Let $G$ be a finite group, $K$ be a $p$-adic field with an uniformizer $\pi$ and residue field $k \cong \Bbb F_q$, and $V$ be an irreducible representation of $G$ over $K$.
Let $X_{V}^G$ be the set of $G$-invariant lattices in $V$ modulo scaling (if $\Lambda$ is invariant so is $a \Lambda$ for any $a \in K^{\times}$).
For any $\Lambda \in X_V^G$, by reduction we get a $k$-representation $\rho_{\Lambda}$ of $G$ (well-defined up to isomorphism). Let $Y_{V}$ be the set of all isomorphism classes of such representations respect to $V$.
It's well-known that $X_V^G \not = \varnothing$ and $\rho_{\Lambda}^{ss}$ (semisimplification) is independent of $\Lambda$. My question is,
- How do we compute $\# X^G_V$ and $\# Y_V$?
- How do we classify the shape of $X_{V}^G$ (a little vague, maybe inside Bruhat-Tits building as below)?
If $\text{dim}V=2$, then we can consider the relation $\pi L_1\subseteq L_1 \subseteq L_2$ to make $X_V$ the set of all lattices module scaling in $V$ a tree i.e the Bruhat-Tits tree. Using such tool, one proves that $X_V^G$ is a finite set (convex inside $X_V$) and is a singleton, or a segment, or a graph that all neighbors of any interior point in $X_{V}$ lie in $X_{V}^G$.
And the reduction representations behaves differently according the shape of $X_V^G$ and their positions, in particular we can use one reduction representation to detect the shape of all reduction representations. From this we can construct some reduction representations (reducible but not semi-simple), which is probably due to Ribet in his proof of the converse to Herbrand theorem (However, the generalized Ribet lemma follows a different approach).
Example: $G=S_3$, $V$ the unique $2$-dim irreducible representation over $\Bbb Q_p$, if $p \not= 3$ then the invariant lattice is unique. If $p=3$ there are only two invariant lattices $\mathbb Z_p^2$ and $\Bbb Z_p(1,1)+\Bbb Z_p(3,0) $, and the two mod $p$ representations are reducible and not isomorphic (the semisimplification is $\text{trivial} \oplus \text{sgn}$).
What if the dimension is large than 2? $X^G_V$ is still a finite set by a compactness argument. Moreover, the global field analogue (assuming absolute irreducibility)is still true by local-global principle for lattices. This can be also seen from Jordan–Zassenhaus theorem.
Example: If some $\rho_{\Lambda_0}$ is irreducible, then it's not hard to see $X_V^G=\{ \Lambda_0 \}$: irreducibility means there is no invariant lattice between $\pi \Lambda_0$ and $\Lambda_0$, so if there is other any invariant lattice $\Lambda$, we can assume $\Lambda \subseteq \Lambda_0$ by scaling, then consider $\pi \Lambda_0+\Lambda$ to get $\Lambda \subseteq \pi \Lambda_0$, replacing $\Lambda$ by $\pi^{-1}\Lambda$ eventually we get a contradiction. , $S_n$ act on the $\mathbb Q_p$ vector space $V=\{(x_i) \in \Bbb Q_p^n |\sum_{i=1}^n x_i=0\}$ by permuting the coordinates. How do we compute all invariant lattices?
Example: let $S_n$ act on the $\mathbb Q_p$ vector space $V=\{(x_i) \in \Bbb Q_p^n |\sum_{i=1}^n x_i=0\}$ by permuting the coordinates. Then we can compute all invariant lattices--precisely those lying between the natural two lattices i.e $\mathbb Z^n$ and its dual, see Feit, Walter. Integral representations of Weyl groups rationally equivalent to the reflection representation, J. Group Theory 1, 1998, no. 3, 213–218.
Edit: I thought about the problem for a while and some original problems are solved. To my limited knowledge, not many examples are known. For example what about $S_5$? I don't know how to compute all invariant lattices of the irreducible 5-dimensional representations and the 6-dimensional representation.
