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Appendix on Double Cosets

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1968)

Abstract

We now discuss a double coset decomposition for the symplectic group GSp (2n, F), which in the case n = 2 was found by Schröder [81]. Let F be a local non-Archimedean field of residue characteristic not equal to 2, let oF be its ring of integers, and let πF denote a prime element. LetG(F) = GSp(2n, F) ⊆ Gl(2n, F) be the group of symplectic similitudes. Hence, gG(F) iff gJg = λ(g) · J for a scalar λ(g) ∈ F*, where

$$ J = \left( \begin{array}{l}0{E} \\ {- E0} \\\end{array} \right)$$

and where E denotes the unit matrix. Then gG(F) ⇐⇒ (g′)?1G(F) ⇐⇒ g′ ∈ G(F) and J′ = J?1 = −JG(F). Let G(oF) = GSp(2n, oF ) denote the group of all unimodular symplectic similitudes.

Keywords

  • Parabolic Subgroup
  • Double Coset
  • Standard Type
  • Elementary Divisor
  • Integral Matrix

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Correspondence to Rainer Weissauer .

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© 2009 Springer-Verlag Berlin Heidelberg

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Weissauer, R. (2009). Appendix on Double Cosets. In: Endoscopy for GSp(4) and the Cohomology of Siegel Modular Threefolds. Lecture Notes in Mathematics(), vol 1968. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89306-6_12

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