On finite symplectic modules arising from supercuspidal representations

Let $F$ be a non-Archimedean local field with finite residue field. Let $\mathcal{A}^{et}_n(F)$ be the collection of isomorphism classes of essentially tame irreducible supercuspidal representations of $\mathrm{GL}_n(F)$ studied by Bushnell-Henniart. It is known that we can parameterize $\mathcal{A}^{et}_n(F)$ by the collection $P_n(F)$ of equivalence classes of admissible pairs $(E, \xi)$ consisting of a tamely ramified extension $E/F$ of degree $n$ and an $F$-admissible character $\xi$ of $E^\times$. We are interested in a finite symplectic module $V = V(\xi)$ arising from the construction of the supercuspidal representation from the character $\xi$. This module $V$ is known to admit an orthogonal decomposition with respect to a symplectic form depending on $\xi$. We work with a fixed ambient module $U$ containing $V$ and show that $U$ decomposes in a way analogous to the root space decomposition of the Lie algebra $\mathfrak{gl}_n(F)$. We then obtain a complete orthogonal decomposition of the submodule $V$ by restriction. Such decomposition relates the finite symplectic module of a supercuspidal and certain admissible embedding of L-groups. This relation provides a different interpretation on the essentially tame local Langlands correspondence.