Let /mathfrak{g} be a complex semi-simple Lie algebra with a parabolic subalgebra /mathfrak{p}. A module X of /mathfrak{g}  is called discretely decomposable if there exists a filtration of /mathfrak{g}-modules of finite length, the union of which is X. Now let /tau be an involutive automorphism of /mathfrak{g} with the set of fixed point /mathfrak{g}'. Moreover, denote by G the adjoint group Int(/mathfrak{g}) of /mathfrak{g}, and G' and P the corresponding subgroup of /mathfrak{g}' and /mathfrak{p} respectively. The main result is that the followings are equivalent: 1) G'P is closed in G; 2) For any simple /mathfrak{g}-module X in the generalized BGG category O^/mathfrak{p}, the restriction of X to /mathfrak{g}' contains at least one simple /mathfrak{g}'-module. 3) For any simple /mathfrak{g}-module X in the generalized BGG category O^/mathfrak{p}, the restriction of X to /mathfrak{g}' is discretely decomposable as a /mathfrak{g}'-module. This job is due to KOBAYASHI Toshiyuki (小林 俊行 さん). In this talk, I shall assume that you already fully understand finite dimensional Lie algebra with its representation theory,  and generalized BGG category. Also, I shall assume that you are familiar with the definition and some fundamental properties of linear algebraic group and Lie group.