Normalizations of Eisenstein integrals for reductive symmetric spaces

We construct minimal Eisenstein integrals for a reductive symmetric space G/H as matrix coefficients of the minimal principal series of G. The Eisenstein integrals thus obtained include those from the \sigma-minimal principal series. In addition, we obtain related Eisenstein integrals, but with different normalizations. Specialized to the case of the group, this wider class includes Harish-Chandra's minimal Eisenstein integrals.Comment: 66 pages. Minor revisions. To be published in Journal of Functional Analysi

KK-invariant cusp forms for reductive symmetric spaces of split rank one

Let $G/H$ be a reductive symmetric space of split rank $1$ and let $K$ be a maximal compact subgroup of $G$. In a previous article the first two authors introduced a notion of cusp forms for $G/H$. We show that the space of cusp forms coincides with the closure of the $K$-finite generalized matrix coefficients of discrete series representations if and only if there exist no $K$-spherical discrete series representations. Moreover, we prove that every $K$-spherical discrete series representation occurs with multiplicity $1$ in the Plancherel decomposition of $G/H$.Comment: 12 page

The infinitesimal characters of discrete series for real spherical spaces

Let $Z=G/H$ be the homogeneous space of a real reductive group and a unimodular real spherical subgroup, and consider the regular representation of $G$ on $L^2(Z)$. It is shown that all representations of the discrete series, that is, the irreducible subrepresentations of $L^2(Z)$, have infinitesimal characters which are real and belong to a lattice. Moreover, let $K$ be a maximal compact subgroup of $G$. Then each irreducible representation of $K$ occurs in a finite set of such discrete series representations only. Similar results are obtained for the twisted discrete series, that is, the discrete components of the space of square integrable sections of a line bundle, given by a unitary character on an abelian extension of $H$.Comment: To appear in GAF

On the little Weyl group of a real spherical space

In the present paper we further the study of the compression cone of a real spherical homogeneous space $Z=G/H$. In particular we provide a geometric construction of the little Weyl group of $Z$ introduced recently by Knop and Kr\"otz. Our technique is based on a fine analysis of limits of conjugates of the subalgebra $\mathrm{Lie}(H)$ along one-parameter subgroups in the Grassmannian of subspaces of $\mathrm{Lie}(G)$. The little Weyl group is obtained as a finite reflection group generated by the reflections in the walls of the compression cone

A note on LpL^p-factorizations of representations

In this paper we give an overview on $L^p$-factorizations of Lie group representations and introduce the notion of smooth $L^p$-factorization.Comment: This article is dedicated to the fond memories of Gerrit van Dij

SQUID developments for the gravitational wave antenna MiniGRAIL

We designed two different sensor SQUIDs for the readout of the resonant mass gravitational wave detector MiniGRAIL. Both designs have integrated input inductors in the order of 1.5 muH and are planned for operation in the mK temperature range. Cooling fins were added to the shunt resistors. The fabricated SQUIDs show a behavior that differs from standard DC-SQUIDs. We were able to operate a design with a parallel configuration of washers at reasonable sensitivities. The flux noise saturated to a value of 0.84 muPhi0/radicHz below a temperature of 200 mK. The equivalent noise referred to the current through the input coil is 155 fA/radicHz and the energy resolution yields 62 h

A Paley-Wiener theorem for Harish-Chandra modules

We formulate and prove a Paley-Wiener theorem for Harish-Chandra modules for a real reductive group. As a corollary we obtain a new and elementary proof of the Helgason conjecture.Comment: Submitted version; with two appendices on the Helgason conjecture and an applicatio