Let G be a connected reductive group over an algebraically closed field k and \(\lambda\) : \(G_ m\to G\) be a one-parameter subgroup of G. The centralizer of \(\lambda\) in G is another reductive group \(G'\). Let B be a Borel subgroup of G such that \(\lim_{t\to 0}\lambda (t)\cdot b(\lambda (t))^{-1}\) exists in G for all b in B. Then \(B'=G'\cap B\) is a Borel subgroup of \(G'\). Let \(\chi\) be a dominant character of B where \(\chi (\lambda (t))=t^ n\) for some integer n. Let \(\chi '\) be the restriction of \(\chi\) to \(B'\). Restriction of functions gives a \(G'\)-homomorphism \(\rho\) : \(V_ G(\chi)\to V_{G'}(\chi ')\) where \(V_ G(\chi)\) denotes the G-module induced from the character \(\chi\) of B and similarly with the prime. It is proved Theorem 1. \(\rho\) induces an isomorphism \(V_ G(\chi)[n]\to V_{G'}(\chi ')\) where [n] denotes the eigensubspace where \(\lambda\) (t) acts by multiplication by \(t^ n\). If \(W_ G(\chi)\) is the irreducible sub-G-module of \(V_ G(\chi)\), then the same theorem holds with W replacing V.