Related papers: The Plancherel decomposition for a reductive symme…
We introduce a generalization of the Stirling numbers via symmetric functions involving two weight functions. The resulting extension unifies previously known Stirling-type sequences with known symmetric function forms, as well as other…
In respect of b-linear functional, Riesz representation theorem in n-Hilbert space have been proved. We define b-sesquilinear functional in n-Hilbert space and establish the polarization identities. A generalized form of the Schwarz…
We give a Hahn-Jordan decomposition in Riesz spaces which generalizes that of [{{\sc B. A. Watson}, {An And\^o-Douglas type theorem in Riesz spaces with a conditional expectation,} {\em Positivity,} {\bf 13} (2009), 543 - 558}] and a…
This paper focuses on a wide class of Collatz-type arithmetic dynamics, and presents a systematic derivation of recursive formulas and functional equations satisfied by the associated generating functions. The main tools belong to complex…
We discuss various forms of the Plancherel Formula and the Plancherel Theorem on reductive groups over local fields.
The popular cubic smoothing spline estimate of a regression function arises as the minimizer of the penalized sum of squares $\sum_j(Y_j - {\mu}(t_j))^2 + {\lambda}\int_a^b [{\mu}"(t)]^2 dt$, where the data are $t_j,Y_j$, $j=1,..., n$. The…
We provide a simple analytic formula in terms of elementary functions for the Laplace transform j_{l}(p) of the spherical Bessel function than that appearing in the literature, and we show that any such integral transform is a polynomial of…
Littlewood-Richardson rule gives the decomposition formula for the multiplication of two Schur functions, while the decomposition formula for the multiplication of two Hall-Littlewood functions or two universal characters is also given by…
It is shown that a covariant derivative on any d-dimensional manifold M can be mapped to a set of d operators acting on the space of functions on the principal Spin(d)-bundle over M. In other words, any d-dimensional manifold can be…
We introduce the notions of symmetric and symmetrizable representations of $\text{SL}_2(\mathbb{Z})$. The linear representations of $\text{SL}_2(\mathbb{Z})$ arising from modular tensor categories are symmetric and have congruence kernel.…
We consider the (projective) representations of the group of holomorphic automorphisms of a symmetric tube domain $V\oplus i\Omega$ that are obtained by analytic continuation of the holomorphic discrete series. For a representation…
We consider three uniqueness theorems: one from the theory of meromorphic functions, another one from asymptotic combinatorics, and the third one about representations of the infinite symmetric group. The first theorem establishes the…
The Springer resolution of the nilpotent cone is used to give a geometric construction of the irreducible representations of Weyl groups. Borho and MacPherson obtain the Springer correspondence by applying the decomposition theorem to the…
We introduce the space $X$ of quaternion hermitian forms of size $n$ on a ${\mathfrak p}$-adic field with odd residual characteristic, and define typical spherical functions $\omega(x;s)$ on $X$ and give their induction formula on sizes by…
We study Gelfand pairs for locally compact quantum groups. We give an operator algebraic interpretation and show that the quantum Plancherel transformation restricts to a spherical Plancherel transformation. As an example, we turn the…
We prove the Plancherel formula for Whittaker functions on a reductive p-adic group. This a sequel to our work on Paley-Wiener theorem. Our proof is close to the proof written by Waldspurger of the Harish-Chandra Plancherel formula for…
In this paper, we explore the spectral representation in Klein space, which is the split $(2,2)$ signature flat spacetime. The Klein space can be foliated into Lorentzian $\mathrm{AdS}_3 /\mathbb{Z}$ slices, and its identity resolution has…
Consider an unitary highest weight representation of a group U(p,q) in holomorphic functions on the symmetric space U(p,q)/U(p)\times U(q). Consider its restriction \rho to the subgroup O(p,q). This restriction has a complicated spectrum…
Integral representation for the eigenfunctions of quantum periodic Toda chain is constructed for N-particle case. The multiple integral is calculated using the Cauchy residue formula. This gives the representation which reproduces the…
In this paper, we present an approach to the fractional Dunkl Laplacian in a framework emerging from certain reflection symmetries in Euclidean spaces. Our main result is pointwise formulas, Bochner subordination, and an extension problem…