Let $P$ be a pro-p group. Assume that there is a filtration of $P$ by normal subgroups $P_i$ such that $P_0=P$ and $P_{i+1} < P_i(i\in\mathbb N)$. Let $V$ be an $l$-adic representation of $P$, where the prime $l$ is prime to $p$.

Question: Is there a direct sum decomposition $V=\bigoplus_{i\geq 0,i\in\mathbb Z}V_i$ by $P$-submodules such that the $P_r$-fixed part is given by $V^{P_i}=\bigoplus_{i\leq r} V_i$ ?

  • 1
    $\begingroup$ Profinite groups are in particular compact, so all complex representations are unitarizabile and decompose uniquely into irreducible. For your particular problem, it suffices to define $V_0 = V^P$ and $$ V_i = V^{P_i} \ominus V_{P_{i-1}},$$ where the minus is the orthogonal difference in the sense of Hilbert spaces. In some sense, you need to copy this to the $\ell$-adic reps. $\endgroup$ – Marc Palm Apr 24 '13 at 13:58
  • $\begingroup$ Is the category of $\ell$-adic reps of pro-p-groups not semisimple? $\endgroup$ – Marc Palm Apr 24 '13 at 13:59
  • $\begingroup$ at least if $p \neq \ell$? $\endgroup$ – Marc Palm Apr 24 '13 at 14:00
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    $\begingroup$ The category of $l$-adic reps of pro-$p$ groups is indeed semisimple. This is because it factors through $GL_n(\mathbb Z_l)$, but the Sylow $p$-subgroups of that all inject into $GL_n(\mathbb F_l)$, so there's a bound on their size. $\endgroup$ – Will Sawin Apr 24 '13 at 17:33
  • $\begingroup$ Do you mean that l-adic representations of pro-p groups factors through a finite quotient.. so they are semisimple? By the way, Is there any good reference which discuss about this kind problems? $\endgroup$ – Int Apr 24 '13 at 23:15

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