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\(p\)-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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References:

[1] M.F. Atiyah and R. Bott , A Lefschetz fixed point formula for elliptic complexes: II applications , Ann. of Math. 88 (1968), 451-491. · Zbl 0167.21703
[2] I.N. Bernstein and A.V. Zelevinsky , Induced representations of reductive p-adic groups I , Ann. Sci. Ec. Norm. Sup. 10 (1977), 441-472. · Zbl 0412.22015
[3] R. Bott , Homogeneous vector bundles , Ann. Math. 66 (1957), 203-247. · Zbl 0094.35701
[4] P. Cartier , Representations of p-adic groups: A survey , in: Automorphic forms, representations and L-functions , Proc. Symp. Pure Math., 1979, pp. 111-155. · Zbl 0421.22010
[5] W. Casselman , Introduction to the theory of admissible representations of p-adic reductive groups (unpublished manuscript).
[6] W. Casselman , The unramified principal series of p-adic groups I ., Comp. Math. 40 (1980), 387-406. · Zbl 0472.22004
[7] W. Casselman and J. Shalika , The unramified principal series of p-adic groups II ., Comp. Math. 41 (1980), 207-231. · Zbl 0472.22005
[8] M. Demazure , A very simple proof of Bott’s theorem , Inven. Math. 33 (1976), 271-272. · Zbl 0383.14017
[9] V. Ginsburg , Deligne-Langlands conjecture and representations of affine Hecke algebras (preprint).
[10] N. Iwahori and H. Matsumoto , On some Bruhat decompositions and the structure of the Hecke ring of the p-adic groups , Inst. Hautes Études Sci. Publ. Math. 25 (1965), 5-48. · Zbl 0228.20015
[11] J.-S. Li , Some results on the unramified principal series of p-adic groups (preprint).
[12] D. Kazhdan and G. Lusztig , Proof of the Deligne-Langlands conjecture for Hecke algebras , Invent. Math. 87 (1987), 153-215. · Zbl 0613.22004
[13] G. Kempf , The Grothendieck-Cousin complex of an induced representation , Adv. Math. 29 (1978), 310-396. · Zbl 0393.20027
[14] C.D. Keys , On the decomposition of reducible principal series representations of p-adic Chevalley groups , Pac. Jn. Math. 101 (1982), 351-388. · Zbl 0438.22010
[15] G. Lusztig , Affine Hecke algebras and their graded version , Jn. AMS 2 (1989), 599-635. · Zbl 0715.22020
[16] I.G. Macdonald , The Poincaré series of a Coxeter group , Math. Ann. 199 (1972), 161-174. · Zbl 0286.20062
[17] M. Reeder , On certain Iwahori invariants in the unramified principal series , Pac. Jn. Math. (to appear). · Zbl 0804.22010
[18] M. Reeder , Old forms on GL(n) , Amer. Jn. Math. vol. 113 (1991), 911-930. · Zbl 0758.11027
[19] F. Rodier , Whittaker models for admissible representations of real algebraic groups, in: Harmonic Analysis on Homogeneous Spaces , Proc. Symp. Pure Math., 1973, pp. 425-430. · Zbl 0287.22016
[20] F. Rodier , Sur les représentations non ramifiées des groupes reductifs p-adiques: l’exemple de GSp(4) , Bull. Soc. Math. France, 116 (1988), 15-42. · Zbl 0662.22011
[21] J. Rogawski , On modules over the Hecke algebra of a p -adic group , Invent. Math. 79 (1985), 443-465. · Zbl 0579.20037
[22] T. Shintani , On an explicit formula for class-1 Whittaker functions , Proc. Japan Acad. 52 (1976), 180-182. · Zbl 0387.43002
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