  In the late nineties T. Kobayashi wrote a series of papers in which he
established a criterion for the discrete decomposablity of restrictions of
unitary representations of reductive Lie groups to reductive subgroups. A key
tool in the proof of sufficiency of his criterion was the use of the theory of
hyperfunctions to study the microlocal behavior of characters of restrictions
to compact subgroups. In this paper we show how to replace this tool by
microlocal analysis in the $C^\infty$ category.
