Distinguished positive regular representations

Document Type: Special Issue of BIMS in Honor of Professor Freydoon Shahidi

Author

Abstract

Let $G$ be a tamely ramified reductive $p$-adic‎ ‎group‎. ‎We study distinction of a class of irreducible admissible representations‎ ‎of $G$ by the group of fixed points $H$ of an involution‎
‎of $G$‎. ‎The representations correspond to $G$-conjugacy classes of‎ ‎pairs $(T,\phi)$‎, ‎where $T$ is a‎ ‎tamely ramified maximal torus of $G$ and $\phi$ is a quasicharacter‎ ‎of $T$ whose restriction to the maximal pro-$p$-subgroup‎ ‎satisfies a regularity condition‎.

‎Under mild restrictions on the residual characteristic of‎ ‎$F$‎, ‎we derive necessary conditions for $H$-distinction of‎ ‎a representation corresponding to $(T,\phi)$‎, ‎expressed in terms of properties of $T$ and $\phi$‎ ‎relative to the involution‎.

‎We prove that if an $H$-distinguished representation arises from‎ ‎a pair $(T,\phi)$ such that $T$ is stable under the involution and‎ ‎compact modulo $(T\cap H)Z$ (here‎, ‎$Z$ is the centre of‎
‎$G$)‎, ‎then the representation is $H$-relatively supercuspidal‎.

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