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Let $(\rho,V)$ be a continuous finite dimensional representation of the Weil group $W_F$ over a local field $F$. If $V$ decomposes as a direct sum $V_1 \oplus V_2$ of representations, then

$$L(s,\rho) = L(s,\rho|_{V_1})L(s,\rho|_{V_2})$$

$$\epsilon(s,\rho,\psi) = \epsilon(s,\rho|_{V_1},\psi)\epsilon(s,\rho|_{V_2},\psi)$$

On the other hand, what if we have a subrepresentation $W$ of $V$, and consider the representations $\rho|_W$ and $\overline{\rho}$ of $W$ and $V/W$. Can we still say that $L(s,\rho)$ and $\epsilon(s,\rho,\psi)$ are the products of the analogous objects for $\rho|_W$ and $\overline{\rho}$? Is there some way of relating the L-functions of $\rho$ and its $\Phi$-semisimplification to the reduce to the case of a split exact sequence?

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  • $\begingroup$ Have you considered the case of a 2-dimensional representation $V$ on which $N$ and $I_F$ act trivially and Frobenius acts as $\begin{pmatrix} 1& 1\\0 & 1 \end{pmatrix}$? $\endgroup$
    – David Loeffler
    Jun 10, 2018 at 21:52
  • $\begingroup$ In this case it seems to be true: taking $V_1 = \{ (x,0) \}$, $\rho$ induces the trivial representation on $V_1$ and $V/V_1$, and $$L(s,\rho) = (1- q^{-s})^{-2}, L(s,\rho|_{V_1}) = L(s,\overline{\rho}) = (1-q^{-s})^{-1}$$ $\endgroup$
    – D_S
    Jun 11, 2018 at 18:30
  • $\begingroup$ Okay, I think I see what's going on here. If $(\rho,V)$ is a representation of $W_F$, then by Jordan decomposition, $\rho$-stable subspaces are the same as $\rho_{ss}$-stable subspaces, where $\rho_{ss}$ is the $\Phi$-semisimplification of $\rho$, and $L(s,\rho) = L(s,\rho_{ss}), \epsilon(s,\rho,\psi) = \epsilon(s,\rho_{ss},\psi)$. This reduces us to the case of split exact sequences. $\endgroup$
    – D_S
    Jun 12, 2018 at 5:42
  • $\begingroup$ Sorry, I gave the wrong counterexample by mistake. For representations of $W_F$, as opposed to Weil-Deligne representations (so you do not allow monodromy), and with all your representations on complex vector spaces, it's true. $\endgroup$
    – David Loeffler
    Jun 12, 2018 at 6:37

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