$\SL_n$, Orthogonality Relations and Transfer

http://dx.doi.org/10.4153/CJM-2007-019-8
Canad. J. Math. 59(2007), 449-464

  Published:2007-06-01
 Printed: Jun 2007

Let $\pi$ be a square integrable representation of $G'=\SL_n(D)$, with $D$ a central division algebra of finite dimension over a local field $F$ \emph{of non-zero characteristic}. We prove that, on the elliptic set, the character of $\pi$ equals the complex conjugate of the orbital integral of one of the pseudocoefficients of~$\pi$. We prove also the orthogonality relations for characters of square integrable representations of $G'$. We prove the stable transfer of orbital integrals between $\SL_n(F)$ and its inner forms.