On the Construction of a Finite Siegel Space

J. Pantoja ; J. S. Andrade ; J. A. Vargas

Journal of Lie theory, Tome 25 (2015) no. 4, pp. 1045-1071

Résumé

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.

Détail

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