The two-dimensional Hecke algebra for GL_2(F_q)

By [F1] applied to $S_2=\{1,s\}$ the algebra $H$ has the $\mathbb{C}$-basis $T_1=e_B$ and $T_s=q{e}_B\dot{s}{e}_B=\mathrm{\mid }B{\mathrm{\mid }}^{-1}{\sum }_{x\in B\dot{s}B}x$, so ${\dim }_{\mathbb{C}}H=2$.

The algebra $H$ is generated by $T_s$, because the unital algebra generated by $T_s$ contains the unit $T_1$ and every element is a combination of the two basis elements; it is commutative because $T_s^2=(q-1)T_s+q{T}_1$ lies in the span of $1$ and $T_s$ by [F2]. Hence evaluation $\mathbb{C}[T]\to H$, $T\mapsto T_s$, is a surjective unital algebra homomorphism whose kernel contains $T^2-(q-1)T-q$; the induced map $\mathbb{C}[T]\mathrm{/}(T^2-(q-1)T-q)\to H$ is a surjection from a $2$-dimensional algebra onto a $2$-dimensional algebra, hence an isomorphism. In particular the displayed multiplication rule of [F2] is the complete multiplication table of $H$ in this basis, matching the presentation of [F3].

Factoring $T^2-(q-1)T-q=(T-q)(T+1)$ and noting that $q\mathrm{\ne }-1$ because $q\ge 2$, the ideals $(T-q)$ and $(T+1)$ of $\mathbb{C}[T]$ are comaximal, so [F5] gives an isomorphism $\mathbb{C}[T]\mathrm{/}((T-q)(T+1))\cong \mathbb{C}[T]\mathrm{/}(T-q)\times \mathbb{C}[T]\mathrm{/}(T+1)\cong \mathbb{C}\oplus \mathbb{C}$, hence $H\cong \mathbb{C}\oplus \mathbb{C}$ by step 2.1. Under this isomorphism the two factors are the two simple $H$-modules and the two orthogonal idempotents of $H$ are the corresponding projections; for $q=2$ the factorization reads $(T-2)(T+1)$.

By [F4] right multiplication identifies $H$ with ${\mathrm{End}}_G(\mathbb{C}[G\mathrm{/}B]{)}^{\mathrm{o}\mathrm{p}}$ and $\mathbb{C}[G\mathrm{/}B]\cong \mathbb{C}[{\mathbb{P}}^1({\mathbb{F}}_q)]$, so ${\dim }_{\mathbb{C}}{\mathrm{End}}_G(\mathbb{C}[{\mathbb{P}}^1({\mathbb{F}}_q)])={\dim }_{\mathbb{C}}H=2$ by step 1.1; since $H$ is commutative by step 2.1, the opposite algebra is $H$ itself and the opposite-algebra ambiguity is invisible.

Step 1.1 gives the basis $T_1=e_B$, $T_s$ and the dimension, step 2.1 gives the quadratic multiplication and the presentation $\mathbb{C}[T]\mathrm{/}(T^2-(q-1)T-q)$, step 3.1 gives the splitting $H\cong \mathbb{C}\oplus \mathbb{C}$, and step 3.2 gives the endomorphism-algebra dimension and the opposite-algebra statement. All algebras are finite-dimensional over $\mathbb{C}$, all sums are finite, and no choice principle is used.