Ph.D Thesis
Ph.D Student Kemarsky Alexander Distinguished Representations of GLn(c) Department of Mathematics Professor Omer Offen Professor Ehud Moshe Baruch

Abstract

Let G be a group and H be a subgroup of G. We say that a representation  (π,V) of G is H -distinguished if there exists a non-zero continuous H-invariant linear functional L : V → C. In this work we investigate GL(n,R)- distinguished representations of GL(n,C). We consider only admissible, smooth representations of a moderate growth on a Fréchet space. We give a necessary condition for the distinction of an irreducible representation of GL(n,C). The condition is given in terms of Langlands data of the representation. The theorem follows from an investigation of orbits of the standard action of the standard Borel subgroup of GL(n,C) on the symmetric space GL(n,C)/GL(n,R). Let P be the standard mirabolic subgroup in GL(n,R) consisting of matrices with last row equal to (0,0,...,0,1). Let (π,V) be a GL(n,R)-distinguished, irreducible, admissible representation of GL(n,C). We prove that every P-invariant linear functional on V is automatically GL(n,R)-invariant. Let (π,V) be a GL(n,R)-distinguished, irreducible, admissible representation of GL(n,C), let π0 be an irreducible, admissible, GL(m,R)-distinguished representation of GL(m,C), and let ψ be a non-trival character of C which is trivial on R. From a purely formal argument it is known that the value of Ranking-Selberg gamma factor γ(1/2,π?π0;ψ) is 1 or −1. This value is an analogy of the sign of the classical Gauss sum. As a consequence of our necessary condition for distinction we prove that the Rankin-Selberg gamma factor at  s = 1/2 is γ(1/2,π?π0;ψ) = 1. For unitary, irreducible representations of GL(n,C) we give a complete characterization of distinguished representations. For such representations we use Bernstein’s method of  meromorphic continuation of integrals and build an invariant functional explicitly. We prove also a converse type theorem for gamma factors. Let π be a ﬁxed unitary representation of GL(n,C) and suppose γ(1/2,π?π0;ψ) = 1 for “suﬃciently many” distinguished representations π0. Then we prove that π  satisﬁes the necessary condition for distinction. In many cases, for example for generic representations, it follows that π is distinguished. _____________________________________________________________________________