  We study the intermediate extension of the character sheaves on an adjoint
group to the semi-stable locus of its wonderful compactification. We show that
the intermediate extension can be described by a direct image construction. As
a consequence, we show that the ``ordinary'' restriction of a character sheaf
on the compactification to a boundary piece inside the semi-stable locus is a
shift of semisimple perverse sheaf and is closely related to Lusztig's
restriction functor (from a character sheaf on a reductive group to a direct
sum of character sheaves on a Levi subgroup). We also provide a (conjectural)
formula for the boundary values inside the semi-stable locus of an irreducible
character of a finite group of Lie type, which gives a partial answer to a
question of Springer \cite{Sp2}. This formula holds for Steinberg character and
characters coming from generic character sheaves. In the end, we verify
Lusztig's conjecture \cite[12.6]{L3} inside the semi-stable locus of the
wonderful compactification.

Representation Theory;Algebraic Geometry