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We investigate the mod-$p$ supersingular representations of $GL_2(D)$, where$D$ is a division algebra over a $p$-adic field with characteristic 0, bycomputing a basis for the vector space of the pro-$p$ Iwahori subgroupinvariants of a certain quotient of a compact induction. This work generalizesthe results of Hendel and Schein.

On supersingular representations of $GL_2(D)$ with a division algebra $D$ over a $p$-adic field