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The reflection character of $D_n(q)$

Published online by Cambridge University Press:  24 February 2005

ANTHONY C. KABLE
Affiliation:
Department of Mathematics, Oklahoma State University, Stillwater OK 74078, U.S.A. e-mail: akable@math.okstate.edu and nilabh.sanat@wku.edu
NILABH SANAT
Affiliation:
Department of Mathematics, Oklahoma State University, Stillwater OK 74078, U.S.A. e-mail: akable@math.okstate.edu and nilabh.sanat@wku.edu

Abstract

We evaluate the character of the reflection representation of the split special orthogonal group over a finite field $F$ of odd characteristic. The value of the character at $g$ is expressed in terms of the dimensions of the eigenspaces of $g$ for eigenvalues in $F$ and in the norm 1 torus in the quadratic extension of $F$, and the behavior of the restriction of the quadratic form preserved by the group to the 1 and ${-}1$ eigenspaces of $g$.

Type
Research Article
Copyright
2005 Cambridge Philosophical Society

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