Dear all,
The "Bernstein center" of a $p$-adic reductive group appears frequently in the literature of automorphic forms, often without a precise definition. For example, in page 233 of Moeglin-Waldspurger's classic "Spectral decomposition and Eisenstein series" , the couple tell us :
"...in particular the centre of enveloping algebra acts on $\delta$ via a character at the infinite places and the Bernstein centre does so at the finite places... "
So one may guess that it is some analogy of "the centre of enveloping algebra" at fintie places.
My questions are:
- What is the definition of the Bernstein centre of a p-adic reductive group.
- What is the original motivation to introduce it ?
- What role does it play in the theory of automorphic forms ?
- Could you explain these in some concrete example ,say $GL_2$ ?
AnyPlease feel free to choose part of the questions to reply. Any comments and references (in English) will also be very welcome !
Thank you very much in advance!