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Recently I have found a need to learn more about p-adic group representations over a p-adic vector space. Generally, this motivates a study of representations $\left( V, \rho \right)$ for some group $G$ where $V$ is a vector space over $\mathbb{Q}_p$. Since I'm only familiar with the theory for representations acting on a complex vector space, I was hoping for references where I could find more information?

This may seem like a question suited more for google, however, I don't seem to have enough knowledge to prompt google in the correct direction.

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Dear jdmorgan, Could you be more precise about the kind of representations you want to consider? Do you mean continuous representations of a topological group (say) on a $p$-adic vector space, or just abstract representations of a discrete group? In the latter case, the only difference between the case of $\mathbb Q_p$ and $\mathbb C$ is that the former is not algebraically closed. In the former case, there is a lot more to say, but I will hold off saying anything until I know better what you actually want. Regards, –  Matt E Jan 6 '11 at 0:50
I'm interested in the case when $G$ is a topological group and $\rho$ is continuous. –  jdmorgan Jan 6 '11 at 1:09
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1 Answer

There are a few basic situations to consider that I know about.

The first case is when the group $G$ is compact. Then if we are given a continuous rep. $\rho: G \to GL_n(\mathbb Q_p)$, the image is compact, so lies in a maximal compact subgroup of $GL_n(\mathbb Q_p)$. Any such is conjugate to $GL_n(\mathbb Z_p)$, so after changing basis, we may assume that $\rho: G \to GL_n(\mathbb Z_p)$. (Another way to phrase this is that $G$ must leave some $\mathbb Z_p$-lattice invariant, and there are lots of ways to prove this, without having to mention the concept of "maximal compact", if that makes you at all nervous.)

At this level of generality, there is not that much more to say. But there are various subcases of interest in which one can say more.

E.g. if $G$ is a Galois group, then there is an enormous literature about $p$-adic Galois representations of Galois groups. If this is the case you are interested in, you might want to ask another more specific question about it.

There are two other basic subcases, but before introducing them, I have to mention a basic fact about $GL_n(\mathbb Z_p)$, namely that there is a quotient map $GL_n(\mathbb Z_p) \to GL_n(\mathbb F_p)$, whose kernel, which I'll denote by $K(1)$, is easily seen to be a pro-p-group (i.e. a projective limit of $p$-groups). (The reason for the $(1)$ is that we could also consider $K(n)$, the kernel of the "reduction modulo $p^n$'' map, for each $n\geq 1$.)

Here now are two other interesting subcases of the compact case.

The first is when the group $G$ is profinite, and is virtually pro-prime-to-$p$, i.e. contains an open subgroup that is the projective limit of finite groups of order prime to $p$.

Examples of such $G$ are $GL_n(\mathbb Z_{\ell})$.

In this case, let $H$ be the open subgroup that is pro-prime-to-$p$. Since $K(1)$ is pro-$p$, we see that $\rho(H)$ and $K(1)$ must have trivial intersection, and so $\rho(H)$ injects into $GL_n(\mathbb F_p)$.
Thus $\rho(H)$ is finite, and hence $\rho(G)$ is finite too. Thus in this case, the continuous $\rho$s all factor through some finite quotient of $G$, and we reduce to standard representation theory (i.e. rep'n theory of finite groups over a field of char. $0$).

The second, and more interesting, case is when $G$ is virtually pro-$p$, i.e. contains an open subgroup $H$ which is pro-$p$. (E.g. if $G$ is itself $GL_n(\mathbb Z_p)$, or a closed subgroup thereof.) In this case the theory is more genuinely $p$-adic, i.e. it doesn't just reduce to the classical rep'n theory of finite groups. A good place to learn about some aspects of this is Lazard's opus Groupes analytiques $p$-adiques, in Publ. Math. IHES vol. 26 (1965), where among other things he studies the continuous cohomology of such representations (when the group $G$ is $p$-adic analytic, e.g. a matrix group), explains the relationship to Lie theory and Lie algebra cohomology, and proves various Poincare duality-type results for the cohomology.

More recent discussions of Lazard's work and related ideas can be found in some of the literature surrounding non-abelian Iwasawa theory, e.g. Venjakob's article On the structure theory of the Iwasawa algebra of a $p$-adic Lie group in J. Eur. Math. Soc. vol. 4 (2002).

Finally, let me mention that if $G$ is not compact, but is just locally compact, e.g. $GL_n(\mathbb Q_{\ell})$, then there usually won't be many interesting finite-dimensional representations in which $G$ preserves a lattice, and so it is natural to consider $p$-adic Banach space representations in which $G$ preserves a lattice instead.

If $G$ contains a profinite open subgroup that is pro-prime-to-$p$, then this theory is not so novel — one can see for example Vigneras's article Banach $\ell$-adic representations of $p$-adic groups in Astérisque vol. 330 (2010).

On the other hand, if $G$ contains a pro-$p$ open subgroup, then the theory is much more involved, and is the subject of a lot of recent work, especially by people thinking about $p$-adic Langlands. You can see some of the papers of Schneider and Teitelbaum, and of Breuil and Colmez, as well as some of the papers on my web-page. (I'm Emerton at Chicago.)

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Dear Matt, I added a few links to your answer and updated your location info. I hope that's okay with you. Best wishes, Theo –  t.b. Nov 7 '11 at 16:05
Dear Theo, That's fine; thanks very much! Best wishes, –  Matt E Nov 7 '11 at 16:11
This is a really good survey. @MattE: Could you give a reference for a prove that we could assume the $p$-adic representation to lie in $GL_n(\Bbb Q_p)$? –  BIS HD Nov 23 '13 at 17:22
@BISHD: Dear BIS, I'm not sure what context you're thinking about. Are you asking why a representation that into $GL_n(\overline{\mathbb Q}_p)$ can always be defined over a finite extension of $\mathbb Q_p$? Regards, –  Matt E Nov 23 '13 at 21:32
@BISHD: Dear BIS, Here is a sketch: since $G$ and $\mathbb Z_p$ are compact, the $\mathbb Z_p$-submodule of $\mathbb Q_p^n$ generated by $G\cdot \mathbb Z_p^n$ is compact, thus is isomorphic to $\mathbb Z_p^n$. Conjugating the image of $G$ by the change of basis matrix coming from this isomorphism, we find that the conjugated image of $G$ preserves $\mathbb Z_p^n$, and so lies in $GL_n(\mathbb Z_p)$. Regards, –  Matt E Nov 24 '13 at 6:41
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