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Consider $G = PGL(2)$ and $PSL(2)$.

Let $F$ P$be a non archimedean field with ring parabolic subgroup of integers$o$. Let GL(n)$ with Levi decomposition $K$ be a maximal compact in P =MN$, where$G(F)$, that N$ is the normalizer unipotent radical.

Let $NI$ \pi$be an irreducible representation of the Iwahori subgroup$I$or$G(o)$. Given a standard Borel$B \subset G$with$G(F) = B(F) K$, and a character$\chi : B(o) \rightarrow \mathbb{C}^1$. How does$ Ind_{B(o)}^K \chi$decompose into irreducibles? As an intermediate step one has M(\mathbf{Z}_p)$ inflated to know:

How does $Ind_{B(o)}^I \chi$ decompose into irreducibles?

This probably easier, since we can rely on the Iwahori decomposition.

I know P(\mathbf{Z}_p)$, how the situation behaves for$PGL(2)$for$K = PGL(2)$in full generality (Casselman, Silberger), and for$K=NI$only in odd residue characteristic (Silberger), I know nothing about$SL(2)$, but$does $Res_{K \cap PSL(2)} Ind_{B(o)}^K \chi \cong Ind_{B(o) \cap PSL(2)}^{K \cap PSL(2)} Res_{B(o) \cap PSL(2)Ind_{P(\mathbf{Z}_p)}^{GL_n(\mathbf{Z}_p)} \chi$$should give a bijection, which gives the results for SL(2) from the results of GL(2). Let \chi have conductor p^k. Is (Ind_{B(o)}^K \chi)^{\Gamma(p^k)} irreduciblepi decompose? Is (Ind_{B(o)}^K \chi)^{\Gamma(p^N)} \ominus (Ind_{B(o)}^K \chi)^{\Gamma(p^{N-1})} irreducible, It would be sufficient for N>k? I do not me to know what the filtration \Gamma(p^k) of NI should be, but for G(o) consider result in the standard filtration of matricessimplest case, which are the identity modulo p^k. Does the last result generalize to where P parabolic in PGL(n) or PSL(n)?is a Borel subgroup 4 edited title # Parabolic induction ontheleveloftheopencompactGL(n,Zp) 3 added 116 characters in body; added 14 characters in body Consider G = PGL(2) and PSL(2). Let F be a non archimedean field with ring of integers o. Let K be a maximal compact in G(F), that is the normalizer NI of the Iwahori subgroup I or G(o). Given a standard Borel B \subset G with G(F) = B(F) K, and a character \chi : B(o) \rightarrow \mathbb{C}^1. How does Ind_{B(o)}^K \chi decompose into irreducibles? As an intermediate step one has to know: How does Ind_{B(o)}^I \chi decompose into irreducibles? This probably easier, since we can rely on the Iwahori decomposition. I know how the situation behaves for PGL(2) for K = PGL(2) in full generality (Casselman, Silberger), and for K=NI only in odd residue characteristic (Silberger), I know nothing about SL(2), but$$ Res_{K \cap PSL(2)} Ind_{B(o)}^K \chi \cong Ind_{B(o) \cap PSL(2)}^K PSL(2)}^{K \cap PSL(2)} Res_{B(o) \cap PSL(2)} \chi$$should give a bijection, which gives the results for$SL(2)$from the results of$GL(2)$. Let$\chi$have conductor$p^k$. Is$(Ind_{B(o)}^K \chi)^{\Gamma(p^k)}$irreducible? Is$(Ind_{B(o)}^K \chi)^{\Gamma(p^N)} \ominus (Ind_{B(o)}^K \chi)^{\Gamma(p^{N-1})}$irreducible, for$N>k$? I do not know what the filtration$\Gamma(p^k)$of$NI$should be, but for$G(o)$consider the standard filtration of matrices, which are the identity modulo$p^k$. Does the last result generalize to$P$parabolic in$PGL(n)$or$PSL(n)\$?

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