## $$p$$-adic Whittaker functions and vector bundles on flag manifolds.(English)Zbl 0819.22012

Let $$G$$ be a split reductive group over a $$p$$-adic field $$F$$. The author considers unramified principal series representations of $$G$$ and their Whittaker model. The image, in a Whittaker model, of the $$K$$-spherical vector of an unramified principal series representation is described by the Casselman-Shalika formula which links it to the character of a representation of the dual group to $$G$$. By the Borel-Weil theorem, this representation is realized in the global sections of a certain line bundle on the flag manifold associated to the dual group. In the same spirit, the author investigates the images of certain Iwahori fixed vectors and relates them to the Lefschetz traces of various cohomology groups of sheaves on the flag variety. As applications of his results he gives necessary and sufficient conditions for injectivity of the Whittaker map, generalizing results of Bernstein-Zelevinsky and extending his own previous results. The other main result is a non-vanishing result on the maximal $$F$$-split torus of $$G$$ for the image of Iwahori fixed vectors in the Whittaker model.

### MSC:

 22E50 Representations of Lie and linear algebraic groups over local fields 22E35 Analysis on $$p$$-adic Lie groups 43A85 Harmonic analysis on homogeneous spaces 14M17 Homogeneous spaces and generalizations 14M15 Grassmannians, Schubert varieties, flag manifolds
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