Reference-coset-reps

3136 days ago by jimlb

var('x,y,z') 
       
var('a11,a12,a21,a22,b11,b12,b21,b22,d11,d12,d21,d22'); 
       

A typical element of the Siegel parabolic is given by $g$.  I could get write the $d_{ij}$ in terms of $A$, but I think it would make it messier to look at.

g=Matrix(SR,[[a11,a12,b11,b12],[a21,a22,b21,b22],[0,0,d11,d12],[0,0,d21,d22]]);g 
       

The first coset rep is just the identity, so that just leaves $g$.  The second set are of the following form with $z \in \mathbb{F}_{p}$.

Z=Matrix(SR,[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,-1,0,z]]);Z 
       

Multiplied by a typical parabolic element:

g*Z 
       

The third set is given by reps of the following form with $y,z \in \mathbb{F}_p$.

Y = Matrix(SR,[[0,1,0,0],[0,0,1,0],[0,0,y,1],[-1,y,z,0]]);Y 
       

Multiplied by a typical parabolic element:

g*Y 
       

Finally, the last type is given by the following form with $x,y,z \in \mathbb{F}_p$.

X= Matrix(SR,[[0,0,0,1],[0,0,1,0],[0,-1,y,x],[-1,0,z,y]]);X 
       

Multiplied by a typical parabolic element:

g*X