Projected Richardson varieties and affine Schubert varieties

He, Xuhua; Lam, Thomas

Annales de l'Institut Fourier, Tome 65 (2015) no. 6, pp. 2385-2412.

Soit $G$ un groupe algébrique complexe quasi-simple et $G/P$ une variété de drapeaux partiels. La projection sur $G/P$ des variétés de Richardson (de la variété des drapeaux complets) forment une stratification de $G/P$. Nous montrons que les relations d’adhérence des variétés de Richardson projetées correspondent à celles d’un certain sous-ensemble de variétés de Schubert sur la variété de drapeaux affine de $G$. Nous comparons aussi les classes de cohomologie équivariante et de $K$-théorie de ces deux stratifications. Notre travail généralise celui de Knutson, Lam et Speyer pour la grassmannienne de type $A$.

Let $G$ be a complex quasi-simple algebraic group and $G/P$ be a partial flag variety. The projections of Richardson varieties from the full flag variety form a stratification of $G/P$. We show that the closure partial order of projected Richardson varieties agrees with that of a subset of Schubert varieties in the affine flag variety of $G$. Furthermore, we compare the torus-equivariant cohomology and $K$-theory classes of these two stratifications by pushing or pulling these classes to the affine Grassmannian. Our work generalizes results of Knutson, Lam, and Speyer for the Grassmannian of type $A$.

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