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Collisions of Calogero-Moser particles and an adelic Grassmannian (with an appendix by I. G. Macdonald). (English) Zbl 0906.35089
A 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.

##### MSC:
 35Q53 KdV equations (Korteweg-de Vries equations) 14M15 Grassmannians, Schubert varieties, flag manifolds
##### Keywords:
KP equations; Calogero-Moser systems; Schur functions
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