  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 unipo- tent 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 adjoint
groups of type G2, F4, E6, E7 and E8. We also pro- vide an appendix which gives
a complete list of the cuspidal character sheaves on all quasi-simple groups.
