The topology of Birkhoff varieties Luke Gutzwiller and Stephen A . Mitchell

The Schubert cells eλ are the orbits of the Iwahori subgroup B̃, while the Birkhoff strata Sλ are the orbits of the opposite Iwahori subgroup B̃ . The cells and the strata are dual in the sense that Sλ ∩ eλ = {λ}, and the intersection is transverse. The closure of eλ is the affine Schubert variety Xλ. It has dimension l (λ), where l is the minimal length occuring in the coset λW̃I , and its cells are indexed by the lower order ideal generated by λ in the Bruhat order on W̃/W̃I . Dually, the closure of Sλ is the Birkhoff variety Zλ. It is an infinitedimensional irreducible ind-variety with codimension l(λ). Its Birkhoff strata are indexed by the upper order ideal generated by λ. Thus the Birkhoff varieties may be viewed as analogous to the dual Schubert varieties from the classical setting, in which the role of F is played by a finite-dimensional flag variety. More generally, let I denote an upper order ideal in the Bruhat poset W̃/W̃I . Then ZI = ∪λ∈ISλ is a finite union of Birkhoff varieties. Our main theorem shows that in one respect, the classical and affine cases differ dramatically.