28 research outputs found
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
-invariant cusp forms for reductive symmetric spaces of split rank one
Let be a reductive symmetric space of split rank and let be a
maximal compact subgroup of . In a previous article the first two authors
introduced a notion of cusp forms for . We show that the space of cusp
forms coincides with the closure of the -finite generalized matrix
coefficients of discrete series representations if and only if there exist no
-spherical discrete series representations. Moreover, we prove that every
-spherical discrete series representation occurs with multiplicity in
the Plancherel decomposition of .Comment: 12 page
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 . In particular we provide a geometric
construction of the little Weyl group of introduced recently by Knop and
Kr\"otz. Our technique is based on a fine analysis of limits of conjugates of
the subalgebra along one-parameter subgroups in the
Grassmannian of subspaces of . The little Weyl group is
obtained as a finite reflection group generated by the reflections in the walls
of the compression cone
The infinitesimal characters of discrete series for real spherical spaces
Let be the homogeneous space of a real reductive group and a
unimodular real spherical subgroup, and consider the regular representation of
on . It is shown that all representations of the discrete series,
that is, the irreducible subrepresentations of , have infinitesimal
characters which are real and belong to a lattice. Moreover, let be a
maximal compact subgroup of . Then each irreducible representation of
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 .Comment: To appear in GAF
A note on -factorizations of representations
In this paper we give an overview on -factorizations of Lie group
representations and introduce the notion of smooth -factorization.Comment: This article is dedicated to the fond memories of Gerrit van Dij
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