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Journal of Lie Theory 16 (2006), No. 2, 251--270
Copyright Heldermann Verlag 2006

Symmetry of Arthur Parameters under Aubert Involution

Dubravka Ban
Universität Münster, Mathematisches Institut, Einsteinstr. 62, 48149 Münster, Germany
Permanent Address: Dept. of Mathematics, Southern Illinois University, Carbondale, IL 62901, U.S.A.


For a generic irreducible representation $\pi$ of the odd orthogonal group SO$(2n+1,F)$ over a $p$-adic field $F$, we compute the Aubert involution $\hat{\pi}$ and the corresponding $L$-parameter. We show that, among generic representations, only tempered representations are base points attached to $A$-parameters and prove that in this case the $A$-parameters of $\pi$ and $\hat{\pi}$ are symmetric. In addition, we consider $A$-parameters $\psi$ of SO$(2n+1, F)$ corresponding to certain nontempered representations and prove that $\psi$ and $\hat{\psi}$ are symmetric.

Keywords: Arthur parameters, Aubert involution, odd orthogonal groups over $p$-adic fields.

MSC: 22E50, 11F70

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