TITLE: Questions about the Bernstein center of a $p$-adic reductive group 
QUESTION [24 upvotes]: 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 $GL_2$ ?

Please 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!

REPLY [2 votes]: Ralf Meyer develops part of the theory from scratch for locally compact groups via projective systems of Lie groups in section 5 of http://arxiv.org/abs/math/0310094
This theory speciales to the Bernstein center in the totally disconnected case and to the theory of the center of the universal envelopping algebra for complex Lie groups with trivial center. Also he relates the center to distributions.
He works on bornological vector-spaces, but states in the introduction that the theory includes smooth representation on Frechet-spaces. So the notion bornological and topological can be interchanged with no harm in this case. (I assume that many of us are more comfy with topological vectorspaces than with bornological ones. At least I am.)