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Let $G$ be a connected reductive group over a non archimedean local field $k$. Let $P \subset G$ be a parabolic subgroup with Levi decomposition $P=MN$, $Z_M \subset L$ be the center of $M$ and $S_M \subset Z_M$ be the maximal split torus.

It is commonly used that there exist element $s \in S_M(k)$ whose action by conjugation on $N(k)$ is strictly contracting, meaning that for any compact open subgroup $N_0 \subset N(k)$, the intersection $\bigcap_{n \geq 0} s^n N_0 s^{-n}$ is trivial.

This is equivalent to $v(\alpha(s))>0$ for all root $\alpha$ appearing in $\rm{Lie}(N)$, $v$ being the valuation on $k$.

Unfortunately I could not find a reference for this fact.

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Let $G$ be a connected reductive group of an arbitrary field $k$, $P$ a parabolic $k$-subgroup, and $U = \mathscr{R}_u(P)$ the unipotent radical of $P$ (so $U$ is what is unfortunately traditionally denoted as $N$).

For any Levi $k$-subgroup $M$ of $P$ there is a 1-parameter $k$-subgroup $\lambda:{\rm{GL}}_1 \rightarrow G$ such that $P = P_G(\lambda)$, $M = Z_G(\lambda)$, and $U = U_G(\lambda)$. In particular, $\lambda$ factors through the maximal split central $k$-torus $S_M$ in $M$, so $\lambda(t) \in S_M(k)$ for any $t \in k^{\times}$. By the Borel-Tits structure theory, if $S$ is a maximal split $k$-torus in $M$ containing $S_M$ then it is also maximal as such in $G$ and the set $\Phi(G,S)$ of nontrivial $S$-weights on ${\rm{Lie}}(G)$ is a root system (spanning ${\rm{X}}(S)_{\mathbf{Q}}$), possibly non-reduced.

The $k$-group $U$ is directly spanned in any order by the root groups $U_a$ for non-multipliable $a \in \Phi(G,S)$ such that $\langle a, \lambda \rangle > 0$. (Of course, $\dim U_a$ might be very large, and $U_a$ might be non-commutative if $a$ is divisible.)

If $a$ is not divisible in $\Phi(G,S)$ then $U_a$ is a vector group admitting a linear structure relative to which the effect of $\lambda(t)$-conjugation on $U_a$ is scaling by $t^{\langle a, \lambda\rangle}$, and is an extension of such by another using $2a$ when $a$ is divisible. In this way we see that any $t \in k^{\times}$ with $|t| < 1$ yields $s := \lambda(t) \in S_M(k)$ that does the job (as $\langle a, \lambda \rangle > 0$ precisely for those $a$ occurring as $S$-weights on ${\rm{Lie}}(U)$).

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  • $\begingroup$ Thank you. I was looking for such $\lambda$. Is there an explicit description (given $G$, $P$ and a maximal split torus $S \subset G$) ? $\endgroup$
    – Arkandias
    Commented Mar 12, 2014 at 12:58
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    $\begingroup$ @Arkandias: You seek $\lambda$ without specifying $M$ (so with $P_G(\lambda)=P$ you could choose $Z_G(\lambda)$ to be $M$). Let $B$ be a minimal parabolic $k$-subgroup of $G$ contained in $P$ and containing $S$, so $\Phi(B,S)$ is a positive system of roots in $\Phi(G,S)$ contained in the parabolic set of roots $\Phi(P,S)$. Let $\Delta$ be the base of $\Phi(B,S)$, $\Delta^{\ast}$ the dual basis of X$_{\ast}(S\cap\mathscr{D}(G))_{\mathbf{Q}}$. Then $P=P_I$ with $I\subset\Delta$ as usual. Use $\lambda=N\cdot \sum_{\chi \not\in I}\chi$ for $N>0$ divisible to make $\lambda \in{\rm{X}}_{\ast}(S)$. $\endgroup$
    – user76758
    Commented Mar 12, 2014 at 13:09
  • $\begingroup$ Typo in above comment: in the final sum indexed by characters $\chi \in \Delta - I$, should be summing the associated (isogeny category) "cocharacters" $\chi^{\ast}$ in the dual basis $\Delta^{\ast}$. $\endgroup$
    – user76758
    Commented Mar 13, 2014 at 5:33

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