Harmonic Locus and Calogero-Moser Spaces

Abstract We study the harmonic locus consisting of the monodromy-free Schrödinger operators with rational potential and quadratic growth at infinity. It is known after Oblomkov that it can be identified with the set of all partitions via the Wronskian map for Hermite polynomials. We show that the harmonic locus can also be identified with the subset of Wilson’s Calogero–Moser space that is fixed by the symplectic action of $$\mathbb C^\times .$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>×</mml:mo> </mml:msup> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> As a corollary, for the multiplicity-free part of the locus we effectively solve the inverse problem for the Wronskian map by describing the partition in terms of the spectrum of the corresponding Moser matrix. We also compute the characters of the $$\mathbb C^\times $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>×</mml:mo> </mml:msup> </mml:math> -action at the fixed points, proving, in particular, a conjecture of Conti and Masoero. In the Appendix written by N. Nekrasov there is an alternative proof of this result, based on the space of instantons and the ADHM construction.