I came across a notation while I am reading in an article but I am not famialr by what it means. And it is like the following:
Let $B$ be the upper triangular subgroup of $SL(2,\mathbb C)$ and define the character $\chi $ on $B$ by $$\chi^n\begin{bmatrix} t & s \\ 0 & 1/t \end{bmatrix}=t^n$$ Now let $\begin{bmatrix} x & y \\ z & t \end{bmatrix}\in SL(2,\mathbb C)$ then Gram-Schmidt process we have $$\begin{bmatrix} x & y \\ z & t \end{bmatrix}=k\begin{bmatrix} \sqrt{ |x|^2+|z|^2} & 0 \\ 0 & \dfrac{1}{\sqrt{ |x|^2+|z|^2}} \end{bmatrix}\begin{bmatrix} 1 & \eta \\ 0 & 1 \end{bmatrix}$$ Where $\eta =\dfrac{y\bar x+t\bar z}{ \sqrt{ |x|^2+|z|^2}}$ and $k\in SU(2)$.
Writing $\chi=\chi^n$ then we have $SL(2,\mathbb C)\times_\chi\mathbb C \stackrel{\Theta}{\cong}SU(2)\times_{\chi_{S^1}}\mathbb C$.
Can someone explain to me what is going on? What does $\times_\chi$ mean? And why this works like this?