Double coset decomposition of symplectic group over a quadratic extension - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T20:40:11Z http://mathoverflow.net/feeds/question/91933 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/91933/double-coset-decomposition-of-symplectic-group-over-a-quadratic-extension Double coset decomposition of symplectic group over a quadratic extension jy 2012-03-22T17:26:05Z 2012-03-22T22:37:48Z <p>I'm trying to understand the double coset decomposition of $G(F)\setminus G(E)/K_E$ , where $G = \mathrm{GSp}_{2n}$ is the rank $n$ group of symplectic similitudes, $E/F$ is a quadratic extension of $p$-adic fields and $K_E=G(\mathcal{O}_E)$ is the ring of integers in $E$. My knowledge of buildings is limited at present, but as I understand it, $G(E)$ acts transitively on the vertices of the affine building over $E$, and $K_E$ is the stabilizer of a hyperspecial point, so the quotient $X(E) = G(E)/K_E$ can be identified with the vertices of this building.</p> <p>If this is correct, then the double coset decomposition becomes a question about orbits of $G(F)$ on $X(E)$. </p> <blockquote> <p>What is known about the orbits of $G(F)$ on $X(E)$?</p> </blockquote> <p>Specifically, I would like to find representatives that give some idea as to the geometry of the situation. (I.e., Sage or Matlab might be able to find coset representatives, but I would still have no idea <em>why</em> these are the representatives.)</p> <p>(I'm reading through Garrett's book on buildings and trying to work through Tits' article in Corvallis, but any other direction for sources would be appreciated as well.)</p> http://mathoverflow.net/questions/91933/double-coset-decomposition-of-symplectic-group-over-a-quadratic-extension/91958#91958 Answer by Paul Broussous for Double coset decomposition of symplectic group over a quadratic extension Paul Broussous 2012-03-22T22:37:48Z 2012-03-22T22:37:48Z <p>This question is answered for a large number of reductive groups (including the symplectic) in a paper by P. Delorme and V. Sécherre (e.g. Math Arxiv, An analogue of the Cartan decomposition for p-adic reductive symmetric spaces). </p>