Geodesic
On the Construction of a Finite Siegel Space
José Pantoja1 ; Jorge Soto Andrade2 ; Jorge A. Vargas3
1Instituto de Matemáticas, Universidad Catolica, Blanco Viel 596, Valparaíso, Chile
2Dep. de Matemáticas, Universidad de Chile, Las Palmeras 3425, Santiago, Chile
3FAMAF-CIEM, Ciudad Universitaria, 5000 Córdoba, Argentina
Journal of Lie Theory, Tome 25 (2015) no. 4, pp. 1045-1071

Voir la notice de l'article provenant de la source Heldermann Verlag

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.
Classification : 20G40, 11E16, 14M20, 17B10
Mots-clés : Finite Siegel half space, star-analogue

José Pantoja  1   ; Jorge Soto Andrade  2   ; Jorge A. Vargas  3

1 Instituto de Matemáticas, Universidad Catolica, Blanco Viel 596, Valparaíso, Chile
2 Dep. de Matemáticas, Universidad de Chile, Las Palmeras 3425, Santiago, Chile
3 FAMAF-CIEM, Ciudad Universitaria, 5000 Córdoba, Argentina 
José Pantoja; Jorge Soto Andrade; Jorge A. Vargas. On the Construction of a Finite Siegel Space. Journal of Lie Theory, Tome 25 (2015) no. 4, pp. 1045-1071. http://geodesic.mathdoc.fr/item/JOLT_2015_25_4_a5/
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     author = {Jos\'e Pantoja and Jorge Soto Andrade and Jorge A. Vargas},
     title = {On the {Construction} of a {Finite} {Siegel} {Space}},
     journal = {Journal of Lie Theory},
     pages = {1045--1071},
     year = {2015},
     volume = {25},
     number = {4},
     url = {http://geodesic.mathdoc.fr/item/JOLT_2015_25_4_a5/}
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