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The main message is that there is "certain relation" (described below) between standard Gelfand-Tsetlin maximal commutative subalgebra in $U(gl_N)$ and the maximal commutative subalgebra in $C[S_N]$ generated by Jucys-Murphy elements. The relation consists of two steps which can be seen as generalized Schur-Weyl duality and generalized $gl_M - gl_N$ duality. Both steps involves an intermediate object - "bending flow" commutative in $U(gl_N \oplus ... \oplus gl_N)$ (sum contains $M$ terms). Briefly speaking these generalized dualities say that: images in certain representations of these commutative subalgebras coincide.

So we get certain commutative subalgebra in $U(gl_N \oplus ... \oplus gl_N)$ such that it is "Schur-Weyl dual" to JM subalgebra, meaning that the images of these subalgebras in $End(V)$ coincide.

Theorem: the images in $End(V)$ of GT and "bending flows" coincide.

The main message is that there is "certain relation" (described below) between standard Gelfand-Tsetlin maximal commutative subalgebra in $U(gl_M)$ and the maximal commutative subalgebra in $C[S_M]$ generated by Jucys-Murphy elements. The relation consists of two steps which can be seen as generalized Schur-Weyl duality and generalized $gl_M - gl_N$ duality. Both steps involve an intermediate object - "bending flow" commutative subalgebra in $U(gl_N \oplus ... \oplus gl_N)$ (sum contains $M$ terms). Briefly speaking these generalized dualities say that: images in certain representations of these commutative subalgebras coincide.

So we get certain commutative subalgebra in $U(gl_N \oplus ... \oplus gl_N)$ such that it is "Schur-Weyl dual" to JM subalgebra, meaning that the images of these subalgebras in $End(V)$ coincide. Such a commutative subalgebra is called "generalized bended flows" or just "bending flows", by reason commented below.

Theorem: the images in $End(W)$ of GT and "bending flows" coincide.

There is a relation between JM in S_n and Gelfand-Tsetlin in U(gl). JM elements will be mapped into quadratic Casimir operators (if I remember correctly, not sure), so it might be possible to get the answer in this way, although, it would NOT be the shortest one:) But the dualities above might be of interest on their own. Consider representation of S_n in $V\oplus ... V$, JM gives commutative subalgebra there.

Remark. It is interesting to mention that in case V=R^3 it is the same as "bending flows" integrable system - very nice thing - bending the polygon along non intersecting diagonals you see a set of commuting hamiltonians on the moduli space of polygons... Commutativity is obvious from the geometry, lemma is to show that we get JM.

Return to JM: consider $V=C^k$ so $V\oplus ... \oplus V = C^k \otimes C^n$. The point is that if one considers representation of $U(gl_k)$ in $C^k \otimes C^n$ then image of Gelfand-Tsetlin subalgebra in $U(gl_k)$ will coincide with JM image. This was first observed by Flashka and Millson. Another treatment is given in our paper: http://arxiv.org/abs/0710.4971

Our point was to show that JM and bending flows are limit of the so-called Gaudin commutative subalgebra in $U(gl_n)\otimes ... \otimes U(gl_n)$, which is interesting from variaous points of view from Langlands to Bethe ansatz. Since we know that spectrum of JM is simple we can conclude that spectrum of Gaudin is simple in certain cases.

[EDIT] @Darij

"JM elements will be mapped into quadratic Casimir operators". By what?

By the construction I tried to described below. I will try to describe it more clearer again and happy to answer any questions...

Schur-Weyl duality?

No it is not that simple. It is composition of Shur-Weyl with GL_n-GL_m dualities, but both are being generalized from their classical form to more strong statements.

How exactly?

Let me try to say it again:

Consider representation of $gl_k$ in $C^k\otimes C^n$ (where we act only the first space). Consider the representation of $S_n$ in the $C^k\oplus ... \oplus C^k$ (here $n$-times is summed). Note that there is isomorphism of vector spaces: $C^k\otimes C^n = C^k\oplus ... \oplus C^k$.
  So we see that two algebras $U(gl_k)$ and $S_n$ act on the same vector space $C^k\otimes C^n = C^k\oplus ... \oplus C^k$.

Fact 1. The image in $End( C^k\otimes C^n ) $ of the JM commutative subalgebra in $S_n$ coincide with the image of GT in the same space.

Is this formulation clear ?

The fact above was about the subalgebras (i.e. vector spaces), we might want to ask for more strong fact - take some particular element in JM subalegbra i.e. $X_k$ and ask what is the particular element in GT such the their images coincide under the representations above ?

"quadratic Casimir" - means the quadratic element of the center of U(gl), it is not unique, but almost unique: any such element have form $a C_2 + b C_1^2$, where a,b are complex numbers and $C_1$ - generator of the center of gl itself (i.e. identity matrix) and $C_2$ is any quadratic generator of the center.

As far as I rememeber one should take quadratic Casimirs in GT of U(gl_k) and "they will be mapped into JM elements X_l" (meaning that the images of these elements will coincide under representations above). How to fix "a,b" I do not remember. The point is that the number $l$ in index of JM element $X_l$ will be the same as index of $gl$ subalgebra in $gl_l \in gl_n$ in construction of GT subalgebra in U(gl_k).

Is the claim clear ?

1. I am sorry: my previous posts were incorrect, I will correct below.

2. I would suggest you guys insert the statement and may be proof to Wiki article, it is quite worth and since it was mainly written by me, imho I might give such a suggestion.

The main message is that there is "certain relation" (described below) between standard Gelfand-Tsetlin maximal commutative subalgebra in $U(gl_N)$ and the maximal commutative subalgebra in $C[S_N]$ generated by Jucys-Murphy elements. The relation consists of two steps which can be seen as generalized Schur-Weyl duality and generalized $gl_M - gl_N$ duality. Both steps involves an intermediate object - "bending flow" commutative in $U(gl_N \oplus ... \oplus gl_N)$ (sum contains $M$ terms). Briefly speaking these generalized dualities say that: images in certain representations of these commutative subalgebras coincide.

Since I forget some details I would NOT make again the claim that "JM elements go to "quadratic Casimirs"", which might give another (but very long) way to answer Igor's question. Just simply describe the relation which might be interesting on its own.

Step 1. Generalized Schur-Weyl from JM to "bending flows". (Rather trivial step).

Consider $V=C^N \otimes ... \otimes C^N$ ($M$ terms in tensor product). $C[S_M]$ acts here in a natural way. $U(gl_N \oplus ... \oplus gl_N)$ surjects on $End(V)$. Since it surjects we can find certain elements in $U(gl_N \oplus ... \oplus gl_N)$ which are mapped to JM elements, moreover we require such elements to be quadratic in generators of $U(gl_N \oplus ... \oplus gl_N)$, and it would fix these elements. The basic idea is that the permutation operator (12) acting in $C^N\otimes C^N$ is OBVIOUSLY an image of $\sum_{ij} E_{ij}\otimes E_{ji} \in U(gl_N)\otimes U(gl_N)=U(gl_N\oplus gl_N)$ and nothing more than that.

By $E_{ij}$ denoted the matrix with $1$ at position $(ij)$ and zeros everywhere else.

So we get certain commutative subalgebra in $U(gl_N \oplus ... \oplus gl_N)$ such that it is "Schur-Weyl dual" to JM subalgebra, meaning that the images of these subalgebras in $End(V)$ coincide.

Step 2. $GL_M-GL_N$-duality from "bending flows" to Gelfand-Tsetlin. (This step is not so trivial). It is mainly due to Flaschka and Millson - section 8 of http://arxiv.org/abs/math.SG/0108191

Consider the vector space $W = S(C^N\otimes C^M) = S(C^N \oplus ... \oplus C^N)$ (M-terms in sum) and $S$ denotes symmetric algebra of the vector space. Lie algebras $gl_M$ and $U(gl_N \oplus ... \oplus gl_N)$ acts on $W$ in a natural way.

Theorem: the images in $End(V)$ of GT and "bending flows" coincide.

In such a form it is Theorem 2 page 9 in our paper: http://arxiv.org/abs/0710.4971

Why the name "bending flows" ? If we make similar considerations for $U(so_3 \oplus ... \oplus so_3)$ or more precisely its associated grade Poisson algebra $S(so_3 \oplus ... \oplus so_3)$ we get a (Poisson) commutative subalgebra there.

The beautiful fact is that "JM" type generators have very nice geometric interpretation. We can identify $so_3=R^3$ and so elements of $(so_3 \oplus ... \oplus so_3)$ can be seen as $M$-gons in $R^3$. The statement is that if we "bend" polygon along the non-intersecting diagonals then such flows will be hamiltonian and will be defined by JM-type generators in $S(so_3 \oplus ... \oplus so_3)$. Well, I omitted some details and may be comment is not so clear, one should draw simple pictures in order to see what is going on.

Bending flows were proposed for $S(so_3 \oplus ... \oplus so_3)$ in paper M. Kapovich, J. Millson, The symplectic geometry of polygons in Euclidean space,J. Differ. Geom. 44, 479–513 (1996)

Generalized further in several papers in particular in Gregorio Falqui, Fabio Musso, Gaudin Models and Bending Flows: a Geometrical Point of View, J. Phys. A 36 (2003), no. 46,11655–11676. nlin.SI/0306005

http://arxiv.org/abs/nlin/0306005

[EDIT] @Darij

"JM elements will be mapped into quadratic Casimir operators". By what?

By the construction I tried to described below. I will try to describe it more clearer again and happy to answer any questions...

Schur-Weyl duality?

No it is not that simple. It is composition of Shur-Weyl with GL_n-GL_m dualities, but both are being generalized from their classical form to more strong statements.

How exactly?

Let me try to say it again:

Consider representation of $gl_k$ in $C^k\otimes C^n$ (where we act only the first space). Consider the representation of $S_n$ in the $C^k\oplus ... \oplus C^k$ (here $n$-times is summed). Note that there is isomorphism of vector spaces: $C^k\otimes C^n = C^k\oplus ... \oplus C^k$.
So we see that two algebras $U(gl_k)$ and $S_n$ act on the same vector space $C^k\otimes C^n = C^k\oplus ... \oplus C^k$.

Fact 1. The image in $End( C^k\otimes C^n ) $ of the JM commutative subalgebra in $S_n$ coincide with the image of GT in the same space.

Is this formulation clear ?

The fact above was about the subalgebras (i.e. vector spaces), we might want to ask for more strong fact - take some particular element in JM subalegbra i.e. $X_k$ and ask what is the particular element in GT such the their images coincide under the representations above ?

"quadratic Casimir" - means the quadratic element of the center of U(gl), it is not unique, but almost unique: any such element have form $a C_2 + b C_1^2$, where a,b are complex numbers and $C_1$ - generator of the center of gl itself (i.e. identity matrix) and $C_2$ is any quadratic generator of the center.

As far as I rememeber one should take quadratic Casimirs in GT of U(gl_k) and "they will be mapped into JM elements X_l" (meaning that the images of these elements will coincide under representations above). How to fix "a,b" I do not remember. The point is that the number $l$ in index of JM element $X_l$ will be the same as index of $gl$ subalgebra in $gl_l \in gl_n$ in construction of GT subalgebra in U(gl_k).

Is the claim clear ?