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:

1. What is the definition of the Bernstein centre of a p-adic reductive group.
2. What is the original motivation to introduce it ?
3. What role does it play in the theory of automorphic forms ?
4. 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!