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, g ∈ G(F) iff g′Jg = λ(g) · J for a scalar λ(g) ∈ F*, where
and where E denotes the unit matrix. Then g ∈ G(F) ⇐⇒ (g′)?1 ∈ G(F) ⇐⇒ g′ ∈ G(F) and J′ = J?1 = −J ∈ G(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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© 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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DOI: https://doi.org/10.1007/978-3-540-89306-6_12
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Publisher Name: Springer, Berlin, Heidelberg
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Online ISBN: 978-3-540-89306-6
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