There is a Lattice model for affine Grassmannians of type A, due to Lusztig. It describes affine Grassmannians of type A as the moduli space of certain subspaces in an infinite-dimensional $\mathbb{C}-$vector space.

Is there any similar pictorial models for affine Grassmannian of other types? Any references?

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    $\begingroup$ There are lattice models for Bruhat-Tits buildings associated to groups of classical type: P. Abramenko, G. Nebe: Lattice chain models for affine buildings of classical type. Math. Ann. 322 (2002), no. 3, 537–562. Given that affine Grassmannians and Bruhat-Tits buildings are fairly close (both related to linear groups over discretely valued fields) it may be possible to turn the lattice models for the Bruhat-Tits buildings into lattice models for the affine Grassmannians for classical types. $\endgroup$ Oct 24, 2014 at 13:23
  • $\begingroup$ Beilinson-Drinfeld Quantization of Hitchin's system. There is also a survey by Goertz on Affine springer fibers and afffine Deligne-Lusztig. $\endgroup$
    – prochet
    Nov 2, 2014 at 6:36

1 Answer 1


For any $ G$, you can always embed the affine Grassmannian $Gr_G$ into the space of $\mathbb C [[t]]$-lattices in $ \mathfrak{g} \otimes \mathbb C ((t))$. I'm pretty sure that this appears in Lusztig's paper.

Alternatively, for any $ G $, there is an embedding $ Gr_G \rightarrow \prod_{\lambda} \{ \text{lattices in } V(\lambda) \otimes \mathbb C ((t)) \}$. The image of this embedding will be those sequences of lattices which are compatible with morphisms $ V(\lambda) \otimes V(\mu) \rightarrow V(\nu) $. See section 10.3 in Finkelberg-Mirkovic for more details.

Unfortunately, neither of these embeddings are as helpful as the one for $ GL_n$.


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