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The basic tool is the decomposition ofN pairs of free charged bosons with respect toglN and the commuting withglN Lie algebra of infinite matrices?l.
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Recently, there is a renewed interest in wonderful varieties of rank two since they were shown to hold a keystone position in the theory of spherical varieties, see [L], [BP], and [K].
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It was then observed independentely by Lusztig and Ginzburg-Vasserot (see [L1], [GV]) that this construction admits an affine analogue in terms of periodic flags of lattices.
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As a consequence we prove a recent conjecture of Lusztig (see [L1]).
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The aim of this paper is to begin a study of the cohomology modules Hi(X(w),Lλ)
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