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                <term xml:lang="en">homogeneous ind-variety</term>
                <term xml:lang="en">generalized flag</term>
                <term xml:lang="en">Schubert decomposition</term>
                <term xml:lang="en">Bruhat decomposition</term>
                <term xml:lang="en">Classical ind-group</term>
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              <p>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].</p>
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