### A characterization of bi-invariant Schwartz space multipliers on nilpotent Lie groups

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It is an interesting open problem to establish Paley-Wiener theorems for general nilpotent Lie groups. The aim of this paper is to prove one such theorem for step two nilpotent Lie groups which is analogous to the Paley-Wiener theorem for the Heisenberg group proved in [4].

We prove a restriction theorem for the class-1 representations of the Heisenberg motion group. This is done using an improvement of the restriction theorem for the special Hermite projection operators proved in [13]. We also prove a restriction theorem for the Heisenberg group.

Let N be a simply connected, connected non-commutative nilpotent Lie group with Lie algebra of dimension n. Let H be a subgroup of the automorphism group of N. Assume that H is a commutative, simply connected, connected Lie group with Lie algebra . Furthermore, assume that the linear adjoint action of on is diagonalizable with non-purely imaginary eigenvalues. Let $\tau =In{d}_{H}^{N\u22caH}1$. We obtain an explicit direct integral decomposition for τ, including a description of the spectrum as a submanifold of (+)*, and a...

The asymptotics of spherical functions for large dimensions are related to spherical functions for Olshanski spherical pairs. In this paper we consider inductive limits of Gelfand pairs associated to the Heisenberg group. The group K = U(n) × U(p) acts multiplicity free on 𝓟(V), the space of polynomials on V = M(n,p;ℂ), the space of n × p complex matrices. The group K acts also on the Heisenberg group H = V × ℝ. By a result of Carcano, the pair (G,K) with G = K ⋉ H is a Gelfand pair. The main results...

We introduce a new class of connected solvable Lie groups called $H$-group. Namely a $H$-group is a unimodular connected solvable Lie group with center $Z$ such that for some $\ell $ in the Lie algebra $h$ of $H$, the symplectic for ${B}_{\ell}$ on $h/z$ given by $\ell \left(\right[x,y\left]\right)$ is nondegenerate. Moreover, apart form some technical requirements, it will be proved that a connected unimodular Lie group $G$ with center $Z$, such that the center of $G/\mathrm{Rad}\phantom{\rule{0.166667em}{0ex}}G$ is finite, has discrete series if and only if $G$ may be written as $G=H{S}^{\text{'}}$, $H\cap S={Z}^{0}$, where $H$ is a $H$-group with...