On character sheaves and characters of reductive groups at unipotent classes

With a view to determining character values of finite reductive groups at unipotent elements, we prove a number of results concerning inner products of generalised Gelfand-Graev characters with characteristic functions of character sheaves, here called Lusztig functions. These are used to determine projections of generalised Gelfand-Graev characters to the space of unipotent characters, and to the space of characters with a given wave front set. Such projections are expressed largely in terms of Weyl group data. We show how the values of characters at their unipotent support or wave front set are determined by such data. In some exceptional groups we show that the projection of a generalised Gelfand-Graev character to a family with the same wave front set is (up to sign) the dual of a Mellin transform. Using these results, in certain cases we are able to determine roots of unity which relate almost characters to the characteristic functions. In particular we show how to compute the values of all unipotent characters at all unipotent classes for the exceptional groups of type $G_2$, $F_4$, $E_6$, ${}^2 E_6$, $E_7$ and $E_8$ by a method different from that of [L86, K2]; we therefore require weaker restrictions on $p$ and $q$. We also provide an appendix which gives a complete list of the cuspidal character sheaves on all quasi-simple groups.

Keywords

reductive group, character sheaf, Gelfand-Graev

2010 Mathematics Subject Classification

Primary 20C33. Secondary 20G05, 20G40.

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Published 19 November 2014