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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

Jorge Soto Andrade
Dep. de Matemáticas, Universidad de Chile, Las Palmeras 3425, Santiago, Chile

Jorge A. Vargas
FAMAF-CIEM, Ciudad Universitaria, 5000 Córdoba, Argentina


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

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