J Fr Éd

Éric Bonnans

AND Justina Gianatti

Francisco J Silva

ON THE CONVERGENCE OF THE SAKAWA-SHINDO IN STOCHASTIC CONTROL

We analyze an algorithm for solving stochastic control problems, based on Pontryagin's maximum principle, due to Sakawa and Shindo in the deterministic case and extended to the stochastic setting by Mazliak. We assume that either the volatility is an affine function of the state, or the dynamics are linear. We obtain a monotone decrease of the cost functions as well as, in the convex case, the fact that the sequence of controls is minimizing, and converges to an optimal solution if it is bounded. In a specific case we interpret the algorithm as the gradient plus projection method and obtain a linear convergence rate to the solution.

Introduction

In this work we consider an extension of an algorithm for solving deterministic optimal control problems introduced by Sakawa and Shindo in [START_REF] Sakawa | On global convergence of an algorithm for optimal control[END_REF], and analyzed by Bonnans [START_REF] Bonnans | On an algorithm for optimal control using Pontryagin's maximum principle[END_REF]. This algorithm has been adapted to a class of stochastic optimal control problems in Mazliak [START_REF] Mazliak | An algorithm for solving a stochastic control problem[END_REF]. We extend here the analysis of the latter to more general situations appearing naturally in applications, and obtain stronger results regarding the convergence of iterates.

We introduce now the problem and the algorithm. Let (Ω, F, F, P) be a filtered probability space on which an m-dimensional standard Brownian motion W (•) is defined. We suppose that F = {F t } t∈[0,T ] (T > 0) is the natural filtration, augmented by all the P-null sets in F, associated to W (•) and we recall that F is right-continuous. Let us consider the following controlled Stochastic Differential Equation (SDE):

(1. [START_REF] Backhoff | Some sensitivity results in stochastic optimal control: A Lagrange multiplier point of view[END_REF] dy(t) = f (y(t), u(t), t, ω)dt + σ(y(t), u(t), t, ω)dW (t) t ∈ (0, T ), y(0) = y 0 ∈ R n , where f : R n × R r × [0, T ] × Ω → R n and σ : R n × R r × [0, T ] × Ω → R n×m are given maps. In the notation above y ∈ R n denotes the state function and u ∈ R r the control. We define the cost functional (1.2) J(u) = E T 0 (y(t), u(t), t)dt + g(y(T )) .

where : R n × R r × [0, T ] × Ω → R and g : R n × Ω → R are given. Precise definition of the state and control spaces, and assumptions over the data will be provided in the next sections. Let U ad be a non-empty closed, convex subset of R r and

(1.3) U := u ∈ (H 2 ) r ; u(t, ω) ∈ U ad , for almost all (t, ω) ∈ (0, T ) × Ω , where

H 2 := v ∈ L 2 ([0, T ] × Ω); the process (t, ω) ∈ [0, T ] × Ω → v(t, ω) is F-adapted .
The control problem that we will consider is (1.4) Min J(u) subject to u ∈ U.

The Hamiltonian of the problem is defined by (1.5)

H : R n × R r × R n × R n×m × [0, T ] × Ω → R, (y
, u, p, q, t, ω) → (y, u, t, ω) + p • f (y, u, t, ω) + q • σ(y, u, t, ω),

Date: December 28, 2015. The first and third authors thank the support from project iCODE :"Large-scale systems and Smart grids: distributed decision making" and from the Gaspar Monge Program for Optimization and Operation Research (PGMO). The second author was supported by CIFASIS -Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas and INRIA.

where p • f (y, u, t, ω) is the scalar product in R n and q • σ(y, u, t, ω) := m j=1 q j • σ j (y, u, t, ω), where by q j and σ j we denote the columns of the matrices q and σ. Finally, given (y, u) satisfying (1.1) let us define the adjoint state (p(•), q(•)) ∈ (H 2 ) n × H 2 n×m as the unique solution of the following Backward Stochastic Differential Equation (BSDE) (1.6) dp(t) = -∇ y H(y(t), u(t), p(t), q(t), t, ω)dt + q(t)dW (t) t ∈ (0, T ), p(T ) = ∇ y g(y(T )).

Given ε > 0, we define the augmented Hamiltonian as

(1.7) K ε : R n × R r × R r × R n × R n×m × [0, T ] × Ω → R, (y
, u, v, p, q, t, ω) → H(y, u, p, q, t, ω) + 1 2ε |u -v| 2 . We consider the following algorithm to solve (1.4). Algorithm:

(1) Let some admissible control u 0 (•) and a sequence {ε k } of positive numbers be given. Set k = 0. Using (1.1), compute the state y 0 (•) associated to u 0 . (2) Compute p k and q k , the adjoint state, solution of (1.6), associated to u k and y k .

(3) Set k = k + 1. Compute u k and y k such that y k is the state corresponding to u k and

(1.8) u k (t, ω) = argmin K ε k (y k (t, ω), u, u k-1 (t, ω), p k-1 (t, ω), q k-1 (t, ω), t, ω) ; u ∈ U ad ,
for almost all (t, ω) ∈ [0, T ] × Ω. We will see in Section 3 that u k is well defined if ε k is small enough. (4) Stop if some convergence test is satisfied. Otherwise, go to [START_REF] Bensoussan | Lectures on stochastic control[END_REF]. The main idea of the algorithm is to compute at each step a new control that minimizes the augmented Hamiltonian K ε which depends on H and on a quadratic term that penalizes the distance to the current control. We can prove that this is a descent method and that the distance to a gradient and projection step tends to zero. Consequently, in the convex framework the algorithm is shown to be globally convergent in the weak topology of (H 2 ) r . Step 3 in the algorithm reveals its connection with the extension of the Pontryagin maximum principle [START_REF] Pontryagin | The mathematical theory of optimal processes[END_REF] to the stochastic setting. We refer the reader to Kushner and Schweppe [START_REF] Kushner | A maximum principle for stochastic control systems[END_REF], Kushner [START_REF] Kushner | On the stochastic maximum principle: Fixed time of control[END_REF][START_REF] Kushner | Necessary conditions for continuous parameter stochastic optimization problems[END_REF], Bismut [START_REF] Bismut | Conjugate convex functions in optimal stochastic control[END_REF][START_REF] Bismut | An introductory approach to duality in optimal stochastic control[END_REF], Haussmann [START_REF] Haussmann | Some examples of optimal stochastic controls or: the stochastic maximum principle at work[END_REF] and Bensoussan [START_REF] Bensoussan | Lectures on stochastic control[END_REF][START_REF] Bensoussan | Stochastic maximum principle for distributed parameter systems[END_REF] for the the initial works in this area. Afterwards, general extensions were established by Peng [START_REF] Peng | A general stochastic maximum principle for optimal control problems[END_REF] and by Cadenillas and Karatzas [START_REF] Cadenillas | The stochastic maximum principle for linear convex optimal control with random coefficients[END_REF]. The stochastic maximum principle usually involves two pairs of adjoint processes (see [START_REF] Peng | A general stochastic maximum principle for optimal control problems[END_REF]). Nevertheless, the gradient of the cost function depends only on one pair of adjoint processes and since we suppose that U is convex, the first order necessary condition at a local optimum u * depends only on ∇J(u * ) (see [20, p. 119-120] for a more detailed discussion).

In this paper we work with two types of assumptions. The first one supposes that σ in (1.1) does not depend on u and that the cost functions and g are Lipschitz. In the second assumption we suppose that the functions f and σ involved in (1.1) are affine with respect to (y, u). Thus, our results are a significant extension of those of [START_REF] Mazliak | An algorithm for solving a stochastic control problem[END_REF]. Let us explain now our main improvements, referring to Remark 2.1(i) for other technical differences. In [START_REF] Mazliak | An algorithm for solving a stochastic control problem[END_REF] the author studies a restricted form of our first assumption, he shows that if in addition σ is independent of the state y and the problem is convex, then, except for some subsequence, the iterates u k converges weakly to a solution of (1.4). If σ depends on the state y, it is proved in [START_REF] Mazliak | An algorithm for solving a stochastic control problem[END_REF]Theorem 5] that given ε > 0, the algorithm can be suitably modified in such a way that every weak limit point û of u k is an ε-approximate optimal solution, i.e. J(û) ≤ J(u) + ε for all u ∈ U. We show in Theorem 4.6 that such a modification is unnecessary as we prove that the sequence of iterates u k , generated by the algorithm described above, satisfies that each weak limit point of u k solves (1.4). Moreover, as we said before, in our second assumption we suppose that σ can depend in an affine manner on the control u and the state y and the Lipschtiz assumption on the cost terms and g are removed. This implies that the sequences of adjoint states p k are not bounded almost surely, which is the basic ingredient in the proof of the main results in [START_REF] Mazliak | An algorithm for solving a stochastic control problem[END_REF]. Finally, let us underline that in Corollary 4.4 we prove that some weak forms of optimality conditions are satisfied for both assumptions. In the convex case, this allows us to prove that the iterates u k form a minimizing sequence, a result that is absent in [START_REF] Mazliak | An algorithm for solving a stochastic control problem[END_REF] and also in the deterministic case studied in [START_REF] Bonnans | On an algorithm for optimal control using Pontryagin's maximum principle[END_REF]. Of course, this implies that if in addition J is strongly convex, we have strong convergence of the sequence u k .

The article is organized as follows: in Section 2 we state the main assumptions that we make in the entire paper. In Section 3, we prove that the algorithm is well-defined. In Section 4 we analyse the convergence of the method. We show that the sequence of costs generated by the algorithm is decreasing and convergent. Under some convexity assumptions, we can prove that every weak limit point of the sequence of iterates solves (1.4). Finally, in the last section we consider a quadratic cost with respect to the control, we study the relation with the gradient plus projection method and we prove that the rate of convergence is linear in this case.

Main assumptions

Let us first fix some standard notations. Endowed with the natural scalar product in L 2 ([0, T ]× Ω) denoted by (•, •), H 2 is a Hilbert space. We denote by • the L 2 ([0, T ] × Ω) norm on (H 2 ) l , for any l ∈ N. As usual and if the context is clear, we omit the dependence on ω of the stochastic processes. We set S 2 for the subspace of H 2 of continuous processes x satisfying that Let us now fix the standing assumptions that will be imposed from now on. (H1) Assumptions for the dynamics:

E sup t∈[0,T ] |x(t)| 2 < ∞.
(a) The maps

ϕ = f, σ are B (R n × R r × [0, T ]) ⊗ F T -measurable. (b) For all (y, u) ∈ R n × R r the process [0, T ] × Ω (t, ω) → ϕ(y, u, t, ω) is F-adapted. (c) For almost all (t, ω) ∈ [0, T ] × Ω the map (y, u) → ϕ(y, u, t, ω) is C 2
and there exist a constant L > 0 and a process ρ ϕ ∈ H 2 such that for almost all (t, ω) ∈ [0, T ] × Ω and for all y, ȳ ∈ R n and u, ū ∈ U ad we have

(2.1)        |ϕ(y, u, t, ω)| ≤ L [|y| + |u| + ρ ϕ (t, ω)] , |ϕ y (y, u, t, ω)| + |ϕ u (y, u, t, ω)| ≤ L, |ϕ yy (y, u, t, ω) -ϕ yy (ȳ, ū, t, ω)| ≤ L (|y -ȳ| + |u -ū|) , |ϕ yy (y, u, t, ω)| + |ϕ yu (y, u, t, ω)| + |ϕ uu (y, u, t, ω)| ≤ L.
(H2) Assumptions for the cost: (a) The maps and g are respectively

B (R n × R r × [0, T ]) ⊗ F T and B (R n ) ⊗ F T mea- surable. (b) For all (y, u) ∈ R n × R r the process [0, T ] × Ω (t, ω) → (y, u, t, ω) is F-adapted. (c) For almost all (t, ω) ∈ [0, T ] × Ω the map (y, u) → (y, u, t, ω) is C 2 ,
and there exist L > 0 and a process ρ (•) ∈ H 2 such that for almost all (t, ω) ∈ [0, T ] × Ω and for all y, ȳ ∈ R n and u ∈ U ad we have

(2.2)        | (y, u, t, ω)| ≤ L [|y| + |u| + ρ (t, ω)] 2 , | y (y, u, t, ω)| + | u (y, u, t, ω)| ≤ L [|y| + |u| + ρ (t, ω)] , | yy (y, u, t, ω)| + | yu (y, u, t, ω)| + | uu (y, u, t, ω)| ≤ L, | yy (y, u, t, ω) -yy (ȳ, u, t, ω)| ≤ L |y -ȳ| . (d)
For almost all ω ∈ Ω the map y → g(y, ω) is C 2 and there exists L > 0 such that for all y, ȳ ∈ R n and almost all ω ∈ Ω,

(2.3)        |g(y, ω)| ≤ L [|y| + 1] 2 , |g y (y, ω)| ≤ L [|y| + 1] , |g yy (y, ω)| ≤ L, |g yy (y, ω) -g yy (ȳ, ω)| ≤ L |y -ȳ| .
(H3) At least one of the following assumptions holds true: (a) For all (y, u) ∈ R n × R r and almost all (t, ω) ∈ [0, T ] × Ω we have (2.4) σ u (y, u, t, ω) ≡ 0 and σ yy (y, t, ω) ≡ 0.

Moreover, the following Lipschitz condition holds true: there exists L ≥ 0 such that for almost all (t, ω) ∈ [0, T ] × Ω, and for all y, ȳ ∈ R n and u, ū ∈ U ad , (2.5)

| (y, u, t, ω) -(ȳ, ū, t, ω)| ≤ L(|y -ȳ| + |u -ū|), |g(y, ω) -g(ȳ, ω)| ≤ L |y -ȳ| .
(b) For ϕ = f, σ and for almost all (t, ω) ∈ [0, T ] × Ω the map (y, u) → ϕ(y, u, t, ω) is affine.

Remark 2.1. (i) Our assumptions (H1)-(H2)-(H3)(a) are weaker than those in [START_REF] Mazliak | An algorithm for solving a stochastic control problem[END_REF], where it is supposed that and g are bounded and, in the statement of the main results, σ y ≡ 0. In addition, the data , g, f , σ do not depend explicitly on ω and the set U is assumed to be bounded.

(ii) Under assumption (H1), for any u ∈ U the state equation (1.1) admits a unique strong solution in (S D y σ j (y(t), t) q j (t) dt + q(t)dW (t)

p(T ) = ∇ y g(y(T ))
and under (H1)-(H2), it has a unique solution (p, q) ∈ (S 2 ) n ×(H 2 ) n×m (see [START_REF] Bismut | Linear quadratic optimal stochastic control with random coefficients[END_REF] and [20, Theorem 2.2, p. 349]).

Well-posedness of the algorithm

The aim of this section is to prove that the iterates of the Sakawa-Shindo algorithm are well defined. We need first the following lemma. Lemma 3.1. Under assumptions (H1)-(H2) and (H3)-(a), there exists C > 0 such that the solution (p, q) of (2.6) satisfies

(3.1) |p(t)| ≤ C, ∀t ∈ [0, T ], P -a.s.
Proof. By Itô's formula and (2.6), we have that (3.2)

|p(t)| 2 = 2 T t p(s) • [∇ y (y(s), u(s), s) + f y (y(s), u(s), s) p(s) + m j=1 D y σ j (y(s), s) q j (s)]ds - T t m j=1 |q j (s)| 2 ds -2 T t p(s) • q(s)dW (s) + |p(T )| 2 , ∀t ∈ [0, T ], P -a.s.
Using that p(T ) = ∇ y g(y(T )), our assumptions and the Cauchy-Schwarz and Young inequalities imply that for any ε > 0 we have

(3.3) |p(t)| 2 ≤ 2 T t [L |p(s)| + L |p(s)| 2 + m j=1 L |p(s)| |q j (s)|]ds - T t m j=1 |q j (s)| 2 ds -2 T t p(s) • q(s)dW (s) + L 2 ≤ L 2 T + T t (1 + 2L) |p(s)| 2 + m j=1 [ L 2 ε |p(s)| 2 + ε |q j (s)| 2 ]ds - T t m j=1 |q j (s)| 2 ds -2 T t p(s) • q(s)dW (s) + L 2 = L 2 (T + 1) + 1 + 2L + mL 2 ε T t |p(s)| 2 ds + (ε -1) m j=1 T t |q j (s)| 2 ds -2 T t p(s) • q(s)dW (s) Choosing ε < 1, we get (3.4) |p(t)| 2 ≤ C 1 + C 2 T t |p(s)| 2 ds -2 T t p(s) • q(s)dW (s).
for some constants C 1 , C 2 > 0. Now fix t ∈ [0, T ] and define r(t) := E |p(t)| 2 |Ft ≥ 0 for all t ≥ t. Combining (3.4) and [1, Lemma 3.1] we have

(3.5) r(t) ≤ C 1 + C 2 T t r(s)ds.
Thus, by Gronwall's Lemma there exists C > 0 independent of (t, ω) and t such that r(t) ≤ C for all t ≤ t ≤ T and so in particular |p( t)| 2 = r( t) ≤ C for a.a. ω. Since t is arbitrary and p admits a continuous version, the result follows.

Lemma 3.2. Consider the mapping

(3.6) u ε : R n × P × R n×m × U ad × [0, T ] × Ω → R r
where P := B(0, C) (ball in R n ) in case (a), and P = R n in case (b), defined by (3.7) u ε (y, p, q, v, t, ω) := argmin{K ε (y, u, v, p, q, t, ω) ; u ∈ U ad }.

Under assumptions (H1)-(H2) and (H3), there exist ε 0 > 0, α > 0 and β > 0 independent of (t, ω), with β = 0 if (H3)(a) is verified, such that, if ε < ε 0 , u ε is well defined and for a.a.

(t, ω) ∈ [0, T ] × Ω and all (y i , p i , q i , v i ) ∈ R n × P × R n×m × U ad , i = 1, 2: (3.8) u ε (y 2 , p 2 , q 2 , v 2 , t) -u ε (y 1 , p 1 , q 1 , v 1 , t) ≤ 2 v 2 -v 1 + α y 2 -y 1 + p 2 -p 1 + β q 2 -q 1 .
Proof. We follow [START_REF] Bonnans | On an algorithm for optimal control using Pontryagin's maximum principle[END_REF][START_REF] Mazliak | An algorithm for solving a stochastic control problem[END_REF]. Setting w := (y, p, q, v), element of E := R n × P × R n×m × U ad , we can rewrite K ε as K ε (u, w). We claim that for ε small enough:

(3.9)

D 2 uu K ε (u, w)(u , u ) ≥ (1/ε -2C 3 )|u | 2 ≥ 1 2ε |u | 2 for all w ∈ E and u in R m .
This holds if (H3)(a) holds since p, f uu and uu are bounded, and also if (H3)(b) since f and σ are affine and uu is bounded. By (3.9), K ε is a strongly convex function of u with modulus 1/(2ε), and hence, for all u 1 , u 2 ∈ U ad :

(3.10) D u K ε (u 2 , w) -D u K ε (u 1 , w) u 2 -u 1 ≥ 1 2ε u 2 -u 1 2 .
On the other hand, for i = 1, 2, take w i = (y i , p i , q i , v i ) ∈ E and denote u i := u ε (y i , p i , q i , v i ). Then

(3.11) D u K ε (u i , w i ) u 3-i -u i ≥ 0.
Summing these inequalities for i = 1, 2 with (3.10) in which we set w := w 1 , we obtain that (3.12)

1 2ε u 2 -u 1 2 ≤ D u K ε (u 2 , w 1 ) -D u K ε (u 2 , w 2 ) u 2 -u 1 , ≤ D u K ε (u 2 , w 1 ) -D u K ε (u 2 , w 2 ) u 2 -u 1 . Since D u K ε (u, w) = (1/ε)(u -v) + H u (y, u, p, q), (3.13) u 2 -u 1 ≤ 2 v 2 -v 1 + 2ε ∇ u H(y 1 , u 2 , p 1 , q 1 ) -∇ u H(y 2 , u 2 , p 2 , q 2 ) .
Inequality (3.2) easily follows from (3.13) and our assumptions.

Theorem 3.3. Under assumptions (H1)-(H3), there exists ε 0 > 0 such that, if ε k < ε 0 for all k, then the algorithm defines a uniquely defined sequence u k of admissible controls.

Proof. Given u 0 , let us define f : R n × [0, T ] × Ω → R n and σ : R n × [0, T ] × Ω → R n×m as (3.14) f (y) := f (y, u ε 1 (y, p 0 , q 0 , u 0 )), σ(y) := σ(y, u ε 1 (y, p 0 , q 0 , u 0 )).

Assumption (H1) and Lemma 3.2 imply that the SDE (3.15) dy(t) = f (y)dt + σ(y)dW (t) t ∈ [0, T ],

y(0) = y 0 ,
has a unique strong solution (e.g. [20, Theorem 6.16, p. 49]). Therefore u 1 := u ε 1 (y, p 0 , q 0 , u 0 ) is uniquely defined; so is u k by induction.

Convergence

In this section we prove our main results. If sup k ε k ≤ ε 0 where ε 0 is small enough, then the cost function decreases with the iterates (see Theorem 4.2 and Theorem 4.3). Moreover, if the problem is convex, then any weak limit point of the sequence u k solves the problem (see Theorem 4.6). We will need the following elementary Lemma. Lemma 4.1. Under assumption (H1), there exists C > 0 such that for any u, u ∈ U,

(4.1) E sup 0≤t≤T y(t) -y (t) 2 ≤ C u -u 2 .
where y and y are respectively the associated states to u and u .

Proof. Define δy := y -y and δu := u -u .

By Theorem 4.2. Under assumptions (H1)-(H3), there exists α > 0 such that any sequence generated by the algorithm satisfies

(4.4) J(u k ) -J(u k-1 ) ≤ - 1 ε k -α u k -u k-1 2 .
Proof. We drop the variable t when there is no ambiguity. We have, (4.5)

J(u k ) -J(u k-1 ) = E T 0 H(y k , u k , p k-1 , q k-1 ) -H(y k-1 , u k-1 , p k-1 , q k-1 ) -p k-1 • (f (y k , u k ) -f (y k-1 , u k-1 )) -q k-1 • (σ(y k , u k ) -σ(y k-1 , u k-1 )) dt +g(y k (T )) -g(y k-1 (T )) .
Define δy k := y k -y k-1 and δu k := u k -u k-1 . By Itô's formula, almost surely we have (4.6)

p k-1 (T ) • δy k (T ) = p k-1 (0) • δy k (0) + T 0 p k-1 • f (y k , u k ) -f (y k-1 , u k-1 ) -H y (y k-1 , u k-1 , p k-1 , q k-1 )δy k +q k-1 • σ(y k , u k ) -σ(y k-1 , u k-1 ) dt + T 0 p k-1 • σ(y k , u k ) -σ(y k-1 , u k-1 ) + q k-1 • δy k dW (t).
Then, replacing in (4.5) and using that δy k (0) = 0 we get (4.7)

J(u k ) -J(u k-1 ) = E T 0 H(y k , u k , p k-1 , q k-1 ) -H(y k-1 , u k-1 , p k-1 , q k-1 ) -H y (y k-1 , u k-1 , p k-1 , q k-1 , t)δy k (t) dt -p k-1 (T ) • δy k (T ) + g(y k (T )) -g(y k-1 (T )) .
Moreover, we have (4.8)

∆ := H(y k , u k , p k-1 , q k-1 ) -H(y k-1 , u k-1 , p k-1 , q k-1 ) = H(y k , u k , p k-1 , q k-1 ) -H(y k , u k -δu k , p k-1 , q k-1 ) +H(y k-1 + δy k , u k-1 , p k-1 , q k-1 ) -H(y k-1 , u k-1 , p k-1 , q k-1 ) = ∆ y -∆ u + H y (y k-1 , u k-1 , p k-1 , q k-1 )δy k + H u (y k , u k , p k-1 , q k-1 )δu k , where (4.9) ∆ u := H(y k , u k -δu k , p k-1 , q k-1 ) -H(y k , u k , p k-1 , q k-1 ) +H u (y k , u k , p k-1 , q k-1 )δu k and (4.10) ∆ y := H(y k-1 + δy k , u k-1 , p k-1 , q k-1 ) -H(y k-1 , u k-1 , p k-1 , q k-1 ) -H y (y k-1 , u k-1 , p k-1 , q k-1 )δy k .
Replacing in (4.7) we obtain (4.11)

J(u k ) -J(u k-1 ) = E T 0 H u (y k , u k , p k-1 , q k-1 )δu k -∆ u + ∆ y dt -p k-1 (T ) • δy k (T ) + g(y k (T )) -g(y k-1 (T )) .
Since in both cases in (H3) we have σ uu ≡ 0, we get (4.12)

∆ u = 1 0 (1 -s)H uu (y k , u k + sδu k , p k-1 , q k-1 )(δu k , δu k )ds = 1 0 (1 -s) uu (y k , u k + sδu k )(δu k , δu k ) + p k-1 • f uu (y k , u k + sδu k )(δu k , δu k ) ds. If (
∆ u ≥ - L 2 δu k 2 - CL 2 δu k 2 .
On the other hand, if (H3)-(b) holds true, we have f uu ≡ 0, and so (H2) implies

(4.14) ∆ u ≥ - L 2 δu k 2 .
Now, for both cases in (H3) we have σ yy ≡ 0. Thus, (4.15) If (H3)-(b) holds, then f yy ≡ 0 and so (H2) implies ∆ y ≤ L 2 δy k 2 . In conclusion, there exists

∆ y = 1 0 (1 -s)H yy (y k-1 + sδy k , u k-1 , p k-1 , q k-1 )(δy k , δy k )ds = 1 0 (1 -s)[ yy (y k-1 + sδy k , u k-1 )(δy k , δy k ) +p k-1 • f yy (y k-1 + sδy k , u k-1 )(δy k , δy k )]ds. If (
C 4 > 0 such that (4.17) ∆ u ≥ -C 4 δu k 2 and ∆ y ≤ C 4 δy k 2 .
Then, combining (4.11), and (4.17), we deduce

(4.18) J(u k ) -J(u k-1 ) ≤ E T 0 H u (y k , u k , p k-1 , q k-1 )δu k + C 4 δu k 2 + C 4 δy k 2 dt -p k-1 (T ) • δy k (T ) + g(y k (T )) -g(y k-1 (T )) . Since u k minimizes K ε k we have, (4.19) D u K ε k (y k , u k , u k-1 , p k-1 , q k-1 )δu k ≤ 0, a.e. t ∈ [0, T ] , P-a.s. then, (4.20) H u (y k , u k , p k-1 , q k-1 )δu k = D u K ε k (y k , u k , u k-1 , p k-1 , q k-1 )δu k -1 ε k δu k 2 ≤ -1 ε k δu k 2 , a.e. t ∈ [0, T ] , P-a.s.
By assumption (H2)-(d) and the definition of p k-1 (T ), there exists C 5 > 0 such that (4.21)

-p k-1 (T ) • δy k (T ) + g(y k (T )) -g(y k-1 (T )) ≤ C 5 δy k (T ) 2 .
Then, by (4.20) and (4.21) we obtain

(4.22) J(u k ) -J(u k-1 ) ≤ E T 0 C 4 - 1 ε k δu k (t) 2 + C 4 δy k (t) 2 dt + C 5 δy k (T ) 2 .
Then, the conclusion follows from Lemma 4.1.

Now we consider the projection map P U : (H 2 ) r → U ⊂ (H 2 ) r , i.e. for any u ∈ (H 2 ) r ,

P U (u) := argmin { u -v ; v ∈ U} .
By [8, Lemma 6.2], (4.23) P U (u)(t, ω) = P U ad (u(t, ω)), a.e. t ∈ [0, T ], P -a.s., where P U ad : R r → U ad ⊂ R r is the projection map in R r . We have the following result:

Theorem 4.3. Assume that J is bounded from below and that assumptions (H1)-(H3) hold true.

Then there exists ε 0 > 0 such that, if ε k < ε 0 , any sequence generated by the algorithm satisfies:

(1)

J(u k ) is a nonincreasing convergent sequence, (2) u k -u k-1 → 0, (3)
u k -P U (u k -ε k ∇J(u k )) → 0.
Proof. The first two items are a consequence of Theorem 4.2 and the fact that J is bounded from below. Since u k minimizes K ε k we have

(4.24) D u K ε k (y k , u k , u k-1 , p k-1 , q k-1 )(v -u k ) ≥ 0, ∀v ∈ U ad , a.e. t ∈ [0, T ], P -a.s. then, (4.25) u k -u k-1 + ε k ∇ u H(y k , u k , p k-1 , q k-1 ), v -u k ≥ 0, ∀v ∈ U ad , a.e. t ∈ [0, T ], P -a.s.
and so (4.26)

u k = P U u k-1 -ε k ∇ u H(y k , u k , p k-1 , q k-1 )
.

By [8, Proposition 3.8] we know that

(4.27) ∇J(u k-1 ) = ∇ u H(y k-1 , u k-1 , p k-1 , q k-1 ) in H 2 ,
then, by (4.26), (4.28)

u k-1 -P U (u k-1 -ε k ∇J(u k-1 )) = u k-1 -u k + P U u k-1 -ε k ∇ u H(y k , u k , p k-1 , q k-1 ) -P U u k-1 -ε k ∇ u H(y k-1 , u k-1 , p k-1 , q k-1 )
.

As P U is non-expansive in H 2 , we obtain

(4.29) u k-1 -P U u k-1 -ε k ∇J(u k-1 ) ≤ u k-1 -u k +ε k ∇ u H(y k , u k , p k-1 , q k-1 ) -∇ u H(y k-1 , u k-1 , p k-1 , q k-1 ) .
Now, let us estimate the last term in the previous inequality. By (H3), considering any of the two cases, we have σ uy ≡ σ uu ≡ 0. Therefore, for a.e. t ∈ [0, T ] there exist (ŷ, û) ∈ R n × U ad such that (4.30)

∇ u H(y k , u k , p k-1 , q k-1 ) -∇ u H(y k-1 , u k-1 , p k-1 , q k-1 ) = H uy (ŷ, û, p k-1 , q k-1 )(y k -y k-1 ) + H uu (ŷ, û, p k-1 , q k-1 )(u k -u k-1 ) = uy (ŷ, û) + p k-1 f uy (ŷ, û) (y k -y k-1 ) + uu (ŷ, û) + p k-1 f uu (ŷ, û) (u k -u k-1 ) ≤ C y k -y k-1 + u k -u k-1 ,
where in the last inequality, if (H3)-(a) holds we use Lemma 3.1 and assumptions (H1)-(H2), and if (H3)-(b) holds we use the fact that f uy ≡ f uu ≡ 0 and (H2). We conclude that (4.31)

∇ u H(y k , u k , p k-1 , q k-1 ) -∇ u H(y k-1 , u k-1 , p k-1 , q k-1 ) 2 ≤ 2C 2 u k -u k-1 2 + y k -y k-1 2 .
Using (4.29)-(4.31), Lemma 4.1 yields the existence of C > 0 such that (4.32)

u k-1 -P U u k-1 -ε k ∇J(u k-1 ) ≤ C u k-1 -u k ,
which proves the last assertion.

Corollary 4.4. Assume that the sequence generated by the algorithm u k is bounded, ε k < ε 0 and lim inf ε k ≥ ε > 0. Then, for every bounded sequence v k ∈ U we have that

(4.33) lim inf k→∞ ∇J(u k ), v k -u k ≥ 0.
In particular

(4.34) lim inf k→∞ ∇J(u k ), v -u k ≥ 0 ∀ v ∈ U
and in the unconstrained case U = (H 2 ) r Proof. We define for each k, (4.36)

w k := P U (u k -ε k ∇J(u k )),
then we have

(4.37) 0 ≤ w k -u k + ε k ∇J(u k ), v k -w k , ∀v k ∈ U.
By Theorem 4.3(3), we know u k -w k → 0. Using the fact that ∇J(u k ) = ∇ u H(y k , u k , p k , q k ), and the boundedness of u k , which in particular yields the boundedness of (p k , q k ) in (S Under convexity assumptions we obtain a convergence result.

w k -u k + ε k ∇J(u k ), v k -w k = ε lim inf k→∞ ∇J(u k ), v k -u k ,
Theorem 4.6. Let J(u), defined as above, be convex and bounded from below. Moreover, suppose that ε k < ε 0 , where ε 0 is given by Theorem 4.3, and lim inf ε k > 0. Then any weak limit point ū of u k is an optimal control and J(u k ) → J(ū).

Proof. Consider a subsequence u k i i∈N that converges weakly to ū, then u k i is bounded. Using the convexity of J and (4.34), for all v ∈ U we obtain (4.39)

J(v) ≥ lim inf i→∞ J(u k i ) + ∇J(u k i ), v -u k i ≥ lim inf i→∞ J(u k i ).
By Theorem 4.3 (1), J(u k ) is a convergent sequence, then we obtain

(4.40) J(v) ≥ lim k→∞ J(u k ) ≥ J(ū),
where the last inequality holds because of the weak lower semi-continuity of J, which is implied by the convexity and continuity of J. Since (4.40) holds for all v ∈ U, we can conclude that ū is an optimal control and J(u k ) → J(ū).

Remark 4.7. The fact that any weak limit point is an optimal control can also be obtained with Theorem 4.3(3) and the arguments in [START_REF] Bonnans | On an algorithm for optimal control using Pontryagin's maximum principle[END_REF].

We also want to note that the previous result is one of the main differences with respect to [START_REF] Mazliak | An algorithm for solving a stochastic control problem[END_REF], where it is only shown that the weak limit points are ε-approximate optimal solutions.

If J is strongly convex in (H 2 ) r we obtain strong convergence of the iterates. Corollary 4.8. If in addition J(u) is strongly convex, then the whole sequence converges strongly to the unique optimal control. Thus, (5.8)

(I + εN )(u k -u k-1 ) + ε(N u k-1 + B p k-1 + m j=1 B j q k-1 j ), v -u k ≥ 0,
and this is equivalent to (5.9)

M ε u k -u k-1 + εM -1 ε ∇J(u k-1 ) , v -u k ≥ 0.
By the previous Remark and (5.5), the inequality (5.9) implies that u k is the point obtained by the gradient plus projection method when we consider (H 2 ) r endowed with the metric defined by (5.5).

Under additional assumptions, the gradient plus projection method has a linear rate of convergence. In our framework, we will need the following result.

∇J(u) = H u (y, u, p, q) = N u + B p + m j=1 B j q j .
in H 2 , where u ∈ U, y is the state associated to u, given by (1.1), and p, q are the corresponding adjoint states, given by (1.6). Now, let u and u in U, and y, p, q and y , p , q the associated states and adjoint states, respectively. Define p := p -p and q := q -q , then (p, q) solves the following BSDE,

(t) = -∇ y 1 (y, t) -∇ y 1 (y , t) + A(t) p(t) + m j=1 A j (t) qj (t) dt + q(t)dW (t) t ∈ (0, T ), p(T ) = ∇ y g(y(T )) -∇ y g(y (T )). (5.11) dp
By [20, Now we can prove the main result of this section:

Theorem 5.4 (Linear convergence). Assume that there exists γ > 0 such that for a.a. (t, ω) we have v N (t, ω)v ≥ γ|v| 2 for all v ∈ R m . Moreover, suppose that the assumptions of Lemma 5.3 hold true and that 1 (•, t, ω) and g(•, ω) are convex for a.a. (t, ω). Then, J is strongly convex (in both norms), (1.4) has a unique solution u * and for ε small enough, there exists α ∈ (0, 1) and C > 0 such that

(5.14) u k -u * ≤ Cα k u 0 -u * , ∀k ∈ N,
where u k is the sequence generated by the algorithm.

Proof. By our assumptions J is strongly convex in ((H 2 ) r , • ), and it is easy to check that J is also strongly convex in ((H 2 ) r , • ε ) with a strong convexity constant β = γ/δ > 0, independent of ε small enough. Then, there exists a unique u * solution of (1.4). Since the norms • ε and • are equivalent, by (5.6) and Lemma 5.3 we can deduce that ∇ ε J is Lipschitz in (H 2 ) r endowed with the norm • ε and the Lipschitz constant L is independent of ε small enough. It is known that the rate of convergence of the gradient plus projection method is linear when the gradient is Lipschitz, but we write the proof in a few lines for the reader's convenience. By Theorem 5.2 we know that (5.15) u k+1 = P ε U (u k -ε∇ ε J(u k )) where P ε U is the projection map in ((H 2 ) r , • ε ). By the optimality of u * we have u * = P ε U (u * -ε∇ ε J(u * )), then for any k we obtain (5.16) u k+1 -u * 2 ε = P ε U (u k -ε∇ ε J(u k )) -P ε U (u * -ε∇ ε J(u * ))

2 ε ≤ u k -u * -ε ∇ ε J(u k ) -∇ ε J(u * ) 2 ε = u k -u * 2 ε -2ε u k -u * , ∇ ε J(u k ) -∇ ε J(u * ) ε + ε 2 ∇ ε J(u k ) -∇ ε J(u * ) 2 ε ≤ (1 -2εβ + ε 2 L 2 ) u k -u * 2 ε . If ε < 2β/L 2 we have 1 -2εβ + ε 2 L 2 < 1.
Using again the equivalence between • ε and • we deduce that (5.14) holds true.

  Finally, given an Euclidean space R l , we denote by | • | the Euclidean norm and by B(R l ) the Borel sigma-algebra.

  (4.3) Assumption (H1)-(c) and the Cauchy-Schwarz inequality imply directly (4.1).

  H3)-(a) holds true, Lemma 3.1 and assumptions (H1)-(H2) imply (4.13)

  H3)-(a) holds, Lemma 3.1 and (H1)-(H2)

which proves ( 4 .Remark 4 . 5 .

 445 33). Inequality (4.34) follows from (4.33) by taking v k ≡ v, for any fixed v ∈ U, and identity (4.35) follows by letting v k = u k -∇J(u k ), which is bounded in (H 2 ) r . In the unconstrained case, by (4.36) we have w k = u k -ε k ∇J(u k ), and by Theorem 4.3 (3) we know u k -w k → 0. It follows that (4.35) holds true even if u k is not bounded.

Lemma 5 . 3 .

 53 Under assumptions (H1), (H2), (H3)-(b) and (H4), the mapping u ∈ (H 2 ) r → ∇J(u) ∈ (H 2 ) r is Lipschitz, when H 2 is endowed with the L 2 -norm.Proof. By [8, Proposition 3.8], (H4) and (5.3) we have(5.10)

T 0 1 y 2 .

 12 Theorem 2.2, p. 349], there exists a constant C > 0 such that (5.12)E sup t∈[0,T ] |p(t)| 2 + T 0 |q(t)| 2 dt ≤ CE g y (y(T )) -g y (y (T )) 2 + (y, t) -1 y (y , t) 2 dt .By (H2)(c)-(d) and Lemma 4.1 we can conclude that there exists a positive constant, that we still denote by C, such that (5.13) E sup t∈[0,T ] |p(t)| 2 + T 0 |q(t)| 2 dt ≤ C u -u Now, by assumptions (H1)-(c) and (H2)-(c), the functions N, B and B j are bounded for a.a. (t, ω). The result follows by combining (5.13) with (5.10).

  CIFASIS -CENTRO INTERNACIONAL FRANCO ARGENTINO DE CIENCIAS DE LA INFOR-MACI ÓN Y DE SISTEMAS -CONICET -UNR -AMU, S2000EZP ROSARIO, ARGENTINA INSTITUT DE RECHERCHE XLIM-DMI, UMR-CNRS 7252, FACULT É DES SCIENCES ET TECH-NIQUES, UNIVERSIT É DE LIMOGES, 87060 LIMOGES, FRANCE

  2 ) n , see[START_REF] Mou | A variational formula for stochastic controls and some applications[END_REF] Proposition 2.1]. Also, by the estimates in [16, Proposition 2.1] and assumption (H2) the function J is well defined. Moreover, equation (1.6) can be written as

	(2.6)	  	dp(t) = -∇ y (y(t), u(t), t) + f y (y(t), u(t), t) p(t) +	m j=1
		 

  2 ) n × (H 2 ) n×m (see e.g.[START_REF] Mou | A variational formula for stochastic controls and some applications[END_REF] Proposition 3.1] or[START_REF] Yong | Stochastic controls, Hamiltonian systems and HJB equations[END_REF] Chapter 7]), we can deduce that ∇J(u k ) is a bounded sequence in (H 2 ) r . Finally, since the sequence v k is bounded by assumption, we can conclude

	(4.38)	0 ≤ lim inf k→∞

Acknowledgments. We thank the anonymous referee for her/his useful comments and the suggestion of studying the convergence rate.

Proof. Since J is strongly convex, there exists a unique u * such that J(u * ) = min u∈U J(u). Moreover, since Theorem 4.3 [START_REF] Backhoff | Some sensitivity results in stochastic optimal control: A Lagrange multiplier point of view[END_REF] implies that J(u k ) ≤ J(u 0 ), for all k ∈ N, the strong convexity of J implies that the whole sequence u k is bounded and by the previous Theorem, J(u k ) → J(u * ). By the strong convexity of J and the optimality of u * , there exists β > 0 such that (4.41)

Since J(u k ) → J(u * ) this implies u k -u * → 0.

Rate of Convergence

In this section, as in [START_REF] Bonnans | On an algorithm for optimal control using Pontryagin's maximum principle[END_REF], we present a particular case where, endowing with a new metric the space of controls, the algorithm is equivalent to the gradient plus projection method [START_REF] Goldstein | Convex programming in Hilbert space[END_REF]. The iterates of the method, for the minimization of J over the set U, can be written as (5.1) ûk+1

where ε k is the step size. Now, we consider the interesting case where the cost function is quadratic with respect to the control, so we make the additional assumption: (H4) The function has the form

where 1 and the matrix-valued N process are such that (H2) is satisfied. In particular, N is essentially bounded. Moreover, we assume that N (t, ω) is symmetric for a.a. (t, ω). In what follows we assume that (H1) and (H2) are satisfied, and, since the cost is quadratic in u, we will assume (H3)-(b). Let us point out that instead of (H3)-(b), we can assume that and g satisfy (H3)-(a) with the additional assumption that U ad is bounded, but, for the sake of simplicity, we will only assume (H3)-(b), i.e. that the state equation can be written as (5.3) dy(t) = [A(t)y(t) + B(t)u(t) + C(t)]dt + m j=1 [A j (t)y(t) + B j (t)u(t) + C j (t)]dW j (t), t ∈ (0, T ), y(0) = y 0 where A, B, C, A j , B j and C j , for j = 1, ..., m are matrix-valued processes such that (H1) holds.

We also assume that the sequence {ε k } is constant and equal to ε > 0. For this ε let us set

Note that if ε is small enough, then, since N is essentially bounded, there exist δ, δ > 0 such that δ |v| 2 ≥ v M ε (t, ω)v ≥ δ|v| 2 for a.a. (t, ω) and all v ∈ R r . Thus, defining

the norm • ε is equivalent to the L 2 -norm in (H 2 ) r and so ((H 2 ) r , • ε ) is a Hilbert space.

Remark 5.1. If we note ∇ ε J the gradient of the functional J in ((H 2 ) r , • ε ), we obtain the relation

where ∇J is the gradient of J in (H 2 ) r endowed with the L 2 -norm.

Theorem 5.2. The algorithm coincides with the gradient plus projection method for solving (1.4) in ((H 2 ) r , • ε ).

Proof. Let u k be the sequence generated by the algorithm, then by (4.25) and all the assumptions, we obtain skipping time arguments (5.7) u k -u k-1 + ε(N u k + B p k-1 + m j=1 B j q k-1 j ), v -u k ≥ 0, ∀v ∈ U ad , a.e. t ∈ [0, T ], P -a.s.


Paul-Emile Paradan
email: paradan@math.univ-montp2.fr

Michèle Vergne
email: michele.vergne@imj-prg.fr

Admissible coadjoint orbits for compact Lie groups

Introduction

Let K be a real Lie group with Lie algebra k. The notion of admissible coadjoint orbit was introduced by M. Duflo [START_REF] Duflo | Construction de représentations unitaires d'un groupe de Lie, CIME[END_REF] in order to study the unitary dual of K. This notion is also important in the work of D.Vogan (see [START_REF] Vogan | The method of coadjoint orbits for real reductive groups, Representation Theory of Lie Groups[END_REF]). Indeed, quoting him, "the orbit method provides some light in a very dark room". In this article, we restrict ourselves to the case where K is a compact connected Lie group. In this case, everything seems clear, admissible orbits are very easy to describe as well as K. Nevertheless, we discovered some remarkable properties of admissible coadjoint orbits which (we believe) were unnoticed before. Furthermore, as we briefly explain below, these properties are fundamental in the study of the equivariant index of twisted Dirac operators on general K-manifolds. It thus became necessary to return to this subject. Let us describe our motivations, and some of our results.

Choose a K-invariant scalar product on k ˚, a Cartan subalgebra, and a system of positive roots and denote by }ρ K } the norm of ρ K " 1 2 ř αą0 α. Coadjoint orbits can be grouped in different subsets that we call Dixmier sheets. If a coadjoint orbit Kξ varies in a Dixmier sheet, the stabilizer subgroup K ξ of ξ P k ˚remains in a fixed conjugacy class. The coadjoint orbits of maximal dimension form the regular sheet. Furthermore, there is a map Q spin K (see Definition 3.8) associating to an admissible coadjoint orbit P a virtual representation Q spin K pPq of K. By this correspondence, regular admissible coadjoint orbits parameterize the set p K of classes of unitary irreducible representations of K. For example, the coadjoint orbit of ρ K is regular admissible and parameterize the trivial representation of K.

We associate to a Dixmier sheet S a subset I S of p K, a positive number }ρ S } and a conjugacy class s S of semi-simple subalgebras of k: the set I S is the set of irreducible representations of K of the form Q spin K pPq where P is an admissible orbit belonging to S, s S is the conjugacy class of the semi-simple part of the infinitesimal stabilizer k ξ for any orbit Kξ in S, and }ρ S } is the norm of the ρ-element for the subgroup K ξ of k (thus }ρ S } " 0 exactly when S is the regular sheet).

Let S k be the set of conjugacy classes of the subalgebras s S where S varies over all Dixmier sheets. The set S k and the particular subset I S of K associated to a Diximier sheet S occur in the following theorem, that we prove in [START_REF] Paradan | Equivariant Dirac operators and differentiable geometric invariant theory[END_REF] and announced in [START_REF] Paradan | The multiplicities of the equivariant index of twisted Dirac operators[END_REF].

Assume K is acting on a connected Spin manifold M (or more generally a Spin c manifold). Let L be a K-equivariant line bundle on M . Consider the twisted Dirac operator D L and its equivariant index QpM, Lq. Then the following holds.

' If the semi-simple part s of the generic infinitesimal stabilizer k M of the action of K in M does not belong to S k , then QpM, Lq " t0u for any equivariant line bundle L on M .

' If s is equal to s S , then QpM, Lq is a virtual sum of irreducible representations of K belonging to I S .

The proof of this result uses an inequality on distance between coadjoint orbits, that we call the magical inequality. We prove this inequality in this article (Theorem 4.3). In this introduction, let us just state two of its consequences.

Theorem 1.1 Let O be a regular admissible coadjoint orbit, and let P be a coadjoint orbit in a Dixmier sheet S. Then the distance between O and P is greater or equal to }ρ S }.

Assume that P is an admissible coadjoint orbit. We prove in Theorem 5.1 that Q spin K pPq is either 0, or irreducible. If Q spin K pPq is irreducible, then Q spin K pPq " Q spin K pOq for a unique admissible regular orbit O. We then say that P is an ancestor of O. We prove the following theorem (Theorem 5.4). In general O has a unique ancestor, O itself. But for example the orbit of ρ K has 2 r ancestors, where r is the rank of rK, Ks.

Notations

Throughout the paper :

• K denotes a compact connected Lie group with Lie algebra k.

• T is a maximal torus in K with Lie algebra t.

• Λ Ă t ˚is the weight lattice of T : every µ P Λ defines a 1-dimensional T -representation, denoted C µ , where t " exppXq acts by t µ :" e ixµ,Xy .

• If h is a subalgebra of k, we denote by phq its conjugacy class.

• We fix a K-invariant Euclidean inner product p¨, ¨q on k. This allows us to identify k and k ˚when needed.

We denote by x¨, ¨y the natural duality between k and k ˚.

• We denote by RpKq the representation ring of K : an element E P RpKq can be represented as finite sum E "

ř λP p K m λ V K λ , with m λ P Z.
Here we have denoted by V K λ the irreducible representation of K indexed by λ.

• We denote by RpKq the space of Z-valued functions on K. An element E P RpKq can be represented as an infinite sum E " ř λP p K mpλqV K λ , with mpλq P Z. An element of RpKq will be called a virtual representation of K.

• If H is a closed subgroup of K, the induction map Ind K H : RpHq Ñ RpKq is the dual of the restriction morphism RpKq Ñ RpHq. It is given by the Frobenius reciprocity formula: the multiplicity mpλq of the irreducible representation

V K λ in Ind K H V H µ is the multiplicity of V H µ in the restriction of the representation V K λ to H.
• If E is a complex representation space of H, we denote simply by Ź ' E the virtual representation of H which is the difference of the representations of H in Ź even E and Ź odd E.

• The stabilizer of ξ P k ˚is denoted by K ξ . It is a connected subgroup of K. The Lie algebra of K ξ is denoted by k ξ . The element ξ P k ˚is called regular if K ξ is a Cartan subgroup of K.

• In general, the notation P is for any coadjoint orbit, while the notation O is reserved for regular coadjoint orbits.

Coadjoint orbits and Dixmier sheets

Let K be a compact connected Lie group with Lie algebra k.

We first define the ρ-orbit. Let T be a Cartan subgroup of K. Then t is imbedded in k ˚as the subspace of T -invariant elements. Roots will be considered as elements of t ˚(the corresponding character of T being e iα ). Choose a system of positive roots ∆ `Ă t ˚, and let ρ K " 1 2 ř αą0 α. The definition of ρ K requires the choice of a Cartan subgroup T and of a positive root system. However a different choice leads to a conjugate element. Thus we can make the following definition. Definition 2.1 We denote by opkq the coadjoint orbit of ρ K P k ˚. We call opkq the ρ-orbit. We denote by }ρ K } the norm of any point on opkq.

Let H k be the set of conjugacy classes of the reductive algebras k ξ , ξ P k ˚.

The set H k contains the conjugacy class ptq formed by the Cartan subalgebras. It contains also pkq (stabilizer of 0). Remark 2.2 If h " k ξ , then h C is the Levi subalgebra of the parabolic subalgebra determined by ξ. Parabolics are classified by subsets of simple roots. Thus there are 2 r conjugacy classes of parabolics if r is the rank of rk, ks. However, different conjugacy classes of parabolics might give rise to the same conjugacy class of Levi subalgebras (as seen immediately for type A n ).

We group the coadjoint orbits according to the conjugacy class phq P H k of the stabilizer. Definition 2.3 The Dixmier sheet k phq is the set of orbits Kξ with k ξ conjugated to h.

We also say that an element of k phq is a coadjoint orbit of type phq.

If phq " ptq, the corresponding Dixmier sheet is the set of regular coadjoint orbits.

If phq " pkq, the corresponding Dixmier sheet is the set of 0-dimensional orbits, that is the orbits tξu of the elements ξ of k ˚vanishing on rk, ks.

We denote by S k the set of conjugacy classes of the semi-simple parts rh, hs of the elements phq P H k .

Lemma 2.4

The map phq Ñ prh, hsq induces a bijection between H k and S k .

Proof. Assume that rh, hs " rh 1 , h 1 s " s. Let N be the normalizer of s, and n its Lie algebra. Thus h and h 1 are contained in n. Let t, t 1 be Cartan subalgebras of h, h 1 . Then t and t 1 are conjugated by an element of N . As h " s `t, we see that h is conjugated to h 1 . l

The connected Lie subgroup with Lie algebra h is denoted H. Thus if h " k ξ , then H " K ξ . We write h " z ' rh, hs where z is the center and rh, hs is the semi-simple part of h. Thus h ˚" z ˚' rh, hs ˚and z ˚is the set of elements in h ˚vanishing on the semi-simple part of h. We write k " h ' rz, ks, so we embed h ˚in k ˚as a H-invariant subspace, that is we consider an element ξ P h ˚also as an element of k ˚vanishing on rz, ks.

We consider the ρ-orbit ophq in h ˚. Remark that ophq is contained in rh, hs ˚. We denote by }ρ H } the norm of any point in ophq.

If S is the Dixmier sheet associated to phq, we denote by }ρ S } the norm of ρ H .

Admissible coadjoint orbits

We will be interested in admissible coadjoint orbits. This notion is defined in [START_REF] Duflo | Construction de représentations unitaires d'un groupe de Lie, CIME[END_REF] for any real Lie group. Let us now describe the set of admissible coadjoint orbits in concrete terms when K is a compact connected Lie group.

Consider ξ P k ˚. We have xξ, rk ξ , k ξ sy " 0. If iθ : k ξ Ñ iR is the differential of a character of K ξ , we denote by C θ the corresponding 1dimensional representation of K ξ , and by rC θ s " KˆK ξ C θ the corresponding line bundle over the coadjoint orbit Kξ Ă k

˚. Definition 3.1 An element ξ P k ˚is integral if iξ is the differential of a 1-dimensional representation of K ξ .
This notion of integrability is invariant under the coadjoint action, so a coadjoint orbit is called integral when any of its element is integral.

The following notion of admissibility is necessary for defining the spin quantization Q spin K pPq of an orbit P. Let Kξ be a coadjoint orbit. The quotient space k{k ξ is equipped with the symplectic form Ω ξ p X, Ȳ q :" xξ, rX, Y sy, and with a unique K ξ -invariant complex structure J ξ such that Ω ξ p´, J ξ ´q is a scalar product. We denote by q ξ the space k{k ξ considered as a complex vector space via the complex structure J ξ . Any element X P k ξ defines a complex linear map adpXq : q ξ Ñ q ξ . Definition 3.2 For any ξ P k ˚, we denote by ρpξq the element of k ξ such that xρpξq, Xy "

1 2i
Tr q ξ adpXq, X P k ξ .

Note that ρpξq vanishes on rk ξ , k ξ s. As explained in the preceding section, we consider also ρpξq as an element of k ˚.

Notice that 2ρpξq is integral since 2iρpξq is the differential of the character k P K ξ Þ Ñ det q ξ pkq.

Let t be a Cartan subalgebra and ξ P t ˚. Then if ∆ is the root system with respect to t, we have ρpξq " 1 2 ř αP∆,pα,ξqą0 α.

Definition 3.3 We say that the coadjoint orbit Kξ is admissible if ipξ ρpξqq is the differential of a 1-dimensional representation of K ξ .
Remark that if the coadjoint orbit Kξ is admissible, then 2ξ is integral. It is easy to see that, for a compact connected Lie group K, this definition of admissibility is equivalent to Duflo definition via the metaplectic correction.

In particular the orbit opkq is admissible. Indeed if ξ " ρ K , then ξ ρpξq " 0. Remark 3.4 When the group K is simply connected, then a regular admissible orbit Kξ is integral. This is not the case when K is not simply connected. For example the orbit opkq (which is regular admissible) is not integral when K " SOp3q. However, even if the group K is simply connected, non regular admissible coadjoint orbits are not necessary integral. See the example 3.7 below. Definition 3.5 We denote by Apphqq the set of admissible orbits of type phq.

We now define the shift of an orbit. If µ P k ˚, recall that opk µ q Ă k μ is the coadjoint orbit of the ρ-element for the group K µ . Definition 3.6 To any coadjoint orbit P Ă k ˚, we associate the coadjoint orbit spPq Ă k ˚which is defined as follows : if P " Kµ, take spPq " Kξ with ξ P µ `opk µ q. We call spPq the shift of the orbit P.

If O is regular, spOq " O, and if P " t0u, then spPq " opkq.

Beware that we have two elements of h ˚" rh, hs ˚' z ˚associated to an element µ P k ˚such that k µ " h: the element ρpµq P z ˚(defined canonically) and the element ρ Kµ P rh, hs ˚(defined up to H-conjugacy). Remark that ρpµq `ρKµ is conjugated to ρ K . More concretely, if we choose a Cartan subgroup T of H with Lie algebra t, and a positive root system ∆ `for roots of k with respect to t, then, when µ is dominant,

ρpµq " 1 2 ÿ αP∆ `,pα,µqą0 α and ρ Kµ " 1 2 ÿ αP∆ `,pα,µq"0 α.
The orbit Kµ is admissible if µ ´ρpµq is in the weight lattice of T . The shift of the orbit Kµ is Kpµ `ρKµ q.

Example 3.7 Consider the group K " SU p3q. Then there are 3 different sheets, determined (for this case) by the dimensions of orbits. The regular sheet consists of coadjoint orbits of dimension 6, The subregular sheet consists of coadjoint orbits of dimension 4, finally we have the orbit t0u of dimension 0.

We now parameterize the subregular sheet. Let h be the Lie algebra of H " SpU p1q ˆU p2qq. Then the stabilizer k f of a coadjoint orbit Kf of dimension 4 is conjugated to h. Let us give representatives of the orbits in k phq .

We consider the Cartan subalgebra of diagonal matrices and choose a Weyl chamber. Let ω 1 , ω 2 be the two fundamental weights. Thus

ρ K " ω 1 `ω2 . Let σ 1 , σ 2 be the half lines R ą0 ω 1 , R ą0 ω 2 .
Then k phq is the set of orbits Kptω 1 q, with t ‰ 0. Here we have privileged an element f on the orbit P P k phq such that k f " h. We could also privilege representatives of orbits P P k phq belonging to the chosen closed Weyl chamber. As ´ω1 conjugated to ω 2 , we obtain the description of k phq as the set of orbits tKpt 1 ω 1 q, Kpt 2 ω 2 qu with t 1 ą 0 and t 2 ą 0 (orbits of points in the boundary of the Weyl chamber, except t0u).

Let us now describe the set Apphqq of admissible coadjoint orbits of type phq.

We obtain, using the first description, that the set Apphqq is equal to the collection of orbits K ¨p 1`2n

2 ω 1 q, n P Z (see Figure 1). Remark that orbits in Apphqq are admissible, but not integral. Using the second description, with representatives in the chosen closed Weyl chamber, we see that the set Apphqq is equal to the collection of orbits K ¨p 1`2n 2 ω i q, n P Z ě0 , i " 1, 2. We have also depicted in this description the shifted orbits. Remark that the shift of the admissible elements in A phq are admissible, except for the two elements 1 2 ω 1 and 1 2 ω 2 with shifts ω 2 and ω 1 respectively, elements which are not admissible.

Finally remark that the orbit Kρ K is obtained as the shift of the admissible orbits t0u, Kp 3 2 ω 1 q, Kp 3 2 ω 2 q, and Kρ K . We now associate to an admissible coadjoint orbit an element of RpKq.

Let P " Kξ be an admissible coadjoint orbit. We consider the Z{2Zcomplex space Ź ' q ξ b C ξ´ρpξq " E `' E ´with E `" Ź even q ξ b C ξ´ρpξq and E

´" Ź odd q ξ b C ξ´ρpξq . We still denote by Ź ' q ξ b C ξ´ρpξq the element E `´E ´in RpHq.

Definition 3.8 We define Q spin K pPq P RpKq by the formula:

(3.1) Q spin K pPq " Ind K K ξ ´ľ' q ξ b C ξ´ρpξq ¯. Thus Q spin K pPq is the virtual representation Ind K K ξ E `´Ind K K ξ E ´.
The fact that Q spin K pPq is a finite signed sum of representations of K is easy to see using Frobenius reciprocity formula.

We now interpret Q spin K pPq as the equivariant index of a twisted Dirac operator on P. Consider the K-equivariant graded complex vector bundle S P " K ˆKξ E. It defines a Spin c -bundle on P (see [START_REF] Paradan | Equivariant Dirac operators and differentiable geometric invariant theory[END_REF]). The determinant line bundle of S P is the line bundle rC 2ξ s and its corresponding moment map (as defined in [START_REF] Paradan | Equivariant Dirac operators and differentiable geometric invariant theory[END_REF]) is the canonical injection P Ñ k ˚. Consider the Dirac operator D P associated to this Spin c -bundle. Then by Atiyah-Bott [START_REF] Atiyah | A Lefschetz Fixed Point Formula for Elliptic Complexes: II. Applications[END_REF], the equivariant index of D P is equal to Ind

K K ξ E `´Ind K K ξ E ´. Thus Q spin
K pPq coincides with the equivariant index of the twisted Dirac operator D P (and so belongs to RpKq). For this reason Q spin K pPq is called the spin quantization of P.

The following proposition is well known. We will recall its proof in Lemma 4.1 in the next subsection.

Proposition 3.9

• The map O Þ Ñ π O :" Q spin K pOq defines a bijection between the set of regular admissible orbits and p K.

• Q spin K popkqq is the trivial representation of K.

In Section 5, we will describe the virtual representation Q spin K pPq attached to any admissible orbit in terms of regular admissible orbits. It is either equal to 0, or is an element of p K.

The magical inequality

In order to parameterize coadjoint orbits, we choose a Cartan subgroup T of K with Lie algebra t. Let Λ Ă t ˚be the lattice of weights of T . Let W be the Weyl group. Choose a system of positive roots ∆ `Ă t ˚, and let as before ρ K " 1 2

ÿ αą0 α.
If α P t ˚is a root, we denote by H α P t the corresponding coroot (so xα, H α y " 2). Then xρ K , H α y " 1 if and only if α is a simple root. Define the positive closed Weyl chamber by t ě0 " tξ P t ˚; xξ, H α y ě 0 for all α ą 0u, and we denote by Λ ě0 :" ΛXt ě0 the set of dominant weights. Any coadjoint orbit P of K is of the form P " Kξ with tξu " P X t ě0 . We index the set p K of classes of finite dimensional irreducible representations of K by the set ρ K `Λě0 . The irreducible representation π λ corresponding to λ P ρ K `Λě0 is the irreducible representation with infinitesimal character λ. Its highest weight is λ ´ρK . The representation π ρ K is the trivial representation of K. The Weyl character formula for the representation π λ is, for X P t, Tr π λ pe X q " ř wPW pwqe ixwλ,Xy ś αą0 e ixα,Xy{2 ´e´ixα,Xy{2 .

For any µ P t ˚, recall its element ρpµq P k ˚(Definition 3.2).

Lemma 4.1 Let λ P t ě0 be a regular admissible element of k ˚. Then 1. ρpλq " ρ K .

2. λ P ρ K `Λě0 .

3. Q spin K pKλq " π λ .

Proof. Let λ P t ě0 be regular and admissible, then ρpλq " ρ K , so λ P tρ K `Λu X t ą0 . If α is a simple root, then the integer xλ ´ρK , H α y " xµ, H α y ´1 is non negative, as xλ, H α y ą 0. So λ ´ρK is a dominant weight.

Let O " Kλ. Weyl character formula coincide with Atiyah-Bott-Lefschetz formula [START_REF] Atiyah | A Lefschetz Fixed Point Formula for Elliptic Complexes: II. Applications[END_REF] for the index of the Dirac operator D O . Thus we obtain Lemma 4.1 and Proposition 3.9. l

The positive Weyl chamber t ě0 is the cone determined by the equations xλ, H α y ě 0 for the simple roots α ě 0. We denote by F k the set of the relative interiors of the faces of t ě0 . Thus the set F k is parameterized by subsets of the simple roots and has 2 r elements, r being the rank of rk, ks. Thus t ě0 " š σPF k σ, and we denote by t ą0 P F k the interior of t ě0 . Let σ P F k . Thus Rσ, the linear span of σ, is the subspace determined by xλ, H α y " 0 where the α varies over a subset of the simple roots.

The stabilizer K ξ does not depend of the choice of the point ξ P σ : we denote it by K σ . The map σ Ñ k σ induces a surjective map from F k to H k . This map may not be injective: in Example 3.7, σ 1 and σ 2 leads to the same element of H k , as the corresponding groups K σ 1 " SpU p1q ˆU p2qq and K σ 2 " SpU p2q ˆU p1qq are conjugated.

For σ P F k , we have the decomposition k σ " rk σ , k σ s ' zpk σ q with dual decomposition k σ " rk σ , k σ s ˚' Rσ. Let ρ Kσ :" 1 2 ÿ αą0 pα,σq"0 α be the ρ-element of the group K σ associated to the positive root system tα ą 0, pα, σq " 0u for K σ . Then

ρ K ´ρKσ " 1 2 ÿ αą0 pα,σqą0
α, and for any µ P σ, the element ρpµq P k ˚is equal to ρ K ´ρKσ . In particular, ρ K ´ρKσ vanishes on rk σ , k σ s, so ρ K ´ρKσ P Rσ, while ρ Kσ P rk σ , k σ s ˚. The decomposition ρ K " pρ K ´ρKσ q `ρKσ is an orthogonal decomposition.

Figure 3 shows this orthogonal decomposition of ρ for the case SU p3q. The subset ρ K `tě 0 of the positive Weyl chamber will be called the shifted Weyl chamber. It is determined by the inequalities xλ, H α y ě 1 for any simple root α ě 0, and thus xλ, H α y ě 1 for any positive root. Let us list some immediate properties of the shifted Weyl chamber. Proposition 4.2

1. If λ P ρ K `tě 0 , then pλ, λq ě pλ, ρ K q ě pρ K , ρ K q. The equality pλ, λq " pλ, ρ K q holds only if λ " ρ K .

Let

σ P F k .
' The orthogonal projection of ξ P t ą0 onto Rσ belongs to σ.

' We have ρ K ´ρKσ P σ for any σ P F k .

3. For any phq P H k , }ρ K } ě }ρ H }, and }ρ K } " }ρ H } only if H " K.

Proof. If λ " ρ K `c, with c P t ě0 , inequalities pλ, λq ě pλ, ρ K q ě pρ K , ρ K q follows from the fact that pλ, cq and pρ K , cq are non negative, as the scalar product of two elements of t ě0 is non negative. Equality pλ, λq " pλ, ρ K q implies λ " ρ K .

The second point follows from the fact that the dual cone to t ě0 is generated by the simple roots α i , and pα i , α j q ď 0, if i ‰ j.

We have the orthogonal decomposition ρ K " ρ Kσ `pρ K ´ρKσ q: hence ρ K ´ρKσ , which is the orthogonal projection of ρ K on Rσ, belongs to σ.

For the third point, we might choose

H conjugated to K σ , so }ρ K } 2 " }ρ Kσ } 2 `}ρ K ´ρKσ } 2 . l
The following theorem analyze the distance of a point in t ˚to the shifted Weyl chamber. It is illustrated in Figure 4 in the case SU p3q. ' If λ P ρ K `tě 0 and µ P t ˚, then:

(4.3) 1 2 ÿ αą0 pα,µq"0 pλ, αq ě }ρ Kµ } 2 .
' If one of the inequalities (4.2) or (4.3) is an equality, then µ belongs to t ě0 , and λ " µ `ρKσ where σ P F k is the stratum of t ě0 containing µ. Thus the two inequalities (4.2) or (4.3) are equalities. In particular, λ ´ρpλq " µ ´ρpµq.

Proof. Let k µ be the centralizer of µ and let z be the center of k µ . Consider the orthogonal decomposition t ˚" z ˚' a ˚where a is a Cartan subalgebra for rk µ , k µ s, that is a " ř pα,µq"0 RH α . Let ρ Kµ P a ˚be the ρ element for the system ∆ 1

`" tα ą 0, pα, µq " 0u of rk µ , k µ s. Let us write λ " ρ K `c, with c dominant, and decompose ρ K " p 0 `p1 , c " c 0 `c1 , with p 0 , c 0 P z ˚, p 1 , c 1 P a ˚. Thus λ " λ 0 `λ1 , with λ 0 P z ˚and λ 1 " p 1 `c1 . Now p 1 belongs to the shifted Weyl chamber in a ˚. Indeed, for any α ą 0 such that pα, µq " 0, we have xp 1 , H α y " xρ K , H α y ě 1. Similarly c 1 is dominant for the system ∆ 1 `.

As µ P z ˚, we have }λ ´µ} 2 " }λ 0 ´µ} 2 `}p 1 `c1 } 2 . Using the first point of 4.2, we obtain

}λ ´µ} 2 " }λ 0 ´µ} 2 `}p 1 `c1 } 2 ě pp 1 `c1 , ρ Kµ q ě }ρ Kµ } 2 .
As pp 1 `c1 , ρ Kµ q " pλ, ρ Kµ q " 1 2 ÿ αą0 pα,µq"0 pλ, αq we obtain Inequalities (4.2) and (4.3).

If the inequality }λ ´µ} 2 ě pp 1 `c1 , ρ Kµ q is an equality, then

}λ 0 ´µ} 2 ``}p 1 `c1 } 2 ´pp 1 `c1 , ρ Kµ q ˘" 0.
Both terms of the left hand side are non negative. Thus, using again the first point of Proposition 4.2, we see that necessarily c 1 " 0, p 1 " ρ Kµ , and λ 0 " µ. Thus for roots α P ∆ 1 `, xρ Kµ , H α y " xρ K , H α y. As ρ Kµ takes value 1 on simple roots for K µ , it follows that the set S 1 of simple roots for the system ∆ 1 `is contained in the set of simple roots for ∆ `. As a " ' αPS 1 RH α , the orthogonal z of a is Rσ for the face σ of t ˚orthogonal to the subset S 1 of simple roots. We then have K µ " K σ . Furthermore, λ " µ `ρKσ . Thus µ is the projection of λ on Rσ, so µ P σ Ă t ě0 . As ρpλq " ρ K , and ρpµq " ρ K ´ρKσ , we obtain λ ´ρpλq " µ ´ρpµq. So all assertions are proved. l

In this article, we will only use the following obvious corollary of Inequalities (4.2) and (4.3). However, in our application [START_REF] Paradan | The multiplicities of the equivariant index of twisted Dirac operators[END_REF], we will need both inequalities.

Corollary 4.4 (The magical inequality) If λ P ρ K `tě 0 and µ P t ˚, then:

(4.4) }λ ´µ} ě }ρ Kµ }.
The equality holds if and only if µ is dominant and λ ´ρpλq " µ ´ρpµq.

Let us give some consequences of the magical inequality (4.4). Let us define the notion of very regular element. Definition 4.5 Let λ P k ˚be a regular element. Then λ determines a closed positive Weyl chamber C λ Ă k λ. We say that λ is very regular if λ P ρpλq `Cλ .

In other words, a very regular element is an element which is conjugated to an element of the shifted Weyl chamber. Note that regular admissible elements are very regular.

Corollary 4.6 Let λ, µ be two elements of k ˚. Assume that λ is very regular, then }λ ´µ} ě }ρ Kµ }. The equality holds if and only if µ P C λ and λ ´ρpλq " µ ´ρpµq.

Proof. Consider the minimum of }λ ´µ1 } 2 when µ 1 varies in Kµ. Using the differential, we see that a point where the minimum is reached is in the Cartan subalgebra determined by λ. We can then conclude by Corollary (4.4). l Let us now study the admissible coadjoint orbits and their shifts. We see that if µ is in σ, its shift is spKµq " Kpµ `ρKσ q. Furthermore, µ P σ is admissible if and only if µ `ρKσ P ρ K `Λ.

Theorem 4.8 below says in particular that if a shift µ Ñ µ `ρKσ of an admissible element µ P σ is regular, then it is in the shifted Weyl chamber ρ K `tě 0 . Figure 5 illustrate this fact in the case SU p3q. In this figure, the two points ω 1 , ω 2 (which are not admissible) are represented by the empty circles. The shift of the admissible element 1 2 ω 1 is ω 2 , and so is singular and is not admissible.

Theorem 4.8 Let σ be a relative interior of a face of t ě0 .

1. Let µ P σ. If µ `ρKσ is regular admissible, then µ is admissible.

2. If µ P σ is admissible, and µ `ρKσ is regular, then µ `ρKσ is regular admissible. Moreover µ `ρKσ P ρ K `pΛ ě0 X σq. In particular µ `ρKσ is in the shifted Weyl chamber.

3. If µ P σ is admissible, we have

Q spin K pKµq " # 0 if µ `ρKσ is singular, π µ`ρ Kσ if µ `ρKσ is regular.
Proof. Let us prove the two first points. Let µ P σ such that λ " µ `ρKσ is regular admissible. Thus }λ μ} 2 " }ρ Kµ } 2 . The point λ being regular and admissible, λ is very regular. We use the magical inequality. The equality }λ ´µ} 2 " }ρ Kµ } 2 implies λ ´ρpλq " µ ´ρpµq " µ ´pρ K ´ρKσ q. Thus ρpλq " ρ K , so λ P t ě0 . Furthermore µ is admissible.

Conversely, we see that if µ is admissible, and λ " µ `ρKσ is regular, then λ is regular admissible. We have seen that λ P t ě0 . As λ is regular admissible, it is in ρ K `tě 0 . The element λ ´ρK " µ ´pρ K ´ρKσ q is in Rσ. As it is dominant, it is in σ.

Let us prove the last point. Let q µ be the complex space k{k µ equipped with the complex structure J µ . The equivariant index Θ of the Dirac operator D Kµ associated to the Spin c -bundle S Kµ " K ˆKµ p Ź ' q µ b C µ´ρ K `ρKσ q is given by Atiyah-Bott fixed point formula: for X P t, Θpe X q " ř wPW {Wµ w1 e ixµ,Xy ś xα,µyą0 e ixα,Xy{2 ´e´ixα,Xy{2 . Here W µ , the stabilizer of µ in W , is equal to the Weyl group of the group K σ . Using ř wPWσ pwqe wρ Kσ " ś αą0,xα,σy"0 pe α{2 é´α{2 q, we obtain (4.5) Θpe X q " ř wPW pwqe ixwpµ`ρ Kσ q,Xy ś αą0 e ixα,Xy{2 ´e´ixα,Xy{2 . If µ `ρKσ is singular, Θ is equal to zero. If µ `ρKσ is regular, thanks to the first point, µ `ρKσ is in ρ K `Λě0 , so Θ " π µ`ρ Kσ . l

We proved that if µ P σ is admissible and its shift µ `ρKσ is regular, then µ `ρKσ is admissible and dominant. However, as illustrated in the following examples, if µ `ρKσ is singular, ' µ `ρKσ is not necessarily dominant, ' µ `ρKσ might be admissible or not admissible.

Example 4.9 Consider the group K " U p7q and the face σ " tλ 1 ą λ 2 " λ 3 " λ 4 " λ 5 " λ 6 ą λ 7 u of the Weyl chamber. Here ρ K " p3, 2, 1, 0, ´1, ´2, ´3q and ρ Kσ " p0, 2, 1, 0, ´1, ´2, 0q. Then µ :" p1, 0, 0, 0, 0, 0, ´1q is an admissible element of σ, but its shift µ `ρKσ " p1, 2, 1, 0, ´1, ´2, 0q is singular, not admissible and not dominant.

Example 4.10 Consider the group K " U p5q and the face σ " tλ 1 " λ 2 ą λ 3 ą λ 4 " λ 5 u of the Weyl chamber. Then µ " p1{2, 1{2, 0, ´1{2, ´1{2q is an admissible element of σ. Its shift µ `ρKσ " p1, 0, 0, 0, ´1q is singular and admissible.

For application to the equivariant index of Dirac operators on general Spin c K-manifolds, we reformulate Inequality (4.2) independently of a choice of a Weyl chamber using normalized traces. Definition 4.11 Let N be a real vector space and b : N Ñ N a linear transformation, such that ´b2 is diagonalizable with non negative eigenvalues, and let |b| " ? ´b2 . We denote by nTr N |b| " 1 2 Tr N |b|, that is half of the trace of the action of |b| in the real vector space N . We call nTr N |b| the normalized trace of b.

If N is an Euclidean space and b a skew-symmetric transformation of N , then ´b2 is diagonalizable with non negative eigenvalues.

For any b P k and µ P k ˚fixed by b, we may consider the action adpbq : k µ Ñ k µ . The corresponding normalized trace nTr kµ |adpbq| is denoted simply by nTr kµ |b|. If the equality holds, then µ belongs to the positive Weyl chamber C λ and 1. λ´ρpλq " µ´ρpµq, hence λ is admissible if and only if µ is admissible, 2. spKµq " Kλ.

Proof. Indeed, as λ is fixed by b, we see that β belong to k λ. We may assume that k λ " t ˚. Thus β, λ and µ " λ ´β belong to t ˚. The element λ is a very regular element of t ˚. Note that the element of k corresponding to µ acts trivially on k µ . So the inequality of Proposition 4.12 is a restatement of Inequality 4.2. If the equality holds, we apply Theorem 4.3 and we obtain the proposition. l

Admissible coadjoint orbits and associated representations

In this section, we give some more information on the map P Ñ Q spin K pPq. The following proposition follows almost immediately from Theorem 4.8. Theorem 5.1 Let P be an admissible orbit.

• P ˚:" ´P is also admissible and

Q spin K pP ˚q " Q spin K pPq ˚.
• If spPq is not regular, then Q spin K pPq " 0.

• If spPq is regular, then spPq is a regular admissible coadjoint orbit and

Q spin K pPq " Q spin K pspPqq " π spPq .
For the remaining part of this section, we fix a conjugacy class phq. Let k phq be the Dixmier sheet determined by phq. H-invariant). We denote q C the complex H-module pq, J C q, and ρ C the element of z ˚defined by the relation

xρ C , Xy " 1 2i
Tr q C adpXq, X P h.

We define the holomorphic induction map Hol The following proposition explains the interaction between the holomorphic induction map Hol K H and the spin quantization procedure.

Proposition 5.6 Let Hµ be an admissible orbit for H. ' If µ `ρC R h 0 , then Hol K H pQ spin H pHµqq " 0. ' If µ `ρC P h 0 , then µ `ρC is K-admissible and

Hol K H pQ spin H pHµqq " C C 1 Q spin K pKpµ `ρC qq
where C 1 is the connected component of h 0 containing µ `ρC , and C C 1 is the ratio of the orientation opJ C q and opJ C 1 q on q. Assume first that µ 1 :" µ `ρC P h 0 : let C 1 be the connected component of h 0 containing µ 1 . As K µ 1 " H µ 1 " H µ , we have

I C µ " Ind K K µ 1 ´ľ' pk{hq C b ľ ' ph{h µ 1 q µ 1 b C µ 1 ´ρC ´ρH pµ 1 q
¯. Now we use the fact that the graded K µ 1 -module Ź ' pk{hq C is equal to

C C 1 Ź ' pk{hq C 1 b C ρ C ´ρC 1 . It gives that I C µ " C C 1 Ind K K µ 1 ´ľ' pk{hq C 1 b ľ ' ph{h µ 1 q µ 1 b C µ 1 ´ρC 1 ´ρH pµ 1 q " C C 1 Ind K K µ 1 ´ľ' pk{k µ 1 q µ 1 b C µ 1 ´ρpµ 1 q " C C 1 Q spin K pKµ 1 q.
Here we have used that ρpµ 1 q " ρ C 1 `ρH pµ 1 q. Assume now that I C µ ‰ 0. Thus Q spin H pHµq must be non zero. Hence we have Q spin H pHµq " Q spin H pH μq where μ P µ `oph µ q is an H-admissible and H-regular element. 20 Consider the maximal torus T :" H μ, and a Weyl chamber C " t ě0 for K containing μ. Let J C be the corresponding complex structure on k{t. Let ρ K be the ρ-element associated to the choice of Weyl chamber. Let C 1 be the connected component of h 0 that contains the open face t ą0 .

If we use the relation ρ K " ρ C 1 `ρH pμq, one has like before

I C µ " Ind K H ´ľ' pk{hq C b Q spin H pH μq " Ind K T ´ľ' pk{hq C b ľ ' ph{tq μ b C μ´ρ H pμq " C C 1 Ind K T ´ľ' pk{tq C b C μ`ρ C ´ρK ¯.
We see then that I C µ ‰ 0 only if λ :" μ `ρC " µ 1 `ρH µ 1 is a K-regular element.

Here we have }ρ H µ 1 } " }λ ´µ1 }, and on the other hand by the magical inequality we must have }λ ´µ1 } ě }ρ K µ 1 } since λ is K-regular and admissible. It forces }ρ K µ 1 } to be equal to }ρ H µ 1 }, and then K µ 1 " H µ 1 : the element µ 1 " µ `ρC belongs to h 0 .

The proof is completed. l Proposition 5.6 is a very special case of the formula for equivariant indices of twisted Dirac operators obtained in [START_REF] Paradan | The multiplicities of the equivariant index of twisted Dirac operators[END_REF]. Indeed the representation Hol K H pQ spin H pHµqq is the equivariant index for a Spin c -bundle on M " K{H µ . The infinitesimal stabilizer k M is the conjugacy classes of h µ " k µ`ρ C . It is indeed a representation associated to the Dixmier sheet attached to pk µ`ρ C q.

Theorem 1 . 2

 12 Let O be a regular admissible orbit. The ancestors of O in a Dixmier sheet S are all the elements in S at distance }ρ S } of O.

Figure 1 :

 1 Figure 1: A set of representatives of admissible orbits of type phq

Figure 2 :

 2 Figure 2: Another set of representatives of admissible orbits of type phq and their shifts

Figure 3 :

 3 Figure 3: Orthogonal decomposition of ρ K

Figure 4 :

 4 Figure 4: Distance of a singular element µ to an element λ in the shifted Weyl chamber

Figure 5 :

 5 Figure 5: Shifts of admissible orbits

Corollary 4 . 7

 47 The distance between a regular admissible coadjoint orbit O and an orbit P in a Dixmier sheet S is greater or equal to }ρ S }.

Proposition 4 .ě 1 2

 42 12 Let b P k and denote by β the corresponding element in k ˚. Let λ, µ be elements of k ˚fixed by b. Assume that λ is very regular and that µ ´λ " β. Then }β} 2 nTr kµ |b|.

Definition 5 . 2

 52 Let O Ă k ˚be a regular orbit. A K-orbit P in is called a phq-ancestor of O if P P k phq and spPq " O. Lemma 5.3 If P P k phq and spPq is regular, then spPq is admissible if and only P is admissible.

KH:

  RpHq Ñ RpKq by the relation Hol K H pV q " Ind K H ´ľ' q C b V ¯.

  Proof. Let I C µ " Ind K H ´Ź' pk{hq C b Q spin H pHµq ¯. By definition Q spin H pHµq " Ind H Hµ ´Ź' ph{h µ q µ b C µ´ρ H pµq ¯.

Acknowledgments

We wish to thank the Research in Pairs program at Mathematisches Forschungsinstitut Oberwolfach (February 2014), where our work on equivariant indices of Dirac operators was started. The second author wish to thank Michel Duflo for his many comments.

Proof. Let O " spPq. Assume that O is admissible. We may assume that O " Kλ with λ P ρ K `tě 0 regular admissible, and P " Kµ, with µ P t ě0 . Let σ be the stratum of t ě0 containing µ. By Theorem 4.12, λ " µ `ρKσ . We have µ ´ρpµq " λ ´ρK , so µ is admissible. The converse is proved the same way. l Theorem 5.4 Let O Ă k ˚be a regular admissible orbit.

• If P is a phq-ancestor of O, then P is admissible and at distance }ρ H } of O.

• If P is an element in k phq at distance }ρ H } of O, then P is admissible,

Proof. We need only to prove the second point. Assume that the distance between O and P is equal to }ρ H }. We may assume that O " Kλ with λ P ρ K `tě 0 regular admissible. We write P " Kµ, with µ a point in t ˚, such that }λ ´µ} 2 " }ρ Kµ }. This implies that µ belongs to t ě0 , and that

Example 5.5 Let us go back to Example 3.7 for the group K " SU p3q. We see that the orbit of ρ K has one ancestor (itself ) in the regular sheet, two ancestors in the subregular sheet, and one ancestor 0 in the sheet t0u.

In general an orbit O " Kµ has only one ancestor, that is itself. Only the orbits O belonging to the boundary of the shifted Weyl chamber might have lower dimensional ancestors. For example the orbits P σ of the orthogonal projections of ρ K on the 2 r linear spaces Rσ (σ P F k ) are ancestors of opkq. For all these orbits P σ , the representation Q spin K pP σ q is the trivial representation of K. As the number of sheets is usually less than 2 r , some of the ancestors of ρ K lie in the same sheet.

Finally we end this article by an induction formula relating H-admissible coadjoint orbits to K-admissible coadjoint orbits.

Consider the open subset h 0 :" tξ P h ˚| K ξ Ă Hu. Equivalently, the element ξ, identified to an element of h, is such that the transformation adpξq is invertible on q :" k{h, so it determines a complex structure J ξ on q. We see that the complex structure J ξ depends only of the connected component C of h 0 containing ξ. We denote it by J C (remark that J C is