Let $G$ be a unitary group $U(n)$ and $G’$ is a subgroup $U(m)$. (i.e. $m\le n$)
Let $\mathfrak{g},\mathfrak{g}'$ be the complexified Lie algebra of $G,G’$ and $\mathfrak{z},\mathfrak{z}’$ be the center their universal enveloping algebra.
Let $\mathfrak{t}’$ and $\mathfrak{t}$ be their maximal toral subalgebra of $\mathfrak{g}'$ and $\mathfrak{g}$ such that $\mathfrak{t}’ \subset \mathfrak{t}$.
Since $\mathfrak{t}’ \subset \mathfrak{t}$, I guess that there is a natural canonical surjection $\mathfrak{z}, \to \mathfrak{z},’$ which is compatible with domain restriction morphism $C^{\infty}((U(n)) \to C^{\infty}(U(m))$.(Here, we regards elements of $\mathfrak{z}$ and $\mathfrak{z}'$ as differential operators.)
More precisely, I am asking the exitence of canonical surjection $p:\mathfrak{z}, \to \mathfrak{z}’,$ such that for $\phi \in C^{\infty}((U(n))$ and $X \in \mathfrak{z}’$, $X \cdot (\phi|_{U(m)})=(Y \cdot \phi)|_{U(m)}$ where $Y$ is an arbitrary element in $p^{-1}(X)$.
Such $p$ does exists? If so how can we construct it? Some expert has alluded to me to use Harish-Chandra isomorphism. But I have no idea how to use it in this situation.
Thank you in advance!
