In this article we calculate the signature character of certain Hermitian representations of $GL_N(F)$ for a $p$-adic field $F$. We further give a conjectural description for the signature character of unramified representations in terms of Kostka numbers.
Proof of Vogan's conjecture on Arthur packets: simple parameters of p-adic general linear groups
Some Unipotent Arthur Packets for Reductive $p$-adic Groups
Distinguished Representations for $\rm{SL}_n(D)$ where $D$ is a quaternion division algebra over a $p$-adic field
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