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What & Why is Questions about the Bernstein center of a $p$-adic reductive group ?

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user4245
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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 $GL_2$ ?

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!

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 $GL_2$ ?

Any comments and references (in English) will also be very welcome !

Thank you very much in advance!

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!

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user4245
  • 809
  • 1
  • 9
  • 17

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 $GL_2$ ?

Any comments and references (in English) will also be very welcome !

Thank you very much in advance!

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 $GL_2$ ?

Any references (in English) will also be very welcome !

Thank you very much in advance!

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 $GL_2$ ?

Any comments and references (in English) will also be very welcome !

Thank you very much in advance!

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