The Bessel-Plancherel theorem and applications

/Let G be a simple Lie Group with finite center, and let K \subset G be a maximal compact subgroup. We say that G is a Lie group of tube type if G/K is a hermitian symmetric space of tube type. For such a Lie group G, we can find a parabolic subgroup P=MAN, with given Langlands decomposition, such that N is abelian, and N admits a generic character with compact stabilizer. We will call any parabolic subgroup P satisfying this properties a Siegel parabolic. Let ([pi], V) be an admissible, smooth, Fr\'echet representation of a Lie group of tube type G and let P \subset G be a Siegel parabolic subgroup. If [chi] is a generic character of N, let Wh_[chi](V)=[lambda] : V [longrightarrow] mathbb{C} / [lambda]([pi](n)v)=[chi](n)v} be the space of Bessel models of V. After describing the classification of all the simple Lie groups of tube type, we will give a characterization of the space of Bessel models of an induced representation. As a corollary of this characterization we obtain a multiplicity one theorem for the space of Bessel models of an irreducible representation of G. As an application of this results we calculate the Bessel-Plancherel measure of a Lie group of tube type, L²(N\G;[chi]), where [chi] is a generic character of N. Then we use Howe's theory of dual pairs to show that the Plancherel measure of the space L²(O(p-r,q- s)\O(p,q)) is the pullback, under the [Theta] lift, of the Bessel-Plancherel measure L²(N\Sp(m,\mathbb{R});[chi]), where m=r+s and [chi] is a generic character that depends on r and