A Bott-Borel-Weil theory for direct limits of algebraic groups

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 OG/P(E) and their duals OG/P(E). These sheaves are locally free, in general of infinite rank. We prove a general analog of the Bott-Borel-Weil Theorem for OG/P(E), namely that H(G/P;OG/P(E)) is nonzero for at most one index q = q0 and that Hq0 (G/P;OG/P(E)) 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 wP ⊃ P, 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 OG/P(E) we show that there is no analog of Bott’s vanishing theorem, more precisely that OG/P(E) 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