Character sheaves on the semi-stable locus of a group compactification

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.