Let $\mathfrak{g}_1\subset \mathfrak{g}_2$ be a Lie algebra embedding. Assume both are semisimple. For instance take the standard diagonal embedding $\mathfrak{sl}(2, \mathbb{C})\subset \mathfrak{sl}(3, \mathbb{C})$. This lifts to an embedding $U(\mathfrak{sl}(2, \mathbb{C}))\subset U(\mathfrak{sl}(3, \mathbb{C}))$ of the corresponding enveloping algebras. Consider their centers $Z(\mathfrak{sl}(2, \mathbb{C})),Z(\mathfrak{sl}(3, \mathbb{C}))$. It is relatively easy to see that under the previous embedding we have that $Z(\mathfrak{sl}(2, \mathbb{C}))\cap Z(\mathfrak{sl}(3, \mathbb{C})) = \mathbb{C}$. Is there any reasonable way to "connect" these centers? Let me be a bit more clear. For instance, we can pass the same question to the symmetric algebras of $\mathfrak{g}_1,\mathfrak{g}_2$. Here we have $S(\mathfrak{sl}(2, \mathbb{C})^*)^{SL(2,\mathbb{C})}$ the algebra of $SL(2,\mathbb{C})$invariant functions on $\mathfrak{sl}(2, \mathbb{C})$ and $S(\mathfrak{sl}(3, \mathbb{C})^{\*} )^{SL(3,\mathbb{C})}$. There is a natural embedding of $S(\mathfrak{sl}(3, \mathbb{C})^{\*} )^{SL(3,\mathbb{C})}$ into $S(\mathfrak{sl}(3, \mathbb{C})^{\*} )^{SL(2,\mathbb{C})}$ and a projection of $S(\mathfrak{sl}(3, \mathbb{C})^{\*} )^{SL(2,\mathbb{C})}$ onto $S(\mathfrak{sl}(2, \mathbb{C})^{\*} )^{SL(2,\mathbb{C})}$ given by restriction. Are there any results that could describe how these centers behave under restrictions?

Take a look at "Shifted Schur Functions" Andrei Okounkov, Grigori Olshanski http://arxiv.org/abs/qalg/9605042 Section 10: "Coherence property of quantum immanants and shifted Schur polynomials" In particular formulas 10.4, 10.5  they discuss "averaging operators" Z(U(gl(n)) > ZU(gl(N)) , n < N and later prove certain "good" (coherence) property of special generators of the centers Z(U(gl(k)) which has been studied by the authors and M. Nazarov. Hope this helps... What is very interesting for me personally  is try to generalize such things to the case of loop algebras Z(U(\hat gl)). Here certain "good" elements of the centers has been constructed by Talalaev's formula, it is natural to expect that OkounkovOlshanski... story can be generalized to loop algebra case 


I'm doubtful about getting nice relationships between centers of universal enveloping algebras, if you look at embeddings for arbitrary pairs of semisimple Lie algebras; maybe there are subtle connections in special cases, however. For one thing, centers and algebras of invariant polynomial functions for a given semisimple Lie algebra are related only indirectly in the work of Chevalley and HarishChandra. Moreover, embeddings into special linear algebras of unbunded rank occur when you start with a fixed semisimple Lie algebra and consider all of its faithful irreducible representations. But the respective centers of the smaller and larger enveloping algebras are isomorphic to polynomial algebras in the number of indeterminates given by the respective ranks. If there is a useful way to relate centers, it should be visible in diagonal embeddings for Lie algebras both of small rank such as you describe. But it's not immediately visible to me. P.S. A tag liealgebras would be useful. 

