Spectral decomposition of parabolic induced for GL2(Zp) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T05:46:28Z http://mathoverflow.net/feeds/question/60272 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/60272/spectral-decomposition-of-parabolic-induced-for-gl2zp Spectral decomposition of parabolic induced for GL2(Zp) Marc Palm 2011-04-01T09:24:43Z 2012-04-10T08:10:24Z <p>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 $B(F_p)$.</p> <p>How can we describe the irreducible which occur in the restriction of the induced representations $$Res_{GL_2(o_p)} Ind_{B(F_p)}^{GL_2(F_p)} \pi = Ind_{B(o_p)}^{GL_2(o_p)} \pi = lim_r Ind_{B(o/p^r)}^{GL_2(o/p^r)} \pi.$$</p> <p>Is there a nice description of the irreducibles?</p> http://mathoverflow.net/questions/60272/spectral-decomposition-of-parabolic-induced-for-gl2zp/60280#60280 Answer by Paul Broussous for Spectral decomposition of parabolic induced for GL2(Zp) Paul Broussous 2011-04-01T11:38:14Z 2011-04-01T11:53:10Z <p>This is done in :</p> <p>Casselman, William The restriction of a representation of ${\rm GL}<em>2 (k)$ to ${\rm GL}</em>{2}({\mathfrak o})$. Math. Ann. 206 (1973), 311–318.</p> <p>Also see :</p> <p>Silberger, A.: $PGL_2$ over the p-adics. Lecture Notes in Mathematics 166, Berlin- Heidelberg-New York: Springer 1970</p> <p>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 :</p> <p>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. </p>