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Distribution Algebras on p-adic Groups and Lie Algebras

  • Allen Moy (a1)
Abstract

When F is a p-adic field, and is the group of F-rational points of a connected algebraic F-group, the complex vector space of compactly supported locally constant distributions on G has a natural convolution product that makes it into a ℂ-algebra (without an identity) called the Hecke algebra. The Hecke algebra is a partial analogue for p-adic groups of the enveloping algebra of a Lie group. However, has drawbacks such as the lack of an identity element, and the process is not a functor. Bernstein introduced an enlargement . The algebra consists of the distributions that are left essentially compact. We show that the process is a functor. If is a morphism of p-adic groups, let be the morphism of ℂ-algebras. We identify the kernel of in terms of Ker. In the setting of p-adic Lie algebras, with g a reductive Lie algebra, m a Levi, and the natural projection, we show that maps G-invariant distributions on to NG (m)-invariant distributions on m. Finally, we exhibit a natural family of G-invariant essentially compact distributions on g associated with a G-invariant non-degenerate symmetric bilinear form on g and in the case of SL(2) show how certain members of the family can be moved to the group.

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Copyright
References
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[BD] Bernstein, J., Représentations des groupes réductifs sur un corps local. Travaux en Cours., Hermann, Paris, 1984 .
[C] Cartier, P., P. Representations of ℘-adic group: A survey. In: Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., 33, American Mathematical Society, Providence, RI, 1979, pp. 111-155.
[MT] Moy, A. and Tadić, M., Some algebras of essentially compact distributions of a reductive p-adic group. In: Harmonic analysis, group representations, automorphic forms, and invariant theory, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 12, World Sci. Publ., Hackensack, NJ, 2007, pp. 247–276.
[S] A, T.. Springer, Linear algebraic groups. Second ed., Progress in Mathematics, 9, Birkhäuser Boston, Boston, MA, 1998.
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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