Reference-coset-reps

3301 days ago by jimlb

var('x,y,z')
 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(x, y, z\right)$
var('a11,a12,a21,a22,b11,b12,b21,b22,d11,d12,d21,d22');
 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(a_{11}, a_{12}, a_{21}, a_{22}, b_{11}, b_{12}, b_{21}, b_{22}, d_{11}, d_{12}, d_{21}, d_{22}\right)$

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
 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} a_{11} & a_{12} & b_{11} & b_{12} \\ a_{21} & a_{22} & b_{21} & b_{22} \\ 0 & 0 & d_{11} & d_{12} \\ 0 & 0 & d_{21} & d_{22} \end{array}\right)$

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
 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & z \end{array}\right)$

Multiplied by a typical parabolic element:

g*Z
 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} a_{11} & -b_{12} & b_{11} & b_{12} z + a_{12} \\ a_{21} & -b_{22} & b_{21} & b_{22} z + a_{22} \\ 0 & -d_{12} & d_{11} & d_{12} z \\ 0 & -d_{22} & d_{21} & d_{22} z \end{array}\right)$

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
 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & y & 1 \\ -1 & y & z & 0 \end{array}\right)$

Multiplied by a typical parabolic element:

g*Y
 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} -b_{12} & b_{12} y + a_{11} & b_{11} y + b_{12} z + a_{12} & b_{11} \\ -b_{22} & b_{22} y + a_{21} & b_{21} y + b_{22} z + a_{22} & b_{21} \\ -d_{12} & d_{12} y & d_{11} y + d_{12} z & d_{11} \\ -d_{22} & d_{22} y & d_{21} y + d_{22} z & d_{21} \end{array}\right)$

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
 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & y & x \\ -1 & 0 & z & y \end{array}\right)$

Multiplied by a typical parabolic element:

g*X
 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} -b_{12} & -b_{11} & b_{11} y + b_{12} z + a_{12} & b_{11} x + b_{12} y + a_{11} \\ -b_{22} & -b_{21} & b_{21} y + b_{22} z + a_{22} & b_{21} x + b_{22} y + a_{21} \\ -d_{12} & -d_{11} & d_{11} y + d_{12} z & d_{11} x + d_{12} y \\ -d_{22} & -d_{21} & d_{21} y + d_{22} z & d_{21} x + d_{22} y \end{array}\right)$