Holomorphic extensions of representations I: Automorphic functions

Holomorphic extensions of representations : ( I ) automorphic functions

Let G be a connected, real, semisimple Lie group contained in its complexification GC, and let K be a maximal compact subgroup of G. We construct a KC-G double coset domain in GC, and we show that the action of G on the K-finite vectors of any irreducible unitary representation of G has a holomorphic extension to this domain. For the resultant holomorphic extension of K-finite matrix coefficien...

L-Functions and Automorphic Representations*

There are two kinds of L-functions, and they will be described below: motivic L-functions which generalize the Artin L-functions and are defined purely arithmetically, and automorphic L-functions, defined by data which are largely transcendental. Within the automorphic L-functions a special class can be singled out, the class of standard L-functions, which generalize the Hecke L-functions and f...

Automorphic Representations and L-functions

• Decomposition by central characters • Square-integrable cuspforms • Smoothness of cuspforms • Eigen-cuspforms and automorphic representation • Dirichlet series versus zeta and L-functions • L-functions defined via local data • Factoring unitary representations of adele groups • Spherical representations and Satake parameters • Local data, L-groups, higher L-functions References and historical...

Degenerate Principal Series Representations and Their Holomorphic Extensions

Let X = H/L be an irreducible real bounded symmetric domain realized as a real form in an Hermitian symmetric domain D = G/K. The intersection S of the Shilov boundary of D with X defines a distinguished subset of the topological boundary of X and is invariant under H and can also be realized as S = H/P for certain parabolic subgroup P of H . We study the spherical representations Ind P (λ) of ...

Introduction to Automorphic Representations