50 research outputs found
Betti numbers of Springer fibers in type A
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.Comment: 17 page
Schubert decompositions for ind-varieties of generalized flags
Let be one of the ind-groups , ,
and be a splitting parabolic
ind-subgroup. The ind-variety has been identified with
an ind-variety of generalized flags in the paper "Ind-varieties of generalized
flags as homogeneous spaces for classical ind-groups" (Int. Math. Res. Not.
2004, no. 55, 2935--2953) by I. Dimitrov and I. Penkov. In the present paper we
define a Schubert cell on as a -orbit on
, where is any Borel ind-subgroup of
which intersects in a maximal ind-torus. A
significant difference with the finite-dimensional case is that in general
is not conjugate to an ind-subgroup of , whence
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 in "Direct
limits of Schubert varieties and global sections of line bundles" (J. Algebra
320 (2008), 3187--3198).Comment: Keywords: Classical ind-group, Bruhat decomposition, Schubert
decomposition, generalized flag, homogeneous ind-variety. [26 pages