Ph.D Student | Kemarsky Alexander |
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Subject | Distinguished Representations of GLn(c) |

Department | Department of Mathematics |

Supervisors | Professor Omer Offen |

Professor Ehud Moshe Baruch | |

Full Thesis text |

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 π_{0} 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?π_{0};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?π_{0};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?π_{0};ψ) = 1 for
“suﬃciently many” distinguished representations π_{0}. 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. _____________________________________________________________________________