96 research outputs found
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
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