Paths in Weyl chambers and random matrices Thierry Jeulin † Universit´e Paris 7
Philippe Bougerol ∗ Universit´e Paris 6
Abstract Baryshnikov [3] and Gravner, Tracy & Widom [14] have shown that the largest eigenvalue of a random matrix of the G.U.E. of order d has the same distribution as max
1≥t1 ≥···≥td−1 ≥0
[W1 (1) − W1 (t1 ) + W2 (t1 ) − W2 (t2 ) + · · · + Wd (td−1 )] ,
where W = (W1 , · · · , Wd ) is a d-dimensional Brownian motion. We provide a generalization of this formula to all the eigenvalues and give a geometric interpretation. For any Weyl chamber a+ of an Euclidean finite-dimensional space a, we define a natural continuous path transformation T which associates to a path w in a a path T w in a+ . This transformation occurs in the description of the asymptotic behaviour of some deterministic dynamical systems on the symmetric space G/K where G is the complex group with chamber a+ . When a = Rd , a+ = {(x1 , · · · , xd ); x1 > x2 > · · · > xd } and if W is the Euclidean Brownian motion on a then T W is the process of the eigenvalues of the Dyson Brownian motion on the set of Hermitian matrices and (T W )(1) is distributed as the eigenvalues of the G.U.E. Keywords: random matrix, Gaussian Unitary Ensemble, symmetric space, Weyl chamber, Brownian motion, complex semisimple group, representation theory, Pitman’s theorem. AMS Classification: (primary) 15A52, 17B10, 60B99, 60J65; (secondary) 22E30, 22E46, 43A85
∗
Universit´e Paris 6, Probabilit´es et mod`eles al´eatoires (UMR 7599), 4 Place Jussieu, 75232 Paris Cedex 05, France. Email:
[email protected] † Universit´e Paris 7, Probabilit´es et mod`eles al´eatoires (UMR 7599), 2 Place Jussieu, 75251 Paris Cedex 05, France. Email:
[email protected]
1
1. Introduction The Gaussian Unitary Ensemble (G.U.E.) is the probability measure Zd−1 exp(− tr M 2 /2) dM on the linear space of d×d Hermitian matrices, where dM is the Lebesgue measure and Zd is the normalization constant. Baryshnikov [3] and Gravner, Tracy & Widom [14] have given a path representation of the largest eigenvalue λ1 of the G.U.E. : Theorem 1.1 Let W1 , · · · , Wd be independent standard real Brownian motions, then λ1 has the same distribution as max
1≥t1 ≥···≥td−1 ≥0
[W1 (1) − W1 (t1 ) + W2 (t1 ) + · · · − Wd−1 (td−1 ) + Wd (td−1 )] .
The aim of this paper is to give a similar formula for all the eigenvalues, using a geometric approach. The natural general set-up is the following: consider a complex reductive Lie algebra g, by definition g = g1 ⊕ g2 where g1 is semisimple and g2 is the center (see, e.g., Wallach [33]). Let g = k ⊕ p be a Cartan decomposition, a+ a Weyl + chamber of a maximal Abelian subspace a of p (a+ = a+ 1 + g2 where a1 is a Weyl chamber in g1 ). We equip p with an Euclidean structure such that p ∩ g1 and p ∩ g2 are orthogonal and its restriction to p ∩ g1 is given by the Killing form. For any X ∈ p we define rad(X) ∈ a+ as the only element in the closure a+ of a+ such that X ∈ Ad(K) rad(X) where K is the compact group with Lie algebra k. Let C0 ([0, 1], a) be the set of continuous paths w : [0, 1] → a such that w(0) = 0, and C0 ([0, 1], a+ ) be defined analogously. We will introduce a continuous projection T : C0 ([0, 1], a) → C0 ([0, 1], a+ ) which associates to a path w in a a path T w in the closure of the Weyl chamber such that the G.U.E. has the following property: Property 1.2 1. Let Γ be a Gaussian N (0, I) random variable in p then rad(Γ) has the same distribution as (T W )(1) where W is the Euclidean Brownian motion in a. 2. More generally, let Z be the Euclidean Brownian motion on p. Then the process rad(Z) has the same distribution as T W . 2
When g = sl(2, C) then a = R, a+ = R+ and (T w)(t) = w(t) − 2 min w(s). 0≤s≤t
(1)
This path transformation occurs in Littelmann [22] (see also Mathieu [23]) in connection with representation theory of complex groups. It also appears in Pitman [28] where it is shown that when w is the usual Brownian motion then T w is a 3-dimensional Bessel process. Actually our results can be seen as a generalization of Pitman’s results to Weyl chambers and is related with Biane [4]. We define T in the general set-up in Theorem 2.1, using the root system of g. It occurs in the following deterministic setting: Let G = KAN be an Iwasawa decomposition of the simply connected Lie group G with algebra g. We consider the following differential equation on the symmetric space P = G/K εyε0 (t) = Ft (yε (t)), yε (0) = o
(2)
where Ft is an AN -invariant continuous section of the tangent bundle T P , continuous in t and o is the class K in P . Then we will show that under a non-degeneracy assumption on F , lim ε rad(Exp−1 yε (t)) = (T w)(t) ε→0
when w ∈ C0 ([0, 1], a) is defined by w0 (t) = πa Ft (o) and πa : p = T Po → a is the orthogonal projection (recall that Exp : p → P is a diffeomorphism). The transformation T is not linear in the usual sense. But it is linear in the so-called max-plus algebra, and it is thus quite natural that it appears in the study of dynamical systems on the curved space P as in (2). Actually, the analysis of the radial part of the Gaussian measure on p rests on a similar approach. We will imbed this measure in a Euclidean Brownian motion Z on the flat space p. Then we will consider the Brownian motion B on the curved Riemannian symmetric space P . We will show that there is a ground state τ on P such that, if B τ is the Doob h-transform of B then rad(Z) = rad(Exp−1 B τ ) and B τ is AN -invariant. Using the scaling property of Z we will have that, in distribution, 1 1 rad(Γ) = rad(Z1 ) = lim √ rad(Zt ) = lim √ rad(Exp−1 Btτ ) t→+∞ t→+∞ t t 3
By AN -invariance, B τ can be seen as a process with multiplicative independent increments on the solvable group AN and thus can be computed explicitely in terms of stochastic integrals. The computation of rad(Exp−1 B τ ) is more tricky. We use finite dimensional representations of g. Upper bounds are similar to the deterministic case and we prove that rad(Z) is stochastically larger than T W for the order of the dual cone of a+ in full generality. Lower bounds appear to be difficult and we have only succeeded to prove them for minuscule representations, which is enough to obtain the G.U.E. We conjecture that Property 1.2 holds for any reductive complex Lie algebra. It is known that rad(Z) can be interpreted as the Euclidean Brownian motion in a, conditionned to stay in a+ forever (Biane [4], Grabiner [13]). Thus this would provide a path representation of this process. The paper is organized as follows. We first introduce and study the transformation T in section 2 in the deterministic setting. Then in section 3 we study the relations between the Brownian motion Z on p, the Brownian motion B on P and the AN -invariant h-process B τ . The generator of B τ is given by the distinguished Laplacian on AN . In section 4 we first prove that in general rad(Z) is stochastically larger than T W for the order of the dual cone of a+ . Then we prove a criterion based on minuscule representations which in particular implies that Property 1.2 holds (see Theorem 4.7 and Theorem 4.9). We end with some examples. Since a reductive Lie algebra is the orthogonal sum of an Euclidean space and a semisimple Lie algebra, it is enough to consider the semisimple case. For the ease of notations we only consider this case. But for the reader who is mainly interested in the G.U.E. case we state the results explicitely in this reductive case in section 4.3. The study of the asymptotic properties of invariant processes on the solvable group AN can be seen as a kind a central limit theorem on these groups. It would be very interesting to have further results in this direction. Grinceviˇcius began the study of triangular matrices in [15] which is not always easy to follow and a lot remains to be done. Our approach works only in the case of complex Lie algebra, since we need the simple connection between the Brownian motion on p and an h-transform of the Brownian motion on P which does not hold in general. There is apparently no hope that it gives some results in the other classical cases as the G.O.E. or the G.S.E. for instance (see e.g. Mehta [25]). The idea of studying properties of random matrices by using the associated curved symmetric space is not new and appears already more or less explicitely
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in Akuzawa & Wadati [2], Dyson [11], Kuperberg [21], Mehta [25], Zirnbauer [34] for instance. The approach of Akuzawa & Wadati is to make the curvature goes to 0, in our case it rather goes to +∞. By making use of the representation of rad(Z) as a conditionned Brownian motion, O’Connell & Yor [26] have proposed a completely different approach of the study of the path representation of the G.U.E., starting from the Burke reversibility property of queues.
2. The path transform We consider a simply connected semisimple complex group G with Lie algebra g. We will need the following classical notions. Convenient references are Helgason [16] and Humphreys [19]. Let g = k ⊕ p be a Cartan decomposition, a ⊂ p a maximal Abelian subspace, h = a + ia a Cartan subalgebra. Let Σ be the set of roots, and for any α ∈ Σ, gα = {x ∈ g; [h, x] = α(h)x, h ∈ h}. We choose a Weyl chamber a+ . It determines the set Σ+ of positive roots. Let S be the set of simple positive roots. By definition, each α ∈ Σ+ can be written in a unique way P P α = σ∈S kσ σ where kσ ∈ N, S is a basis of the dual a∗ . Let n = α∈Σ+ g−α be the nilpotent Lie algebra spanned by the root subspaces for the negative roots ¯ in the literature). Let A, N, K be the subgroups (notice that n is often denoted by n of G with Lie algebras a, n and k respectively. In the sequel we will use the following notation. We consider an irreducible representation π of the Lie algebra g in a d-dimensional vector space V and π ˜ the representation of the group G such that d˜ π = π. Let P(π) be the set of weights of ˜ π, Λπ be the highest weight and Λπ be the lowest one (which is also the opposite of the highest weight of the contragredient representation). One can write each P µ ∈ P(π) as µ = Λπ − α∈S kα α where kα ∈ N, see Humphreys ([19], 20.2). We P set p(µ) = α∈S kα and r = max{p(µ), µ ∈ P(π)}, ˜ π ). For each α ∈ Σ+ we choose a non-zero element yα in the that is r = p(Λ one-dimensional space g−α . Let D(π) = {(α1 , · · · , αr ) ∈ S r ; π(yαr ) · · · π(yα1 ) 6= 0}. We put the uniform norm on the spaces C0 ([0, 1], a) and C0 ([0, 1], a+ ) of continuous functions, null at 0, with values in a or in a+ . Let us introduce, for each w ∈ 5
C0 ([0, 1], a) and t ≥ 0, (Tπ w)(t) =
max
max
(α1 ,··· ,αr )∈D(π) t≥t1 ≥···≥tr ≥0
[Λπ (w(t)) −
r X
αk (w(tk ))].
(3)
k=1
We will show that: Theorem 2.1 There is a continuous transformation T from C0 ([0, 1], a) to C0 ([0, 1], a+ ) such that, for all t ≥ 0, Λπ ((T w)(t)) = (Tπ w)(t)
(4)
for every irreducible representation π of g. It is sufficient to use the fundamental representations to define T . Let {λα , α ∈ S} 2α be the dual basis of {˜ α, α ∈ S} where α ˜ = hα,αi defined by hλα , σ ˜ i = δα,˜σ for all α, σ ∈ S, the scalar product being given by the Killing form. The fundamental representation π(α) associated with the simple root α ∈ S is the irreducible representation with highest weight λα (see Humphreys [19], 13.1). The transformation T is given explicitely by (T w)(t) = w(t) −
X
Ut (α, w)Hα
(5)
α∈S
where Ut (α, w) =
min
min
(α1 ,··· ,αr )∈D(π(α)) t≥t1 ≥···≥tr ≥0
r X
αk (w(tk ))
k=1
and where {Hα , α ∈ S} is the element of a which represents α ˜ , i.e. hHα , xi = α ˜ (x), for all x ∈ a. Thus the claim of Theorem 2.1 is that if the transformation T is defined by (5), then (4) holds for all irreducible representations. This is a non-trivial abstract property of these representations and will be useful. Notice that T is a projection since when w ∈ C0 ([0, 1], a+ ), α(w(s)) ≥ 0 for all α ∈ S, hence Tπ w = w. The fact that T w is with values in a+ will follow from its construction (as a limit of paths in a+ ). To prove Theorem 2.1, we consider a continuously differentiable function w ∈ C0 ([0, 1], a), a continuous function n : [0, 1] → n and we solve the following differential equation on the group AN : εYε0 (t) = Ft (Yε (t)), Yε (0) = e, 6
where {Ft (x), x ∈ AN } is the left-invariant vector field on AN such that Ft (e) = w0 (t) + n(t) and ε > 0 is a parameter of the singular perturbation problem (i.e. ε → 0). For any x ∈ P we let Rad(x) = rad(Exp−1 x) where, when X ∈ p, rad(X) is the unique point such that rad(X) ∈ Ad(K)X ∩ a+ (see Helgason [16], p.196). For any g ∈ G we set Rad(g) be the unique element in a+ defined by g ∈ KeRad(g) K. The notation is coherent: for all g ∈ G, Rad(g) = Rad(g.o). We will first show that (Condition C is defined in 2.3): Theorem 2.2 Under the non-degeneracy Condition C, (T w)(t) = lim ε Rad(Yε (t)). ε→0
Theorem 2.1 will follow by a density argument in Section 2.3. Remark : Notice that instead of AN one could as well have considered an invariant equation on the Borel subgroup B = M AN . The result is the same (and proved in the same way) if Ft (e) = m(t) + w0 (t) + n(t) where m(t) ∈ m, n(t) ∈ n, w0 (t) ∈ a.
2.1. Upper bound Let π be an irreducible representation of g in V . The weight space V (µ) associated to µ ∈ P(π) is V (µ) = {v ∈ V ; π(h)v = µ(h)v, h ∈ h}. Recall that for all α ∈ Σ, π(gα )V (µ) ⊂ V (µ + α).
(6)
We choose a basis {ei , 1 ≤ i ≤ d} of V adapted to the decomposition V = ⊕µ∈P(π) V (µ) in the following way: e1 is a non-zero element of the one-dimensional subspace V (Λπ ) and each ei , i > 1, can be written as ei = π(yαk ) · · · π(yα1 )e1 .
(7)
for some α1 , · · · , αk ∈ S (see Humphreys [19], 20.2). We let µi be the weight such that ei ∈ V (µi ). We choose the indices in such a way that i ≥ j when 7
˜ π ) is the one-dimensional space spanned by ed and p(µi ) ≥ p(µj ). Notice that V (Λ that all the elements of {π(yαr ) · · · π(yα1 )e1 , (α1 , · · · , αr ) ∈ D(π)} are non-zero multiple of ed . ˜ π and all non-zero v ∈ V (µ), there exist α1 , · · · , αk ∈ S Lemma 2.3 For all µ 6= Λ and c 6= 0 such that π(yαk ) · · · π(yα1 )v = ced . ˜ π ) (see [19], 20.2, Corollary). Proof. We know that π(n)v = 0 implies that v ∈ V (Λ ˜ π ) there is a simple positive root α1 such that π(yα )v 6= Therefore, if v is not in V (Λ 1 0. Arguing recursively, the lemma is obtained by choosing a maximal k for which π(yαk ) · · · π(yα1 )v is non-zero (notice that all these vectors are in distinct weight subspaces by (6)). Lemma 2.4 Let σ1 , · · · , σm ∈ Σ+ and µ ∈ P(π) such that π(yσm ) · · · π(yσ1 )V (µ) 6= {0}. There exists (α1 , · · · , αr ) ∈ D(π) and 1 ≤ i1 ≤ i2 ≤ · · · ≤ im ≤ im+1 ≤ r with the following property: µ = Λπ −
iX 1 −1 j=1
αj , σ1 =
iX 2 −1
im+1 −1
αj , · · · , σm =
j=i1
X
αj , µ −
j=im
m X n=1
˜ σn − Λ(π) =
r X
αj .
j=im+1
We say that (α1 , · · · , αr ) refines (µ, σ1 , · · · , σm ). Proof. We choose a basis vector ek ∈ V (µ) such that π(yσm ) · · · π(yσ1 )ek 6= 0. By (7) there exists α1 , · · · , αi1 ∈ S such that ek = π(yαi1 −1 ) · · · π(yα1 )e1 . P 1 −1 αj . Since n is generated by {yα , α ∈ S}, each π(yσ ), σ ∈ Σ+ , Then µ = Λπ − ij=1 is in the algebra generated by the π(yα ), α ∈ S. Thus π(yσ ) is in the linear span of {π(yαl ) · · · π(yα1 ); α1 , · · · , αl ∈ S, α1 + · · · + αl = σ}. This implies easily that there exists αi1 , · · · , αim+1 −1 such that, for 1 ≤ n ≤ m, Pin+1 −1 αj and σn = j=i n π(yαim+1 −1 ) · · · π(yαi1 )ek 6= 0.
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Using Lemma 2.3 one can complete the sequence α1 , · · · , αim+1 −1 in an element (α1 , · · · , αr ) of D(π), finishing hence the proof. Let π ˜ be the representation of G with differential π. Then Xε (t) = π ˜ (Yε (t)) is the solution of the non-autonomous linear differential equation in End(V ): εXε0 (t) = Xε (t)π(w0 (t) + n(t)), Xε (0) = Id.
(8)
One may suppose that π ˜ (k) is unitary for each k ∈ K, that the weight spaces are orthogonal and that End(V ) is equipped with the operator norm. Lemma 2.5 For all g ∈ G, log k˜ π (g)k = Λπ (Rad(g)). Proof: Since Λπ is the highest weight, for any µ in P(π), Λπ (H) ≥ µ(H) when H ∈ a+ . Therefore log k˜ π (g)k = log k˜ π (exp(Rad(g)))k = Λπ (Rad(g)). Proposition 2.6 Let us write X
n(t) =
nα (t)yα , nα (t) ∈ C.
α∈Σ+
For all 1 ≤ i ≤ d, Xε (t)i,i = exp( 1ε µi (w(t))) and, when i > j, Xε (t)i,j =
i−j X
1
X
[π(yαm ) · · · π(yα1 )]i,j e ε µj (w(t))
m=1 {α1 ,··· ,αm ∈Σ+ }
Z
1
e− ε
×
Pm
k=1
αk (w(tk )) −m
ε
nα1 (t1 ) · · · nαm (tm ) dt1 · · · dtm .
t>t1 >···>tm >0
Proof. In the basis (ei ) the matrices π(w0 (t) + n(t)) are lower triangular. The solution of the linear differential equation (8) is given by Xε (t)i,j =
i−j X
X
m=1 i>lm−1 >···>l1 >j
Z t>t1 >···>tm >0
1
e ε [µi (w(tm ))+µlm−1 (w(tm−1 )−a(tm ))+···+µl1 (w(t1 )−a(t2 ))+µj (w(t)−a(t1 ))] × ε−m π(n(tm ))i,lm−1 · · · π(n(t1 ))l1 ,j dt1 · · · dtm . If, for any 1 ≤ j ≤ d and α ∈ Σ+ we project the relation π(n(t))ej =
d X i=j+1
9
π(n(t))i,j ei
on V (µj − α) one obtains X
nα (t)π(yα )ej =
π(n(t))i,j ei ,
{i>j;µj −µi =α}
hence, by induction, for all α1 , · · · , αm ∈ Σ, nαm (tm )π(yαm ) · · · nα1 (t1 )π(yα1 )ej = X =
π(n(tm ))im ,im−1 · · · π(n(t1 ))i1 ,j ei1 .
{im >···>i1 >i0 =j; µik −µik−1 =αk ,∀k=1,··· ,m}
The lemma is obtained by grouping together in the expression of Xε all the terms corresponding to the same µl1 , · · · , µlm −1 . Proposition 2.7 One always has lim sup ε log kXε (t)k ≤ Tπ w(t). ε→0
Proof. Using Proposition 2.6 it suffices to remark that, for all σ1 , · · · , σm ∈ Σ+ , if [π(yσ1 ) · · · π(yσm )]i,j 6= 0 then, for all t > t1 > · · · > tm > 0, µj (w(t)) −
m X
σk (w(tk )) ≤ Tπ w(t).
k=1
This is clear since, if (α1 , · · · , αr ) ∈ D(π) is the sequence refining (µj , σ1 , · · · , σm ) given by Lemma 2.4, then µj (w(t)) −
m X
σk (w(tk )) = Λπ (w(t)) −
r X
αn (w(sn )),
n=1
k=1
where s1 = · · · = si1 −1 = t, si1 = · · · = si2 −1 = t1 , · · · , sim +1 = · · · = sr = tm . Corollary 2.8 For every highest weight Λ, lim sup Λ(ε Rad(Yε (t))) ≤ (Tπ w)(t). ε→0
The corollary follows from Lemma 2.5. It will imply that lim sup ε Rad(Yε (t)) ≤ (T w)(t) ε→0
for the order of the dual cone of a+ , once we have proved Theorem 2.1.
10
(9)
2.2. Lower bound We consider a fixed t > 0. Let us introduce the following condition: Condition C(π): • C1. For each (α1 , · · · , αr ) ∈ D(π) the minimum M (α1 , · · · , αr ) of the funcP tion rk=1 αk (w(tk )) on the set Cr (t) = {(t1 , · · · , tr ); t ≥ t1 ≥ · · · ≥ tr ≥ 0} is reached at a unique point t(α) and, for each i ∈ {1, · · · , r − 1}, t(α)i 6= t(α)i+1 unless αi = αi+1 . • C2. The map M : D(π) → R is one to one. • C3. For each (α1 , · · · , αr ) ∈ D(π), nα1 (t1 ) · · · nαr (tr ) 6= 0 at the point t(α). Proposition 2.9 Under Condition C(π) lim ε log kXε (t)k = (Tπ w)(t).
ε→0
Proof. Consider Rε (σ1 , · · · , σm ) = [π(yσm ) · · · π(yσ1 )]d,1 Z Pm 1 × e ε [Λπ (w(t))− k=1 σk (w(tk ))] ε−m nσ1 (t1 ) · · · nσm (tm ) dt1 · · · dtm . t>t1 >···>tm >0
Then, by Proposition 2.6, using the definition of r, Xε (t)d,1 =
r X
X
Rε (σ1 , · · · , σm ).
m=1 {σ1 ,··· ,σm ∈Σ+ }
It follows from (9) that lim sup ε log |Rε (σ1 , · · · , σm )| ≤ ε→0
max
{t≥t1 ≥···≥tm ≥0}
Λπ (w(t)) −
m X
σk (w(tk )).
k=1
One has max
{t≥t1 ≥···≥tm ≥0}
Λπ (w(t)) −
m X
σk (w(tk )) ≤ Λπ (w(t)) − M (α1 , · · · , αr ),
k=1
11
(10)
where (α1 , · · · , αr ) ∈ D(π) is the sequence refining (Λπ , σ1 , · · · , σm ) given by Lemma 2.4. Let us show that if there is equality in (10) then m = r. With the notation of Lemma 2.4, and i1 = 1, im+1 = r + 1, one has m X
σk (w(tk )) =
m ik+1 X X−1
αj (w(tk )).
k=1 j=ik
k=1
If there is equality in (10) and if the maximum of the left hand side is reached at (s1 , · · · , sm ), then it follows by uniqueness of t(α) that, for all 1 ≤ k ≤ m, sk = t(α)ik = · · · = t(α)ik+1 −1 and thus, by C1, that αik = · · · = αik+1 −1 . This implies that σk = (ik+1 − ik )αik , which is possible only when ik+1 − ik = 1 since σk is in Σ+ (see Humphreys [19] 8.4). Therefore (σ1 , · · · , σm ) = (α1 , · · · , αr ) and m = r. Now when m = r and (σ1 , · · · , σr ) ∈ D(π), the Laplace method shows that under C1 and C3, lim ε log |Rε (σ1 , · · · , σr )| = Λπ (w(t)) − M (σ1 , · · · , σr ).
ε→0
Thus it follows from C2 that lim inf ε log |Xε (t)d,1 | ≥ (Tπ w)(t). ε→0
This gives the proposition by making use of Proposition 2.7.
2.3. Proof of Theorems 2.1 and 2.2 We will use the following proposition which is proved in the appendix. Proposition 2.10 Given a finite set I of irreducible representations of g, the set L(I) of w ∈ C0 ([0, 1], a+ ) such that C1 and C2 hold for all π ∈ I is a dense subset of C0 ([0, 1], a+ ). Let π be an arbitrary irreducible representation of g and let I be the set made by π and the fundamental representations of g. For all w in L(I), it follows from Proposition 2.9 applied to the fundamental representations that lim ε Rad(Yε (t)) = (T w)(t),
ε→0
where T w is defined by (5) and limε→0 Λπ (ε Rad(Yε (t))) = Tπ w(t) (one can for instance take each nα to be a non-zero constant to be sure that C3 holds). Thus, Λπ ((T w)(t)) = (Tπ w)(t). 12
Since L(I) is dense, this equality holds for all w in C0 ([0, 1], a+ )) by continuity. This proves Theorem 2.1. We introduce: Condition C: C(π) holds for a set R of irreducible representations such that {Λπ , π ∈ R} span the dual of a. Under this condition it follows from Proposition 2.9 that, for all π ∈ R, lim Λπ (ε Rad(Yε (t))) = Λπ ((T w)(t)).
ε→0
Thus limε→0 ε Rad(Yε (t)) = (T w)(t) which proves Theorem 2.2.
3. Ground state processes on P We still consider a complex semisimple Lie algebra g. We equip p with the scalar product given by the Killing form K R (X, Y ) on g considered as a real Lie algebra. Let us recall some standard facts on the Brownian motion on P = G/K (see e.g. [1]). Let ∆P be the Laplace Beltrami operator on P and {Bt , t ≥ 0} be the Brownian motion on P with generator ∆P /2 and semigroup {Qt , t ≥ 0}, starting at the class o of K in P . The bottom of the L2 -spectrum of −∆P /2 is λ0 = kρk2 /2 P where ρ = − α∈Σ+ α. Any positive C 2 -function h on P such that 1 − ∆P h = λ0 h 2 is called a ground state. Consider the second order elliptic operator ∆hP on P defined by ∆hP f =
1 ∆P (f h) + 2λ0 f = ∆P f + 2∇P log h · ∇P f h
where ∇P is the gradient on P . Following Doob we introduce the h-transform B h of B as the minimal Markov process associated with ∆hP /2. It has no spectral gap. Its semigroup (Qht ) is given by Qht f (x) = eλ0 t Qt (f h)(x)/h(x)
(11)
for all measurable f : P → R+ and x ∈ P (c.f. Pinsky [27] or [1]). There exist many ground states and two of them will play a role. For any g ∈ G, we denote by H[g] the a-component of g in the Iwasawa decomposition G = K(exp a)N . The basic spherical function φ0 of Harish–Chandra is Z −1 φ0 (g) = e−ρ(H[g k]) dk, g ∈ G, K
13
where dk is the normalized Haar measure on K. We also consider τ (g) = e−ρ(H[g
−1 ])
,
g ∈ G.
The functions φ0 and τ are right K-invariant. Hence they define functions on P = G/K. These two functions are ground states on P . The function φ0 is the only one which is K-invariant. Let B 0 be the h-process B h when h = φ0 and B τ be the h-process B h when h = τ . It is shown in Anker, Bougerol & Jeulin [1] that B 0 can be considered as the infinite Brownian loop on P . Let us check that the ground state τ defines an AN -invariant process. The map f : AN → P defined by f (g) = g.o is a diffeomorphism. We identify B τ with the process f −1 (B τ ) on AN . Let Q be the scalar product on a ⊕ n for which dfe : a ⊕ n → To P = p is an isometry. If θ is the Cartan involution on g, then 1 Q(X1 + Y1 , X2 + Y2 ) = K R (X1 , X2 ) − K R (Y1 , θY2 ) 2 for all X1 , X2 ∈ a, Y1 , Y2 ∈ n and if β is the Euclidean Brownian motion on a ⊕ n, B τ is the solution of the following stochastic differential equation on AN , dBtτ = Btτ ◦ dβt , B0τ = e (see, e.g. Malliavin & Malliavin [24], Bougerol [6], Taylor [32], Cornwall [8]). Its generator is the distinguished left-invariant Laplacian on the solvable group AN studied by Cowling, Gaudry, Giulini & Mauceri [9]. Let {Zt , t ≥ 0} be the Euclidean Brownian motion on p starting from 0 (with generator 12 ∆p ). The hypothesis that g is complex is crucial for the next theorem. Theorem 3.1 The three processes rad(Z), Rad(B 0 ) and Rad(B τ ) have the same distribution. Proof. This is detailed in Anker, Bougerol & Jeulin [1] but for the convenience of the reader we present here the main arguments: since g is complex, it follows from Helgason ([17], Proposition II.3.13) that the radial part of ∆p on a+ is rad(∆p ) = ∆a + 2∇a log π · ∇a , where ∆a is the Euclidean Laplacian on a, ∇a is its gradient and π = By Helgason [17], II.3, Remark 1, Rad(∆P ) = ∆a + 2 ∇a log δ 1/2 · ∇a ,
14
(12) Q
α∈Σ+
α.
Q where δ(x) = α∈Σ+ sinh2 α(x). The function φ0 is K-invariant. Thus the radial part Rad(B 0 ) of the φ0 -transform of B is the φ0 -transform of Rad(B) and its generator is 1 1 ∆a + ∇a log δ 1/2 · ∇a + ∇a log φ0 · ∇a = ∆a + ∇a log(δ 1/2 φ0 ) · ∇a . 2 2 When G is complex φ0 = δ −1/2 π (cf. Theorem IV.4.7 in Helgason [17]) thus it follows from (12) that rad(Z) and Rad(B 0 ) have the same distribution. Let T > 0 and F : C(R+ , a+ ) → R+ be a FT -measurable function for the canonical filtration. Then, by (11), E[F (Rad(B τ ))] = eλ0 T E[F (Rad(B))τ (BT )]. Notice that
Z τ (k.x)dk = φ0 (x). K
We use this equality, the invariance of the Brownian motion B under the isometries of K and the relation Rad(k.x) = Rad(x) for k ∈ K and x ∈ P to write Z τ λ0 T E[F (Rad(B ))] = e E[F (Rad(B))τ (k.BT )] dk K
= eλ0 T E[F (Rad(B))φ0 (BT )] = E[F (Rad(B 0 ))]. This proves that the process Rad(B τ ) has the same distribution as Rad(B 0 ). Let {Btτ,ε , t ≥ 0} be the process on AN solution of εdBtτ,ε = Btτ,ε ◦ dβt , B0τ,ε = e.
(13)
Proposition 3.2 For all ε > 0, the processes rad(Z) and ε Rad(B τ,ε ) have the same distribution. Proof. The scaling property of the Euclidean Brownian motion Z on p tells us that for each fixed ε > 0, {Zt , t ≥ 0} has the same distribution as {εZt/ε2 , t ≥ 0}. Since rad(λx) = λ rad(x) for each λ > 0, it follows from Theorem 3.1 that the three processes τ {rad(Zt ), t ≥ 0}, {ε rad(Zt/ε2 ), t ≥ 0}, {ε Rad(Bt/ε 2 ), t ≥ 0}
have the same law. On the other hand, it follows from the scaling property of β τ , t ≥ 0} has the same distribution as the process {B τ,ε , t ≥ 0}. that {Bt/ε 2 t
15
4. Path representation in the random case For each α ∈ Σ+ we choose yα ∈ g−α such that Q(yα , yα ) = 1. Then we can write the Brownian motion {βt , t ≥ 0} on a ⊕ n considered in (13) as βt = Wt +
X
βtα yα
α∈Σ+
where W is the Euclidean Brownian motion on a and the β α , α ∈ Σ+ , are independent complex standard Brownian motions, independent of W . Since ε dBtτ,ε = Btτ,ε ◦ dβt , B0τ = e, we see that for any irreducible representation π ˜ of G with values in V , Xε (t) = π ˜ (Btτ,ε ) is the solution of the stochastic differential equation on End(V ), ε dXε (t) = Xε (t) ◦ π(dβt ), Xε (0) = I, where π = d˜ π . As in Proposition 2.6 we have the following explicit expression α (since β is a complex Brownian motion, there is no need to make a distinction between Ito and Stratonovitch stochastic integrals). Proposition 4.1 For all 1 ≤ i ≤ d, Xε (t)i,i = exp[ 1ε µi (W (t))] and, when i > j, Xε (t)i,j =
i−j X
1
X
[π(yα1 ) · · · π(yαm )]i,j ε−m e ε µj (W (t))
m=1 (α1 ,··· ,αm )∈Σm +
Z
1
e− ε
×
Pm
k=1
αk (W (tk ))
dβ α1 (t1 ) · · · dβ αm (tm ).
t>t1 >···>tm >0
4.1. Upper bound The next proposition says that rad(Z) ≤ T W for the stochastic order of the dual cone of a+ . Proposition 4.2 We can define Z and W on the same probability space in such a way that for all t ≥ 0, and all highest weight Λ, Λπ (rad(Z(t))) ≤ Λπ (T W (t)).
16
Proof. It suffices to show that for any t1 ≤ t2 ≤ · · · ≤ tn , for any highest weight Λ1 , · · · , Λn , if f : Rn → R is bounded continuous and increasing component-wise then E[f (Λ1 (rad(Z(t1 )), · · · , Λn (rad(Z(tn )))] ≤ E[f (Λ1 (T W (t1 )), · · · , Λn (T W (tn )))] (see, e.g., Kamae, Krengel & O’Brien [20]). For 1 ≤ i ≤ n, let πi be the irreducible representation with highest weight Λi . By Lemma 2.5 and Proposition 3.2, E[f (Λ1 (rad(Z(t1 )), · · · , Λn (rad(Z(tn )))] = E[f (ε log k˜ π1 (Btτ,ε )k, · · · , ε log k˜ πn (Btτ,ε )k)]. n 1 Thus, using Fatou’s lemma, it is enough to prove that, almost surely, lim sup ε log k˜ πi (Btτ,ε )| ≤ (Tπi W )(ti ) i
(14)
ε→0
when ε−1 ∈ N, since Λi (T W ) = Tπi W . This follows by the same arguments as in Proposition 2.7 using Proposition 4.1 and the lemma below. Lemma 4.3 For all t > 0, almost surely, when 1/ε ∈ N, Z Pm 1 lim sup ε log | e ε [µj (W (t))− k=1 αk (W (tk ))] dβ α1 (t1 ) · · · dβ αm (tm )| ε→0
t>t1 >···>tm >0
≤
max
t>t1 >···>tm >0
µj (W (t)) −
m X
αk (W (tk ))
k=1
Proof. It suffices to notice that if Mt =
max
t>t1 >···>tm >0
µj (W (t)) −
m X
αk (W (tk )),
k=1
then the L2 norm of the stochastic integral Z Pm 1 e ε [µj (W (t))−Mt − k=1 αk (W (tk ))] dβ α1 (t1 ) · · · dβ αm (tm ) t>t1 >···>tm >0
is bounded. Hence, multiplied by ε2 , it tends to 0 almost surely since 1/ε ∈ N.
4.2. Lower bound In order to deal with the lower bound, we will need two lemmas. Let J be a finite set and {Ui , i ∈ J} be a family of independent complex standard Gaussian random √ variables (i.e. each Ui = Xi + −1Yi where Xi and Yi are independent N (0, 1) 17
random variables). For a fixed r ∈ N∗ we denote J(r) the set of subsets of J with exactly r elements. For all j = {j1 , · · · , jr } ∈ J(r) let U (j) = Uj1 Uj2 · · · Ujr . Lemma 4.4 For each j ∈ J(r) let c(j) ∈ C. For all a > 0, P(|
X
c(j)U (j) | ≤ a max |c(j)|) ≤ ra1/r . j∈J(r)
j∈J(r)
Proof. It is first easy to see that if (A, B) ∈ C2 is a random variable independent of Ui , then for all x > 0 P(|AUi + B| ≤ x) ≤ P(|AUi | ≤ x).
(15)
Let j 0 = {j10 , · · · , jr0 } ∈ J(r) such that |c(j0 )| = max{|c(j)|, j ∈ J(r)}. We see that X X X c(j)U (j) = c(j)U (j) + c(j)U (j) {j∈J(r);j10 ∈j}
j∈J(r)
{j∈J(r);j10 6∈j}
can be written as AUj10 + B where (A, B) is independent of Uj10 . Thus (15) gives P(|
X
c(j)U (j) | ≤ a|c(j0 )|) ≤ P(|
X
c(j)U (j) | ≤ a|c(j0 )|).
{j∈J(r);j10 ∈j}
j∈J(r)
Now X
c(j)U (j) =
{j∈J(r);j10 ∈j}
X
X
c(j)U (j) +
{j∈J(r);j10 ,j20 ∈j}
c(j)U (j)
{j∈J(r);j10 ∈j,j20 6∈j}
can be written as AUj20 + B with (A, B) independent of Uj20 , hence by (15), P(|
X
X
c(j)U (j) | ≤ a|c(j0 )|) ≤ P(|
j∈J(r)
c(j)U (j) | ≤ a|c(j0 )|).
{j∈J(r);j10 ,j20 ∈j}
Carrying on, we arrive at P(|
X
c(j)U (j) | ≤ a|c(j0 )|) ≤ P(|c(j0 )U (j0 ) | ≤ a|c(j0 )|) = P(|U (j0 ) | ≤ a).
j∈J(r)
Finally P(|U (j0 ) | ≤ a) ≤ rP(|Ui | ≤ a1/r ) ≤ ra1/r .
18
Proposition 4.5 Let α1 , · · · , αn ∈ S, W be a Brownian motion on a and 1 ≤ i < j ≤ n. If, with a positive probability, min
0≤tn ≤···≤t1 ≤1
n X
αk (W (tk ))
k=1
is reached at a (random) point s where si = si+1 = · · · = sj , then αi = αi+1 = · · · = αj . Proof. Let s0 = 1 and sn+1 = 0. We can find 1 ≤ i0 ≤ i < j ≤ j0 ≤ n and non-random 0 < u < v < 1 such that, with positive probability, sj0 +1 < u < sj0 = sj0 −1 = · · · = si0 < v < si0 −1 . Let c = v − u. If we replace W (t) by c−1/2 (W (ct + u) − W (u)) and {1, · · · , n} by {i0 , · · · , j0 } one can and will suppose that i = i0 = 1 and j = j0 = n. Let P a = nk=1 αk , the real Brownian motion β(t) = a(W (t))/||a|| reaches its minimum at a unique point σ ∈ (0, 1). Conditionally on (σ, β (σ)) the distribution of (β (t + σ) − β (σ))0≤t≤1−σ is equivalent to the distribution of a 3-dimensional Bessel process (see, e.g., Biane and Yor [5]). We fix some m ∈ {1, · · · , n − 1}. By hypothesis, with positive probability, σ = sl for all 1 ≤ l ≤ n. Thus, for all t ∈ [σ, 1], m X k=1
hence
αk (W (σ)) +
n X
αk (W (σ)) ≤
k=m+1 n X
m X k=1
αk (W (σ)) ≤
n X
αk (W (σ)) +
n X
αk (W (t)) ,
k=m+1
αk (W (t)) .
(16)
k=m+1
k=m+1
P λ a+b where λ ∈ R and b is orthogonal to a. Let us show We write nk=m+1 αk = kak that b = 0. If b 6= 0, η(t) = b(W (t))/kbk is a real Brownian motion, independent of β, and with positive probability, for all t ∈ [0, 1 − σ], by (16) λ (β (t + σ) − β (σ)) + kbk (η (t + σ) − η (σ)) ≥ 0. This is impossible since, when R is a 3-dimensional Bessel process and B is a real Brownian motion, independent of R, then for all x ∈ R, P [inf {t > 0 | xRt + Bt < 0} = 0] = 1. 19
Indeed, when x ≤ 0, this follows from the fact that almost surely, inf {t > 0; Bt < 0} = 0. When x > 0, let Lt = sup {s > 0 | Rs = t} . Then {Lt , t ≥ 0} is a stable process of index 12 (cf, e.g., Getoor [12]) and thus C = BL is a Cauchy process. Then (see, e.g. Rogozin [31]) : P [inf {t > 0 | Ct + xt < 0} = 0] = 1. P This proves that b = 0, hence nk=m+1 αk is proportional to a for all j. This implies that α1 = · · · = αn , since αi are simple positive roots. We will only establish the lower bound for minuscule representations. An irreducible representation π is called minuscule if all its weights are in the same orbit of the Weyl group, see Bourbaki [7], VIII.7.3. A complete list of the minuscule representations is given there and its Proposition 6, (iii), implies that: Lemma 4.6 When π is minuscule, for each (α1 , · · · , αr ) ∈ D(π) there is no i < r such that αi = αi+1 . Theorem 4.7 If π is minuscule, the processes Λπ (rad(Z)) and Tπ W have the same distribution. Proof. By Proposition 3.2, we know that rad(Z) has the same distribution as ε Rad(B τ,ε ). Since Λπ T = Tπ and log k˜ π (g)k = Λπ (Rad(g)), it follows from Proposition 4.2 (see (14)) that it suffices to show that, for each t > 0, almost surely, when ε−1 ∈ N, lim inf ε log k˜ π (Btτ,ε )k ≥ Tπ W (t). ε→0
Let Xε (t) = π ˜ (Btτ,ε ) and for α1 , · · · , αm ∈ Σ+ , Rε (α1 , · · · , αm ) = [π(yαm ) · · · π(yα1 )]d,1 Z Pm 1 × e ε [Λπ (W (t))− k=1 αk (W (tk ))] dβ α1 (t1 ) · · · dβ αm (tm ). t>t1 >···>tm >0
Then, by Proposition 4.1, using the definition of r, Xε (t)d,1 =
r X
X
m=1 {σ1 ,··· ,σm ∈Σ+ }
20
ε−m Rε (σ1 , · · · , σm )
For each m < r, it follows from Lemma 4.3 that, almost surely, lim sup ε log |Rε (σ1 , · · · , σm )| ≤ ε→0
max
{t≥t1 ≥···≥tm ≥0}
Λπ (W (t)) −
m X
σk (W (tk )).
k=1
By Lemma 4.6 and Proposition 4.5 this is strictly smaller than max
{t≥t1 ≥···≥tr ≥0}
Λπ (W (t)) −
r X
αk (W (tk ))
k=1
where (α1 , · · · , αr ) ∈ D(π) is the sequence refining (Λπ , σ1 , · · · , σm ) given by Lemma 2.4 and thus is strictly smaller than Tπ W (t). We now consider the case m = r. Let us write Z Z Pr 1 [Λ (W (t))− α (W (t ))] α α π r 1 k k k=1 eε dβ (t1 ) · · · dβ (tr ) = · · · dβ αr (tr ) Dε
t>t1 >···>tr >0
Z
X
+
{(n1 ,··· ,nr )∈Nr ;0 ed (H)} and the corresponding set of simple positive roots is {ei − ej , ei + ej , 1 ≤ i < j ≤ d}. We apply Theorem 4.7 to the natural representation π(M ) = M of so(2d) which is minuscule ([7], VIII.7.3). Consider U (t) =
max
0≤t2d−2 ≤···≤t1 ≤t
[W1 (t) + W2 (t1 ) − W1 (t1 ) + · · · + Wd (td−1 ) − Wd−1 (td−1 )
−Wd−1 (td ) − Wd (td ) − Wd−2 (td+1 ) + Wd−1 (td+1 ) + · · · − W1 (t2d−2 ) + W2 (t2d−2 )] , V (t) =
max
0≤t2d−2 ≤···≤t1 ≤t
[W1 (t) + W2 (t1 ) − W1 (t1 ) + · · · + Wd−1 (td−2 ) − Wd−2 (td−2 )
−Wd (td−1 ) − Wd−1 (td−1 ) − Wd−1 (td ) + Wd (td ) − Wd−2 (td+1 ) + Wd−1 (td+1 ) + · · · − W1 (t2d−2 ) + W2 (t2d−2 )] , We obtain the following Proposition 5.2 Let Z be the Brownian motion on the set of skew symmetric hermitian matrices of order 2d. Then the process λ1 (Z) has the same distribution as max(U, V ). The natural representation of so(2d + 1) is not minuscule. It happens that one can treat this particular representation by an ad hoc variant of the method presented here. But this is maybe anecdotic and we don’t present the proof here. We do not know for instance how to deal with one of the two fundamental representations of the rank-two exceptional Lie algebra G2 .
5.4. Conclusion We conjecture that the lower bound always hold. An important step would be to prove that (see Lemma 4.3) in distribution Z 1 Pn lim ε log | e ε k=1 αk (W (tk )) dβ α1 (t1 ) · · · dβ αn (tn )| ε→0
1>t1 >···>tn >0
=
max
n X
1≥t1 ≥···≥tn ≥0
αk (W (tk ))
k=1
for all α1 , · · · , αn ∈ Σ+ . A kind of Laplace method for such iterated Wiener stochastic integrals would be interesting in itself. 27
6. Appendix Proof of Proposition 2.10: It follows easily from the two next lemmas that the set where C2 and C1, except the uniqueness of the minimun, hold is open and dense. By replacing w(t) by w(t)−f (t)v where v ∈ a+ and f ≥ 0 is small and with a small support we can also obtain the uniqueness of the minimum on a dense Gδ subset, since α(v) > 0 for all α ∈ S. Lemma 6.1 Let α1 , · · · , αn be simple positive roots, and F = ∪n−1 i=1 {(t1 , · · · , tn ); 1 ≥ t1 ≥ · · · ≥ tn ≥ 1; ti = ti+1 , αi 6= αi+1 }. Then the set of w ∈ C0 ([0, 1] , a) such that n X
min
1≥s1 ≥···≥sn ≥0
αk (w (sk ))
si = si+1 = · · · = sj > sj+1 contains an open set. Let us show that this implies that αi = αi+1 = · · · = αj . Let for 0 ≤ a < b ≤ 1, U (a, b) be the set of w ∈ C0 ([0, 1] , a) such that j X
min
a≤sj ≤···≤si ≤b
αk (w (sk )) = min
a≤s≤b
k=i
j X
αk (w (s)) .
k=i
Since U is the union of the U (a, b)’s, when U has an interior point then some U (a, b) has also an interior point, say w0 , by e.g. the Baire property. We may suppose P that w0 is C 1 and that mina≤s≤b jk=i αk (w0 (s)) is reached at s∗ ∈ (a, b). Let Pj 1 us suppose that v ∈ a is such that k=i αk (v) = 0. Consider a C -function ϕ : [0, 1] → [−1, 1] with ϕ (0) = 0 and ϕ0 (s∗ ) 6= 0. If γ > 0 is small enough, wγ = w0 − γϕv belongs to U (a, b) and min
a≤sj ≤···≤si ≤b
since
Pj
∗ k=i ϕ (s ) αk (v)
j X
αk (wγ (sk )) =
k=i
j X
αk (w0 (s∗ ))
k=i
= 0. Thus for all i ≤ k ≤ j, (αk ◦ wγ )0 (s∗ ) = 0. Hence αk w00 (s∗ ) − γϕ0 (s∗ ) αk (v) = 0, 28
for all γ > 0 small enough, therefore αk (v) = 0. Thus, we have shown that Pj k=i αk (v) = 0 implies that αk (v) = 0 for all i ≤ k ≤ j, therefore all the positive roots αk are proportional, and thus equal. Lemma 6.2 Let α1 , · · · , αm and β1 , · · · , βm be simple positive roots. Then the set of w ∈ C0 ([0, 1] , a) such that min
m X
0≤sm ≤···≤s1 ≤1
αk (w (sk )) 6=
k=1
min
0≤sm ≤···≤s1 ≤1
m X
βk (w (sk ))
k=1
is open and dense in C0 ([0, 1] , a) unless αi = βi for all 1 ≤ i ≤ m. Proof. This set is clearly open. Suppose that there is equality on an open P 0 set. Let m(α) = min0≤s1 ≤···≤sm ≤1 m k=1 αk (w (sk )) be reached at s and m(β) = Pm min0≤t1 ≤···≤tm ≤1 k=1 βk (w (tk )) be reached at t0 . If s0 = t0 , then min
0≤s1 ≤t1 ≤···≤sm ≤tm ≤1
m X
(αk (w (sk )) + βk (w (tk )))
k=1
is reached when si = ti for all 1 ≤ i ≤ m, which implies that αi = βi by the previous lemma. Else, for some i, s0i 6= t0j for all 1 ≤ j ≤ m. Clearly we can modify slightly w on a neighbourhood of this point s0i in such a way that m(α) decreases and m(β) does not change, thus w is not in the interior.
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