We determine the Betti numbers of the Springer fibers in type A.To do this, we construct a cell decomposition of the Springer fibers. The codimension of the cells is given by an analogue of the Coxeter length. This makes our cell decomposition well suited for the calculation of Betti numbers.

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Let B u be the Springer fiber over a nilpotent endomorphism u ∈ End(C n ). Let J (u) be the Jordan form of u regarded as a partition of n. The irreducible components of B u are all of the same dimension. They are labelled by Young tableaux of shape J (u). We study the question of the singularity of the components of B u and show that all the components of B u are nonsingular if and only if

Abstract. Let G be one of the ind-groups GL(∞), O(∞), Sp(∞) and P ⊂ G be a splitting parabolic ind-subgroup. The ind-variety G/P has been identified with an ind-variety of generalized flags in [4]. In the present paper we define a Schubert cell on G/P as a B-orbit on G/P, where B is any Borel ind-subgroup of G which intersects P in a maximal ind-torus. A significant difference with the finite-dimensional case is that in general B is not conjugate to an ind-subgroup of P, whence G/P admits many non-conjugate Schubert decompositions. We study the basic properties of the Schubert cells, proving in particular that they are usual finite-dimensional cells or are isomorphic to affine ind-spaces.We then define Schubert ind-varieties as closures of Schubert cells and study the smoothness of Schubert ind-varieties. Our approach to Schubert ind-varieties differs from an earlier approach by H. Salmasian [12].