 
Summary: Distinguished nonArchimedean representations
U. K. Anandavardhanan
Abstract. For a symmetric space (G, H), one is interested in un
derstanding the vector space of Hinvariant linear forms on a rep
resentation of G. In particular an important question is whether
or not the dimension of this space is bounded by one. We cover
the known results for the pair (G = RE/F GL(n), H = GL(n)), and
then discuss the corresponding SL(n) case. In this paper, we show
that (G = RE/F SL(n), H = SL(n)) is a Gelfand pair when n is
odd. When n is even, the space of Hinvariant forms on can have
dimension more than one even when is supercuspidal.
The latter work is joint with Dipendra Prasad.
1. Introduction
Let G be a group, and H the group of fixed points of an involution
on G. A representation of G is said to be distinguished with respect
to H, if the space of Hinvariant linear forms on is nonzero. More
generally, if the space HomH(, ) is nonzero for a character of H, we
say that is distinguished with respect to H. In this paper we are
interested in the case when (G, H) is defined over a padic field.
The initial impetus for much of the research in this field came from
