Let $H = L^{2}(S^{1},\mathbb{C}^{n})$, $H\_{0}\subseteq H$ the subset of maps that extend holomorphically to the unit disc, and $H\_{m} = z^{m}H\_{0}$. Consider the affine Grassmannian for $GL\_{n}$ in the lattice model: 


$\mathcal{Gr}\_{n} = \left\{ V\subseteq H:\, zV\subseteq V\right\}$


Let me describe two possible filtrations of this object. The first, is by the subspaces whose lattice diagrams have all their sticks and dots concentrated in a band of width $2i$ about zero:


$\mathcal{Gr}\_{n}^{i} = \left\{V \in \mathcal{Gr}\_{n}:\, H\_{i}\subseteq V\subseteq H\_{-i}\right\}$


I think this can also be identified as the union of orbits of elements $f \in \mathrm{GL}\_{n}(\mathcal{K})$ such that $f(z) = \sum\_{\vert k\vert \leq m} A\_{k}z^{k}$.


The second filtration I can think of comes from the decomposition of the affine Grassmannian into $\mathcal{Gr}\_{\lambda}$'s. For each coroot $\lambda = (\lambda\_{1},\dots,\lambda\_{n})$ satisfying $\lambda\_{1}\geq \lambda\_{2}\geq\dots\geq \lambda\_{n}$, we get a finite dimensional variety $\mathcal{Gr}\_{\lambda}$ from the orbit of $GL\_{n}(\mathcal{O})$ through $\text{diag}(z^{\lambda\_{1}},\dots,z^{\lambda\_{n}})\cdot H\_{0}$.


If a coroot $\lambda = (\lambda\_{1},\dots,\lambda\_{n})$, define its height by $\text{ht}(\lambda) = \sum\_{i} \lambda\_{i}$. Let $\check{T}/W$ denote the set of Weyl orbits of the coroot lattice, for which I always pick representatives in the positive Weyl chamber, and define:


$X\_{i} = \coprod\_{\lambda \in \check{T}/W,\, \text{ht}(\lambda) \leq i} \mathcal{Gr}\_{\lambda}$


My question: Is $X\_{i}$ the same space as $\mathcal{Gr}\_{m}^{i}$? I think the main difficulty I'm having with answering this for myself is that I'm not sure how to compute to which $\mathcal{Gr}\_{\lambda}$ a given lattice $V$ belongs. Any reference or explanation for this would be much appreciated!


Thanks!



Do the following two filtrations of the affine Grassmannian agree?
This statement isn't true; as you've currently defined everything, your $X\_i$ are infinite dimensional (in fact they are unions of connected components.) You want instead to take the $X\_i$ unions of $\lambda$ with $i\geq\lambda\_1$ and $\lambda\_n\geq -i$.


Let $B$ be the Borel for $GL\_n(\mathcal{K})$. Both $\mathcal{Gr}\_n^i$ and $X\_i$ are $B$ invariant (as they are $GL\_n(\mathcal{O})$ invariant), so by Bruhat decomposition we just need to check that they contain the same set of points of the form $\operatorname{diag}(z^{\lambda\_1},\cdots,z^{\lambda\_n})\cdot H\_0.$ $\mathcal{Gr}\_n^i$ contains such a point if and only if all the $\lambda\_i$ are from $-i$ to $i$, and $\mathcal{Gr}\_{\lambda'}$ contains such a point if and only if $\lambda$ and $\lambda'$ are Weyl conjugate. This immediately gives what you want.


(As for an explicit description of which $\mathcal{Gr}\_{\lambda}$ a given lattice is contained in: This is determined by the "dimensions" of the intersections of the lattice with $H\_i$ (where by dimension I really mean valuation of $z$ in the determinant.) This is because this is a $G(\mathcal{O})$ invariant property and then we use Bruhat decomposition as above.)


If you would prefer to avoid using Bruhat decomposition, for $GL\_n$ all of this can be done explicitly by explicit diagonalization starting from dimensions of intersections with $H\_i$. Sorry if any of this is sketchy; I don't have much time right now, if necessary I can give more details later.


