Collisions of Calogero-Moser particles and an adelic Grassmannian (with an appendix by I. G. Macdonald).

*(English)*Zbl 0906.35089A rational solution of the KP equation is known to correspond to a rational Calogero-Moser system, which is a classical particle system with inverse square potential. This paper reformulates such a correspondence in a geometrical way. Let \(\overline{C}_n\) be the space of all pairs \((X,Z)\) of \(n \times n\) complex matrices such that \([X,Z] + I\) has rank one, and let \(C_n\) be the quotient space \(\overline{C}_n / GL(n, \mathbb C)\), where \(GL(n, \mathbb C)\) acts on \(\overline{C}_n\) by simultaneous conjugation of \(X\) and \(Z\). Let \(C_n'\) be the subspace of \(C_n\) represented by pairs \((X,Z)\) with \(X\) diagonalizable. Then \(C_n'\) can be identified with the phase space for classical \(n\)-particle systems.

With this identification the Calogero-Moser flows on \(C_n'\) are quotients of certain \(GL(n,\mathbb C)\)-invariant flows that are defined on the whole of \(\overline{C}_n\), and \(C_n\) can be regarded as the completed phase space for the complex Calogero-Moser system. On the other hand, the rational solutions to the KP hierarchy are parametrized by a certain adelic Grassmannian \(\text{Gr}^{\text{ad}}\), and there is a map from each \(C_n'\) into \(\text{Gr}^{\text{ad}}\) commuting with the Calogero-Moser and KP flows of these spaces.

In this paper the author proves that these maps extend to a bijection \(\beta: \bigcup_{n \geq 0} C_n \to \text{Gr}^{\text{ad}}\) and that the image of \(C_n\) under \(\beta\) is the union of all the open cells of dimension \(n\) in \(\text{Gr}^{\text{ad}}\). The paper has also an appendix by I. G. Macdonald which gives an elementary proof of a formula for Schur functions.

With this identification the Calogero-Moser flows on \(C_n'\) are quotients of certain \(GL(n,\mathbb C)\)-invariant flows that are defined on the whole of \(\overline{C}_n\), and \(C_n\) can be regarded as the completed phase space for the complex Calogero-Moser system. On the other hand, the rational solutions to the KP hierarchy are parametrized by a certain adelic Grassmannian \(\text{Gr}^{\text{ad}}\), and there is a map from each \(C_n'\) into \(\text{Gr}^{\text{ad}}\) commuting with the Calogero-Moser and KP flows of these spaces.

In this paper the author proves that these maps extend to a bijection \(\beta: \bigcup_{n \geq 0} C_n \to \text{Gr}^{\text{ad}}\) and that the image of \(C_n\) under \(\beta\) is the union of all the open cells of dimension \(n\) in \(\text{Gr}^{\text{ad}}\). The paper has also an appendix by I. G. Macdonald which gives an elementary proof of a formula for Schur functions.

Reviewer: Min Ho Lee (Cedar Falls)