Let K be a closed Lie subgroup of the unitary group U(n) acting by automorphisms on the (2n+1)-dimensional Heisenberg group $H_n$. We say that $(K,H_n)$ is a Gelfand pair when the set $L_K^1\left(H_n\right)$ of integrable K-invariant functions on $H_n$ is an abelian convolution algebra. In this case, the Gelfand space (or spectrum) for $L_K^1\left(H_n\right)$ can be identified with the set $\Delta (K,H_n)$ of bounded K-spherical functions on $H_n$. In this paper, we study the natural topology on $\Delta (K,H_n)$ given by uniform convergence on compact subsets in $H_n$. We show that $\Delta (K,H_n)$ is a complete...

We present a new method for establishing the ‘‘gap” property for finitely generated subgroups of $SU\left(2\right)$, providing an elementary solution of Ruziewicz problem on $S^2$ as well as giving many new examples of finitely generated subgroups of $SU\left(2\right)$ with an explicit gap. The distribution of the eigenvalues of the elements of the group ring $𝐑[SU(2\left)\right]$ in the $N$-th irreducible representation of $SU\left(2\right)$ is also studied. Numerical experiments indicate that for a generic (in measure) element of $𝐑[SU(2\left)\right]$, the “unfolded” consecutive spacings...

We study spectral multipliers for a distinguished Laplacian on certain groups of exponential growth. We obtain a stronger optimal version of the results proved in [CGHM] and [A].

Let G be a Lie group, Xj right invariant vector fields on G, which generate (as a Lie algebra) the Lie algebra of G,L = -Σ Xj2.(...) In this paper we consider L1(G) boundedness of F(L) for (some) metabelian G and a distinguished L on G. Of the main interest is that the group is of exponential growth, and possibly higher rank. Previously positive results about higher rank groups were only about Iwasawa type groups. Also, our groups may be unimodular, so it is the second positive result (after [13])...

For locally compact, second countable, type I groups G, we characterize all closed (two-sided) translation invariant subspaces of L²(G). We establish a similar result for K-biinvariant L²-functions (K a fixed maximal compact subgroup) in the context of semisimple Lie groups.

Spectrum generating technique introduced by Ólafsson, Ørsted, and one of the authors in the paper (Branson, T., Ólafsson, G. and Ørsted, B., Spectrum generating operators, and intertwining operators for representations induced from a maximal parabolic subgroups, J. Funct. Anal. 135 (1996), 163–205.) provides an efficient way to construct certain intertwinors when $K$-types are of multiplicity at most one. Intertwinors on the twistor bundle over $S^1\times S^{n-1}$ have some $K$-types of multiplicity 2. With some additional...

We study the action of a real-reductive group $G=Kexp\left(𝔭\right)$ on a real-analytic submanifold $X$ of a Kähler manifold. We suppose that the action of $G$ extends holomorphically to an action of the complexified group $G^{ℂ}$ on this Kähler manifold such that the action of a maximal compact subgroup is Hamiltonian. The moment map induces a gradient map ${\mu }_{𝔭}:X\to 𝔭$. We show that ${\mu }_{𝔭}$ almost separates the $K$–orbits if and only if a minimal parabolic subgroup of $G$ has an open orbit. This generalizes Brion’s characterization of spherical...

We propose a generalization of the theory of spherical harmonics to the context of symmetric subgroups of reductive groups acting on flag manifolds. We give some sample results for the case of the orthogonal group acting on Grassmann manifolds, especially the case of 2-planes.

The aim of this paper is to study mean value operators on the reduced Heisenberg group Hn/Γ, where Hn is the Heisenberg group and Γ is the subgroup {(0,2πk): k ∈ Z} of Hn.

Let $F$ be a local non-archimedean field. The set of all equivalence classes of irreducible spherical representations of $GL(n,F)$ is described in the first part of the paper. In particular, it is shown that each irreducible spherical representation is parabolically induced by an unramified character. Bernstein’s result on the irreducibility of the parabolically induced representations of $GL(n,F)$ by irreducible unitary ones, and Ol’shanskij’s construction of complementary series give directly a description of all...