On the construction of a finite Siegel space

TL;DR

This paper constructs a finite analogue of classical Siegel's Space by describing it as a space of Lagrangians over a quadratic extension of a finite field, providing a new finite geometric perspective.

Contribution

It introduces a finite version of Siegel's Space using non-commutative geometry and describes its structure via Lagrangians and symplectic group actions.

Findings

Finite Siegel Space described as Lagrangians over a quadratic extension

Orbits of symplectic group actions characterized as homogeneous spaces

Siegel's Space represented as anti-involutions of the symplectic group

Abstract

In this note we construct a finite analogue of classical Siegel's Space. Our approach is to look at it as a non commutative Poincare's half plane. The finite Siegel Space is described as the space of Lagrangians of a $2n$ dimensional space over a quadratic extension $E$ of a finite base field $F$. The orbits of the action of the symplectic group $Sp(n,F)$ on Lagrangians are described as homogeneous spaces. Also, Siegel's Space is described as the set of anti-involutions of the symplectic group.22