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Let $F$ be a $p$-adic field and $G$ be split over $F$ plus all other usual adjectives. Let $\pi$ be an admissible tempered representation with Iwahori-fixed vectors. Let $P\supset I$ be some parahoric subgroup of $G$. What is known about the circumstances for which $\pi$ has $P$-fixed vectors?

For example, principal series will admit fixed vectors under maximal parahorics, and the Steinberg representation of $G$ doesn't admit any $P$-fixed vectors for $P\supsetneq I$. What goes on in between?

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It turns out that one can give a necessary criterion (most simply stated for tempered representations) for $P$-fixed vectors in terms of the unipotent part of the parameter and the dimension of the corresponding Springer fibre. I do so in Theorem 19 here (arXiv). Namely, if $w_{P}$ is the longest element in the parahoric subgroup of the affine Weyl group defined by $P$, and $\ell(w_P)>\dim_{\mathbb{C}}(\mathcal{B}_u)$, then $\pi^P=\{0\}$ for any standard module $\pi=K(u,s,\rho)$, where $\mathcal{B}$ is the flag variety of $G^\vee$.

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