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Consider Schur polynomials - symmetric polynoms of variables x_1, ... x_n, indexed by $d_1 \ge d_2 \ge ... \ge d_n$. I wonder what is known about the differential operators which have them as eigenfunctions ?

First, I am not sure such operators exist, but almost sure, by the reasons below.

Second, it seems to me (see reason below) that these operators have one parameter deformation which makes them trigonometric Calogero-Moser operators (may be modula some change of variables). Schur polynomials have also one parameter deformation - Jack polynomials, so it might be these deformations compatible, i.e. Jack polynoms - eigenfunctions of trigonometric Calogero-Moser.

Motivation 1

It is known (see Wikipedia) that Schur polynoms are specializations of Jack polynomials. Which are eigenfunctions of the so-called Sekeguchi operators (see e.g. first formula in Okounkov-Olshanski "SHIFTED JACK POLYNOMIALS, BINOMIAL FORMULA, AND APPLICATIONS" ). So I guess specialization of these operators should give desired operators for Schur polynomials.

I cannot remember clearly, but some time ago Evgeny Sklyanin showed me something like how to make change of variables from Sekeguchi operators to Calogero-Moser operators, unfortunately I cannot find the notes.

Motivation 2

The followings steps seems can give the desired operators.

1) Schur polynomials can be obtained as characters of the represenations of GL_n, restricted on diagonal matrices.

2) Characters are eigenfunctions for the center of universal enveloping acting by differential operators on functions on the group GL_n.

So we need to "restrict" these differential operators on diagonal matrices - and that should be the answer to the question. It seems to me if it is correct that should be well-known. Naive word "restrict" means "calculate the radial part", i.e. we should take into account that characters are functions invariant for conjugation action of GL_n on itself, center of U(gl_n) also invariant differential operator, so acting by inv.oper. on inv. func. we get inv. func. So we get desired operators.

Why I think this is related to the trigonometric Calogero-Moser. By Etingof-Ginburg rational Calogero-Moser can be obtained by "deformation" of similar procedure applied to invariant differential operators with constant coefficients. Center of U(gl_n) I think plays the same role, but for trigonometric Calogero-Moser. Probably this is too long to put details here...

Some related questions:

representation theoretic interpretation of Jack polynomials

Jack polynomials as determinants

"The" Harish-Chandra homomorphism (see Etingof-Ginzburg) for invariant dif. opers. on gl_n - what are images of higher order Laplacians e.g. Tr(D^3) = d_ijd_jkd_ki ?

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1 Answer

up vote 6 down vote accepted

Schur polynomials are defined to be the characters of irreducible representations of $G=GL_n$. By the Weyl character formula, $$s_\lambda=\frac{1}{\delta}\sum_{w\in W}(-1)^{l(w)}e^{w(\lambda+\rho)},$$ where $\delta$ is the Vandermonde determinant.

Clearly, the numerator is an eigenfunction of the Laplace operator on the Cartan $\mathfrak{h}$: $$L_1(\delta s_\lambda):=\Delta (\delta s_\lambda) = (\lambda+\rho, \lambda+\rho) \delta s_\lambda.$$

So, the Schur polynomials themselves are eigenfunctions of $M_1=\delta^{-1} L_1\delta$.

The operators $L_1$ and $M_1$ admit one-parameter deformations $L_k$ and $M_k=\delta^{-k} L_k\delta^k$ to the quadratic Sutherland (trigonometric Calogero-Moser) and the Sekiguchi operators. Eigenfunctions of $M_k$ are the Jack polynomials.

The degeneration Jack -> Schur corresponds to taking the coupling constant $k=1$ in which case the Sutherland Hamiltonians are all free.

You can see formulas for a general root system in hep-th/9403168: the conjugated quadratic Hamiltonian is given by (2.4).

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Thank you for the answer! –  Alexander Chervov Apr 7 '12 at 6:02
    
I clarified a little how things work for the quadratic Hamiltonian. –  Pavel Safronov Jun 12 '12 at 20:29
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