We prove some new results on the Bruhat and Birkhoff decompositions of the affine Grassmannian in Lie type An. These decompositions generalize the classical Schubert and opposite Schubert decompositions of a flag variety. The Bruhat or Schubert decomposition of the affine Grassmannian into finite-dimensional cells has been studied extensively, but the Birkhoff decomposition into finite-codimensional strata has received less attention. The closure of a cell yields an (affine) Schubert variety, and the closure of a stratum yields a Birkhoff (ind-)variety. The main result is that, despite their singularities, Schubert and Birkhoff varieties possess tubular neighborhoods similar to those of smooth embedded submanifolds, and these allow us to compute their cohomologies. We also use the Birkhoff stratification to construct open affine neighborhoods of distinguished points of the Schubert, Birkhoff, and closely-related Richardson varieties, and develop a method for obtaining explicit defining polynomial equations for these neighborhoods. With a combination of these techniques we prove a number of results on the topology and geometry of Birkhoff and Richardson varieties, specializing to the case n = 1. We begin with a self-contained introduction to the affine Grassmannian in Lie type An, starting with a brief review of the theory of compact and complex Lie groups, their associated flag varieties, and loop groups.
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