# Characters of simply connected semsimple algebraic groups over local fields

Let $G$ be a semisimple algebraic group over $\mathbb{Q}_p$. Then by definition $G$ admits no non-trivial algebraic characters, i.e. homomorphisms $G \to \mathbb{G}_m$.

However, it is quite possible that $G(\mathbb{Q}_p)$ admits topological characters. E.g. take $G=\mathrm{PGL}_n$ and consider the composition $$\mathrm{PGL}_n(\mathbb{Q}_p) \to \mathbb{Q}_p^*/\mathbb{Q}_p^{*n} \to S^1, \quad g \mapsto \chi(\det(g)),$$ where $\chi: \mathbb{Q}_p^*/\mathbb{Q}_p^{*n} \to S^1$ is some character.

In this special case $G$ is adjoint, however. I can also do similar constructions for other adjoint groups. So I'm wondering whether this can also happen for simply connected $G$.

Let $G$ be a simply connected semisimple algebraic group over $\mathbb{Q}_p$. Is any continuous homomorphism $$G(\mathbb{Q}_p) \to S^1$$ trivial?

• What do you mean by $g\mapsto \text{det}(g)$? For every linear representation that I can think of, the image of $\textbf{PGL}_n$ is contained in $\textbf{SL}_M$. – Jason Starr May 19 '16 at 9:55
• Are you taking $n=p-1$, so that the reduction of the determinant on $\textbf{GL}_{p-1}$ is a well-defined homomorphism $\textbf{PGL}_{p-1}(\mathbb{Z}_p) \to \mathbb{F}_p^\times$? If so, how do you extend this to $\textbf{PGL}_n(\mathbb{Q}_p)$? – Jason Starr May 19 '16 at 9:58
• For PGL_n, det takes values in H^1(k,mu_n)=k*/(k*)^n. But det is surjective and its target admits nontrivial homomorphisms to the circle when k is the p-adics so you've still got your topological character. – Peter McNamara May 19 '16 at 10:15
• @Jason and Peter: Yes sorry, I forgot to compose det with a character; I have changed the statement. – Daniel Loughran May 19 '16 at 10:35
• A version of @YCor's comment: Let $G$ be a simply connected absolutely simple $k$-group, where $k$ is a nonarchimedean local field (a $p$-adic field or the field of rational functions in one variable over a finite field). Assume that $G$ is isotropic (i.e., not isomorphic to $\mathrm{SL}(1,D)$ of a central division algebra $D$ over $k$). Then any nontrivial normal subgroup of $G(k)$ is central, hence finite. For a proof see the book by Platonov and Rapinchuk. Therefore, $G(k)$ admits no nontrivial homomorphisms into abelian groups. – Mikhail Borovoi May 19 '16 at 11:16

## 1 Answer

As I have written in a comment, the answer is YES (any abstract homomorphism into an abelian group is trivial) when $$G$$ is an isotropic, simply connected, simple algebraic group over a nonarchmedean local field $$k$$. For a proof see the book by Platonov and Rapinchuk, Section 7.2, Theorems 7.1 and 7.6. Note that any simply connected anisotropic simple group is isomorphic to $$\mathrm{SL}(1,D)$$, where $$D$$ is a central simple algebra over a finite separable extension $$K$$ of $$k$$.

However, the answer is NO when $$k=\mathbb{Q}_2$$, $$G=\mathrm{SL}(1,D)$$, and $$D$$ is the quaternion division algebra over $$k$$. EDIT of 18.11.2018: As Arkandias explains in his comments below, for the group $$G=\mathrm{SL}(1,D)$$ as above, it follows from the Corollary to Theorem 21 of Carl Riehm's paper The norm 1 group of a p-adic division algebra that the abelianization $$G^{\rm ab}:=G/[G,G]$$ is a group of order 3. Since the commutator subgroup $$[G,G]$$ is open and hence, closed in $$G$$, we see that $$G$$ admits a non-trivial continuous homomorphism to $$S^1$$.

• This example seems to contradict the Corollary to Theorem 21 in Riehm "The norm 1 group of a $p$-adic division algebra", which implies that the abelianization of $\mathrm{SL}(1,D)$ has order $q+1$. Am I missing something? – Arkandias Nov 15 '18 at 16:02
• @Arkandias: In Theorem 21 Riehm assumes that $D$ is not a dyadic quaternion algebra. – Mikhail Borovoi Nov 15 '18 at 18:58
• Yes, but the Corollary includes that case (see just above the statement, and most of the proof is about the dyadic quaternion algebra case). – Arkandias Nov 15 '18 at 21:36
• @Arkandias: Could you please explain how the corollary to Theorem 21 implies that the abelianization of ${\rm SL}(1,D)$ has order $q+1$? – Mikhail Borovoi Nov 16 '18 at 13:44
• @Arkandias: The maps $G(k)\to\mathbb{Z}_2/2\mathbb{Z}_2$ that I constructed is not a homomorphism! I will edit my answer tomorrow. – Mikhail Borovoi Nov 16 '18 at 20:00