Self-dual representations of 𝑆𝐿(𝑛,𝐹)

Balasubramanian, Kumar

doi:10.1090/proc12739

Self-dual representations of $\operatorname {SL}(n,F)$
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by Kumar Balasubramanian

Proc. Amer. Math. Soc. 144 (2016), 435-444

DOI: https://doi.org/10.1090/proc12739

Published electronically: May 28, 2015

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Abstract:

Let $F$ be a non-Archimedean local field of characteristic $0$ and $G=\operatorname {SL}(n,F)$. Let $(\pi ,W)$ be an irreducible smooth self-dual representation $G$. The space $W$ of $\pi$ carries a non-degenerate $G$-invariant bilinear form $( , )$ which is unique up to scaling. The form $( , )$ is easily seen to be symmetric or skew-symmetric and we set $\varepsilon ({\pi })=\pm 1$ accordingly. In this article, we show that $\varepsilon {(\pi )}=1$ when $\pi$ is an Iwahori spherical representation of $G$.

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