5
$\begingroup$

In the theory of automorphic forms we often right away reduce the study to admissible representations, and I wonder how much everything breaks when not. If I understood well, admissible representations are nice because

  1. They have finite-dimensionality properties (of the $K$-types).
  2. The contragredient of an admissible representation still is admissible.
  3. The parabolic induction of an admissible representation still is admissible.

I feel 1 is great in practice and 3 is important to classify representations. But are there less philosophical reasons to consider only admissible representations? E.g., do the properties above break for non-admissible representations? Or are there other very important properties we can prove of them and not of other representations?

I would like clear counter-examples, if any.

Improve this question
$\endgroup$
4
  • 1
    $\begingroup$ Counterexamples to what? My understanding is that considering admissible representations is like considering reductive groups: they are numerous enough that they have nice heredity properties, like your (3), and small enough that they are accessible through more familiar means, like your (1). Of course the set of all, say, smooth representations satisfies your (2)–(3), but it's too big to hope to get a handle on it. (You'll find non-admissible counterexamples to (1) in abundance, because that's practically the definition of admissibility.) $\endgroup$
    – LSpice
    Commented Nov 1, 2020 at 3:43
  • 4
    $\begingroup$ Isn't it mainly because we're interested in the irreducible smooth representations, and the admissible representations have all the useful finiteness properties of irreducible smooth representations but are closed under more operations? $\endgroup$
    – Will Sawin
    Commented Nov 1, 2020 at 4:11
  • 1
    $\begingroup$ You won't probably have a nice theory of new/test vectors or nice models, like unique Whittaker models. $\endgroup$
    – Kimball
    Commented Nov 1, 2020 at 4:23
  • 1
    $\begingroup$ @Kimball Indeed, local newforms lie in a $K$-type that occurs with multiplicity one: arxiv.org/abs/2009.08571 $\endgroup$
    – Peter Humphries
    Commented Nov 1, 2020 at 7:24

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .