨ ∨
rescalingthat α (β)=β (α), without changing P and P . We use the notations
∨ ∨ α β
1 1
ρ=− α∨(β)=− β∨(α), X(s)=α∨(π(s)), Y(s)=β∨(π(s)).
2 2
Theorem 2.5. Let n be a positive integer, if ρ≥cosπ, then one has
n
n 1
−
(P P P ...)π(t) = π(t)− inf T (ρ)Z(i)(s ) α
α β α i i
nterms t≥s0≥s1≥...≥sn−1≥0(cid:0)Xi=0 (cid:1)
n 2
(2.4) | {z } − inf − T (ρ)Z(i+1)(s ) β
i i
t≥s0≥s1≥...≥sn−2≥0 i=0
(cid:0)X (cid:1)
whereZ(k) =X ifkisevenandZ(k) =Y ifkisodd. TheT (x)aretheTchebycheff
k
polynomials defined by T (x) = 1, T (x) = 2x, and 2xT (x) = T (x)+T (x)
0 1 k k 1 k+1
−
for k ≥1.
The Tchebycheff polynomials satisfy T (cosθ) = sin(k+1)θ and, in particular,
k sinθ
under the assumptions on ρ and n, one has T (ρ)≥0 for all k ≤n−1.
k
Assuming Theorem 2.5 we obtain Theorem 2.4.
ProofofTheorem2.4. Letα (β)=β (α)=−2cosπ,thenonehasT (ρ)=
∨ ∨ n n 1
−
0 and the last term in the coefficient of α in the right hand side of (2.4) vanishes.
It follows by inspection that this term equals the coefficient of α in the analogous
formula for P P P ...π(t). A similar argument works for the coefficient of β. ♦
β α β
nterms
The proof of Theorem 2.5 will be by induction on n. It is easy to check the
| {z }
formula for n=1 or 2. We shall do the induction in sections 2.5 and 2.6.
6 PHILIPPEBIANE, PHILIPPEBOUGEROL,ANDNEILO’CONNELL
2.5. Two intermediate lemmas.
Lemma 2.6. Let X : [0,t] → R be a continuous functions with X(0) = 0 and
let t =sup{s≥0|X =inf X }, then for all u≤t one has
0 s s u 0 u 0
≥ ≥
inf (X(s)−2 inf X(w))=− inf X(v).
t s u s w 0 u v 0
≥ ≥ ≥ ≥ ≥ ≥
Proof. This is obtained as a byproduct of the proof in section 6. Again it is
perhaps more convincing to stare at Fig. 1 than to give a formal proof. ♦
Elaborating on this we obtain the next result.
Lemma 2.7. LetX andY becontinuousfunctions, suchthat X(0)=Y(0)=0,
then
inf X(s)+inf Y(u) =inf X(s)+
t s 0 s u 0 t s 0
≥ ≥ ≥ ≥ ≥ ≥
inf X(s)−2(cid:0)inf X(u)+inf (cid:1) Y(u)+inf X(v)
t s 0 s u 0 s u 0 u v 0
≥ ≥ ≥ ≥ ≥ ≥ ≥ ≥
(cid:16) (cid:0) (cid:1)(cid:17)
Proof. The first term is I =inf (X(s)+Y(u)). Let t be, as in Lemma
t s u 0 0
≥ ≥ ≥
2.6, the last time when X reaches its minimum over [0,t], then
I =inf inf (Y(u)+X(t )); inf (Y(u)+X(s))
0
t0≥u≥0 t≥s≥u≥t0≥0
(cid:0) (cid:1)
Let J be the second term in the identity to be proved, then
J = inf X(s)−2 inf X(u)+ inf (Y(u)+ inf X(v)) +X(t )
0
t s 0 s u 0 s u 0 u v 0
≥ ≥ ≥ ≥ ≥ ≥ ≥ ≥
(cid:2) (cid:3)
Introduce again the time t , then
0
J = inf X(s)−2 inf X(w)+Y(u)+ inf X(v) +X(t )
0
t s u 0 s w 0 u v 0
≥ ≥ ≥ ≥ ≥ ≥ ≥
(cid:0) (cid:1)
= inf inf Y(u)+X(s)−2 inf X(w)+ inf X(v)+X(t ) ;
0
t s u 0 s w 0 u v 0
(cid:16) ≥t0≥≥u≥ (cid:0) ≥ ≥ ≥ ≥ (cid:1)
inf Y(u)+X(s)−2 inf X(w)+ inf X(v)+X(t )
0
t≥s≥u≥t0(cid:0) s≥w≥0 u≥v≥0 (cid:1)(cid:17)
but if u ≤ t then by lemma 2.6 one has inf (X(s) − 2inf X(w)) =
0 t s u s w 0
≥ ≥ ≥ ≥
−inf X(v). If t ≤u then inf X(w)=X(t ), therefore
u v 0 0 s w 0 0
≥ ≥ ≥ ≥
J = inf inf Y(u)+X(t ) ; inf Y(u)+ inf X(s)
0
= inf(cid:16)t0i≥nuf≥0((cid:0)Y(u)+X(t ))(cid:1); 0≥ui≥ntf0(cid:0) (Y(u)t≥+s≥Xu(s)) (cid:1)(cid:17)
0
t0≥u≥0 t≥s≥u≥t0≥0
= I. (cid:0) (cid:1)
2.6. EndofproofofTheorem2.5. AssumetheresultoftheTheoremholds
for some n with n even. Then P P P ...=P P P ...P , and one has
α β α α β α α
n+1terms nterms
α (P π(s|)) ={z X}(s)−|2 i{nzf X}(u)
∨ α
s u 0
≥ ≥
β (P π(s)) = Y(s)+2ρ inf X(u)
∨ α
s u 0
≥ ≥
LITTELMANN PATHS AND BROWNIAN PATHS 7
therefore, by induction hypothesis
P P P ...π(t) = P P P ...(P π)(t)
α β α α β α α
n+1terms nterms
n 1
| {z } = π|(t)−{z inf} X(s)α− inf − Zˆ(i)(s ) α
i
t≥s≥0 t≥s0≥s1≥...≥sn−1≥0(cid:16)Xi=0 (cid:17)
n 2
− inf − Zˆ(i+1)(s ) β
i
t≥s0≥s1≥...≥sn−2≥0(cid:16)Xi=0 (cid:17)
where
X(s)−2inf X(u) for i even
Zˆα(i)(s)= Y(s)+2ρinfs≥u≥0 X(u) for i odd.
s u 0
(cid:26) ≥ ≥
The coefficient of α in the above expression has the form
H =− inf T (ρ)X(s)
α 0
t s 0
≥ ≥
− inf T (ρ)X(s)−2 inf T (ρ)X(u)+ inf Γ(u)+ inf T (ρ)X(v)
0 0 0
t s 0 s u 0 s u 0 u v 0
≥ ≥ (cid:16) ≥ ≥ ≥ ≥ (cid:0) ≥ ≥ (cid:1)(cid:17)
where
Γ(u) = T (ρ)Y(u)+2ρT (ρ) inf X(v)+
1 1
u v 0
≥ ≥
n 1
inf − T (ρ)Zˆ(i)(u ) −T (ρ) inf X(v)
i i 0
u≥u2≥u3≥...≥un−1≥0(cid:16)Xi=2 (cid:17) u≥v≥0
= T (ρ)Y(u)+T (ρ) inf X(v)+
1 2
u v 0
≥ ≥
n 1
inf − T (ρ)Zˆ(i)
i
u≥u2≥u3≥...≥un−1≥0(cid:16)Xi=2 (cid:17)
so that we can apply lemma 2.7 to transform it into
H =− inf T (ρ)X(s)+ inf Γ(u)
α 0
t s 0 s u 0
≥ ≥ ≥ ≥
(cid:0) (cid:1)
Let us prove by induction on k that
2k
H = − inf T (ρ)Z(i)(u )+W (u )
α i i k 2k 1
t≥u0≥u1≥...≥u2k(cid:16)Xi=0 − (cid:19)
with
n 1
W (v) = inf − T (ρ)(Zˆ(i)(u )
k i i
v≥u2k≥u2k+1≥...≥un−1≥0(cid:16)iX=2k (cid:17)
8 PHILIPPEBIANE, PHILIPPEBOUGEROL,ANDNEILO’CONNELL
Indeed the formula holds for k = 1 by the computation above. Assume this holds
for some k then one has
2k
H =− inf T (ρ)Z(i)(u )+W (u )
α i i k 2k 1
t≥u0≥u1≥...≥u2k(cid:16)Xi=0 − (cid:19)
2k 1
−
=− inf T (ρ)Z(i)(u )+ inf T (ρ)X(v)+
i i 2k
t≥u1≥u2≥...≥u2k−1(cid:18)i=0 u2k−1≥v≥0
X
inf (T (ρ)X(v)−2 inf T (ρ)X(w))+
2k 2k
u2k−1≥v≥0(cid:16) v≥w≥0
inf R (z)+ inf T (ρ)X(τ)
k 2k
w z 0 z τ 0
≥ ≥ (cid:0) ≥ ≥ (cid:1)(cid:17)(cid:19)
where
R (z) = T (ρ)Y(z)+2ρT (ρ) inf X(τ)+
k 2k+1 2k+1
z τ 0
≥ ≥
n 1
inf − Zˆ(i)(u ) − inf T (ρ)X(τ)
i 2k
z≥u2k+2≥...un−1 i=2k+2 z≥τ≥0
(cid:0) X (cid:1)
n 1
= T (ρ)Y(z)+T (ρ) inf X(τ)+ inf − Zˆ(i)(u )
2k+1 2k+2 i
z≥τ≥0 z≥u2k+2≥...un−1 i=2k+2
(cid:0) X (cid:1)
where we used 2ρT (ρ)−T (ρ)=T (ρ). Applying Lemma 2.1. we get
2k+1 2k 2k+2
2k 1
−
H = − inf T (ρ)Z(i)(u )+
α i i
t≥u1≥u2≥...≥u2k−1(cid:16)Xi=0
inf T (ρ)X(v)+ inf R (z)
2k k
u2k−1≥v≥0(cid:0) 2k+2 w≥z≥0 (cid:1)(cid:17)
= − inf T (ρ)Z(i)(u )+W (u )
i i k+1 2k+1
t≥u0≥u1≥...≥u2k+2(cid:18)i=0 (cid:19)
X
Taking k = n gives the required formula for H . For the coefficient of β, remark
α
that
P P P ...π(t)=P (P P P ...π)(t)
α β α α β α β
n+1terms nterms
and the formula for n + 1 follows immediately from the formula at step n for
| {z } | {z }
P P P .... The case where n is odd is treated in a similar way. ♦
β α β
nterms
| 2{.z7. P}itman transformations for Coxeter and Weyl groups. Let W be
a Coxeter group, i.e. W is generated by a finite set S of reflections of a real vector
space V, and (W,S) is a Coxeter system (see [8], [18]). For each s∈S, let α ∈V
s
and α ∈V , where V is the dual space of V, such that s=s is the reflection
∨s ∨ ∨ αs
associatedtoα (see(2.1)). Thenα iscalledthesimplerootassociatedwiths∈S
s s
and α its coroot.
∨s
Denote by P the Pitman transform associated with the pair (α ,α ). By the
s s ∨s
resultsofthe precedingsections,the P ;s∈S forma representationofthe monoid
s
generated by idempotents satisfying the braid relations. Such a monoid occurs in
LITTELMANN PATHS AND BROWNIAN PATHS 9
the theory of Hecke algebras for q =0, and in the calculus of Borel orbits (see e.g.
[20] where this monoid is called Richardson-Springermonoid).
Let H be the closed half space H = {v ∈ V|α (v) ≥ 0}. Let w ∈ W and
s s ∨s
let w = s ...s be a reduced decomposition of w, where l = l(w) is the length of
1 l
w. By Theorem 2.4 and a fundamental result of Matsumoto ([8] Ch. IV, no 1.5,
Proposition 5) the operator P ...P depends only on w, and not on the chosen
s1 sl
reduced decomposition. We shall denote by P this operator.
w
Proposition 2.8. Let w ∈ W, L = {s ∈ S|l(sw) < l(w)},R = {s ∈
w w
S|l(ws)<l(w)}. For any path π, the path P π lies in the convex cone ∩ H ,
w s∈Lw s
one has P P =P for all s∈L and P P =P for all s∈R .
s w w w w s w w
Proof. If l(sw) < l(w) then w has a reduced decomposition w = ss ...s
1 k
thereforeP =P P ...P andP π =P (P ...P π)liesinH byProposition
w s s1 sk w s s1 sk s
2.2(ii). FurthermoreonehasP P =P sinceP isaninvolution(seeProposition
s w w s
2.2 (ii) ). Similarly P P =P when l(ws)<l(w). ♦
w s w
Corollary 2.9. If W is finite and w is the longest element, then P π takes
0 w0
values in the closed Weyl chamber C =∩ H , furthermore P is an idempotent
s∈S s w0
and P P =P P =P for all w ∈W.
w w0 w0 w w0
Assume now that W is a finite Weyl group, associated with a weight lattice
in V. Recall that paths taking values in the Weyl chamber C are called dominant
paths in [22], and that the set Bπ of all (nonzero) paths obtained by applying
products of Littelmann operators to a dominant path π is called the Littelmann
module. From the connection between Pitman’s andLittelmann’s operators,given
in section 2.2, we deduce the following (see also [23]).
Corollary 2.10. Let π be a dominant integral path, then a path η belongs to
the Littelmann module Bπ if and only if η is integral and π =P η.
w0
IndeedforanypathηandxsuchthatExη 6=0onehasP Exη =P η,therefore
α α α α
P Exη = P P Exη = P η. It follows that the set of paths whose image by
w0 α w0 α α w0
P is π is stable underthe actionofLittelmann operators. Ifη is anintegralpath
w0
suchthatP η =π,andw =s ...s isareduceddecomposition,thenbysection
w0 0 1 n
2.2 the sequence η,P η,P P η,...,π is obtained by successive applications
αn αn−1 αn
ofLittelmannoperatorsthereforethey allbelong to the Littelmann module Bπ. ♦
Let us come back to the general case of a finite Coxeter group. We shall
now study the set of all paths η such that P η is a given dominant path. Let
w
w =s ...s be a reduced decomposition. Let η be a path such that η(0)=0 and
1 q
π = P η is a dominant path. Denote η = π,η = η, and η = P ...P η
w 0 q j sj+1 sq q
for j = 1,2,...,q−1, then by Proposition 2.2 (iv) for all j = 1,2,...,q the path
η is uniquely specified among paths γ such that P γ = η , by the number
j sj j−1
x = −inf α (η (t)) ∈ [0,α (η (T))]. It follows that η = η is uniquely
j 0≤t≤T ∨sj j ∨sj j−1 q
specified, among all paths γ such that P γ = π by the sequence x ,x ,...,x .
w0 1 2 q
These coordinates are subject to the inequalities 0≤x ≤α (η (T)). From
j ∨sj j−1
η (T)=η (T)+x α
j−1 j j sj
one obtains
j
π(T)=η (T)=η (T)+ x α
0 j l sl
l=1
X
10 PHILIPPEBIANE, PHILIPPEBOUGEROL,ANDNEILO’CONNELL
therefore the inequality 0≤x ≤α (η (T)) reads
j ∨sj j−1
j 1
−
0≤x ≤α (π(T))− x α (α ).
j ∨sj l ∨sj sl
l=1
X
It follows that the set of all paths η such that P η = π can be parametrized by a
w
subset of the convex polytope
j 1
−
K ={(x ,...,x )∈Rq|0≤x ≤α (π(T))− x α (α );j =1,...,q}.
π 1 q j ∨sj l ∨sj sl
l=1
X
The path η corresponding to the point (x ,...,x ) is specified by the equalities
1 q
η (T)=η (T)+x α
j−1 j j sj
where η = P ...P η. In the case of a Weyl group, it follows from [23] that
j sj+1 sq
the subset ofK correspondingto paths η suchthat P η =π is the intersectionof
π w
K with a certain convex cone which does not depend on π. This convex cone is
π
quite difficult to describe, see [3]. Also we do not know if a similar result holds for
allfinite Coxeter groups. We hope to come back to these questions in future work.
3. A representation theoretic formula for P
w
3.1. Semisimplegroups. Werecallsomestandardterminology. Weconsider
asimplyconnectedcomplexsemisimpleLiegroupG,associatedwitharootsystem
R. Let H be a maximal torus, and B+,B be corresponding opposite Borel sub-
−
groups with unipotent radicalsN+,N . Let α ,i∈I, and α ,i∈I, be the simple
− i ∨i
positive roots and coroots, and s the corresponding reflections in the Weyl group
i
W. Let e ,f ,h ,i ∈ I, be Chevalley generators of the Lie algebra of G. One can
i i i
chooserepresentativesw ∈Gforw ∈W byputtings =exp(−e )exp(f )exp(−e )
i i i i
and vw = vw if l(v)+l(w) = l(vw) (see [12] (1.8), (1.9)). The Lie algebra of H,
denoted by h has a Cartan decomposition h = a+ia such that the roots α take
i
realvalues onthe realvectorspacea. Thus ais generatedby α ,i∈I andits dual
∨i
a by α ,i∈I. The setof weightsis the lattice P ={λ∈a ;λ(α )∈Z,i∈I} and
∗ i ∗ ∨i
the set of dominant weights is P+ ={λ∈a ;λ(α )∈N,i∈I}. For each λ∈P+,
∗ ∨i
choose a representationspace V with a highest weightvector v , and an invariant
λ λ
inner product on V for which v is a unit vector.
λ λ
Lemma 3.1. For any dominant weight λ, w ∈W and indices i ,...,i ∈I one
1 n
has
he ...e wv ,v i≥0
i1 in λ λ
Proof. This is an immediate consequence of Lemma 7.4 in [3]. ♦
Let(ω ,i∈I)∈PI be the fundamentalweights,characterizedbythe relations
i
ω (α ) = δ ,j ∈ I. The principal minor associated with ω is the function on G
i ∨j i,j i
given by
∆ωi(g)=hgvωi,vωii
see [2] and [12]. If g ∈ G has a Gauss decomposition g = [g] [g] [g] with [g] ∈
0 +
N , [g] ∈H, [g] ∈N+, then one has − −
− 0 +
(3.1) ∆ωi(g)=[g]ωi =eωi(log[g]0).
0