There are many papers studying branching laws of irreducible admissible complex representations of classical groups over local fields, are there some analogues for $p$-adic representations?

For example, consider an irreducible admissible $p$-adic representation of $GL_2(\mathbb Q_p)$, what are the multiplicities of its restriction to $GL_1(\mathbb Q_p)$ ? How about $GL_n(\mathbb Q_{p^2})$ to $GL_n(\mathbb Q_p)$ ?

More precisely, $p$-adic representations of $G$ mean admissible unitary $L$-Banach representations of $G(\mathbb Q_p)$ where $G$ is a reductive group over $\mathbb Q_p$, and $L$ is a finite extension of $\mathbb Q_p$. They are natural objects in the $p$-adic Langlands program. And we care about dimension of $Hom_H(\pi|_{H}, \sigma)$ as in the classical branching law where $H$ is a reductive subgroup of $G$, and the representations $\pi$ and $\sigma$ are both irreducible.

  • $\begingroup$ Can you clarify whether you are asking about representations defined over $p$-adic fields or over $\mathbb C$? Paul Broussous' answer assumes the latter, though this falls under the heading of "irreducible admissible complex representations of classical groups over local fields" which you said you knew about. $\endgroup$
    – Kimball
    Oct 25 '19 at 9:23
  • $\begingroup$ @Kimball You're right I did not understand the question correctly ... $\endgroup$ Oct 25 '19 at 11:31
  • $\begingroup$ @PaulBroussous Well, I'm not sure---I find the question a bit vague as the term "$p$-adic representation" is. Also, previous questions of the OP make me wonder which is actually meant. $\endgroup$
    – Kimball
    Oct 25 '19 at 14:29
  • $\begingroup$ @Kimball Sorry I shall be more specific, thank you! $\endgroup$
    – sawdada
    Oct 25 '19 at 17:19

If you're asking about admissible p-adic Banach space representations in the sense of Schneider--Teitelbaum, then I think virtually nothing is known in this setting about branching laws, even in the simplest cases like branching from $GL_2(\mathbb{Q}_p)$ to the diagonal maximal torus (a case which we understand extremely well for smooth representations, thanks to Waldspurger's theorem).

  • $\begingroup$ Are there any $p$-adic multiplicity 1 results known, like a (non-reductive) analogue of Whittaker models? $\endgroup$
    – Kimball
    Oct 25 '19 at 18:19
  • 1
    $\begingroup$ Colmez's work on p-adic Langlands uses something he calls a Kirillov model; but I don't know if it has the same uniqueness properties as the Whittaker/Kirillov models in the smooth theory. $\endgroup$ Oct 26 '19 at 9:13

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.