The Fine Structure of Translation Functors
Abstract
Let E be a simple finite dimensional representation of a semisimple Lie algebra with extremal weight nu and choose nonzero e in Enu. Let M(tau) be the Verma module with highest weight tau and vtau in M(tau)tau its canonical generator. We investigate the projection of e vtau in E M(tau) on the central character chi(tau + nu). This is a rational function in tau and we calculate its poles and zeros. We then apply this result in order to compare translation functors.
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