Flags and Bruhat cells for GL_2(F_q)

A complete flag of the two-dimensional space $V$ is a chain $0<F_1<V$ with $\dim F_1=1$, so it is determined by its member $F_1$, which is a line; conversely every line $L$ gives the complete flag $0<L<V$. Hence the complete flags correspond bijectively to the lines in $V$, and by [F3] the coset space $G\mathrm{/}B$ is in $G$-equivariant bijection with the set of lines, that is with ${\mathbb{P}}^1({\mathbb{F}}_q)$. In particular the standard flag $V_{\bullet }$ corresponds to the standard line $⟨e_1⟩$, whose stabiliser in $G$ is $B$.

Every nonzero vector of $V$ is of the form $c_1{e}_1+c_2{e}_2$ with $(c_1,c_2)\mathrm{\ne }(0,0)$, and the line it spans is $⟨e_1⟩$ when $c_2=0$ and $⟨e_2+a{e}_1⟩$ with $a=c_1{c}_2^{-1}\in {\mathbb{F}}_q$ when $c_2\mathrm{\ne }0$; the $q+1$ lines $⟨e_1⟩$ and $⟨e_2+a{e}_1⟩$, $a\in {\mathbb{F}}_q$, are pairwise distinct, because $e_2+a{e}_1$ and $e_2+a^{\mathrm{\prime }}{e}_1$ are proportional only when $a=a^{\mathrm{\prime }}$, and none of them lies in $⟨e_1⟩$. Hence ${\mathbb{P}}^1({\mathbb{F}}_q)$ has exactly $q+1$ points.

The standard line is fixed by $B$, because an invertible upper triangular matrix sends $e_1$ to $\alpha e_1$ with $\alpha \mathrm{\ne }0$; hence $\{⟨e_1⟩\}$ is a $B$-orbit, of size $1$. For $g=\left(\begin{array}{cc}\alpha & \beta \\ 0& \delta \end{array}\right)\in B$ and $a\in {\mathbb{F}}_q$ one has $g(e_2+a{e}_1)=(\beta +\alpha a)e_1+\delta e_2=\delta (e_2+\frac{\alpha a+\beta }{\delta }{e}_1)$, so $g$ sends the line $⟨e_2+a{e}_1⟩$ to the line $⟨e_2+a^{\mathrm{\prime }}{e}_1⟩$ with $a^{\mathrm{\prime }}=\frac{\alpha a+\beta }{\delta }$; given $a,a^{\mathrm{\prime }}\in {\mathbb{F}}_q$ the choices $\alpha =\delta =1$, $\beta =a^{\mathrm{\prime }}-a$ produce such an element of $B$, so $B$ is transitive on the $q$ lines of the complement. Hence the complement of the standard line is a single $B$-orbit of size $q$, and the two $B$-orbits on ${\mathbb{P}}^1({\mathbb{F}}_q)$ have sizes $1$ and $q$.

The $B$-orbits on $G\mathrm{/}B$ are the sets $B\cdot (gB)=\{bgB:b\in B\}$ of left cosets, that is exactly the quotients $BgB\mathrm{/}B$ of the double cosets $BgB$ in $G$. By [F4] the double cosets $BgB$ are exactly $B{P}_{\mathrm{i}\mathrm{d}}B=B$ and $B{P}_{s}B$; so the two $B$-orbits of step 2.1 are the quotients $B\mathrm{/}B$ and $B{P}_{s}B\mathrm{/}B$, the orbit $\{⟨e_1⟩\}$ corresponding to the identity and the complement $\{⟨e_2+a{e}_1⟩:a\in {\mathbb{F}}_q\}$ to $s$.

By [F4] the numbers of left cosets of $B$ in the two cells are $\mathrm{\mid }B{P}_{\mathrm{i}\mathrm{d}}B\mathrm{/}B\mathrm{\mid }=q^{\mathrm{\ell }(\mathrm{i}\mathrm{d})}=q^0=1$ and $\mathrm{\mid }B{P}_{s}B\mathrm{/}B\mathrm{\mid }=q^{\mathrm{\ell }(s)}=q^1=q$ by [F2]; these agree with the orbit sizes $1$ and $q$ computed in step 2.1, and their sum $1+q$ is the number of points of ${\mathbb{P}}^1({\mathbb{F}}_q)$ found in step 1.2. Thus $G\mathrm{/}B$ is the projective line with $q+1$ points, its two $B$-orbits are the standard line of size $1$ and its complement of size $q$, and they are indexed by $\mathrm{i}\mathrm{d}$ and $s$. ∎