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Let $G$ be a $p$-adic reductive group, so by definition as a locally profinite group it's the group of $\mathbb Q_p$ points of a connective reductive group over $\mathbb Q_p$, $K$ be a parahoric subgroup of $G$, and $Z$ be the center of $G$. Conisder another compact open subgroup $K_0$ inside $G$, so $K_0$ acts on $X=G/ZK$, and assume the $K_0$-fixed points are finite.

For some natural filtration of compact open subgroups $K_n$ of $K_0$ (the p-adic Lie filtration, the filtration by intersections with congruence subgroups, and so on), how does the number of fixed points of $K_n$ on $X=G/ZK$ grow with respect to $n$? Can we decide the number of fixed points explicitly in good cases?

Example: Consider $K_0=GL_n(\mathbb Z_p)$ acts on $GL_n(\mathbb Q_p)/\mathbb Q_p^{\times} GL_n(\mathbb Z_p)$, then the problem reduces to decide the number of invariant lattices up to scaling inside the standard representation $\mathbb Q_p^n$. For $K_m=\Gamma(p^m)$, the $K_m$-invariant lattices are precisely those lattices between $p^m\mathbb Z_p^n$ and $\mathbb Z_p^n$.

Motivation: invariant lattices for Galois representations where the quotient Galois group is a compact open subgroup inside a $p$-adic reductive group.

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  • $\begingroup$ @reuns This is a typo, I shall mean parahoric subgroup. $\endgroup$
    – sawdada
    Feb 11, 2020 at 8:32
  • $\begingroup$ For a lattice in $L\subset Z_p^n$ I think you can understand the action of $I+p^k M_n(Z_p)$ on $ L$ in term of a representative $L=A Z_p^n$ such that $v(A_i)\le v(A_{i+1})$ and $(v(A_1),\ldots, v(A_n))$ is minimal for the lexicographic order among all representatives $\in A GL_n(Z_p)$. Equivalently, in term of $\# L / (p^{v(A_1)} Z_p^n \cap L)$ $\endgroup$
    – reuns
    Feb 11, 2020 at 8:57

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