Abstract

We develop a Bott-Borel-Weil theory for direct limits of algebraic groups. Some of our results apply to locally reductive ind-groups G in general, i.e., to arbitrary direct limits of connected reductive linear algebraic groups. Our most explicit results concern root-reductive ind-groups G, the locally reductive ind-groups whose Lie algebras admit root decomposition. Given a parabolic subgroup P of G and a rational irreducible P-module, we consider the irreducible G-sheaves [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] and their duals [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] These sheaves are locally free, in general of infinite rank. We prove a general analog of the Bott-Borel-Weil Theorem for [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] namely that [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /] is nonzero for at most one index q = q0 and that [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="05i" /] is isomorphic to the dual of a rational irreducible G-module V. For q0 > 0 we show that (in contrast to the finite dimensional case) V need not admit an irreducible P-submodule. There, however, one has a larger parabolic subgroup wPP, constructed from P and a Weyl group element w of length q0, such that V is generated by an irreducible wP-submodule. Consequently certain G-modules V can appear only for q0 > 0, never for q0 = 0. For [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="06i" /] we show that there is no analog of Bott's vanishing theorem, more precisely that [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="07i" /] can have arbitrarily many nonzero cohomology groups. Finally, we give an explicit criterion for the projectivity of the ind-variety G/P, showing that G/P is in general not projective.

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