I am studying Schubert variety and I came across a problem understand a particular detail.

Let $G$ be a reductive group, and $\mu\in X_{\bullet}(T)$ a coweight i.e. $\mu\in Hom(\mathbb{G}_m,T)$, where $T$ is the abstract Cartan of $G$. Then the Schubert variety $Gr_{\leq\mu}$ is defined to be

$$Gr_{\leq\mu}:=\{(E,\beta)\in Gr_G|Inv(\beta)\leq\mu\},$$

where $Gr_G$ is the affine Grassmannian of $G$ and $Inv(\beta)\in G(\mathcal{O})\backslash G(F)/G(\mathcal{O})$ via the Cartan decomposition.

I am interested in the following special case. Let $G=GL_n$, for $\mu=(u_1,u_2,\cdots,u_n)$, how can we describe the Schubert variety $Gr_{\leq\mu}$ in terms of lattices?

Thank you in advance for any comments and answers! I also appreciate if you would like to provide some reference which helps to solve this problem.

  • $\begingroup$ I am not sure I understand your question. In the definition you've given for the spherical Schubert variety, there is an implied partially ordered set. Is your question as to whether or not this poset is a lattice? $\endgroup$ – Simone Weil Aug 30 '16 at 3:10
  • $\begingroup$ @Harambe Close to that. Because now we already have a particular value for $\mu$, I would like to see how the Schubert variety look like in terms of lattices which is the more familiar definition for me. In fact, I would be happy if I could know how the Schubert variety look like if $\mu$ is given a simpler value. $\endgroup$ – David Lucien Aug 30 '16 at 6:32
  • $\begingroup$ I think I understand your question better, but unfortunately I don't have a good answer. In the "traditional" Grassmannian, Schubert varieties can be defined based on their intersections with subspaces. In the affine Grassmannian, the role of subspaces are replaced (somewhat) with lattices, but I don't know that Schubert varieties in the affine Grassmannian can be defined using them. I should mention that I have not spent much time working with affine Grassmannians. Perhaps you will be able to find a better answer, I hope so. $\endgroup$ – Simone Weil Sep 1 '16 at 21:51

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