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- W2034170203 abstract "In [7], Raghunathan proved a vanishing theorem for H’(r, V,) when I is a uniform (cocompact) discrete subgroup of a semi-simple Lie group G and V,, is an irreducible real representation of G. In the case G is simple, Raghunathan’s theorem implies H’ (I, I’,,) = 0 unless G is locally isomorphic to SO@, 1) and V, = sj V where V is the standard representation of SO (n, 1) and .@‘j Vdenotes the harmonic (annihilated by the wave operator A 8 /at2) polynomials on Vof degreej or G is locally isomorphic to S U (n, 1) and V, = Sj I’, the symmetric power of the standard representation V or its dual. Here j is a non-negative integer, possibly zero. The purpose of this paper is to prove a complement to Raghunathan’s theorem in the case G = SO (n, 1). Before stating our theorem we establish some notation and terminology. By a two sided hypersurface we mean a hypersurface with trivial normal bundle. Any orientable hypersurface of an orientable manifold is two-sided. D will denote the symmetric space attached to G, however, the symbol H” will be used to denote hyperbolic n-space. If ,VO is a representation of G and I is a uniform, discrete, torsion free subgroup of G then I’, will denote the corresponding flat bundle (local system) over the locally-symmetric space M = rD. Since M is a space of type K(r, 1) we have" @default.
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- W2034170203 date "1985-01-01" @default.
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- W2034170203 title "A remark on Raghunathan's vanishing theorem" @default.
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- W2034170203 doi "https://doi.org/10.1016/0040-9383(85)90018-7" @default.
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