How is George Wilson's adelic Grassmannian from e.g. the paper https://link.springer.com/article/10.1007%2Fs002220050237 related to the adeles or (especially) the affine Grassmannian (a.k.a. the loop Grassmannian)? Is there a more algebraic definition of the adelic Grassmannian than the one presented in the above paper of G. Wilson?
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2 Answers
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One answer is provided in this paper I wrote with Tom Nevins (inspired especially by work of Berest-Wilson). We show Wilson's Grassmannian is precisely analogous to the Beilinson-Drinfeld Grassmannian, which is the "adèlic" form of the affine Grassmannian (as explained eg in Zhu's excellent article cited by dorebell). Namely, it can be identified with the moduli space of ``D-line-bundles" (rank 1 projective D-modules) on a curve (usually $P^1$) equipped with a trivialization (identification with D) outside of finitely many points. Wilson's linear-algebraic picture is obtained from this D-module picture by applying a Riemann-Hilbert correspondence. It forms a factorization ind-scheme -- roughly speaking these play the role of groups to the "Lie algebras" which are vertex algebra (in this case the $\mathcal W_{1+\infty}$ algebra).
answered Aug 13, 2019 at 14:01
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Xinwen Zhu has fantastic notes on all sorts of affine Grassmannians from the point of view of algebraic geometry: see here. (You can take your base field to be $\mathbf{C}$ everywhere, and some of the ind-schemes and non-representable sheaves and things can probably actually be represented in a more concrete way by some infinite-dimensional complex-analytic spaces).
His section 4.3 describes a relationship between "adelic" affine Grassmannians and the adele ring of the corresponding function field (here, this would be the adele ring of the function field $\mathbf{C}(z)$, i.e. the restricted product of $\mathbf{C}((z-\lambda))$ as $\lambda$ ranges through $\mathbf{C} \cup \{\infty\}$).
I think Wilson's $\mathrm{Gr}_\lambda$'s are loosely analogous to the loop affine Grassmannian $\mathrm{Gr}_{\mathrm{GL}_1} = \mathbf{C}((z-\lambda))^\times/\mathbf{C}[[z-\lambda]]^\times$ (note that this latter space, however, is a disjoint union of non-reduced points). The $\mathrm{Gr}_\lambda$'s parametrize $\mathbf{C}$-subspaces of $\mathbf{C}(z)$ which, for some $k$, sit between $(z-\lambda)^k \mathbf{C}[z]$ and $(z-\lambda)^{-k}\mathbf{C}[z]$, with the condition that these inclusions have codimension $k$.
If you drop this codimension condition and require the subspace to be a $\mathbf{C}[z]$-module, you'd get $\mathrm{Gr}_{\mathrm{GL}_1}$. I don't really know how to think about Wilson's definition, or why it's useful for things - it seems related to a more analytic notion of affine Grassmannian. I think the relationship with the usual "loop" affine Grassmannian is mostly an analogy, and the two things arise in rather different contexts, but I'm not an expert here.
This becomes "adelic" when you allow $\lambda$ to vary, and consider a space parametrizing finite sets $\{\lambda_1, \ldots, \lambda_n\}$ inside $\mathbf{C}$. Wilson's $\mathrm{Gr}^{\mathrm{Ad}}$ is analogous to Zhu's $\mathrm{Gr}_{\mathrm{Ran}, \mathrm{GL}_1}$. The latter can be thought of as parametrizing finite subsets $\{\lambda_1, \ldots, \lambda_k\}$ of $\mathbf{C}$, together with $\mathbf{C}[z]$-modules inside of $\mathbf{C}(z)$ which sit between $(z-\lambda_1)^{-k} \cdots (z-\lambda_n)^{-k} \mathbf{C}[z]$ and $(z-\lambda_1)^k \cdots (z-\lambda_n)^k\mathbf{C}[z]$ for some $k$. The points of this space are related to $\mathrm{GL}_1$ of the adeles of $\mathbf{C}(z)$ by "Weil's uniformization theorem": see this answer.
