On some extension of Paley Wiener theorem

Abstract Paley Wiener theorem characterizes the class of functions which are Fourier transforms of ℂ∞ functions of compact support on ℝn by relating decay properties of those functions or distributions at infinity with analyticity of their Fourier transform. The theorem is already proved in classical case : the real case with holomorphic Fourier transform on L2(ℝ), the case of functions with compact support on ℝn from Hörmander and the spherical transform on semi simple Lie groups with Gangolli theorem. Let G be a locally compact unimodular group, K a compact subgroup of G, and δ an element of unitary dual ̑K of K. In this work, we’ll give an extension of Paley-Wiener theorem with respect to δ, a class of unitary irreducible representation of K, where G is either a semi-simple Lie group or a reductive Lie group with nonempty discrete series after introducing a notion of δ-orbital integral. If δ is trivial and one dimensional, we obtain the classical Paley-Wiener theorem.


Introduction
In this section, we shall study Paley-Wiener theorem in classical case. Let f ∈ C + , then the following conditions are equivalent: The classical Paley-Wiener theorem uses the holomorphic Fourier transform on classes of square-integrable functions supported on L (R). Schwartz's Paley-Wiener theorem asserts that the Fourier transform of a distribution of compact support on R n is an entire function on C n and gives estimates on its growth at in nity. This theorem says that : An entire function F is the Fourier-Laplace transform of a distribution of compact support if and only if ∀z ∈ C n , there exists constants α and β such that Let G be a connected semi-simple Lie group with nite center, G = KAN an Iwasawa decomposition for G, where A is an abelian subgroup of G, N a nilpotent subgroup of G, K a compact subgroup of G, and K (G), the convolution algebra of continuous complex functions with compact support which are bi-invariant under K.
Paley-Wiener Theorem exists for spherical functions, and is called Gangolli Theorem. this theorem says that for all f ∈ K (G) which vanishes outside the ball of radius R, the spherical Fourier-Laplace transformf of f de ned by the rulê f (ν) = G f (x)ϕν(x)d G (x), (ϕν ∈ S(G \ K), the set of spherical functions on G with respect to K), exists for all ν = ξ + iη ∈ Fc and is a W-invariant entire holomorphic function of ν (ν ∈ Fc), where W is the Weyl group, Fc the complexi cation of F, and F the real dual of A. A is the lie algebra of A in the Iwasawa decomposition. Moreover, given any integer β ≥ , there exists a constant C β > such that Conversely, if F is a W-invariant entire holomorphic function on Fc with the property that there exists an R > such that for any integer β > , there exists a constant C β > for which then there exists a unique function f ∈ K(G), the algebra of continuous functions with compact support, such thatf = F; f vanishes outside the ball of radius R in A and is given by the formula where c(ν) is the Harish-Chandra's constant. Let δ be a class of unitary irreducible representations of a compact subgroup K of G. Our goal is to establish and to prove the Paley-Wiener theorem with respect to the unitary dual of a compact group, in other words, to characterize functions which are Fourier transforms of type δ of some class of functions. As seen above, this result exists where δ is trivial and one dimensional. We'll start by de ning generalized Abel according to δ.

Preliminaries
Let G be a locally compact unimodular group, K a compact subgroup of G,K the set of all equivalence classes of irreducible unitary representations of K. For all class δ ofK, we denote by ξ δ the character of δ, d(δ) the degree of δ and Ifδ is the class of contragredient representations of δ inK, we have χ δ = χδ and thanks to the Schur Orthogonality relations, we can check that χδ * χδ = χδ. For all function f ∈ K(G), the algebra of continuous functions with compact support, we set (where dk is the normalized Haar measure on K), and K δ (G) is a subalgebra of K(G) and the mapχδ * f *χδ is a projection of K(G) onto K δ (G).
Consider a Banach representation U of G on a Banach space E. Set P(δ) = U(χ δ ) et E(δ) = P(δ)E, E(δ) the closed subspace of E consisting of those vectors in E which transform under K according to δ. If g =χδ * f *χδ, . For all f , g ∈ K(G), we have the following properties We set If K is a compact subgroup of G and U a topological completely irreducible Banach representation of G on E, then the set of all operators If we decompose the representation k −→ U δ (k) of K onto E(δ) to obtain m irreducible equivalent representation, then the centralizer is isomorphic to the algebra Mm(C) of square matrices of order m. Thus, there The algebra K δ (G) is isomorphic to algebra U c,δ (G) of continuous functions ψ with compact support of G onto Fδ = Hom C (Eδ , Eδ) and which verify the relation A seminorm γ on G is a positive lower semicontinuous and bounded function on every compact of G such that If ϕ is a quasi-bounded continuous function on G with values in End C (E) such that Then the function ϕ is spherical function of type δ i As well, if U is an irreducible Banach representation of G over a space E such that δ occurs m times in the restriction of U to K, then there exists a function ϕ U δ de ned on G which is spherical of type δ. The function ϕ U δ is said to be associated to the representation U. .
is called the generalized Gelfand transformation associated to B. If the algebra B is commutative, then irreducible unitary representations of B are one dimensional and thus we identify them with characters of B. We get the usual de nition of Gelfand transformation. Let where ρ is one-half the sum of the positive roots associated to Lie algebra of G.
Let δ ∈K and µ δ ∈ δ, an unitary irreducible representation of K onto the hilbert space E δ . For every f dans K δ (G), let's consider the integral de ned by We shall call the map f → F δ f the Abel transformation on G of type δ. We know that The Abel transformation is linear and one-to-one mapping of the algebra K δ (G) onto K δ (A). Let N K (A) and Z K (A), be respectively normalizer and centralizer of A in K. The Weyl group of the pair (G, A) can be identi ed with quotient

Theorem 3.1. Let G be a connected semi-simple Lie group with nite center, and KAN its Iwasawa decomposition. For all f ∈ K δ (G, End C (E)) which vanishes outside the ball of radius R in A, the spherical Fourier-Laplace transform of type δ,f δ of f de ned by the rulê
exists for all ν = ξ + iη ∈ Fc and is a W-invariant entire holomorphic function of ν (ν ∈ Fc), where W is the Weyl group, Fc the complexi cation of F, and F the real dual of A. Moreover, given any integer β ≥ , there exists a constant C β,δ > such that

Conversely, if F is a W-invariant entire holomorphic function on
Fc with the property that there exists an R > such that for any integer β > , there exists a constant C β,δ > for which || F(ξ + iη) ||≤ C β,δ ( + ||ξ + iη||) −β e R||η|| ξ , η ∈ F then there exists a unique function f ∈ K δ (G, End C (E)) such thatf δ = F; f vanishes outside the ball of radius R in A and is given by the formula Proof. (1) Suppose that f is an element of K δ (G, End C (E)) which vanishes outside the ball of radius R in A. The spherical Fourier-Laplace transform of type δ,f δ of f de ned by the rulê exists for all ν = ξ + iη ∈ Fc and is a W-invariant entire holomorphic function of ν (ν ∈ Fc). Also, we see that the Abel transform F δ f of f has support in the ball of radius R in A; Then, classical Paley-Wiener Theorem tells us that the Euclidean Fourier transformF δ f of F δ f is holomorphic on Fc and has the property that for any given integer β ≥ , there exists a constant C β,δ > such that The result follows from the fact thatF δ f =f δ , indeed We have the following relation (1) The algebra K δ (G, End C (E)) is isomorphic to algebra U c,δ (G) of continuous functions with compact support of G in F δ and which are u δ −sphérical on G. The isomorphism is de ned by Moreover, if δ is trivial and one dimensional, we have ϕ(x) = K e (iν−ρ)H(xk) dk, and the relation (1) Thus, with identi cation, ϕ ν,δ (x) = K uδ(k − )e (iν−ρ)H(xk) dk so : We have the result.
(2) Suppose now that F is a W-invariant entire holomorphic function on Fc with the property that there exists an R > , such that for any integer β > there exists a constant C β,δ > > 0 for which : De ne a function f ∈ K δ (G, End C (E)) by End C (E)) and f vanishes outside the ball of radius R in A. Indeed : because ϕ ν,δ ∈ Jc(G). Moreover, K is unimodular, we also havē And, by analytic continuationf = F on Fc. It remains only to show that f is unique; Thus, suppose that g a function which verify the asserted properties above, thenf Because the is one-to-one of K δ (G) onto K δ (A).

. A Paley-Wiener theorem on reductive Lie groups
Let G be a reductive Lie group with non-empty discrete series, G the Lie algebra of G; J a Cartan compact subgroup of G; U an open subset of G which is completely invariant, γ a regular element of U (γ ∈ Ureg) . One denotes by J the identity component of J and K = γJ K is Cartan compact subgroup of G, K the Lie algebra of K, and the Weyl group W(G; K) acts on G/K × Kreg, dg a measure of G which is invariant on G/K.
As seen previously, the algebra K δ (U) is isomorphic to the algebra U c,δ (U) of continuous functions with compact support ψ of U onto Fδ = Hom C (Eδ , Eδ) for any double Banach representation (uδ , uδ) of K.
If δ is trivial and one dimensional , we obtain the classical orbital integral. We denote by G p,δ the set of spherical functions of type δ and height p. Let H be the factor of Lagrange decomposition of G which contains K, (G = K.H) and H the Lie algebra of H. For all ϕ ∈ G p,δ , ∃ uδ ∈K and ν ∈ H * C such that Thus, we can identify G p,δ withK × H * C . For any ϕ ∈ G p,δ (G), ∃(u δ , ν) ∈K × H * C and we set : ||(u δ , ν)|| = ||u δ || + ||ν||.
We denote by ρ ∆ one-half the sum of the positive roots associated to Lie algebra of G, κ the signature of the Weyl group W of (G, K), and ∆ a positive root system.
If m is a linear combination of roots of K, we denote by ξµ the corresponding character of K. For all r > , we denote by PW δ (K)r the set of all functions F: G p,δ −→ End(E) such that Let put PW δ (K) = r> PW δ (K)r. We equip PW δ (K)r with the topology de ned by norms We denote by I ∞ c,δ (G), the subspace of K δ (G) of in nitely di erentiable functions.
where the map f −→ Ff is the spherical Fourier transform of type δ.
Let x some conjugacy classes of Cartan subgroup of G, K , K , ..., K k ∈ Car(G), and ∆ , ∆ , ..., ∆ k the corresponding positive root system. Then ∀f ∈ I ∞ c,δ (G), we have That prove that F is surjective.