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This feels like a silly question, but it keeps coming up in seminars and even after much searching I still have no idea what it means. I would like to see a definition and some examples of things that are and aren't semistable, as well as why this is an important condition on a representation. Any good references would also be appreciated.

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An E-rep of $G_K$ (E either $Q_p$ or $\bar{Q}_\ell$, K an extension of $Q_p$) being semistable basically means that it looks like a representation coming from the etale cohomology of a variety with semistable reduction. Not every semistable representation​ arises in this way though.

In the $\ell$-adic case, semistable is equivalent to the representation being unipotent, i.e. inertia acts by unipotent operators, i.e. the representation has unramified semisimplification. In fact, Grothendieck's $\ell$-adic monodromy theorem says that every $\ell$-adic Galois rep is potentially semistable, i.e. becomes semistable after restrictive to some finite index normal subgroup.

When the rep $V$ is $p$- adic the analogue is that $V$ should be $\mathbf{B}_{st}$-admissible, where $\mathbf{B}_{st}$ is the ring of semistable periods. You'll have to learn some $p$-adic Hodge theory to understand this; a good source is Fontaine's book which is available here: https://www.math.u-psud.fr/~fontaine/galoisrep.pdf

As for the importance of it: I'm far from an expert, and i actually never really think about p-adic representations, so this might not be the best answer. I like to base it on the analogy with the $\ell$-adic case. There's also a p-adic monodromy theorem, which says that all de Rham representations are potentially semistable; de Rham is a condition which doesn't arise in the $\ell$-adic world as any representation is easily seen to be de Rham. So (potentially) semistable representations form a large class of representations which have nice properties -- somehow they're the representations which aren't so hideously complicated that you have no real chance of studying them with standard tools. For example, if your representation is geometric in the sense that it arises as some etale cohomology group, then it not being semistable means that you have to start considering the cohomology of varieties which aren't very well behaved.

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