Gestur Ólafsson
email: olafsson@math.lsu.edu

Angela Pasquale
email: pasquale@math.univ-metz.fr

THE Cos λ AND Sin λ TRANSFORMS AS INTERTWINING OPERATORS BETWEEN GENERALIZED PRINCIPAL SERIES REPRESENTATIONS OF SL(n + 1, K)

Keywords: 2000 Mathematics Subject Classification. Primary 22E45, 22E46. Secondary 53C35, 43A80 Intertwining operators, representations of semisimple Lie groups, Cos λ and Sin λ -transform, Grassmanian manifolds

In this article we connect topics from convex and integral geometry with well known topics in representation theory of semisimple Lie groups by showing that the Cos λ and Sin λ -transforms on the Grassmann manifolds Gr p (K) = SU(n + 1, K)/S(U(p, K) × U(n + 1 -p, K)) are standard intertwining operators between certain generalized principal series representations induced from a maximal parabolic subgroup P p of SL(n + 1, K). The index p indicates the dependence of the parabolic on p. The general results of Knapp and Stein and Vogan and Wallach then show that both transforms have meromorphic extension to C and are invertible for generic λ ∈ C. Furthermore, known methods from representation theory combined with a Selberg type integral allow us to determine the K-spectrum of those operators.

Introduction

The Cos λ and Sin λ transforms have appeared in several mathematical contexts and have a long history. In the last few years they have been widely studied because of their connection to convex geometry and to some classical integral transforms, like the Funk and Radon transforms on the sphere and their generalizations to Grassmann manifolds.

For the sphere S n , the Cos λ transform C λ is defined on L 2 (S n ) by

C λ (f )(ω) = S n |(x, ω)| λ-ρ f (x) dσ(x) , ω ∈ S n , (0.1)
where dσ is the normalized rotational invariant measure on S n , (•, •) stands for the usual inner product on R n and ρ = (n + 1)/2. The ρ-shift is chosen so that C λ agrees with a standard intertwining operator between certain principal series representations of SL(n + 1, R). The integral kernel of (0.1) is a power of the cosine of the angle between x and ω. This is at the origin of the name cosine transform, given to (0.1) by E. Lutwak in [START_REF] Lutwak | Centroid bodies and dual mixed volumes[END_REF] for the case λ -ρ = 1.

It is clear that C λ is an integral transform on L 2 (B), where B := Gr 1 (R) is the Grassmann manifold of lines in R n+1 . Denote by Gr n (R) the Grassmann manifold of n-dimensional subspaces of R n+1 . The Sin λ -transform S λ maps functions on B to functions on Gr n (R) and is given by S λ (f )(ω) = C λ (f )(ω ⊥ ) .

Note that Gr n (R) B so that one can also view S λ as an integral transform on L 2 (B). We note that C λ (f ) exists for all f ∈ L 2 (S n ) as long as λ ∈ C ρ := {z ∈ C | Re (z) ≥ ρ} and λ → C λ (f ) is holomorphic on C o ρ , the interior of C ρ . The transform C λ : C ∞ (S n ) → C ∞ (S n ) extends meromorphically to C. One way to determine the singularities of this extension is to determine the SO(n + 1)-spectrum of C λ . The group K = SO(n + 1) acts naturally on L 2 (S n ) and, as a representation of K, the space L 2 (S n ) decomposes into a direct sum

L 2 (S n ) K m∈N 0 Y m
where Y m = {p| S n | p a homogeneous polynomial of degree m and ∆p = 0} .

Here ∆ = ∂ 2 1 + • • • + ∂ 2 n+1
denotes the Laplace operator on R n+1 . Likewise, by considering Cos λ as a transform on L 2 (B), we have the decomposition

L 2 (B) K m∈2N 0 Y m .
See for instance [START_REF] Strichartz | The explicit Fourier decomposition of L 2 (SO(n)/SO(n -m))[END_REF]. As these are multiplicity-one decompositions and C λ commutes with rotations, it follows from Schur's Lemma that for each m ∈ 2N 0 there exists a holomorphic function η m : C ρ → C such that

C λ | Y m = η m (λ)id Y m .
Information about η m gives more detailed information about the meromorphic extension of C λ as well as the image and the kernel of the Cos λ -transform, as clearly both of those spaces are K-invariant.

The functions η m (λ) can be for instance determined using a Funk-Hecke type formula [START_REF] Rubin | Spherical Radon transform and related wavelet transforms[END_REF][START_REF]Inversion of fractional integrals related to the spherical Radon transform[END_REF], and one gets for m ∈ 2N 0 η m (λ) = (-1) m/2 Γ(ρ) Γ(1/2)

Γ 1 2 (λ -ρ + 1) Γ 1 2 (-λ + ρ + m) Γ 1 2 (λ + ρ + m) Γ 1 2 (-λ + ρ) . Note that Γ 1 2 (-λ + ρ + m) Γ 1 2 (-λ + ρ) = 1 if m = 0 2 -m/2 (-λ + ρ)(-λ + ρ + 2) • • • (-λ + ρ + m -2) if m ∈ 2N
is a polynomial of degree m/2.

The first pole of C λ is at λ = (n -1)/2 and the Funk transform

F(f )(ω) = (x,ω)=0 f (x) dm(x)
is then given, up to a constant that can be determined, by F(f )(ω) = c Res λ=(n-1)/2 C λ (f )(ω).

A similar discussion relates the Radon transform with the singularities of the Sin λ -transform. All of these integral transforms have been generalized to the Grassmann manifolds over R, C and H. We refer to [START_REF] Alesker | The α-cosine transform and intertwining integrals on real Grassmannians[END_REF][START_REF] Alesker | Range characterization of the cosine transform on higher Grassmannians[END_REF][START_REF] Ournycheva | The composite cosine transform on the Stiefel manifold and generalized zeta integrals[END_REF][START_REF]Intersection bodies and generalized cosine transforms[END_REF][START_REF]The lower dimensional Busemann-Petty problem for bodies with the generalized axial symmetry[END_REF][START_REF] Funk | Sine Transform on Stiefel and Grassmann Manifolds[END_REF][START_REF] Funk | Sine Transform on Stiefel and Grassmann Manifolds[END_REF][START_REF] Zhang | Radon, cosine and sine transforms on Grassmannian manifolds[END_REF] for more details and discussions.

As mentioned earlier, the importance of the Cos λ and Sin λ transforms comes from the connection to convex geometry and classical integral transforms like the Funk and Radon transforms. The problem is to characterize its rank and derive an inversion formula. The injectivity for λ -ρ = 1 was first proved in [START_REF] Schneider | Über die Bestimmung eines konvexen Körpers durch die Inhalte seiner Projektionen[END_REF]. The connection to stochastic geometry was established by G. Matheron in [START_REF] Matheron | Un théorème d'unicité pour les hyperplans poissoniens[END_REF]. The description of the rank for the same case was first achieved by [START_REF] Alesker | Range characterization of the cosine transform on higher Grassmannians[END_REF]. In the unpublished manuscript [START_REF] Alesker | The α-cosine transform and intertwining integrals on real Grassmannians[END_REF], S. Alesker connected the Cos λ -transform with intertwining operators and showed that the image was an irreducible representation of GL(n + 1, R) for suitable parameters λ. We would also like to point out [START_REF] Goodey | Processes of flats induced by higher-dimensional processes[END_REF] where the other Grassmann manifolds were considered, and the work by B. Rubin, A. Koldobsky and coworkers [START_REF] Koldobsky | The Busemann-Petty problem via spherical harmonics[END_REF][START_REF]Fourier Analysis in Convex Geometry[END_REF][START_REF] Koldobsky | Modified Busemann-Petty problem on sections of convex bodies[END_REF][START_REF] Koldobsky | The complex Busemann-Petty problem on sections of convex bodies[END_REF][START_REF]Intersection bodies and generalized cosine transforms[END_REF][START_REF]The lower dimensional Busemann-Petty problem for bodies with the generalized axial symmetry[END_REF][START_REF] Funk | Sine Transform on Stiefel and Grassmann Manifolds[END_REF][START_REF] Funk | Sine Transform on Stiefel and Grassmann Manifolds[END_REF] where the Cos λ and Sin λ transforms are considered in most generality and the connection to the Busemann-Petty problem plays an important role.

The integral transform in (0.1) appeared independently in representation theory of SL(n + 1, K), where K = R, C or H, around the same time as the Cos λ -transform. The real case was studied in [START_REF] Van Dijk | Tensor products of maximal degenerate series representations of the group SL(n, R)[END_REF], the complex case in [START_REF] Dooley | Generalized principal series representations of SL(n+1, C)[END_REF], and finally the quaternionic case in [START_REF] Pasquale | Maximal degenerate representations of SL(n + 1, H)[END_REF]. In these articles it was shown that the integral (0.1) defines an intertwining operator between generalized principal series representations induced from a maximal parabolic subgroup in SL(n + 1, K). The K-spectrum was determined. Among the applications was the composition series, some embeddings of the complementary series and the study of the so-called canonical representations on rank-one Riemannian symmetric spaces of the noncompact type, [START_REF] Van Dijk | Canonical representations related to hyperbolic spaces[END_REF][START_REF]Maximal degenerate representations, Berezin kernels and canonical representations[END_REF][START_REF] Van Dijk | Canonical representations of Sp(1, n) associated with representations of Sp(1)[END_REF]. However the connections to convex geometry and to the Funk and Radon transforms was neither discussed nor mentioned.

As first shown in [START_REF] Zhang | Radon, cosine and sine transforms on Grassmannian manifolds[END_REF], the Sin λ -transform can be viewed as a Knapp-Stein intertwining operator. This was used to construct complementary series representations for GL(2n, R). This transform is then also naturally linked to reflection positivity, which relates complementary series representations of GL(2n, R) to highest weight representations of SU(n, n), [START_REF] Jorgensen | Unitary representations and Osterwalder-Schrader duality[END_REF][START_REF]Unitary representations of Lie groups with reflection symmetry[END_REF].

Let K = R, C or H and let Gr p (K) = SU(n + 1, K)/S(U(p, K) × U(q, K)), with q = n + 1 -p, be the Grassmann manifold of p-dimensional subspaces of K n+1 . Our main aim in this article is to show that the Cos λ and Sin λ -transforms for Gr p (K) are always canonical intertwining operators between generalized principal series representations induced from maximal parabolic subgroups of SL(n+1, K). We then use the spectrum generating method developed in [START_REF] Branson | Spectrum generating operators and intertwining operators for representations induced from a maximal parabolic subgroup[END_REF] to determine the K-spectrum of the Cos λ and Sin λ -transforms.

The article is organized as follows. In Section 1 we recall needed notations and facts on simple Lie groups and Lie algebras. In Section 2 we introduce the generalized spherical principal series representations. These are representations induced from a character χ λ of a parabolic subgroup P = M AN . The elements of the representation space can be viewed as L 2 sections of the G-homogeneous line bundle over B = G/P corresponding to χ λ . We recall the results by Knapp and Stein and Vogan and Wallach on the meromorphic continuation of standard intertwining operators J(λ) between spherical generalized principal series representations, [START_REF] Knapp | Intertwining operators for semisimple groups[END_REF][START_REF]Intertwining operators for semisimple groups. II[END_REF][START_REF] Vogan | Intertwining operators for real reductive groups[END_REF]. The material of this section is presented in the general setting of semisimple Lie groups rather than for the special case considered later in this article. Indeed, the analytic properties of the intertwining operators which we discuss here possess a rather unified structure for the different semisimple Lie groups. On the other hand, we will associate a geometric meaning to these operators only for a specific class of semisimple Lie groups. It is therefore a natural question whether one can provide a geometric meaning to the standard intertwining operators for other groups or, more generally, to intertwining operators between generalized principal series representations induced from other parabolic subgroups. We expect that the answer will be positive and that higher rank Cos λ -transforms are in fact given by such intertwining operators.

In Section 3 we recall the method from [START_REF] Branson | Spectrum generating operators and intertwining operators for representations induced from a maximal parabolic subgroup[END_REF] on how to find the eigenvalues of the intertwining operators on each of the K-types in case P is a maximal parabolic subgroup and the K-types have multiplicity one. As the situation considered in this article is less general then the one in [START_REF] Branson | Spectrum generating operators and intertwining operators for representations induced from a maximal parabolic subgroup[END_REF], the results and their proofs are simpler. The principal result is Theorem 3.5, which provides us with an inductive way to compute the spectrum.

We specialize to the case of the Grassmann manifolds Gr p (K) in Section 4. We introduce the Cos λ -transform C λ and show that C λ = J(λ). This main result has several applications. Using the well-known fact that the generalized principal series are irreducible for almost all λ ∈ C, we obtain that there exists a meromorphic function c

P (λ) = J(λ)1 = C λ (1) such that C -λ • C λ = C λ • C -λ = c P (-λ)c P (λ)id
for almost all λ. Moreover, together with the method from [START_REF] Branson | Spectrum generating operators and intertwining operators for representations induced from a maximal parabolic subgroup[END_REF] outlined in Section 3, the equality C λ = J(λ) allows us to calculate the K-spectrum of C λ . This computation is done in Section 5. The main result is Theorem 5.11. Finally, a similar discussion about the Sin λ -transform is carried out in Theorem 6.4.

Semisimple Lie Groups and Parabolic Subgroups

In this section we introduce the notation and basic facts on semisimple Lie groups, Lie algebras and parabolic subgroups. The material can be found in any standard textbook on semisimple Lie group, e.g. [START_REF] Knapp | Representation theory of semisimple groups. An overview based on examples[END_REF].

From now on G will stand for a noncompact connected semisimple Lie group with finite center. The Lie algebra of G is denoted by g. In general Lie groups will be denoted by upper case Latin letters and their Lie algebras by the corresponding German letter.

An involution θ :

G → G is a Cartan involution if K = G θ = {a ∈ G | θ(a)
= a} is a maximal compact subgroup of G. In that case K is connected. From now on θ will denote a fixed Cartan involution on G and K the corresponding maximal compact subgroup. Our standard example will be G = SL(n + 1, K), K = R, C, or H, and θ(x) = (x t ) -1 where denotes the canonical conjugation in K which fixes R. In this case

K = {x ∈ G | x t = x}.
The general structure theory discussed in this section is specialized to this case in Section 4.

The derived involution θ : g → g and its complex linear extension to g C = g ⊗ R C will for simplicity also be denoted by θ. Then g = k ⊕ s where k = g θ is the Lie algebra of K and s = {X ∈ g | θ(X) = -X}. We have [s, s] ⊆ k, [k, s] ⊆ s, and Ad(K)s = s. We denote the representation of K on s C by Ad s .

Denote by • , • the Killing form on g, given by X, Y = Tr(ad(X)ad(Y )). Then

(X, Y ) → X, Y θ := X, θ(Y )
is an inner product on g. If X ∈ g then ad(X) * = -ad(θ(X)). In particular, if X ∈ s then ad(X) is a symmetric operator and hence diagonalizable over the reals. Let a ⊂ s be abelian. For α ∈ a * let

g α = {X ∈ g | (∀H ∈ a) [H, X] = α(H)X} and let m = {X ∈ g | (∀H ∈ a) [H, X] = 0 and X ⊥ a} .
Then m ⊕ a = g 0 . We will always assume that the Thus B = G/P and G acts on B. To describe this action, write g = k P (g)m P (g)a P (g)n P (g) with (k P (g), m P (g), a P (g), n P (g)) ∈ K × M × A × N . If the role of P is clear, then we leave out the index P . The map G g → (a(g), n(g)) ∈ A × N is analytic and g → a(g) is right M N -invariant. We can therefore view a( • ) as a map G/M N → A. The elements k(g) and m(g) are not uniquely defined. However, the maps

a = z(m) ∩ s = {X ∈ m ⊕ a | [X, m ⊕ a] = {0}} . Let ∆ = ∆(g, a) := {α ∈ a * \ {0} | g α = {0}}. The elements in ∆ are called roots. We have g = m ⊕ a ⊕ α∈∆ g α . As [g α , g β ] ⊆ g α+β for α, β ∈ ∆ ∪ {0} it follows that [m ⊕ a, g α ] ⊆ g α for all α ∈ ∆. Let a r := {H ∈ a | (∀α ∈ ∆) α(H) = 0}. For a fixed H ∈ a r let ∆ + := {α ∈ ∆ | α(H) > 0}. Then ∆ + is system of positive roots, i.e., ∆ = ∆ + ∪ -∆ + , ∆ + ∩ -∆ + = ∅, and (∆ + + ∆ + ) ∩ ∆ ⊂ ∆ + . In particular, n := α∈∆ + g α is a nilpotent subalgebra of g normalized by the Lie algebra p := m ⊕ a ⊕ n. In fact p = n g (n) = {X ∈ g | [X, n] ⊆ n} . p is a parabolic subalgebra of g. It is maximal, if dim a = 1. Let P = N G (p) = {g ∈ G | Ad(g)p ⊂ p}
g → k(g)L ∈ B and g → Lm(g) ∈ L\M
are well defined and analytic. Let b o = eL. Then we can describe the G action on B by g

• (k • b o ) = k(gk) • b o . If b = k • b o ∈ B and g ∈ G, then we set a(gb) := a(gk). Let N := θ(N ). The Lie algebra of N is n = θ(n) = α∈-∆ + g α . Furthermore, g = n ⊕ m ⊕ a ⊕ n = n ⊕ p, and the map N × M × A × N → G, (n, m, a, n) → nman, is an analytic diffeomorphism onto
an open and dense subset N P of G of full measure.

Let P = M AN = θ(P ). Then as a K-manifold B = G/P but the G-action in general is different. Denote this action for the moment by • .

Lemma 1.1. Let g ∈ G and b ∈ B. Then a P (gb) = a P (θ(g)b) -1 , k P (gb) = k P (θ(g)b) and g• b = θ(g) • b. Proof. Let k ∈ K be such that b = k • b o . Then g• b = k P (gk)L. We have gk = θ(θ(g)k) = θ(k P (θ(g)k)m P (θ(g)k)a P (θ(g)k)n P (θ(g)k)) = k P (θ(g)k)θ(m P (θ(g)k))a P (θ(g)k) -1 θ(n P (θ(g)k)) .
The Lemma now follows as M is θ-stable and θ(n P (θ(g)k)) ∈ N .

We note for later use that the the above argument shows that

a P (gk) = a P (θ(g)k) -1 . (1.2)
We normalize the invariant measure on B and compact groups so that the total measure is one. H) , m ∈ M, H ∈ a , and n ∈ N .

If f ∈ C(B) then B f (b) db = K f (k • b o ) dk . For λ ∈ a * C define a homomorphism χ λ : P → C by χ λ (m exp Hn) = e λ(
We will also use the shorthand notation p λ for χ λ (p). For α ∈ ∆ let m α := dim g α and let ρ P := 1 2 α∈∆ + m α α ∈ a * . As P is fixed most of the time we simply write ρ instead of ρ P . We have

p 2ρ = |detAd(p)| n |.
Normalize the Haar measure dn on N such that N a(n) -2ρ dn = 1 and let dn = θ(dn). We normalize the invariant measure on A respectively a such that the inversion formula for the abelian Fourier transform holds without additional constants, i.e., if

f ∈ C ∞ c (A) and f (λ) = A f (a)a -λ da, then f (a) = a * f (iλ)a iλ dλ. Lemma 1.2.
Let the notation be as above.

(1) We can normalize the Haar measure on G and M such that for all

f ∈ L 1 (G) G f (x) dx = K M A N f (kman)a 2ρ dkdmdadn = N M A N f (nman)a 2ρ dndmdadn . (2) The set N • b o ⊂ B is open, dense and of full measure. If f ∈ L 1 (B) then B f (b) db = N f (n • b o ) a(n) -2ρ dn . (1.3) (3) Let f ∈ L 1 (B) and x ∈ G. Then B f (x • b) a(xb) -2ρ db = B f (b) db . (1.4)
Proof. See [START_REF] Knapp | Representation theory of semisimple groups. An overview based on examples[END_REF], pp. 137-140 and p. 170, or [START_REF] Ólafsson | Fourier and Poisson transformation associated to a semisimple symmetric space[END_REF], §7.

Generalized Principal Series Representations

We now introduce the spherical generalized principal series representations and their intertwining operators. We refer to [START_REF] Knapp | Intertwining operators for semisimple groups[END_REF][START_REF]Intertwining operators for semisimple groups. II[END_REF][START_REF] Vogan | Intertwining operators for real reductive groups[END_REF] for proofs. Later we will show that these intertwining operators coincide with the Cos λ -transform.

For λ ∈ a * C define a continuous representation of G on L 2 (B) by [π λ (x)f ](b) := a(x -1 b) -λ-ρ f (x -1 • b) . (2.1)
Then π λ = π P,λ = Ind G P χ λ . It follows easily by (1.4) that π λ is unitary if and only if λ ∈ ia * . Theorem 2.1 (Vogan-Wallach). There exists a countable collection {p n } n of non-zero holomorphic polynomials on a * C such that if λ ∈ a * C and p n (λ) = 0, then π λ is irreducible. In particular, π λ is irreducible for almost all λ ∈ a * C . Proof. This is Lemma 5.3 in [START_REF] Vogan | Intertwining operators for real reductive groups[END_REF].

Replace P by P , realize B as G/P and then induce the character determined by λ from P to G. We denote the corresponding representation by π P ,λ . Lemma 2.2. Let π θ λ := π P,λ • θ. Then π P ,λ = π θ -λ as a representation of G acting on L 2 (B).

Proof. This follows easily from Lemma 1.1, equation (1.2) and ρ P = -ρ P :

[π P ,λ (x)f ](b) = a P (x -1 b) -λ-ρ P f (x -1 • b) = a P (θ(x) -1 b) λ-ρ P f (θ(x) -1 • b) = [π P,-λ (θ(x))f ](b) . For f ∈ C ∞ (B) and b = k • b o define the standard intertwining operator J(λ) = J P |P (λ) by J(λ)f (b) := N a(n) -λ-ρ f (kn • b o ) dn (2.2)
whenever the integral exists. We note that this is well defined. Indeed, if m ∈ L then a(mnm -1 ) = a(n), and the Haar measure on N is invariant under conjugation by m.

Let b = k • b o = k 1 • b o . Then k = k 1 m for some m ∈ L. Thus N a(n) -λ-ρ f (kn • b o ) dn = N a(n) -λ-ρ f (k 1 mn • b o ) dn = N a(mnm -1 ) -λ-ρ f (k 1 (mnm -1 ) • b o ) dn = N a(n) -λ-ρ f (k 1 n • b o ) dn .
Recall that the space of smooth vectors for π λ is C ∞ (B). In particular, it is independent of λ. If X ∈ g then we define as usually π ∞ λ (X) :

C ∞ (B) → C ∞ (B) by lim t→0 π λ (exp(tX))f -f t = d dt t=0 π λ (exp(tX))f
where the derivative is taken in L 2 (B). Then π ∞ λ is a representation of g in C ∞ (B) and extends to a representation of the universal enveloping algebra U

(g). If u ∈ U (g) then π ∞ λ (u) : C ∞ (B) → C ∞ (B) is a differential operator. If λ ∈ a * C write λ = λ R + iλ I with λ R , λ I ∈ a * .

Theorem 2.3 (Vogan-Wallach). The following holds:

(1) There exists a constant a P such that if

λ ∈ a * C (a P ) := {µ ∈ a * C | (∀α ∈ ∆ + ) µ R , α ≥ a P } then the integral (2.
2) converges absolutely. Furthermore, there exists a constant

C > 0 such that for all λ ∈ a * C (a P ) and f ∈ C ∞ (B) J(λ)f ∞ ≤ C f ∞ . (2.3) (2) If f ∈ C ∞ (B) then a * C (a P ) λ → J(λ)f ∈ C ∞ (B) is continuous and holomorphic on the interior {µ ∈ a * C | (∀α ∈ ∆ + ) µ R , α > a P }.
(3) There exists polynomial functions b :

a * C → C and D : a * C → U (g) K such that if λ ∈ a * C (a P ) and f ∈ C ∞ (B), then b(λ)J(λ)f = J(λ + 4ρ)π λ+4ρ (D(λ))f .
In particular λ → J(λ)f extends to a meromorphic function on a * C . (4) The operator J(λ) intertwines π λ and π θ -λ . Thus, if x ∈ G and f ∈ C ∞ (B), then Our situation is simpler than the general case considered in [START_REF] Vogan | Intertwining operators for real reductive groups[END_REF], therefore-and for completeness-we describe the main idea behind the proof. Let

J(λ)(π λ (x)f ) = π θ -λ (x)J(λ)f . ( 2
a P := max α∈∆ + α, ρ > 0 . (2.5)
According to Lemma A.3.3 in [START_REF] Vogan | Intertwining operators for real reductive groups[END_REF] we have log a(n), α ≥ 0 for all α ∈ ∆ + . Thus for

λ ∈ a * C (a P ) |a(n) -λ-ρ | ≤ a(n) -2ρ
and n → a(n) -2ρ is integrable. Thus, the integral defining J(λ) exists and λ → J(λ)f is holomorphic and also (2.3) clearly follows. Note that (2.3) says in particular that J(λ)f ∈ C ∞ (B). Part (2) follows by the above estimates (2.3) that show that the integral defining J(λ)f converges uniformly on any domain of the form a * C (a) with a > a P . The proof of (3) uses tensoring with finite dimensional representations of G. Another way of showing that λ → J(λ)f extends to a meromorphic function on a * C is to use the last part of (2.2) and ideas from [START_REF] Brylinski | Vecteurs distributions H-invariants pour les séries principales généralisées d'espaces symétriques réductifs et prolongement méromorphe d'intégrales d'Eisenstein[END_REF][START_REF] Ólafsson | Fourier and Poisson transformation associated to a semisimple symmetric space[END_REF][START_REF] Ólafsson | On the meromorphic extension of the spherical functions on concompactly causal symmetric spaces[END_REF] using the Bernstein polynomial for n → a(n) -λ-ρ and equation (2.2).

We will show that J(λ) is an intertwining operator after having proved Lemma 2.5 below.

Lemma 2.4. Let a P be as in (2.5) and let λ ∈ a * C (a P ). Let 1 ≤ p ≤ ∞ and let f ∈ L p (B). Then the integral (2.2) exists and

|J(λ)f (k)| ≤ f p . In particular, J(λ)f ∈ L p (B) and J(λ)f p ≤ f p .
Proof. The case p = ∞ is (2.3). We can therefore assume that p < ∞. Let q be such that

1 p + 1 q = 1. Then 2ρ = 2 p ρ + 2 q ρ. As |a(n) -λ-ρ | ≤ a(n) -2ρ and N a(n) -2ρ dn = 1, it follows that N a(n) -λ-ρ f (kk(n) • b o ) dn ≤ N a(n) -2ρ |f (kk(n) • b o )| dn ≤ N a(n) -2 p ρ f (kk(n) • b o ) p dn 1/p = N |f (kk(n) • b o )| p a(n) -2ρ dn 1/p = B |f (kb)| p db 1/p = f p .
Here we used (1.3) to transfer the integral over N to an integral over B.

For x ∈ N P write x = nP (x)m P (x)α P (x)n P (x)

with nP (x) ∈ N , m P (x) ∈ M , α P (x) ∈ A, and n P (x) ∈ N . Note that x → α P (x) is right M N -invariant. We leave out the subscript P if it is clear which parabolic subgroup we are using.

Lemma 2.5. Assume that λ ∈ a * C (a P ). Let f ∈ C ∞ (B). Then for k ∈ K J(λ)f (k) = K α P (k -1 h) λ-ρ f (h) dh = B α P (k -1 b) λ-ρ f (b) db .
In particular, J(λ) is a convolution operator on L 2 (B) if λ ∈ a * C (a P ).

Proof. Let n ∈ N . Then n = k(n)m(n)a(n)n(n) = n(k(n))m N (k(n))α(k(n))n N (k(n))m(n)a(n)n(n) = n(k(n))m α(k(n))a(n)n for some m ∈ M and n ∈ N . Thus α(k(n)) = a(n) -1 . Assume that λ ∈ a * C (a P ). Using (1.3) we get N a(n) -λ-ρ f (kk(n) • b o ) dn = N α(k(n)) λ-ρ f (kk(n) • b o )a(n) -2ρ dn = B α(b) λ-ρ f (kb) db = B α(k -1 b) λ-ρ f (b) db as the measure on B is K-invariant.
We now show that J(λ)π λ (x) = π -λ (θ(x))J(λ). For this, let x ∈ G and λ ∈ a * C (a P ). Then

J(λ)(π λ (x)f )(k) = K α(k -1 h) λ-ρ a(x -1 h) -λ-ρ f (k(x -1 h)) dh .
(2.6)

We have

α(k -1 h) = α(k -1 xx -1 h) = α(k -1 xk(x -1 h))a(x -1 h) .
Hence we can rewrite (2.6) as

J(λ)(π λ (x)f )(k) = K α(k -1 xk(x -1 h)) λ-ρ f (k(x -1 h))a(x -1 h) -2ρ dh = K α(k -1 xh) λ-ρ f (h) dh = K α((x -1 k) -1 h) λ-ρ f (h) dh .
(2.7)

Now use Lemma 1.1 to write

x -1 k = k P (x -1 k)ma P (x -1 k)n = k P (θ(x -1 )k)ma P (θ(x -1 )k) -1 n
for some m ∈ M and n ∈ N and insert this into (2.7) to get

J(λ)(π λ (x)f )(k) = a(θ(x) -1 k) λ-ρ K α(k(θ(x) -1 k) -1 h) λ-ρ f (h) dh = π -λ (θ(x))J(λ)f (k) .
As J(λ) is K-intertwining and the only K-invariant functions in L 2 (B) are multiple of the constant function 1(x) = 1, it follows that there exists a meromorphic function c P : a * C → C such that J(λ)1 = c ( λ). For λ ∈ a * C (a P ) the function c P (λ) is given by the converging integral

c P (λ) := J(λ)1 = N a(n) -λ-ρ dn = K α(k) λ-ρ dk .
(2.8)

Theorem 2.6. Let the notation be as above. Then, as an identity of meromorphic functions, J(-λ)J(λ) = J(λ)J(-λ) = c P (λ)c P (-λ) id . Furthermore, if λ ∈ ia * , then the normalized operator c P (λ) -1 J(λ) is unitary.

Proof. We have

J(-λ)J(λ)π λ (x) = J(-λ)π -λ (θ(x))J(λ) = π λ (x)J(-λ)J(λ)
as θ 2 = id. As π λ is irreducible for λ in an open dense subset of a * C it follows that the J(-λ)J(λ) is a scalar multiple of the identity on that set. As J(-λ)J(λ)1 = c P (-λ)c P (λ) it follows that J(-λ)J(λ) = c(-λ)c(λ)id as a meromorphic function. The same argument shows that J(λ)J(-λ) = c(-λ)c(λ)id and (1) follows. The unitarity of c P (λ) -1 A(λ) for λ purely imaginary follows from [c P (λ) -1 J(λ)] * = c P (-λ) -1 J(-λ) as in this case λ = -λ.

The Spectrum Generating Operator

We will now specialize the situation to the case where G is simple and P is a maximal parabolic subgroup. Thus dim a = 1. Based on our application later on, we will also assume that ∆ = {α, -α}. Hence n is abelian. Finally, we will assume that K is semisimple and that (K, L) is a symmetric pair corresponding to an involution τ : K → K. This implies that each irreducible representation of K that occurs in L 2 (B) has multiplicity one.

Denote by K the set of equivalence classes of irreducible representations of K. For simplicity we will often write δ ∈ K where δ is an irreducible representation of K, i.e., we identify the representation with its equivalence class. Denote by V δ the finite dimensional Hilbert space on which δ acts and let d(δ) := dim V δ . Let

V L := {u ∈ V δ | (∀k ∈ L) δ(k)u = u} . Then, as K/L is a symmetric space, either dim V L δ = 0 or dim V L δ = 1. We say that (δ, V δ ) is spherical if V L δ = {0}. Let K L denote the subset of K of spherical representations. To describe K L write k = l ⊕ q where q = {X ∈ k | τ (X) = -X}. Let b be a maximal abelian subspace of q. Denote by ∆ k the set of roots of b C in k C . Note that ∆ k ⊂ ib * . Let ∆ + k
be a set of positive roots and let ρ k be the corresponding half-sum of roots, counted with multiplicities. Denote by W k the Weyl group of ∆ k . Finally we let

Λ + := {µ ∈ ib * | (∀α ∈ ∆ + k ) µ, α α, α ∈ N 0 } .
Then the map π → (highest weight of π) defines an injective map of K L into Λ + . This map is bijective if and only if B is simply connected. In general K L is isomorphic to a sublattice Λ + (B) of Λ + . We refer to [START_REF] Helgason | Groups and Geometric Analysis[END_REF], p. 535, and the discussion in [START_REF] Ólafsson | Fourier transforms of spherical distributions on compact symmetric spaces[END_REF], in particular the appendix, for more details. Let µ ∈ Λ + (B) and denote by π µ the corresponding spherical representation. Let V µ be the space on which π µ acts and let (•, •) Vµ the inner product on V µ making π µ unitary. We also fix a unit vector e µ ∈ V L µ . Define T µ :

V µ → L 2 (B) by T µ (v)(x) := d(µ) 1/2 (v, π µ (x)e µ ) Vµ .
Then T µ is an isometric K-intertwining operator between π µ and the left regular representation

L of K on L 2 (B). Let L 2 µ (B) := Im (T µ ). Then L is an irreducible K-representation on L 2 (B) and L 2 (B) K µ∈Λ + (B) L 2 µ (B) . Lemma 3.1. For µ ∈ Λ + (B) let J µ (λ) := J(λ)| L 2 µ (B)
. There exists a meromorphic function

η µ : a * C → C such that (1) J µ (λ) = η µ (λ)id L 2 µ (B) . (2) η 0 (λ) = c P (λ), (3) η µ (λ)η µ (-λ) = c P (λ)c P (-λ),
Proof. J(λ) is a G-intertwining operator between π λ and π θ -λ by Theorem 2.3 (4). As K-representations, π λ and π θ -λ agree with L. Hence J(λ) is a K-intertwining operator of L with itself. As each L 2 µ (B) is irreducible and the multiplicity of π µ in L 2 (B) is one, it follows by Schur's Lemma that we can define η µ (λ) by [START_REF] Alesker | The α-cosine transform and intertwining integrals on real Grassmannians[END_REF]. The claim now follows from our normalization J(λ)1 = c P (λ) and Theorem 2.6.

We call {η

µ (λ) | µ ∈ Λ + (B)} the K-spectrum of J(λ).
Our aim is to determine the functions η µ using the results from [START_REF] Branson | Spectrum generating operators and intertwining operators for representations induced from a maximal parabolic subgroup[END_REF].

Fix H o ∈ a such that α(H o ) = 1. Fix a G-invariant R-bilinear form • , • on g such that H o , H o = 1.
For H ∈ a we have α(H) = H, H o . Denote the complex bilinear extension to g C by the same symbol. Define a map ω :

s C → C ∞ (B) by ω(Y )(k) := Y, Ad(k)H o and note that ω(Ad(h)Y )(k) = Ad(h)Y, Ad(k)H o = Y, Ad(h -1 k)H o = ω(Y )(h -1 k) .
Thus ω is a K-intertwining operator.

Note that • , • is negative definite on k. Let X 1 , . . . , X dim q be an orthonormal basis of q such that X 1 , . . . , X p , p = dim b, is an orthonormal basis of b. Denote by Ω =j X 2 j the corresponding positive definite Laplace operator on B. The action of Ω on functions on B is-up to a sign-the same as that of the Casimir element of k. Thus Ω| L 2 µ (B) = µ + 2ρ k , µ id. For later reference we state this as a Lemma:

Lemma 3.2. For µ ∈ Λ + (B) let ω(µ) = - p j=1 µ(X j )(µ + 2ρ k )(X j ) = - p j=1 µ(X j ) 2 + 2µ(X j )ρ k (X j ) . Then Ω| L 2 µ (B) = ω(µ)id. For f ∈ C ∞ (B) denote by M (f ) : L 2 (B) → L 2 (B) the multiplication operator g → f g. Theorem 3.3. Let Y ∈ s. Then [Ω, M (ω(Y ))] = 2π 0 (Y ).
Proof. This is Theorem 2.3 in [START_REF] Branson | Spectrum generating operators and intertwining operators for representations induced from a maximal parabolic subgroup[END_REF].

For µ ∈ Λ + (B) define Φ µ : L 2 µ (B) ⊗ s C → L 2 (B) by Φ µ (ϕ ⊗ Y ) := M (ω(Y ))ϕ . Observe that for k ∈ K, Y ∈ s C and ϕ ∈ L 2 µ (B) we have L k M (ω(Y )ϕ = L k ω(Y ) (L k ϕ) = M ω(Ad(k)Y ) (L k ϕ) with Ad(k)Y ∈ s C and L k ϕ ∈ L 2 µ (B). Hence Φ µ is K-equivariant and Im Φ µ is K-invariant. Define a finite subset S(µ) ⊂ Λ + (B) by Im Φ µ K σ∈S(µ) L 2 σ (B) .
Denote by pr σ the orthogonal projection

L 2 (B) → L 2 σ (B). Lemma 3.4. Assume that µ ∈ Λ + (B). Let σ ∈ S(µ), Y ∈ s C and rα ∈ a * C . Let ω σµ (Y ) := pr σ • M (ω(Y ))| L 2 µ (B) : L 2 µ (B) → L 2 σ (B) . (3.1)
Then

pr σ • π rα (Y )| L 2 µ (B) = 1 2 (ω(σ) -ω(µ) + 2r)ω σµ (Y ) . (3.2)
Proof. This is Corollary 2.6 in [START_REF] Branson | Spectrum generating operators and intertwining operators for representations induced from a maximal parabolic subgroup[END_REF].

Theorem 3.5. Let µ ∈ Λ + (B), σ ∈ S(µ) and λ = rα ∈ a * C . Then η σ (λ) η µ (λ) = 2r -ω(σ) + ω(µ) 2r + ω(σ) -ω(µ) . (3.
3)

The K-spectrum {η µ (λ)} µ∈Λ + (B) and hence J(λ) is uniquely determined by (3.3) and the normalization η 0 (λ) = c P (λ).

Proof. Applying J(λ) to (3.2) from the left, using that J(λ) commutes with pr σ and that

J(λ) • π λ (Y ) = π θ -λ (Y ) • J(λ) = -π -λ (Y ) • J(λ), we obtain: (ω(σ) -ω(µ) + 2r)η σ (λ)ω σµ (Y ) = -(ω(σ) -ω(µ) -2r)η µ (λ)ω σµ (Y ) .
As ω σδ (Y ) is non-zero for generic λ it can be canceled out. The last statement follows from the fact that π λ is irreducible for generic λ, hence iterated application of (3.1) will in the end reach all K-types starting from the trivial K-type.

The case of the Grassmann Manifolds

From now on K denotes the one of the fields R or C, or the skew field H of quaternions. We let G = SL(n + 1, K). For discussion about the determinant function on the space of quaternionic matrices see [START_REF] Aslaksen | Quaternionic determinants[END_REF][START_REF] Gelfand | Quaternionic quasideterminants and determinants[END_REF]. We will use that if

x ∈ M(n+1, K) then |det R x| = |det K x| d , where d = dim R K is equal to 1, 2, 4 for K = R, C, H, respectively. Denote by G p (K) the Grassmann manifold of k-dimensional subspaces of K n+1 . Note that G p (K) G n+1-p (K).
So we can assume without loss of generality that 2p ≤ n + 1. We set q = n + 1 -p. Hence q ≥ p.

In this section we show that G p (K) = G/P = K/L, where P is a maximal parabolic subgroup of G. Furthermore K/L is a symmetric space. Therefore the results from the previous sections apply to this case. We introduce the Cos λ -transform for K/L and then show that it agrees with the intertwining operator J(λ). In the next section we then apply the spectrum generating operator to determine the K-spectrum.

Define an invariant R-bilinear form on g by

X, Y := n + 1 pq Re (Tr(XY )) .
Denote by z → z the conjugation in K and for x = (x νµ ) ∈ M(n + 1, K) let x := (x νµ ) and

x * := (x µν ) = x t . The homomorphism θ : G → G, x → (x -1 ) * is a Cartan involution on G.
The corresponding Cartan involution on g is θ(X) = -X * . We have

K = SU(n + 1, K) := {x ∈ G | x * = x -1 } k = {X ∈ M(n + 1, K) | X * = -X and Tr(X) = 0 } s = {X ∈ M(n + 1, K) | X * = X and Tr(X) = 0 }
In the notation in [START_REF]Differential Geometry, Lie groups, and Symmetric Spaces[END_REF] we have G = SL(n + 1, R) and K = SO(n + 1) for K = R, G = SL(n+1, C) and K = SU(n+1) for K = C, and G = SU * (2(n+1)) and K = Sp(n+1) for K = H. For k ∈ N denote by I k the k × k identity matrix. We write matrices as blocks

X Y W Z , X ∈ M p×p (K) , Y ∈ M p×q (K) , W ∈ M q×p (K) and Z ∈ M q×q (K) .
Let

H o = q n+1 I p 0 0 -p n+1 I q ∈ s ,
m ⊕ a = M (X, Y ) = X 0 0 Y X ∈ M(p, K) Y ∈ M(q, K) and TrX + TrY = 0 = s(gl(p, K) × gl(q, K)) , n = N (X) := 0 pp X 0 qp 0 qq X ∈ M (p × q, K) M (p × q, K) M A = m(a, b) := a 0 0 b a ∈ GL(p, K) b ∈ GL(q, K) and deta detb = 1 = S(GL(p, K) × GL(q, K)) N = n(X) = I p X 0 qp I q X ∈ M(p × q, K) P = p(a, b; X) := a X 0 b a ∈ GL(p, K) b ∈ GL(q, K) , deta detb = 1, X ∈ M(p × q, K) L = S(U(p, K) × U(q, K)) .
Let e 1 , . . . , e n+1 be the standard basis for K n+1 and let b o = Ke 1 ⊕ . . . ⊕ Ke p ∈ Gr p (K). Then Gr p (K) = K • b o K/L = G/P . Thus we can introducce the notation B = Gr p (K), which agrees with the one from the previous sections. Note that L = S(O(p) × O(q)) for K = R, L = S(U(p) × U(q)) for K = C, and L = Sp(p) × Sp(q) for K = H. In particular B is simply connected for K = C and H.

Applying θ to n, we see that

n = N (Y ) = 0 pp 0 pq Y 0 qq Y ∈ M q×p (K) M(q × p, K)
and

N = n(Y ) = I p 0 pq Y I q Y ∈ M(q × p, K) .
The action of M A on n, respectively n, is given by

m(a, b)N (X)m(a -1 , b -1 ) = N (aXb -1 ) and m(a, b) N (Y )m(a -1 , b -1 ) = N (bY a -1 ) . Lemma 4.1. N P = A B C D ∈ G A ∈ GL(p, K) . Proof. A simple calculation shows that n(Y )m(a, b)n(X) = a aX Y a Y aX + b , (4.1)
proving one of the inclusions. On the other hand, let The definition is independent of E as we will see in a moment. Recall, if

x = A B C D be such that A is invertible. Define a = A, X = a -1 B, Y = Ca -1 ,
x : b → b is linear then Vol R (x(E)) = |det R (x)| = |det(x)| d . (4.3)
As the elements in K acts as orthogonal transformations it follows that |Cos(k where (x, y) denotes the angle between x and y.

• b, c)| = |Cos(b, k -1 c)|. In particular, if b = k • b o and c = h • b o with k, h ∈ K, then |Cos(b, c)| = |Cos(h -1 k • b o , b o )| = Vol R (P bo (h -1 kE)) 1/d . ( 4
For reasons that will become transparent in a moment, we identify a * C with C by

λ → n + 1 pq λ(H o ) with inverse z → z pq n + 1 α . (4.5)
We have dim R n = dpq. Hence

ρ = dpq 2 α ←→ d(n + 1) 2 ∈ R .
Then ρ is an integer if and only if K = C, H or K = R and n is odd. For a complex number λ, we write from now on α P (x) λ instead of α P (x) λ pq n+1 α .

Theorem 4.2. Fix a convex subset E ⊂ b o containing zero such that Vol R (E) = 1. For x ∈ G and λ ∈ C we have

α P (x) λ = |Vol R (P bo (x(E)))| λ/d = |Cos(x • b o , b o )| λ . (4.6) In particular, if b = k • b o and c = h • b o ∈ B, then α(h -1 k) λ = |Cos(b, c)| λ . (4.7)
Proof. For z = (z 1 , . . . , z p ) t ∈ K p we have This shows in particular that the definition of Cos is independent of the convex set E.

| λ = α(h -1 k) λ = α(k -1 h) λ = |Cos(c, b)| λ ,
which proves the Corollary.

The Cos λ -transform C λ is defined for Re λ > ρ by

C λ (f )(c) := B |Cos(b, c)| λ-ρ f (c) dc = K α(k -1 h) λ-ρ f (h • b o ) dk c = k • b o (4.10)
where we have used Theorem 4.2. Thus, C λ is nothing but the standard intertwining operator J(λ). We would like to point out, that the proof of the intertwining property of J(λ) shows that C λ is an intertwining operator without the reference to the general theory. But the general theory described in Section 2 gives us the following theorem for free and allows us to apply the results from [START_REF] Branson | Spectrum generating operators and intertwining operators for representations induced from a maximal parabolic subgroup[END_REF] described in Section 3.

Theorem 4.5. The following holds:

(1) The Cos λ -transform extends to a meromorphic family of intertwining operators C λ :

C ∞ (B) → C ∞ (B). (2) Let c P (λ) := C λ (1). Then c P : C → C is meromorphic and C -λ • C λ = C λ • C -λ = c P (-λ)c P (λ)id .
(3) If λ ∈ iR, then 1 c P (λ) C λ is unitary.

5.

The K-spectrum of the Cos λ -Transform

In this section we determine the K-spectrum of J(λ) for the Grassmann manifolds G p (K) = SU(n+1, K)/S(U(p, K)×U(q, K)) SL(n+1, K)/P . This then determines the K-spectrum of the cos λ transform. As before we always assume that p ≤ q and p + q = n + 1. We keep the notation from the previous sections.

The following is well known.

Lemma 5.1. The space B is a symmetric space and the corresponding involution is

τ (x) = I p 0 0 -I q x I p 0 0 -I q = A -B -C D for x = A B C D and K τ = L.
The derived involution is given by the same formula. Let l denote the Lie algebra of L. Then k = l ⊕ q where q = Q(X) = 0 pp X -X * 0 qq X ∈ M(p × q, K) . These representations occur with multiplicity one.

Proof. A simple matrix calculation shows that the weights of h in s C are ±( i + j ) for i, j = 1, . . . , p. Next we note that µ ± ( i + j ), j = i, is not in Λ + (B) as the i th and j th entries are not even. It follows now from [START_REF] Kumar | Proof of the Parthasarathy-Ranga Rao-Varadarajan conjecture[END_REF] and the fact that each submodule in V µ ⊗ s C that occur in L 2 (B) is spherical that σ ∈ S(µ) if and only if there exists j = 1, . . . , p and w ∈ W k such that w(µ ± 2 j ) is dominant and σ = w(µ ± 2 j ). Assume that µ + 2 j is not dominant. Then m j-1 < m j + 2. Hence m j = m j-1 . Applying the transposition (j -1, j) shows that µ + 2 j is W k -conjugate to µ + 2 j-1 . Repeating as long as necessary shows that there exists a k, 1 ≤ k < j such that µ + 2 k is dominant and W k -conjugate to µ + 2 j . The case µ -2 j not dominant is handled in similar way.

Lemma 5.5. Let µ = (m 1 , . . . , m p ) ∈ Λ + (B) and λ ∈ C a * C . Then η µ+2 j (λ) η µ (λ) = λ -m j -ρ + d(j -1) λ + m j + ρ -d(j -1) . ( 5
Then Γ(p, d; λ) is meromorphic with singularities on the union of the hyperplanes

H(d, j, n) = {λ ∈ C p | λ j = -n + d 2 (j -1)} , j = 1, . . . , p, n ∈ N 0 .
In the following, if λ ∈ C, we will also indicate by λ the vector (λ, . . . , λ) ∈ C p . It will be clear from the context if we are using λ as a complex number or as a vector.

Theorem 5.6. Let λ ∈ C. Then there exists a meromorphic function G(λ) such that for all µ ∈ Λ + (B) we have

η µ (λ) = (-1) |µ|/2 G(λ) Γ p,d 1
2 (-λ + ρ + µ) Γ p,d 1 2 (λ + ρ + µ) , (5.3)
where |µ| = p j=1 m j . where σ p is a well defined probability measure on the set {η | ω ⊂ η}. Then R p ,p is a K-intertwining operator whose K-spectrum is independent of λ. Furthermore, defining C λ by (4.10), one shows that, with the obvious notation,

C λ p ,p = C λ • R p ,p .
The determination of the K-spectrum of the more general Cos λ -transform is then reduced to the determination of the λ-independent K-spectrum of R p ,p and the determination of the λ-dependent K-spectrum of C λ . We note that obviously R p ,p is zero on all K-types µ ∈ Λ + (B p ) \ Λ + (B p ) ∩ Λ + (B p ), or equivalently, have a nonzero entry m j with p < j ≤ p . As our purpose is to relate the Cos λ -transform with the standard intertwining operators, we do not go further into this interesting topic, but refer to [START_REF] Zhang | Radon, cosine and sine transforms on Grassmannian manifolds[END_REF] and the references therein for further discussion.

The Sin λ -transform

In this section we introduce the Sin λ -transform and determine its K-spectrum. We assume that p = q. This in particular implies that n+1 is even and p = n+1 This transform has been studied by B. Rubin for more general p, see [START_REF] Funk | Sine Transform on Stiefel and Grassmann Manifolds[END_REF][START_REF] Funk | Sine Transform on Stiefel and Grassmann Manifolds[END_REF] and references therein, and by G. Zhang [START_REF] Zhang | Radon, cosine and sine transforms on Grassmannian manifolds[END_REF]. In particular G. Zhang shows, using the N realization of the principal series representations, that the Sin λ -transform is a Knapp-Stein intertwining operator and uses this fact to determine the complementary series for π λ .

As p = (n + 1)/2 we have H o = But this follows from w(x 1 , . . . , x p , 0, . . . , 0) t = (0, . . . , 0, x 1 , . . . , x p ) t .

As for the Cos λ -transform, it follows that for each µ ∈ Λ + (B) there exists a meromorphic function ν µ (λ) such that S λ | L 2 µ (B) = ν µ (λ)I L 2 µ (B) . As w 2 ∈ L it follows that R(w) defines a unitary K-intertwining operator of order two on each of the K-types L 2 µ (B). Thus R(w)| L 2 µ (B) = r(µ)id L 2 µ (B) where r(µ) is a constant independent of λ, r(µ) 2 = ±1 and ν µ (λ) = r(µ)η µ (λ). Proof. This follows from Theorem 5.11 and Lemma 6.3.

Acknowledgements:

The authors would like to thank B. Rubin for helpful discussions about the Cos λ -transform, his work and the historical development of the subject. We would like to thank S. Alesker for sending us his manuscript [START_REF] Alesker | The α-cosine transform and intertwining integrals on real Grassmannians[END_REF] and O. Warnaar for pointing out [START_REF] Macdonald | Some conjectures for root systems[END_REF]. The first named author would also like to thank the harmonic analysis students at LSU to give him the opportunity to give an extended series of lectures on the subject in their student seminar. This was very helpful to clarify the ideas and the presentation.

.

  Then P is a closed subgroup of G with Lie algebra p. Let A = exp a respectively N = exp n denote the analytic subgroup of G with Lie algebra a respectively n. Then A and N are closed, A is abelian and N is nilpotent. Let M o denote the analytic subgroup of G with Lie algebra m and M := Z K (A)M o . Then M is a closed subgroup of G with finitely many connected components, Z G (A) = M A and the product map M × A × N → P , (m, a, n) → man, is an analytic diffeomorphism. Let L = K ∩ M and B := K/L. Then G = KP and K ∩ P = L . (1.1)

  and define a := RH o , and m = {X ∈ z g (a) | X, H o = 0}. Then z(m) ∩ s = a. We have ∆ = {α, -α} where α(H o ) = 1. As H o , H o = 1 it follows that • , • agrees with the invariant form from last section.Let ∆ + = {α}. Then

  and b = D -Ca -1 B. Then m(a, b) ∈ M A and n(Y )m(a, b)n(X) = x. Let b ∈ Gr p (K) and view b as a dp-dimensional real vector space. Fix a convex subset E ⊂ b containing the zero vector, such that the volume of E is one. For c ∈ Gr p (K) let P c : K n → c denote the orthogonal projection onto c. Define |Cos(b, c)| := Vol R (P c (E)) 1/d (4.2)

. 4 )

 4 As a motivation for the definition assume that p = 1 andK = R. If b = Rx, c = Ry ∈ Gr 1 (R), then |Cos(b, c)| = |(x, y)| x y = | cos( (x, y))|

=

  |det R A| λ/d . (4.8) Let n(Y )m(a, b)n(X) ∈ N P . Write m(a, b) = m o exp(xH 0 ) with m o ∈ M and x ∈ R. Then |det R a| λ = exp λ xdpq n + 1 = exp(xH o ) λ dpq n+1 α = α P exp(xH 0 ) dλ . Thus α P (n(Y )m(a, b)n(X)) λ = |det R (a)| λ/d = |deta| λ(4.9) and (4.6) follows now from (4.8).

Lemma 4 . 3 .

 43 Let k ∈ K. Then α P (k) = α P (k -1 ). Proof. Write k = nman. Then, as θ(k) = k and θ(a) -1 = a, we have k -1 = θ(k) -1 = θ(n) -1 θ(m) -1 aθ(n) -1and the claim follows.

Corollary 4 . 4 .

 44 Let b, c ∈ B. Then |Cos(b, c)| λ = |Cos(c, b)| λ . Proof. Write b = k • b o and c = h • b o . By (4.7) and Lemma 4.3 |Cos(b, c)

Lemma 5 . 4 .

 54 Let µ = (m 1 , . . . , m p ) ∈ Λ + (B). Then S(µ) = {µ ± 2 j | j = 1, . . . , p} ∩ Λ + (B) .

. 1 )

 1 Proof. By Lemma 5.3 we getω(µ + 2 j ) -ω(µ) = 2pq n + 1 (m j + ρ -d(j -1)) . The claim now follows from (3.3) and our identification C a * C , z → z pq n+1 α. Introduce the Siegel or Gindikin Γ-function Γ p,d : C p → C by Γ p,d (λ) = p j=1 Γ(λ j -d 2 (j -1)) .

Remark 5 . 13 .

 513 One can give a more general definition of the Cos λ -transform. Note first that the definition of |Cos(b, c)| λ in (4.2) is valid for all b ∈ Gr p (K), p ≥ p. For p ≤ p one uses the symmetry relation |Cos(b, c)| λ = |Cos(c, b)| λ . For p ≤ p define the Radon transform R p ,p : C ∞ (Gr p ) → C ∞ (Gr p ) by R p ,p (f )(ω) := ω⊂η f (η) dσ p ,ω (η)

2 .

 2 If ω ∈ Gr p (K) then ω ⊥ denotes the orthogonal complement of ω. Define the Sin λ -transformS λ : C ∞ (B) → C ∞ (B) by S λ (f )(ω) := C λ (f )(ω ⊥ ) .

2 . 1 . 6 . 2 .

 2162 Then wH o w -1 = Ad(w)H 0 = -1. It follows that Ad(w)N = N . Lemma 6.1.For f ∈ C ∞ (B) define A(λ)f (k • b o ) := J(λ)(f )(kw • b o ) = R(w)J(λ)f (k • b o ),where R(w) denotes the right translation by w. Then A(λ) :C ∞ (B) → C ∞ (B). Then λ → A(λ) is a meromorphic family of intertwining operators: A(λ) • π λ (x) = π -λ (x)A(λ).Proof. We have w -1 Lw = L as Ad(w) normalizes a. Thus kmw • b o = kw • b o if m ∈ L and k ∈ K. Thus A(λ) is well defined. The intertwining property follows from Theorem 2.3 and the fact that w -1 N w = N and Ad(w)| a = -Theorem If λ ∈ a * C (a P ), then A(λ) = S λ . Proof. We have by definitionA(λ)f (k • b o ) = K |Cos(h • b o , kw • b o )| λ-ρ f (h) dh . (6.1) The claim follows if we can show that (k • b o ) ⊥ = kw • b o . As SU(n +1, K) acts by orthogonal transformations it is enough to show that b ⊥ o = w • b o .

Lemma 6 . 3 . 2 PΓ p,d λ 2 Γ p,d 1 2 1 2 1 2

 6322111 Let µ ∈ Λ + (B). Then r(µ) = (-1) |µ|/2 . Proof. Let e µ ∈ V L µ be as in Section 2. Then π µ (w)e µ = r(µ)e µ . Let v µ be a highest weight vector such that (v µ , e µ ) Vµ = 1. Thenr(µ) = (v µ , π µ (w)e µ ) = (π µ (w -1 )v µ , e µ ) = e -πi m j (v µ , e µ ) = (-1) |µ|/2 .Theorem 6.4. The K-spectrum of the Sin λ -transform is given byν µ (λ) = Γ p,d (ρ) Γ p,d (ρ/2) (-λ + µ + ρ) Γ p,d (-λ + ρ) Γ p,d (λ + µ + ρ) with ρ = dp.

The research by Ólafsson was supported by NSF grant DMS-0801010 and the University Paul Verlaine of Metz.

Let E (r,s) ν,µ = (δ iν δ jµ ) i,j denote the matrix in M(r × s, K) with all entries equal to 0 but the (ν, µ)-th which is equal to 1. For t = (t 1 , . . . , t p ) t ∈ R p let X(t) = -p j=1 t j E (p,q) q-p+j,j ∈ M(p × q, K) ,

Then b = {Y (t) | t ∈ R p } R p is a maximal abelian subspace of q.

To describe the set K L we note first that if K = R, then B is not simply connected. In all other cases B is simply connected. Define j (Y (t)) := t j . We will identify the element λ = p j=1 λ j j with the corresponding vector λ = (λ 1 , . . . , λ p ). Lemma 5.2. We have

with multiplicities respectively d (and not there in case p = 1), d(q -p) (and not there in case p = q) and d -1 (and absent if d = 1).

Proof. This follows from [START_REF]Differential Geometry, Lie groups, and Symmetric Spaces[END_REF]: the table on page 518, the description of the simple root systems on page 462 ff. and the Satake diagrams on page 532-533. Lemma 5.3. Let the notation be as above.

(1) If K = R and p = q, then

In all other cases,

(

Proof. (1) is well known and (2) follows by a simple calculation using the form of ∆ + k and the multiplicities. For (3) let X 1 , . . . , X p ∈ b be so that j ( X k ) = iδ jk . Then -X k , X k = 2(n + 1)/pq. Thus the X k = pq/[2(n + 1)] X k form an orthonormal basis for b and j (X k ) 2 = pq/[2(n + 1)]δ jk . The claim now follows from Lemma 3.2.

Proof. Denote the right hand side of (5.3) by σ µ (λ). Using Γ(z + 1) = zΓ(z), we get

. Now Lemma 5.5 implies that there exists a meromorphic function G(λ) with the requires property.

The function G(λ) will be explicitly computed in (5.8) below, by comparing the formula for η 0 (λ) = c P (λ). For this, we note the following result which can be found in [START_REF] Andrews | Special functions[END_REF], Theorem 8.1.1.

To complete the calculation we need to evaluate α P (exp(Y (t))). We have exp Y (t) =   diag(cos t 1 , . . . , cos t p ) 0 -diag(sin t 1 , . . . , sin t p ) 0 I q-p 0 diag(sin t 1 , . . . , sin t p ) 0 diag(cos t 1 , . . . , cos t p )   (5.4)

where the matrix I q-p = I n+1-2p does not occur in case p = (n + 1)/2. For the following see also [START_REF] Zhang | Radon, cosine and sine transforms on Grassmannian manifolds[END_REF], Lemma 3.2:

Proof. This follows from (5.4), (4.9) and (4.1).

Theorem 5.9.

(5.5)

Proof. In the following c will denote a positive constant that might change from one line to another. Its value will then be determined at the end. Let

Using that sin(2u) = 2 cos(u) sin(u) and

, we get by the substitution u j = cos 2 (t j ):

.

In the last equation we used that d(q + j) = d(n + 1 -p + j) = 2ρ -dp + dj and that for a complex number z Now evaluate c by using that c P (ρ) = 1 and recalling that ρ = d(n + 1)/2.

Remark 5.10. After circulation of the preprint of the present paper, Ole Warnaar kindly pointed us that the integral in Theorem 5.9 is well known to algebraic combinatorialists and was probably first considered by I. Macdonald in [START_REF] Macdonald | Some conjectures for root systems[END_REF], p. 992.

Theorem 5.11. The K-spectrum the Cos λ -transform is given by:

(5.7)

Proof. By taking µ = 0 in (5.3) and using η 0 = c P , we get

(5.8)

The claim then follows from (5.3) and (5.5).

Remark 5.12. In Section 2 we referred to the result of Vogan and Wallach on the meromorphic continuation of the intertwining operator J(λ). This result is not needed for the computation of η µ (λ). Indeed, it is enough to know that J(λ) is holomorphic on a * C (a P ). Then Theorem 5.11 implies the meromorphic continuation of J(λ) = C λ at least at the level of K-finite functions, cf. [START_REF]Intertwining operators for semisimple groups. II[END_REF].


Vivien Despax
email: vivien.despax@lmpt.univ-tours.fr

Simple random walks on the integers conditioned to stay nonnegative

This note gathers some known results on one-dimensional simple random walks conditioned to stay nonnegative. We especially focus on two different notions of such conditionings in the zero drift case. This is the simplest case of the problem we consider in [D] in a multidimensional setting where we solve it by using tools from representation theory. It is worthy of interest to expose apart this particular one-dimensional case in details since it can be studied by using only elementary arguments.

Introduction

In the positive drift case, where the probability that a given simple random walk on the integers always stays nonnegative is positive, it is quite easy to define a Markov chain that can be naturally regarded as the corresponding random walk conditioned to always stay nonnegative : it only requires basic arguments from probability theory, mainly the strong law of large numbers. It is then not difficult to make explicit its law (see Theorem 1.3). One can then let the drift tends to zero in the transition probabilities obtained and so defines a first notion of conditioning for the centered random walk to always stay nonnegative. Now, if one starts directly with the centered simple random walk, the probability that it always stays nonnegative is zero. Nevertheless, we show that its transition probabilities under the conditioning that it stays nonnegative until an instant n have a limit when we let n tends to infinity. This is done by using a classical argument, namely the reflection principle, to derive an estimate on the tail distribution of the exit time from the nonnegative integers, see Theorem 2.6. This yields a second notion of conditioning for the centered random walk to always stay nonnegative and, at the end, one can observe that it coincides with the first one.

Basics on simple random walks (paragraph 1.1) and a first study of the positive drift case (paragraph 1.2) are included in the present introduction. In Section 2, we introduce the finite-time conditioning which is our main topic. After showing how the reflection principle can be used to recover the first result (paragraph 2.2.1), we use it to prove the convergence of the transition probabilities in the zero drift case (paragraph 2.2.2).

Simple random walks on the integers

Denote by P the lattice Z. In the following, we implicitly endow the denumerable sets with the σ-algebra of all subsets. Let ν 0 be a probability measure on P with full support ν 0 (γ) > 0 γ ∈ P.

Let p -1 , p +1 be two reals in ]0, 1[ such that p -1 + p +1 = 1, set θ = p -1 /p +1 and denote by ν = ν (θ) the probability measure

ν = p -1 δ {-1} + p +1 δ {+1} = θ 1 + θ δ {-1} + 1 1 + θ δ {+1} .
Consider (S (0) , X (1) , X (2) , . . .) a sequence of independent random variables defined on a probability space (Ω, T , P) such that 1. S (0) : Ω → P has law ν 0 . For any γ in P , we write P γ instead of P S(0)=γ or P [ . | S (0) = γ] for the conditional probability defined by

P γ [ . ] = P [ . ∩ S (0) = γ] P [S (0) = γ]
.

2. X = (X (n) : Ω → P ) n≥1 are identically distributed with law ν. We denote by

d = d (θ) = E [X (1)] = p +1 -p -1 = 1 -θ 1 + θ
the common expectation of the random variables X .

We define a sequence of random variables S = (S (n)) n≥1 by setting

S (n) = n i=1 X (i) n ≥ 1.
The sequence of random variables W = (W (n)) n≥0 defined by W (0) = S (0) and

W (n) = S (0) + S (n) n ≥ 1
is then called the simple (or nearest neighbors) random walk on P with initial distribution ν 0 and with law of increments ν. Its drift is d. When it is zero (equivalently : when θ = 1 or when p -1 = p +1 = 1/2), we obtain the centered (or symmetric) simple random walk on P with initial distribution ν 0 . The sequence W is a (time-homogeneous) Markov chain with initial distribution ν 0 , state space P and transition matrix p = p (θ) given by

p (γ, Γ) = P [W (i + 1) = Γ | W (i) = γ] = ν (Γ -γ) =      p +1 = 1 1+θ if Γ -γ = +1 p -1 = θ 1+θ if Γ -γ = -1 0 otherwise γ, Γ ∈ P, i ≥ 0. γ γ + 1 γ -1 p +1 p +1 p -1 p -1 Figure 1: Graph of the Markov chain W
Let P be a denumerable set and p a stochastic matrix on P , that is a map p : P × P → [0, 1] such that y∈P p (x, y) = 1 for any x in P . One can build a measurable space (Ω , T ) such that, for each probability measure ν 0 on (Ω , T ), there exists a probability measure P ν 0 on (Ω , T ) and a sequence of P -valued maps defined on this probability space which is a Markov chain with initial distribution ν 0 , state space P and transition matrix p . For a detailed construction, see [W] for example. We adopt this point of view to define a Markov chain from W that we call "the" random walk conditioned to stay nonnegative.

Conditioning to stay nonnegative : the positive drift case

Denote by C the open interval ]0, +∞[ and by C = [0, +∞[ its closure in R. We also introduce

P + = P ∩ C = Z ≥0 P ++ = P ∩ C = Z >0 .
Let us consider the event

W ∈ C = n≥0 W (n) ∈ C = n≥0 W (n) ∈ P + .
For any λ in P + , we also denote by λ + S ∈ C the event

λ + S ∈ C = n≥1 λ + S (n) ∈ C = n≥1 λ + S (n) ∈ P + .
The set C being stable under addition, notice that the sequence λ + S ∈ C λ∈P + is increasing. Using independence and the previous remark, we have thus

P λ W ∈ C = P λ + S ∈ C ≥ P S ∈ C = P 0 W ∈ C λ ∈ P + (1)
and the total probability formula gives then

P W ∈ C = λ∈P + ν 0 (λ) P λ + S ∈ C . (2)
If the event S ∈ C has a positive probability (as a by-product, Theorem 1.3 will give a necessary and sufficient condition on the drift), so has the event W ∈ C by (1) and (2). Then, from a probabilistic point of view, to take the positive probability event W ∈ C into account leads to introduce the conditional probability P W∈C defined by

P W∈C [ . ] = P . ∩ W ∈ C P W ∈ C .
Let us observe how the transition probability between two states chosen in P + is modified if one replaces P by P W∈C . One can easily derive the following proposition from (5) with Proposition 2.8 -its finite-time equivalent -or prove it directly.

Proposition 1.1 Assume the event S ∈ C has a positive probability. We have

P W∈C [W (i + 1) = Λ | W (i) = λ] = p (λ, Λ) P Λ W ∈ C P λ W ∈ C λ, Λ ∈ P + , i ≥ 0.
This proposition leads us to introduce the function

h : P + -→ R λ -→ P λ W ∈ C = P λ + S ∈ C .
Observe that the transition probabilities of our random walk under P W∈C are obtained by a particular transformation of the restriction of the matrix p to the subset P + . In the general setting, given a pair (P , p ) where P is a denumerable set and p is a substochastic matrix on P , that is a map p : P × P → [0, 1] such that y∈P p (x, y) ≤ 1 for any x in P , and a positive function h : P → R >0 , we can define another matrix p h on P by setting

p h (x, y) = p (x, y) h (y) h (x)
x, y ∈ P .

Then the matrix p h is stochastic on P if and only if h satisfies

h (x) = y∈P p (x, y) h (y) x ∈ P .
In this case, we say that h is a positive harmonic function on P for p . As we will see in the following, in the positive drift case, the function h that we introduced previously is an example of such positive harmonic function on P + for the restriction of the matrix p to P + . Then, Proposition 1.1 shows that the map p + = p + (θ) defined on P + × P + by

p + (λ, Λ) = p (λ, Λ) h (Λ) h (λ) λ, Λ ∈ P +
is a stochastic matrix on P + . Any Markov chain associated to the pair (P + , p + ) is called "the" random walk with law ν conditioned to always stay nonnegative and we will denote it by W + . We prove now two properties of the function h that will be sufficient to explicit it.

Proposition 1.2 1. For any drift d, the function h is harmonic on the set P + for the restriction of the matrix p to P + , that is we have

h (λ) = p +1 h (1) if λ = 0 p +1 h (λ + 1) + p -1 h (λ -1) if λ ∈ P ++ . 2. Assume d is positive (equivalently : the parameter θ is in ]0, 1[ or p +1 > p -1 ). We have h (λ) ---→ λ→∞ 1. (3)
Proof.

1. We write

h (λ) = P λ + S ∈ C = P [X (1) = +1] P X (1)=+1 λ + S ∈ C + P [X (1) = -1] P X (1)=-1 λ + S ∈ C = p +1 P X (1)=+1   n≥2 λ + 1 + n i=2 X (i) ∈ C   + p -1 P X (1)=-1   n≥2 λ -1 + n i=2 X (i) ∈ C   .
The random variables X being independent, we have

P X (1)=+1   n≥2 λ + 1 + n i=2 X (i) ∈ C   = P   n≥2 λ + 1 + n i=2 X (i) ∈ C   P X (1)=-1   n≥2 λ -1 + n i=2 X (i) ∈ C   = P   n≥2 λ -1 + n i=2 X (i) ∈ C  
and, these variables being identically distributed, we have

P   n≥2 λ + 1 + n i=2 X (i) ∈ C   = P   n≥2 λ + 1 + S (n -1) ∈ C   = P λ + 1 + S ∈ C P   n≥2 λ -1 + n i=2 X (i) ∈ C   = P   n≥2 λ -1 + S (n -1) ∈ C   = P λ -1 + S ∈ C .
The result follows.

2. According to the strong law of large numbers, we have

S (n) n ---→ n→∞ d P -a.s.
Since d is assumed to be positive, when the event

S (n)/n ---→ n→∞ d is fulfilled, there must exist a positive integer N such that S (n) is in C for any positive integer n with n ≥ N . So S (n) n ---→ n→∞ d ⊂ N ≥1 n≥N S (n) ∈ C
from what we deduce

P   N ≥1 n≥N S (n) ∈ C   = 1. The sequence of events n≥N S (n) ∈ C N ≥1
being increasing, this shows

P   n≥N S (n) ∈ C   ----→ N →∞ 1. ( 4
)
If N is a positive integer and if the event n≥N S (n) ∈ C is fulfilled there must exist λ in

P + such that λ + S (n) is in C for any positive integer n. So n≥N S (n) ∈ C ⊂ λ∈P + n≥1 λ + S (n) ∈ C = λ∈P + λ + S ∈ C N ≥ 1
and one has then

P   n≥N S (n) ∈ C   ≤ P   λ∈P + λ + S ∈ C   N ≥ 1.
Reminding that the sequence of events λ + S ∈ C λ∈P + is increasing, we deduce

P   n≥N S (n) ∈ C   ≤ lim λ→∞ P λ + S ∈ C N ≥ 1.
Then, the result follows from (4).

As said before, once these properties are established, the function h and, in the positive drift case, the matrix p + can be easily computed explicitly.

Theorem 1.3 1. Assume the drift d is positive (equivalently : the parameter θ is in ]0, 1[ or p +1 > p -1 ). We have h (λ) = 1 -θ λ+1 λ ∈ P + .
In particular, we have P 0 W ∈ C = P S ∈ C = 1 -θ > 0 and the transition matrix p + of W + is given by

p + (λ, Λ) = p (λ, Λ) 1 -θ Λ+1 1 -θ λ+1 =      1 1+θ 1-θ λ+2 1-θ λ+1 if Λ -λ = +1 θ 1+θ 1-θ λ 1-θ λ+1 if Λ -λ = -1 0 otherwise λ, Λ ∈ P + .
2. Assume the drift is zero (equivalently : θ = 1). Then h = 0. Thus we have P λ W ∈ C = 0 for any λ in P + and so P W ∈ C = 0.

Proof. Of course, the first part of Proposition 1.2 tells us that the fonction h is a solution of a linear recurrence of order 2 with constant coefficients and we can solve it with standard methods. As we are dealing with probabilities, one can also prefer to use this proposition in order to write

h (λ) = 1 1 + θ h (λ + 1) + θ 1 + θ h (λ -1) λ ∈ P ++ , that is 1 1 + θ h (λ) + θ 1 + θ h (λ) = 1 1 + θ h (λ + 1) + θ 1 + θ h (λ -1) λ ∈ P ++ , or h (λ + 1) -h (λ) = θ (h (λ) -h (λ -1)) λ ∈ P ++ .
We get then

h (λ + 1) -h (λ) = θ λ (h (1) -h (0)) = θ λ+1 h (0) λ ∈ P ++ .
Notice that the previous relation still holds when λ is 0.

1. If the parameter θ is in ]0, 1[, we deduce h (λ) -h (0) = λ-1 Λ=0 θ Λ+1 h (0) = θ 1 -θ λ 1 -θ h (0) λ ∈ P ++ .
Once again, this relation still holds when λ is 0, so we have

h (λ) = h (0) 1 -θ λ+1 1 -θ λ ∈ P + .
the second part of Proposition 1.2 gives the expected formula for h. Using then Proposition 1.1, we obtain p + .

2. If θ = 1, we must have h (0) = 0 since h is bounded on P + .

Conclusions and a question

When the drift is positive, observe that we have

   p + (λ, λ + 1) ---→ λ→∞ p (λ, λ + 1) p + (λ, λ -1) ---→ λ→∞ p (λ, λ -1) .
This corresponds to the intuitive fact that, in this case, the influence of the conditioning to always stay nonnegative must disappear when one computes the transition probabilities between two states which are far away from the origin. The previous construction of W + is known as the Doob's h-transform approach. Remark that in the positive drift case, the proof of Theorem 1.3 shows that the set of positive harmonic functions on P + for the restriction of p + to P + is reduced to the ch where c is a positive real. This may a posteriori justify our interest on the conditioning for the random walk to always stay nonnegative : when the drift is positive, this is the only one to consider in Doob's sense.

In the nonpositive drift case, Theorem 1.3 tells us that Doob's approach does not work anymore -or at least not so naively -since h = 0. Nevertheless, we observe that the map p + pointwise converges on P + × P + as the parameter θ tends to 1 from the left, that is when the drift d tends to zero from the right : the limit is given by

p + (θ) (λ, Λ) ----→ θ→1 θ∈]0,1[ p (1) (λ, Λ) Λ + 1 λ + 1 =      1 2 λ+2 λ+1 if Λ = λ + 1 1 2 λ λ+1 if Λ = λ -1 0 otherwise λ, Λ ∈ P + .
(Once again, the influence of the conditioning disappears asymptotically.) Can we modify this natural first approach in order to build a definition of a random walk conditioned to always stay nonnegative when d ≥ 0 that will both coincides with the previous one when d > 0 and with its limit when d = 0 ?

A finite-time approach

When the drift is zero, we know that the event W ∈ C has probability zero. We introduce (W (≤ n)) n≥0 the increasing sequence of positive probability events defined by

W (≤ n) ∈ C = n i=0 W (i) ∈ C = n i=1 W (i) ∈ P + n ≥ 0 which limit is W ∈ C.
We also introduce for each λ in P + the increasing sequence

(λ + S (≤ n)) n≥1 defined by λ + S (≤ n) ∈ C = n i=1 λ + S (i) ∈ C = n i=1
λ + S (i) ∈ P + n ≥ 1 which limit is λ + S ∈ C. Observe that we have

P [W (≤ n)] = λ∈P + ν 0 (λ) P λ [W (≤ n)] n ≥ 0, P λ [W (≤ n)] = P [λ + S (≤ n)] λ ∈ P + , n ≥ 1, P λ + S (≤ n) ∈ C ≥ P S (≤ n) ∈ C ≥ P n i=1 X (i) = +1 = p n +1 > 0 λ ∈ P + , n ≥ 1.
Then we can recover the result of Theorem 1.3 by writing

P λ W (≤ n) ∈ C ---→ n→∞ P λ W ∈ C = h (λ) λ ∈ P + (5)
and by exploiting the reflection principle described above to rewrite the probabilities

P λ W (≤ n) ∈ C λ ∈ P + , n ≥ 1
in an advantageous manner. Moreover, this finite-time approach can also be exploited in the zero drift case and allows us to give a positive answer to the question formulated in paragraph 1.3. Let us introduce the following sequence (h n ) n≥1

h n : P + -→ R λ -→ P λ W (≤ n) ∈ C = P λ + S (≤ n) ∈ C n ≥ 1.

The reflection principle

Given γ, Γ in P and a positive integer n, we denote by π n (γ, Γ) the set

(π 0 , π 1 , . . . , π n ) ∈ P n+1 : (π 0 , π n ) = (γ, Γ) , π i+1 -π i ∈ {-1, +1} 0 ≤ i ≤ n -1 ,
that is the set of paths in P of length n with steps in {-1, +1} starting at γ and ending at Γ. We will also intoduce π 0 n (γ, Γ), the subset of paths hitting 0 at some instant, and π + n (λ, Λ), the subset of paths always staying in P + . Let γ, Γ be in P and consider a positive integer n. We make some simple observations about the set π n (γ, Γ). Actually, the second and the third facts are the main ingredients of the reflection principle. The reader is referred to [GZ] for a presentation of this argument in its original setting.

1. It is non-empty if and only if the three integers γ, Γ, n are such that

γ -n ≤ Γ ≤ γ + n γ + n = Γ mod 2 .
2. If a path visits both sets P ++ and -P + = Z ≤0 , then it must visit 0. Thus, when γ is in P ++ , the set π n (γ, Γ) \ π 0 n (γ, Γ) is the subset of paths remaining in P ++ .

3. Suppose γ, Γ are in P ++ . If π 0 n (γ, Γ) is nonempty then we must have n ≥ 2. To each π = (π 0 , π 1 , π 2 , . . . , π n ) in π 0 n (γ, Γ) we can associate i 0 , the greatest integer i in {0, . . . , n} such that π i = 0. Actually, the integer i 0 is in {1, . . . , n -1}. Define then another path π = (π 0 , π 1 , π 2 , . . . , π n ) by setting

π i = -π i if 0 ≤ i ≤ i 0 -1 π i if i 0 ≤ i ≤ n .
The path π is clearly in π 0 n (-γ, Γ) and i 0 is the greatest integer i in {1, . . . , n -1} such that π i = 0. We have

(-π 0 , . . . , -π i 0 -1 , π i 0 , π i 0 +1 , . . . , π n ) = π.
Thus, there exists a bijection between the two finite sets π 0 n (γ, Γ) and π 0 n (-γ, Γ). But -γ being in -P ++ = Z <0 and Γ being in P ++ , we have

π 0 n (-γ, Γ) = π n (-γ, Γ) .
In particular, one has

Card π 0 n (γ, Γ) = Card π n (-γ, Γ) . ( 6
)
Remark that this equality still holds when π 0 n (γ, Γ) is empty since π n (-γ, Γ) is also empty in this case.

Theorem 2.1 (André -Gessel, Zeilberger) We have

π + n (λ, Λ) = π n (λ + 1, Λ + 1) -π n (-λ -1, Λ + 1) λ, Λ ∈ P + , n ≥ 1.
Remark 2.2 As an application of this principle, one may deduce the formula

π + n (λ, Λ) = n n+Λ-λ 2 - n n+Λ+λ+2 2 λ, Λ ∈ P + , n ≥ 1, λ -n ≤ Λ ≤ λ + n λ + n ≡ Λ mod 2 .
We will not need it in the following.

Application to the simple random walks

Corollary 2.3 We have

h n (λ) = P λ+1 [W (n) ∈ C] -θ λ+1 P -λ-1 [W (n) ∈ C] λ ∈ P + , n ≥ 1.
Proof. One has

P λ + S (≤ n) ∈ C = Λ∈P + P λ + S (≤ n) ∈ C ∩ S (n) = Λ = Λ∈P +      π∈π + n (λ,Λ) π=(π 0 ,π 1 ,...,πn) P n i=1 λ + S (i) = π i      . Given Λ in P + and π = (π 0 , π 1 , . . . , π n ) in π + n (λ, Λ), one writes P n i=1 λ + S (i) = π i = p n+(Λ-λ) 2 +1 p n-(Λ-λ) 2 -1
.

As this probability does not depend on the choice of π in π + n (λ, Λ), we have then

π∈π + n (λ,Λ) π=(π 0 ,π 1 ,...,πn) P n i=1 λ + S (i) = π i = Card π + n (λ, Λ) p n+(Λ-λ) 2 +1 p n-(Λ-λ) 2 -1
.

But λ and Λ being in P + , we can use Theorem 2.1 to write

π∈π + n (λ,Λ) π=(π 0 ,π 1 ,...,πn) P n i=1 λ + S (i) = π i = Card π n (λ + 1, Λ + 1) p n+(Λ-λ) 2 +1 p n-(Λ-λ) 2 -1 -Card π n (-λ -1, Λ + 1) p n+(Λ-λ) 2 +1 p n-(Λ-λ) 2 -1 = Card π n (λ + 1, Λ + 1) p n+(Λ+1-(λ+1)) 2 +1 p n-(Λ+1-(λ+1)) 2 -1 -θ λ+1 Card π n (-λ -1, Λ + 1) p n+(Λ+1-(-λ-1)) 2 +1 p n-(Λ+1-(-λ-1)) 2 -1 = P [λ + 1 + S (n) = Λ + 1] -θ λ+1 P [-λ -1 + S (n) = Λ + 1] .
Summing over Λ in P + , we finally obtain the expected equality.

The positive drift case

Lemma 2.4 Assume the drift d is in C (equivalently : the parameter θ is in ]0, 1[ or p +1 > p -1 ). We have

P γ [W (n) ∈ C] ---→ n→∞ 1 γ ∈ P.
Proof. According to the strong law of large numbers, once again, we have

P S (n) n ---→ n→∞ d = 1.
When the event S(n) n ---→ n→∞ d is fulfilled, the drift d being positive, there exists a positive integer N such that γ + S (n) is in C for any positive integer n with n ≥ N . Thus we have the inclusion

S (n) n ---→ n→∞ d ⊂ N ≥1 n≥N γ + S (n) ∈ C. (7)
The sequence of events

n≥N γ + S (n) ∈ C N ≥1
being increasing, one has

P   n≥N γ + S (n) ∈ C   ----→ N →∞ P   N ≥1 n≥N γ + S (n) ∈ C   .
Now inclusions (7) and

n≥N γ + S (n) ∈ C ⊂ γ + S (N ) ∈ C N ≥ 1 finally give P [γ + S (N ) ∈ C] ----→ N →∞ 1.
Corollary 2.5 Assume the drift d is in C (equivalently : the parameter θ is in ]0, 1[ or p +1 > p -1 ). The sequence (h n ) n≥1 pointwise converges on P + and its limit is given by

h n (λ) ---→ n→∞ 1 -θ λ+1 λ ∈ P + .
We recover in particular the the first conclusion of Theorem 1.3.

Proof. It is a straightforward consequence of (5), Corollary 2.3 and Lemma 2.4.

The zero drift case

Theorem 2.6 Assume the drift is zero (equivalently : θ = 1 or p +1 = p -1 = 1/2). We have

h n (λ) ∼ n→∞ 2 π (λ + 1) 1 √ n λ ∈ P + .
Proof. As θ = 1, Corollary 2.3 and Lemma 2.4 show that (h n ) n≥1 pointwise converges to zero on P + . We recover in particular the the second conclusion of Theorem 1.3 in the zero drift case. Nevertheless, one can write

h n (λ) = P [λ + 1 + S (n) ∈ C] -P [-λ -1 + S (n) ∈ C] = P [S (n) ≥ -λ] -P [S (n) ≥ λ + 2] = λ+1 i=-λ P [S (n) = i] λ ∈ P + , n ≥ 1.
Let us fix a positive integer n and λ in P + . Remind that the law of the random variable

S(n)+n 2
is the binomial ditribution with parameters n and 1/2. Then one has

P [S (n) = i] = 1 2 n n n+i 2 if -n ≤ i ≤ n and i = n mod 2 0 otherwise -λ ≤ i ≤ λ + 1.
Let us assume that n ≥ λ + 2. We have

h n (λ) = -λ≤i≤λ+1 -n≤i≤n i=n mod 2 1 2 n n n+i 2 = -λ≤i≤λ+1 i=n mod 2 1 2 n n n+i 2
.

Let us fix an integer i such that -λ ≤ i ≤ λ + 1 and i = n mod 2. By using Stirling's approximation of factorials, we know that there exists a sequence of reals

( m ) m≥1 such that    m! = √ 2π √ m m e m (1 + m ) m ≥ 1 m ----→ m→∞ 0 13
Thus, one can write

1 2 n n n+i 2 = 2 π 1 √ n 1 -i n 1 + i n i exp -n 2 ln 1 -i n 2 1 -i n 2 1 + n 1 + n+i 2 1 + n-i 2 .
Hence, for any positive integer n large enough, we have

h 2n (λ) 2 π 1 √ 2n =    -λ≤i≤λ+1 i even 1 + i 2n -i-1 2 1 -i 2n i-1 2 exp -n ln 1 -i 2 4n 2 1 + 2n+i 2 1 + 2n-i 2    (1 + 2n ) (8)
and

h 2n-1 (λ) 2 π 1 √ 2n-1 =     -λ≤i≤λ+1 i odd 1 + i 2n-1 -i-1 2 1 -i 2n-1 i-1 2 exp 1 2 ln 1 -i 2 (2n-1) 2 exp -n ln 1 -i 2 (2n-1) 2 1 + 2n-1+i 2 1 + 2n-1-i 2     × (1 + 2n-1 ) . (9)
In both cases, there are exactly λ+1 terms in the sums (8) ( 9) and each of them converges to 1 when n goes to infinity. Thus, both subsequences

  h 2n (λ) 2 π 1 √ 2n   n≥1   h 2n-1 (λ) 2 π 1 √ 2n-1   n≥1 converge to λ + 1.
Remark 2.7 As we have

h n (λ) = P [τ λ > n] λ ∈ P + , n ≥ 1 where τ λ = inf n ≥ 1 : λ + S (n) / ∈ C λ ∈ P + ,
this result can be regarded as an estimate of the tail distribution of the exit time from the cone C for the centered simple random walk. Recently, Denisov and Wachtel gave a multidimensional version for more general centered random walks and more general cones. Their approach is completely different from ours, see [DW]. Actually, we use their result in our article [D].

When the drift is zero, we already know by ( 5) and Theorem 1.3 that each sequence P λ W (n) ∈ C n≥0 must converge to zero, but now we have an idea of the speed of this convergence. We exploit it in the following.

As said at the begining of Section 2, we can use positive probability approximations of the event W ∈ C in order to build a notion of conditioning of staying nonnegative in the zero drift case. Indeed, we can introduce the sequence of conditional probabilities P W(≤n)∈C n≥0 defined by

P W(≤n)∈C [ . ] = P . ∩ W (≤ n) ∈ C P W (≤ n) ∈ C n ≥ 0.
Since the sequence P W (≤ n) ∈ C n≥1 converges to P W ∈ C , one has, under the assumption that P W ∈ C is positive (that is when the drift is positive or when θ is in ]0, 1[),

P W(≤n)∈C [E] ---→ n→∞ P W∈C [E] E ∈ T . If we set p + n (λ, Λ, i) = P W(≤n)∈C [W (i + 1) = Λ | W (i) = λ] λ, Λ ∈ P + , n ≥ 0, i ≥ 0,
the previous remark gives in particular

p + n (λ, Λ, i) ---→ n→∞ p + (λ, Λ) λ, Λ ∈ P + , i ≥ 0
whenever the parameter θ is in ]0, 1[, that is in the positive drift case. When the drift is zero, that is when the simple random walk considered is the centered one, it is no longer obvious whether the sequences P W(≤n)∈C [W (i + 1) = Λ | W (i) = λ] n≥0 λ, Λ ∈ P + , i ≥ 0 are convergent, since the denominator P W (≤ n) ∈ C tends to zero when n tends to infinity. As done before in the positive drift case, we start by observing how the transition probability between two states in P + is modified when we replace P by one of the P W(≤n)∈C .

Proposition 2.8 Let λ, Λ be in P + . One has

p + n (λ, Λ, i) = p (λ, Λ) h n-i-1 (Λ) h n-i (λ) if i ≤ n -2 p (λ, Λ) if i > n -2 n ≥ 2, i ≥ 0.
In particular, one has for any nonnegative integer i p + n (λ, Λ, i) = p + n-i (λ, Λ, 0) n ≥ i + 2.

Proof. One has

P

  W(≤n)∈C [W (i + 1) = Λ | W (i) = λ] = P W (i + 1) = Λ ∩ W (i) = λ ∩ W (≤ n) ∈ C P W (i) = λ ∩ W (≤ n) ∈ C .Assume first that i > n -2. The Markov property givesP W (i + 1) = Λ ∩ W (i) = λ ∩ W (≤ n) ∈ C P W (i) = λ ∩ W (≤ n) ∈ C = p (λ, Λ) .Assume now that i ≤ n -2. One writesP W(≤n)∈C [W (i + 1) = Λ | W (i) = λ] = P W (≤ i -1) ∈ C ∩ W (i + 1) = Λ ∩ W (i) = λ ∩ n j=i+2 W (j) ∈ C P W (≤ i -1) ∈ C ∩ W (i) = λ ∩ n j=i+1 W (j) ∈ C = P W (i + 1) = Λ ∩ n j=i+2 W (j) ∈ C | W (≤ i -1) ∈ C ∩ W (i) = λ P n j=i+1 W (j) ∈ C | W (≤ i -1) ∈ C ∩ W (i) = λ .Using Markov property, one getsP W(≤n)∈C [W (i + 1) = Λ | W (i) = λ] = P W (i + 1) = Λ ∩ n j=i+2 W (j) ∈ C | W (i) = λ P n j=i+1 W (j) ∈ C| W (i) = λ = P W (i) = λ ∩ W (i + 1) = Λ ∩ n j=i+2 W (j) ∈ C P W (i) = λ ∩ n j=i+1 W (j) ∈ C = P W (i) = λ ∩ W (i + 1) = Λ ∩ n j=i+2 Λ + j k=i+2 X (k) ∈ C P W (i) = λ ∩ n j=i+1 λ + j k=i+1 X (k) ∈ C .Using independance of the random variables X , one has thenP W(≤n)∈C [W (i + 1) = Λ | W (i) = λ]fact that they are identically distributed, one finally gets P W(≤n)∈C [W (i + 1) = Λ | W (i) = λ] = p (λ, Λ) P Λ + S (≤ n -i -1) ∈ C P λ + S (≤ n -i) ∈ C .

As desired, we finally obtain Theorem 2.9 If the drift is zero (equivalently : if θ = 1 or p +1 = p -1 = 1/2), then

In other words, the sequence (p + n ( . , . , 0)) n≥0 pointwise converges on P + × P + and its limit p + (1) is given by

Thus, we have the pointwise convergence on

Proof. Use Proposition 2.8 and Theorem 2.6.