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Let $F$ be a finite degree extension over $\mathbf{Q}_p$ and consider the locally profinite group $G:=GL_2(\mathbf{Q}_p)$.

P1: Give an interesting example (non-artificial one, i.e., one that arises in real life for a representation theorist) of a non-smooth representation $\rho$ of $G$ on a topological $\mathbf{C}$-vector space $V$.

So here I would like the representation $\rho:G\rightarrow Aut(V)$ to be at least continuous in the following sense: for every $v\in V$ I want the orbit map $\pi^v:G\rightarrow V$, $g\mapsto \pi(g)(v)$ to be continuous.

P2: Give an interesting example of a smooth representation of $G$ on a topological $\mathbf{C}$-vector space which is not admissible.

added: Is it possible to construct such representations by inducing over an appropriate closed subgroup of $G$?

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2 Answers 2

up vote 8 down vote accepted

The "smoothness" prevents taking Hilbert-space completions in general, for example. That is, for example, with $G=GL_n(F)$ for a $p$-adic field $F$ and $n\ge 1$, the Hilbert space $V=L^2(G)$ with right translation by $G$ is a continuous representation in the strong topology but is not smooth. In fact, this would be the case for any unimodular totally-disconnected (non-discrete) group $G$.

The subspace of smooth vectors in $L^2(G)$ is dense, of course.

The smooth vectors in $L^2(G)$ are a natural example of a smooth repn that is (too big to be) admissible. Not hard to check.

Edit: and responding to the further-question above, note that the other good answer about action of G on compactly-supported (complex-valued) functions on the Bruhat-Tits building (or tree, for $SL_2(\mathbb Q_p)$), is indeed the (compactly-supported, smooth) induced representation of the trivial representation on the Iwahori subgroup, up to the whole $SL_n(\mathbb Q_p)$.

Crazily enough, if one lifts a "cuspidal" repn from $SL_n(\mathbb F_p)$ to $K=SL_n(\mathbb Z_p)$, and then induces that to $SL_n(\mathbb Q_p)$, a finite direct sum of supercuspidal repns is obtained, so this induced repn is admissible... in contrast to inducing the trivial repn from $K$.

(Also, of course, principal series inducing admissibles on the Levi components are admissible.)

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4  
@Paul: I erased my answer when I noticed it was the same as you (except that, for political reasons, I had chosen left translations...) –  Alain Valette Oct 7 '12 at 13:42
    
@Alain ... "Political"! :) –  paul garrett Oct 7 '12 at 16:02
    
I see, so I had obvious examples just under my nose! Thanks Paul –  Hugo Chapdelaine Oct 7 '12 at 18:47

Another example, this time of a natural representation which is smooth, but not admissible. The group $G$ acts naturally on its Bruhat-Tits tree $X$. Let $C(X)$ be the space of complex-valued functions on (the set of vertices of) $X$ with finite support. Then $G$ has a natural representation on $C(X)$, which is smooth (the stabilizer of a point is a maximal compact subgroup is a compact maximal, the stabilizer of a function is an intersection of finitely many such maximal compact subgroup hence is open) but not admissible (for example, the space of invariants in $C(X)$ by a compact maximal is the underlying space of the unramified (or spherical) Hecke algebra of $G$, which has infinite dimension over $\mathbb C$.

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Thanks Joel for the nice example. Is there a good (recent) reference on representations of topological groups (locally compact is fine with me) which give a good overview of the various "categories" of representations (smooth, admissible etc....)? –  Hugo Chapdelaine Oct 7 '12 at 18:52
    
@Hugo : A nice reference for learning representations of locally profinite groups is Bushnell-Henniart, The local Langlands conjecture for GL(2). –  François Brunault Oct 7 '12 at 20:37
    
@Francois, I did not see much examples in this book outside admissible representations. After all, their goal is to prove local Langlands correspondence. I would like to have a reference where one can get a feeling of the various types of representations: for example unitary versus non-unitary, continuous versus not strongly continuous, smooth but not admissible. For example if you look at $GL_2(R)$ with the discrete topology, "how many more" representations do you get from looking only at smooth ones. Basically, I want to see various ways of organizing representations of top. groups. –  Hugo Chapdelaine Oct 8 '12 at 13:41

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