# Lie algebra embeddings and the center of their enveloping algrabras

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?

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Take a look at "Shifted Schur Functions"

Andrei Okounkov, Grigori Olshanski http://arxiv.org/abs/q-alg/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 Okounkov-Olshanski-... story can be generalized to loop algebra case

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