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While I'm studying about representation theory of $\mathrm{GL}(2)$ over local fields, I found that there's no one talking about $p$-adic Lie algebra. However, for Lie groups over $\mathbb{R}$ or $\mathbb{C}$, it is common to study the Lie algebra representation first, and then study the representation of Lie group using the results in Lie algebra. I want to know if there's any reference about representation theory of $p$-adic Lie algebra that helps to study the representation of $p$-adic Lie groups, such as $\mathrm{GL}(2, \mathbb{Q}_{p})$.

Maybe there's some technical problem with $p$-adic Lie stuff, since the exponential map can't be defined on whole Lie algebra (since they do not converge), but they still converge locally, as I know. Also, to define $p$-adic Lie algebra of given $p$-adic Lie group, since I don't know much about these, let's just assume that $G = \mathrm{GL}(2, \mathbb{Q}_{p})$ and $\mathfrak{g} = \mathfrak{gl}(2, \mathbb{Q}_{p})$, where the latter one is just $2\times 2$ matrices over $\mathbb{Q}_{p}$ with a Lie bracket given by $[A, B] = AB-BA$.

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    $\begingroup$ Well part of the issue is that when we talk about representations of say $GL_2(\mathbb{Q}_p)$ we are usually still talking about actions on complex vector spaces. A lie algebra representation is first and foremost a linear map, so it's not even clear what it would mean for this $p$-adic Lie algebra to have a complex representation. $\endgroup$
    – Nate
    Feb 8, 2019 at 22:46
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    $\begingroup$ Another problem is that the representations we usually care about (i.e. smooth or admissible representations) have the property that any vector is fixed by a neighborhood of the identity. So we'd expect the "Lie algebra" to act trivially. $\endgroup$
    – Nate
    Feb 8, 2019 at 22:49

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Let $G$ be a $p$-adic Lie group. User Nate raises a good issue in the comments: If we study representations of $G$ on complex (or real, or $\ell$-adic, or abstract discrete) vector spaces which meaningfully respect the Lie (i.e. topological) structure of $G$, the Lie algebra $\mathfrak{g}$ of $G$ is quite useless.

Now I assume from your question that you are actually interested in representations on $p$-adic vector spaces (with the same $p$ as for $G$), and I think this is a valid question then. I do not have a definite answer, but some observations, which however are all to be taken with a grain of salt and will probably be criticised by true experts:

  1. $p$-adic representations of $p$-adic groups as a theory is actually quite young and full of active research in what is called $p$-adic Langlands. See this overview talk from 2006 and this one from 2010 (and certainly a lot happened since then, but is not in textbooks and even more beyond my understanding). Notice that Lie algebras get mentioned in the first one, although one quickly goes to the universal enveloping algebra $U(\mathfrak{g})$ and then various completed rings constructed from that (cf. point 7 below). There are attempts to study $p$-adic analogues of Verma modules (https://arxiv.org/abs/1008.3897) etc., so it is not true that no-one is talking about $p$-adic Lie algebras. So maybe, if you are doing research in that direction, you can actually push for using more Lie algebras, I would cheer for you! However, there are some things one should be aware of:

  2. Algebraic/arithmetic facts that are very different from the standard real/complex setting: $\Bbb R$ has only one true algebraic field extension $\Bbb C$, which further is algebraically closed, so e.g. everything semisimple is "split". "Complexification" is an extremely powerful tool in the classical theory. Further, there is e.g. the remarkable fact that all simple Lie groups/algebras have exactly one compact form over $\Bbb R$, and that is exploited in the representation theory of those groups. None of this works in the $p$-adic world: Although in the right setting one can find, for any given group or Lie algebra, some finite splitting extension, there is no "universal splitting field" (if anything, something like $\overline{\Bbb Q_p}$ or $\Bbb Q_p^{unr}$ or completions thereof might be, and are actually used sometimes, but are not local fields anymore ...). Further: Among the simple groups, only those of type $A_n$ have compact forms in the $p$-adic setting; but for big $n$, they typically have many non-isomorphic ones. So even if we had a Lie-group-Lie-algebra-correspondence, much of the classical theory would still not work.

  3. Topological facts that are very different from the standard real/complex setting: $p$-adic manifolds are totally disconnected w.r.t. their standard topology. So everything that involves arguments of connectedness and simply connectedness in the real theory has no chance to work without adaptations. Which of course mathematicians have tried to develop, e.g there is the ongoing research of "(locally) analytic representations" laid out in the first link in 1. Before that came:

  4. A huge class of interesting $p$-adic Lie groups are in fact ($p$-adic points of linear/affine) algebraic groups, and among them the reductive groups, like your example $GL_n$. An amazing bulk of theory, including algebra-ised notions of connectedness and simply-connectedness, can be made to work over any field, see the standard textbooks by Borel or Springer. Much more can be said if that ground field is local; this theory of (reductive) algebraic groups over local fields was laid out in the big works of Borel-Tits and Bruhat-Tits (part 1, part 2) (plus scheme-theoretic input, SGA3), and these provide mathematicians with tools that go far beyond Lie algebras. Actually, it turns out that the entire root system machinery does generalise nicely and is much more important in itself than in its "easy, linear-algebraic" manifestation in Lie algebras; so that the Lie algebras which we all originally learned as the link between groups and root systems can be sidestepped and then completely fall out of (or become a footnote in) these theories. Which, by the way, are still pursued and extended in ongoing research, see e.g. recent book by Conrad, Gabber, Prasad on Pseudo-Reductive Groups.

  5. A particular reason for that is that, very different from the situation over $\Bbb R$ and $\Bbb C$, in $p$-adic theory integrality issues become both important and intricate. Note e.g. that the $p$-adic exponential has a "small" radius of convergence, which is why even in the basic theory of $p$-adic Lie groups (Lazard (who actually talks a lot about Lie algebras), Dixon, Du Sautoy, Mann, Segal) much room is given to investigating structures over $\Bbb Z_p$. For these, in turn, often the reduction mod $p$ is a tool too fruitful to ignore, however in characteristic $p$, the Lie-algebra-Lie-group-correspondence infamously breaks down (whereas one of the great features of Borel-Tits theory is its almost independence from characteristic).

  6. Connecting points 3 and 5 with just another instance of how different the theories must be: One is often interested in (maximal) compact subgroups. Over $\Bbb R$, a standard compact subgroup of $SL_n(\Bbb R)$ would e.g. be $SO_n(\Bbb R)$. However, an interesting compact subgroup of $SL_n(\Bbb Q_p)$ is $SL_n(\Bbb Z_p)$! We can learn something about the first inclusion with Lie algebras -- about the second one, probably not.

  7. Vaguely in the spirit of Lie algebras, but much better adapted to the $p$-adic seeting, you'll find other algebraic structures whose module theory "envelops" $p$-adic representation theory: buzzwords you can search for are the completed group ring (a.k.a. Iwasawa algebra), distribution algebras, Robba rings, overconvergent Laurent series, Hecke algebras, Fontaine's period rings. In my (poor) understanding, they all are supposed to play roles vaguely similar to $U(\mathfrak{g})$ in the classical theory, but because of some or all of the above points are better suited for that.

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  • $\begingroup$ This is really helpful! Thank you so much. $\endgroup$
    – Seewoo Lee
    Feb 11, 2019 at 6:21

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