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I am currently reading "The local langlands conjecture for the $GL(2)$", chapter two (smooth representations), where the author states:

"Most interesting locally profinite groups posses many non-semisimple smooth representations." But gives no example.

Remember: a locally profinite group is a locally compact, totally disconnected (hausdorff) topological group and a smooth representation is a complex representation of a locally profinite group $G$ $(\pi, V)$, where for every $v\in V$ exists a compact-open subgroup $K \subseteq G$, which fixes $v$.

I was trying to find an "interesting group" (~non discrete locally profinite group, like e.g. $GL_n(\mathbb Q_p)$) and a representation that is not semisimple, but had to admit that its harder than i thought because i am not very familiar with such groups. I thought maybe the Hecke Algebra, or some other function space might do the job, but i couldn't verify because i don't quite understand the topology of such groups.

I hope someone can help me out. I would be very glad to hear an example that is not a discrete, but locally profinite group, or a promising "direction" where i can find some.

Thanks in advance!

Tina
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2 Answers2

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See Section 9.10 of Bushnell-Henniart. When $n=2$, there is a representation $V$ of $G=\mathrm{GL}_2$ consisting of smooth compactly supported functions on $\mathbb P^1_{\mathbb Q_p}$. This has the trivial representation $1_G$ as a sub-representation since the constant function is compactly supported. This gives a short exact sequence $$0\to 1_G\to V\to \mathrm{St}_G\to 0,$$ which does not split. The same construction works for all $n$, where now $V$ is the space of smooth compactly supported functions on $\mathbb P^{n-1}_{\mathbb Q_p}$.

Kenta S
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Consider the homomorphism $\chi: GL_n(\mathbb{Q}_p) \to \mathbb{R}^\times$ given by $g \mapsto | \det(g) |$ ($p$-adic absolute value, of course). Then we get a two-dimensional representation by the rule $$ g \mapsto \begin{pmatrix} 1 & \log(\chi(g)) \\ 0 & 1 \end{pmatrix} $$ which is not semisimple. I stole this idea from the top voted answer here, where the same question is raised about Lie groups: Irreducible finite dimensional complex representation of $GL_2(\Bbb C)$.

You might be interested in infinite-dimensional examples. In the case of $GL_n(\mathbb{R})$, I believe that (at least some of) the induced representations coming from characters on the Borel are not semisimple, although they sit in an exact sequence in between an irreducible representation (the principal series?) and an irreducible finite-dimensional representation. I think the result also holds for the p-adic groups but do not know this.

hunter
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  • This is great, thank you! Unfortunately i can't accept two answers. – Tina Dec 03 '24 at 20:01
  • What do you mean by "Characters on the Borel" ? – Tina Dec 03 '24 at 20:04
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    Given two characters of $\mathbb{Q}_p^\times$, you get representations of the group $B = \begin{pmatrix} \star & \star \ 0 & \star \end{pmatrix}$ by evaluating on the diagonal entries and multiplying (ignore the off-diagonal entries). You can then induce this to a representation of $GL_2$ (similar story for $GL_n$). (There's a classification, and these plays an important part, it's not a totally arbitrary construction.) – hunter Dec 03 '24 at 20:49