Igor Krichever introduced an algebro-geometric construction of solutions of KP equations starting from an algebraic curve with some additional data.

George Wilson introduced the adelic Grassmannian, which is a subspace of the Sato Grassmannian parametrizing all solutions to KP equations.

Question: can every point of Wilson's adelic Grassmannian be obtained by Krichever's algebro-geometric construction?


Yes - the adelic Grassmannian precisely parametrizes "rational solutions of KP", which are the Krichever solutions attached to rank 1 torsion free sheaves on cuspidal genus 0 curves -- i.e. curves with $P^1$ as their bijective normalization (or subrings of the field of rational functions). Its adelic (or factorization) nature is explained by keeping track of the finite subset of $P^1$ which is the location of the cusps - i.e. this space parametrizes "Hecke modifications" (Backlund transformations) supported at given points of $P^1$. This is stated as Corollary 5.21 in https://arxiv.org/abs/math/0212094 though I think was known to Wilson, Berest, Mulase and others.

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    $\begingroup$ It might be worth clarifying that these solutions are not called rational for the evident relation to P^1 (ie for the spectral curve), but rather because they give Lax operators which are rational (meromorphic rather than formal), ie for the “differential” parameter. This “coincidence” is a form of Wilson’s bispectrality for these solutions (a kind of Fourier transform) $\endgroup$ – David Ben-Zvi May 8 at 13:57

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