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By John Casey

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Extra info for A treatise on the analytical geometry of the point, line, circle, and conic sections, containing an account of its most recent extensions, with numerous examples

Example text

Ho] R. B. Howlett, Normalizers of parabolic subgroups of reflection groups, J. London Math. Soc. 2 1 (1980), 62–80. [KL] D. Kazhdan, G. Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math. 8 7 (1987) 153–215. [KNS] D. Kazhdan, V. Nistor, P. Schneider, Hochschild and cyclic homology of finite type algebras, Sel. , New ser. 4 (1998), 321–359. -L. Kim, Hecke algebras of classical groups over p-adic fields and supercuspidal representations, Amer. J. Math. 1 2 1 (1999), 967–1029.

KL] D. Kazhdan, G. Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math. 8 7 (1987) 153–215. [KNS] D. Kazhdan, V. Nistor, P. Schneider, Hochschild and cyclic homology of finite type algebras, Sel. , New ser. 4 (1998), 321–359. -L. Kim, Hecke algebras of classical groups over p-adic fields and supercuspidal representations, Amer. J. Math. 1 2 1 (1999), 967–1029. -L. Kim, Hecke algebras of classical groups over p-adic fields II, Compositio Math. 1 2 7 (2001), 117–167.

Also, an indecomposable VΛ,t is qK -pure of weight j if and only if, for each finite extension L of K, the restriction VΛ,t |L of VΛ,t to W DL ⊆ W DK is qL -pure of (the same) weight j. To see this, note since the condition is obviously stable under passage from K to L, it is enough to show the descent statement from an L, as above, such that Λ|L is unramified. In this case, if f = f (L/K) is the degree of the residue field extension, then Φf is a Frobenius element for WL , and, in the above notation, χ(Φf ) = χ0 (Φ)f .

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