I'm trying to figure out/find in literature the details concerning Gelfand-Tsetlin algebras for $\mathfrak{gl}\_n(\mathbb C)$ (Okounkov-Vershik style, if you wish).


Consider the chain $$\mathcal U(\mathfrak{gl}\_1)\subset\ldots\subset\mathcal U(\mathfrak{gl}\_n),$$ denote the centers $Z\_i\subset\mathcal U(\mathfrak{gl}\_i)$ and define the commutative algebra $$GT\_n=\langle Z\_1,\ldots,Z\_n\rangle.$$


One sees that elements of $GT\_n$ act diagonally in the Gelfand-Tsetlin basis in any finite-dimensional irrep of $\mathfrak{gl}\_n$. Beyond that little is clear to me. It'd be great if someone could provide an answer or reference for any of the following. (Of course, 3) is the key question here.)


1) Is $GT\_n$ *precisely* the subalgebra of $\mathcal U(\mathfrak{gl}\_n)$ acting diagonally in any GT basis? Is it a maximal commutative subalgebra?


2) Is the centralizer $Z(\mathcal U(\mathfrak{gl}\_n),\mathcal U(\mathfrak{gl}\_{n-1}))$ commutative? Do we have $$Z(\mathcal U(\mathfrak{gl}\_n),\mathcal U(\mathfrak{gl}\_{n-1}))=\langle Z\_{n-1},Z\_n\rangle?$$


3) Are there generators of $GT\_n$ analogous in some way to the Jucys-Murphy elements? How do they act on an element of a GT basis in terms of the corresponding GT pattern (or SSYT)?


4) Is $GT\_n$ at all the natural object to consider here?



Gelfand-Tsetlin algebras and "Jucys-Murphy elements" for $\mathfrak{gl}_n$
1-2) Both of these seem likely, but without doing more reference hunting than I have time for, I won't swear to it. For 2), it's better to think about the $GL\_{n-1}$ invariants in $U(\mathfrak{gl}\_n)$, which one can understand using the PBW theorem. 


3) If you think about the combinatorics here, this just doesn't seem like quite the right framework. The algebra $GT\_n$ really has $\binom{n+1}{2}$ algebraically independent generators, whereas the JM elements are just a finite extension of the center (this is clearer if you think about the degenerate affine Hecke algebra). The obvious candidate is the Casimir of the different $\mathfrak{gl}\_m$'s, but I don't think this separates the different constituents of the restricted representation in the way you want; I don't think any element of the centralizer will (note that there is a quite interesting connection between the Casimir and JM elements in Schur-Weyl duality, but that's a bit different); it's basically obvious that it can't since the number of generators is just too small. You should just pick your favorite basis of the center of $\mathfrak{gl}\_n$; the resulting $\binom{n+1}{2}$ elements will be a free set of generators for $GT\_n$. If you were smart in how you chose your basis, then you understand exactly how it acts, based on the GT pattern (if you don't, you need to get a better favorite basis). 


4) Sure. What else are you going to consider?


