Categories of Bott-Samelson Varieties

  • Vladimir ShchigolevEmail author


We consider all Bott-Samelson varieties BS(s) for a fixed connected semisimple complex algebraic group with maximal torus T as the class of objects of some category. The class of morphisms of this category is an extension of the class of canonical (inserting the neutral element) morphisms BS(s)↪BS(s), where s is a subsequence of s. Every morphism of the new category induces a map between the T-fixed points but not necessarily between the whole varieties. We construct a contravariant functor from this new category to the category of graded \(\phantom {\dot {i}\!}H^{\bullet }_{T}(\text {pt})\)-modules coinciding on the objects with the usual functor \(\phantom {\dot {i}\!}H_{T}^{\bullet }\) of taking T-equivariant cohomologies. We also discuss the problem how to define a functor to the category of T-spaces from a smaller subcategory. The exact answer is obtained for groups whose root systems have simply laced irreducible components by explicitly constructing morphisms between Bott-Samelson varieties (different from the canonical ones).


Bott-Samelson variety Equivariant cohomology Category 

Mathematics Subject Classification (2010)



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Supported by the Russian Foundation for Basic Research grant no. 16-01-00756.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Financial University Under the Government of the Russian FederationMoscowRussia

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