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.
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.
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...
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