Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Lie Theory 25 (2015), No. 4, 1045--1071Copyright 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.