MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $F$ be a number field and let $o$ be its ring of integers. Let $o_p$ resp. $F_p$ be the completion at a prime ideal $p$ in $o$. Let $B$ be the group of upper triangular matrices in $GL_2$. Let $\pi $ be a character of $B(F_p)$.

How can we describe the irreducible which occur in the restriction of the induced representations $$Res_{GL_2(o_p)} Ind_{B(F_p)}^{GL_2(F_p)} \pi = Ind_{B(o_p)}^{GL_2(o_p)} \pi = lim_r Ind_{B(o/p^r)}^{GL_2(o/p^r)} \pi.$$

Is there a nice description of the irreducibles?

share|cite|improve this question
up vote 3 down vote accepted

This is done in :

Casselman, William The restriction of a representation of ${\rm GL}_2 (k)$ to ${\rm GL}_{2}({\mathfrak o})$. Math. Ann. 206 (1973), 311–318.

Also see :

Silberger, A.: $PGL_2$ over the p-adics. Lecture Notes in Mathematics 166, Berlin- Heidelberg-New York: Springer 1970

Very roughly speaking, the idea is the following. When you restrict an irreducible smooth representation of ${\rm GL}(2)$ to ${\rm GL}(2,{\mathfrak o})$ you get two types of constituents. The first ones (infinitely many) are uninteresting : they appear in the restriction of many other representations. These representations are described by Casselman. The second ones are more interesting : if an ireeducible representation contains such constituents then it must belong to a single component of the Bernstein decomposition of the category. They are called typical. Actually the situation is a bit more complicate. I've made it simpler. For more detail, you may read the paper by Henniart :

Breuil, Christophe; Mézard, Ariane Multiplicités modulaires et représentations de ${\rm GL}_2({\bf Z}_p)$ et de ${\rm Gal}(\overline{\bf Q}_p/{\bf Q}_p)$ en $l=p$. (French) [Modular multiplicities and representations of ${\rm GL}_2({\bf Z}_p)$ and ${\rm Gal}(\overline{\bf Q}_p/{\bf Q}_p)$ at $l=p$] With an appendix by Guy Henniart. Duke Math. J. 115 (2002), no. 2, 205–310.

share|cite|improve this answer
The article of Casselman exactly provides what I need. Thanks for this nice reference. – Marc Palm Apr 2 '11 at 11:28

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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