Harish-Chandra's characterisation of the discrete series using cusp forms: can it be generalized from semisimple Lie groups to reductive symmetric spaces?
Mogens Flensted-Jensen, The Royal Veterinary and Agricultural University, Frederiksberg, Denmark
Abstract
Harish-Chandra's characterisation of the discrete series for a semisimple Lie group
G using cusp forms states, that a rapidly decreasing
K-finite L2-function belongs
to the discrete series if and only if it is a cusp form. Here
K is any maximal compact subgroup of G, and a cusp form is a
rapidly decreasing L2-function, which is
annihilated by integration over the translates of the unipotent part
N of any parabolic subgroup different from the group itself.
I shall reformulate this result in the framework of the Lie group considered as a symmetric space. Then it is seen in relation to Harish-Chandra's Plancherel formula for the group and to the more general formulation of the Plancherel formula for reductive symmetric spaces proved by Delorme, Ban and Schlichtkrull.
Then I shall present an attempt to define a cusp form in the general setting of a reductive symmetric space and to generalize Harish-Chandra's characterisation of the discrete series using cusp forms.
A key problem is that for the seemingly relevant parabolic subgroups the integrals over the corresponding N-parts, which should define a cusp form, do not converge in general for a rapidly decreasing L2-function. The basic idea is to choose an appropriate subgroup of N.