2008-04-18
Dihua Jiang, University of Minnesota
On the Local-Global Principle for Automorphic Forms
It is well known that an irreducible unitary automorphic
representation is a restricted tensor product of irreducible unitary
admissible representations over all corresponding local fields. It is
fundamental to understand how much of the properties of the local
representations can be carried over to the automorphic representations and
hence properties of some local representations imply the same properties for
other local representations occurring in the given automorphic
representation.
In this talk, I will explain our recent work toward such a local-global
principle. For example, we prove that under a certain assumption, for an
irreducible cuspidal automorphic representation of an orthogonal group, if
one local component is generic, then at almost all local places, the local
components are generic. If time permits, we will explain that for a given
irreducible cuspidal automorphic representation, a certain quasi-invariant
property for one local component implies the range of holomorphy of the
standard L-function for a certain right half-plane.