A Paley-Wiener theorem for reductive symmetric spaces

Let X = G/H be a reductive symmetric space and K a maximal compact subgroup of G. The image under the Fourier transform of the space of K-finite compactly supported smooth functions on X is characterized.Comment: 31 pages, published versio

Paley-Wiener spaces for real reductive Lie groups

We show that Arthur's Paley-Wiener theorem for K-finite compactly supported smooth functions on a real reductive Lie group G of the Harish-Chandra class can be deduced from the Paley-Wiener theorem we established in the more general setting of a reductive symmetric space. In addition, we formulate an extension of Arthur's theorem to K-finite compactly supported generalized functions (distributions) on G and show that this result follows from the analogous result for reductive symmetric spaces as well.Comment: Latex2e, 28 pages, change of definition of space P^* on p. 17 + minor correction

Multiplicities in the Plancherel decomposition of a semisimple symmetric space

Let G=H be a semisimple symmetric space, where G is a connected semisimple Lie group provided with an involution ?; and H = G ? is the subgroup of xed points for ?: Assume moreover that G is linear (for the purpose of the introduction, the assumptions on G and H are stronger than necessary). Then G has a ?-stable maximal compact subgroup K; the associated Cartan involution commutes with ?: Let g = h + q and g = k +p be the decompositions of the Lie algebra g induced by ? and , then h is the Lie algebra of H and k is the Lie algebra of K

Convexity for invariant differential operators on semisimple symmetric spaces

Let X = G=H be a homogeneous space of a Lie group G, and let D : C 1 (X) ! C 1 (X) be a non-trivial G-invariant dierential operator. One of the natural questions one can ask for the operator D is whether it is solvable, in the sense that DC 1 (X) =C 1 (X). If G is the group of translations of X = R n and H is trivial, then D has constant coecients, and it is a well known result of Ehrenpreis and Malgrange that hence D is solvable

Strong wavefront lemma and counting lattice points in sectors

We compute the asymptotics of the number of integral quadratic forms with prescribed orthogonal decompositions and, more generally, the asymptotics of the number of lattice points lying in sectors of affine symmetric spaces. A new key ingredient in this article is the strong wavefront lemma, which shows that the generalized Cartan decomposition associated to a symmetric space is uniformly Lipschitz

The notion of cusp forms for a class of reductive symmetric spaces of split rank one

We study a notion of cusp forms for the symmetric spaces G/H with G = SL(n,R) and H = S(GL(n-1,R) x GL(1,R)). We classify all minimal parabolic subgroups of G for which the associated cuspidal integrals are convergent and discuss the possible definitions of cusp forms. Finally, we show that the closure of the direct sum of the discrete series of representations of G/H coincides with the space of cusp forms

The Asymptotic distribution of circles in the orbits of Kleinian groups

Let P be a locally finite circle packing in the plane invariant under a non-elementary Kleinian group Gamma and with finitely many Gamma-orbits. When Gamma is geometrically finite, we construct an explicit Borel measure on the plane which describes the asymptotic distribution of small circles in P, assuming that either the critical exponent of Gamma is strictly bigger than 1 or P does not contain an infinite bouquet of tangent circles glued at a parabolic fixed point of Gamma. Our construction also works for P invariant under a geometrically infinite group Gamma, provided Gamma admits a finite Bowen-Margulis-Sullivan measure and the Gamma-skinning size of P is finite. Some concrete circle packings to which our result applies include Apollonian circle packings, Sierpinski curves, Schottky dances, etc.Comment: 31 pages, 8 figures. Final version. To appear in Inventiones Mat