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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?

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  • $\begingroup$ did you figure out anything more about this? Do you know where to look in the literature? $\endgroup$ – arjun Sep 20 '18 at 3:27
  • $\begingroup$ @arjun By the way, I replied to your email. Just in case. $\endgroup$ – imakhlin Sep 22 '18 at 12:30
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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?

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  • $\begingroup$ Thanks! Is it obvious that the $Z_i$ for various $i$ are algebraically independent? $\endgroup$ – imakhlin Dec 26 '16 at 3:53
  • $\begingroup$ Ok, I think I've got it. A nonzero $p\in Z_1\otimes\ldots\otimes Z_n$ may not annihilate every finite-dimensional $\mathfrak{sl}_n$ irrep. Roughly, because the constant by which it acts on a given element of a GT basis depends polynomially (for a fixed $p$) on the corresponding GT pattern and the set of all GT patterns for all dominant integral highest weights is Zariski dense. $\endgroup$ – imakhlin Dec 26 '16 at 22:03
  • $\begingroup$ Deleted two comments, one best just stick with the $\mathfrak{gl}_n$ setting here, no need to consider $\mathfrak{sl}_n$ (including the above argument). $\endgroup$ – imakhlin Dec 28 '16 at 0:20
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GT commutative subalgebra is certainly maximal commuative.

In our paper https://arxiv.org/abs/0710.4971 We observed certain relation of GT and Jucys-Murphy. We should work not universally but in very specific representation C^n \otimes C^m. In that specific case we might identify the images of the two subalgebras. I have not thought on that few years so might forget a bit some ideas.

So the basic line of the ideas is the following

There is classical gln-glm duality which in partical says that centers of both ugl maps to the same subalgebra acting on S(C^n\otimes C^m). The fact above we observed is of the same spirit.

The high level picture is that GT and JM are both very very degenerated case of Of more geneneral maximal commutative subalgebras Which are related to Gaudin subalgebra Which should be thought coming from center of affine version of gl. Certain facts can be thought as affine versions of Gln-glm duality and their certain degenerations In c^n \otimes c^m case manifest themselves as GT=JM if i remember things correctly...

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