Affine Schubert variety

Definition

Let $G$ be a split connected reductive group over an algebraically closed field, choose a split maximal torus, and let $\lambda$ be a dominant coweight. The positive loop group $L^+G$ acts on the affine Grassmannian, and the orbit through $t^\lambda$ is the affine Schubert cell $\operatorname{Gr}_G^\lambda$ .

The affine Schubert variety

\operatorname{Gr}_G^{\leq\lambda} =\overline{\operatorname{Gr}_G^\lambda}

is its reduced closure. It is a finite-dimensional projective variety, and its boundary is the union of cells indexed by dominant coweights below $\lambda$ in the dominance order.

References

Ivan Mirković and Kari Vilonen, “Geometric Langlands duality and representations of algebraic groups over commutative rings,” Annals of Mathematics 166 (2007), 95–143. arXiv.