Dear all,

Could you kindly tell me what is the precise definition of "Bernstein center" in the theory of automorphic forms ? This appear, for example, in page 233 of Moeglin-Waldspurger's book "Spectral decomposition and Eisenstein series" :

"...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 could guess that it is some analogy of "the centre of enveloping algebra" at fintie places. I hope to know its precise meaning.

Any references (in English) and examples are very welcome ! Thank you !