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EDIT: I would guess it's a wild problem in general to decompose this, even if $P$ is the Borel. Namely if you could decompose this representation, then you would certainly also knowthe $GL_n(\mathbb Z/p^r)$-endomorphism ring $\Pi^{K(r)}$. But by a standard Frobenius reciprocity argument (also used in Casselman) this is the space of functions$f: GL_n(\mathbb Z/p^r) \to \mathbb C$ such that $f(b k b') = \pi(b) f(k) \pi(b')$ for all $k \in GL_n(\mathbb Z/p^r)$ and $b, b' \in P(\mathbb Z/p^r)$. In particular you shouldunderstand the double coset space $P(\mathbb Z/p^r) \backslash GL_n(\mathbb Z/p^r) / P(\mathbb Z/p^r)$ (and this is exactly what's required when $\pi$ is trivial).But this may involve wild classification problems according to http://mathoverflow.net/questions/57922/bruhat-decomposition-for-gr-r-local-ring-or-r-mathbbz-mathfrakpr/57931#57931.[You even need this for all $r$, so you'd need $P(\mathbb Z_p) \backslash GL_n(\mathbb Z_p) / P(\mathbb Z_p)$. Maybe this is known to be wild for $n$ bigger than some bound?]

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If $P$ is the Borel and $\pi$ is admissible, then $\pi$ is 1-dimensional and let $\Pi$ denote the induction. Let $K(r)$ be the principal congruence subgroup of elements in $GL_n(\mathbb Z_p)$ that are 1 modulo $p^r$. Then $\Pi$ is a direct limit of the representations $\Pi^{K(r)}$. Now it's easy to see that $\Pi^{K(r)} \cong Ind_{P(\mathbb Z/p^r)}^{GL_n(\mathbb Z/p^r)} \pi^{T \cap K(r)}$. It suffices to decompose these representations.

When $n = 2$ this was done by Casselman in his paper "The Restriction of a Representation of $GL_2(k)$ to $GL_2(o)$." (Math. Ann., 1973) Link: http://www.digizeitschriften.de/dms/img/?PPN=PPN235181684_0206&DMDID=dmdlog53 (see prop. 1 on p. 312). Basically the complement of each $\Pi^{K(r)}$ in $\Pi^{K(r+1)}$ is irreducible (once $r$ is large enough so that these spaces are nonzero). You can easily compute their dimensions, incidentally, because $P(\mathbb Z/p^r)\backslash GL_2(\mathbb Z/p^r) \cong \mathbb P^1(\mathbb Z/p^r)$, which has $p^{r-1}(p+1)$ elements.

Note that Casselman assumes that $\pi$ is of the form $\epsilon_0 \otimes 1$, but we can reduce to this case by twisting $\pi$ by $\chi \circ \det$ for characters $\chi$ of $\mathbb Z_p^\times$.

It's possible that this can be generalised to $GL_n$, I haven't tried to think about that.