Adelic Grassmannian: A Bispectral Moduli Space

• Adelic Grassmannian is a proalgebraic moduli space characterizing bispectral wave functions tied to the KP hierarchy and integrable systems.
• It constructs bispectral solutions by assigning to each point a unique wave function that satisfies differential equations in both spatial and spectral variables.
• Generalizations to vector and noncommutative settings unify concepts in the Calogero–Moser correspondence and bispectral Darboux transformations.

The adelic Grassmannian is a fundamental proalgebraic moduli space that arises at the confluence of integrable systems, bispectrality, noncommutative algebra, and the geometry of infinite-dimensional Grassmannians. It encapsulates and generalizes Wilson's classifying space for rank-one commutative bispectral rings of ordinary differential operators, with far-reaching connections to the KP hierarchy, Calogero–Moser spaces, and the representation theory of differential and difference operators. Modern formulations extend the adelic Grassmannian beyond the scalar and matrix cases to general finite-dimensional algebras, revealing its unifying role as the parameter space for bispectral Darboux transformations, decorated modules, and the spectral theory of integrable equations.

1. Definitions and Foundational Constructions

The original construction of the (scalar) adelic Grassmannian, denoted $\mathrm{Gr}^{\rm ad}$, is as a subvariety of Sato's big Grassmannian $\mathrm{Gr}$ of closed subspaces $W\subset\mathbb{C}((z^{-1}))$ of “half-infinite” dimension. Each point $W$ corresponds to a rational solution of the KP hierarchy whose associated KP wave function $\psi_W$ is bispectral in $x$ and $z$, i.e., it satisfies differential equations in both variables with polynomial coefficients.

Wilson's explicit realization is as follows: Fix the space $$\mathcal{D} = \operatorname{Span}\{\mathrm{ev}(k, \alpha): f(x)\mapsto f^{(k)}(\alpha)\mid k\in\mathbb{N}_0,\alpha\in\mathbb{C}\}$$ of one-point differential functionals on $\mathbb{C}[x]$. For every finite-dimensional homogeneous (adelic) subspace $C\subset\mathcal{D}$, define the kernel $\mathrm{Gr}$0 and the monic polynomial $\mathrm{Gr}$1 where $\mathrm{Gr}$2 are the supports of $\mathrm{Gr}$3. The associated subspace $\mathrm{Gr}$4 defines a point of $\mathrm{Gr}$5. Thus,

$\mathrm{Gr}$6

This construction can be characterized equivalently via support gap conditions or as the locus classifying ordinary bispectral commutative differential rings (Kasman et al., 2020, Casper et al., 2020).

Vector and noncommutative generalizations, such as the vector adelic Grassmannian $\mathrm{Gr}$7 and the $\mathrm{Gr}$8-valued adelic Grassmannian $\mathrm{Gr}$9 for a finite-dimensional complex algebra $W\subset\mathbb{C}((z^{-1}))$0, are obtained by replacing the scalar functionals, rings, and submodules with appropriate module-theoretic and matrix analogues, imposing primary decomposition or “adelic” support conditions (Wilson, 2015, Horozov et al., 2024).

2. Bispectrality, Calogero–Moser Correspondence, and Classification

A central property of the adelic Grassmannian is its role as a parameter space for bispectral wave functions. For any $W\subset\mathbb{C}((z^{-1}))$1, one constructs a unique (up to normalization) wave function

$W\subset\mathbb{C}((z^{-1}))$2

which is an eigenfunction of commuting ordinary differential operators in both $W\subset\mathbb{C}((z^{-1}))$3 and $W\subset\mathbb{C}((z^{-1}))$4. Specifically, there exist $W\subset\mathbb{C}((z^{-1}))$5 and $W\subset\mathbb{C}((z^{-1}))$6 such that

$W\subset\mathbb{C}((z^{-1}))$7

where $W\subset\mathbb{C}((z^{-1}))$8 are monic polynomials determined by $W\subset\mathbb{C}((z^{-1}))$9. The anti-isomorphism between the $W$0- and $W$1- Fourier algebras arises precisely from this bispectral structure (Casper et al., 2020).

For the vector case, the adelic Grassmannian is set-theoretically the disjoint union

$W$2

where $W$3 are (generalized) Calogero–Moser spaces consisting of quadruples $W$4 in a prescribed moment map level set modulo $W$5 conjugation, encoding positions, momenta, and residue data. The Gibbons–Hermsen system and multi-component KP flows linearize to translations on this moduli space, with explicit relations between the dynamical and geometric data via the stationary (and dynamical) Baker wave function (Wilson, 2015).

In the noncommutative $W$6-valued setting, decorated bispectral Darboux transformations of the $W$7-valued exponential function $W$8 are classified via nondegenerate, adelically decomposed right $W$9-submodules of