We recall the notion of cuspform for Γ = SL2(Z) acting on the upper half-plane, and prove that the orthogonal complement in L(Γ\H) to cuspforms is spanned by pseudo-Eisenstein series, which in turn are expressible as wave packets of Eisenstein series Es. Further we have a Plancherel theorem for this space. The non-trivial harmonic analysis resides in the application of the Fourier transform on ...

This is the last in a series of papers in which we generalized the Tannaka-Krein duality to compact groupoids. In [A1] we studied the representation theory of compact groupoids. In particular, we showed that irreducible representations have finite dimensional fibres. We also proved the Schur’s lemma, Gelfand-Raikov theorem and Peter-Weyl theorem for compact groupoids. In [A2] we studied the Fou...

We study the eigenvalues of a Laplace–Beltrami operator defined on set symmetric polynomials, where are expressed in terms partitions integers. To behaviors these eigenvalues, we assign with restricted uniform measure, Jack or Plancherel measure. first obtain new limit theorem Then, by using it together known results other three measures, prove that global distribution is asymptotically $$\mu $...

It is proved in [BOO], [J2] and [Ok1] that the joint distribution of suitably scaled rows of a partition with respect to the Plancherel measure of the symmetric group converges to the corresponding distribution of eigenvalues of a Hermitian matrix from the Gaussian Unitary Ensemble. We introduce a new measure on strict partitions, which is analogous to the Plancherel measure, and prove that the...

In this paper we discuss one dimensional scattering and inverse scattering for the Helmholtz equation on the half line from the point of view of the layer stripping. By full or nonlinear scattering, we mean the transformation between the index of refraction (actually half of its logarithmic derivative) and the reflection coefficient. We refer to this mapping as nonlinear scattering, because the...

We obtain the Plancherel decomposition for a reductive symmetric space in the sense of representation theory. Our starting point is the Plancherel formula for spherical Schwartz functions, obtained in part I. The formula for Schwartz functions involves Eisenstein integrals obtained by a residual calculus. In the present paper we identify these integrals as matrix coefficients of the generalized...

In HC], Harish-Chandra derived the Plancherel formula on p-adic groups. However, to have an explicit formula, one will have to compute the measures appearing in the formula. Here, we compute Plancherel measures on Sp 4 over p-adic elds explicitly.

We prove the Plancherel formula for spherical Schwartz functions on a reductive symmetric space. Our starting point is an inversion formula for spherical smooth compactly supported functions. The latter formula was earlier obtained from the most continuous part of the Plancherel formula by means of a residue calculus. In the course of the present paper we also obtain new proofs of the uniform t...

So there 5 partitions for 4. Assign each partition with a probability, we then get a probability measure on them. Different ways of assigning probabilities: Uniform measure Plancherel measure Jack measure Restricted uniform measure Restricted Jack measure. We will study the properties of r.v.’s (k1, k2, . . .), a random partition of n. Example 2. As n→∞, under the Plancherel measure, k1 − 2 √ n...

Growth of Young diagrams, equipped with Plancherel measure, follows the automodel equation Kerov. Using technology unitary matrix model we show that such growth process is exactly same as gap-less phase in Gross-Witten and Wadia (GWW) model. The limit shape asymptotic diagrams corresponds to GWW transition point. Our analysis also offers an alternate proof theorem Vershik-Kerov Logan-Shepp. con...