Journal of Lie Theory 25 (2015), No. 4, 1045--1071
Copyright Heldermann Verlag 2015
On the Construction of a Finite Siegel Space
José Pantoja
Instituto de Matemáticas, Universidad Catolica, Blanco Viel 596, Valparaíso, Chile
jpantoja@ucv.cl
Jorge Soto Andrade
Dep. de Matemáticas, Universidad de Chile, Las Palmeras 3425, Santiago, Chile
sotoandrade@u.uchile.cl
Jorge A. Vargas
FAMAF-CIEM, Ciudad Universitaria, 5000 Córdoba, Argentina
vargas@famaf.unc.edu.ar
[Abstract-pdf]
We construct a finite analogue of classical Siegel's Space. This is made by generalizing Poincar\'{e} half plane construction for a quadratic field extension $E\supset F$, considering in this case an involutive ring $A$, extension of the ring fixed points $A_0=A^{\Gamma}$, ($\Gamma$ an order two group of automorphisms of $A$), and the generalized special linear group $SL_*(2,A)$, which acts on a certain $\ast-$ plane $\cal P_A$. Classical Lagrangians for finite dimensional spaces over a finite field are related with Lagrangians for $\cal P_A$. We show $SL_*(2,A)$ acts transitively on $\cal P_A$ when $A$ is a $\ast-$ euclidean ring, and we study extensibly the case where $A=M_n(E)$. The structure of the orbits of the action of the symplectic group over $F$ on Lagrangians over a finite dimensional space over $E$ are studied.
Keywords: Finite Siegel half space, star-analogue.
MSC: 20G40; 11E16, 14M20, 17B10
[ Fulltext-pdf (404 KB)] for subscribers only.