The Bruhat double-coset basis of the finite Hecke algebra

Keep $G={\mathrm{GL}}_n({\mathbb{F}}_q)$ with upper triangular Borel $B$ and the idempotent $e_B=\mathrm{\mid }B{\mathrm{\mid }}^{-1}{\sum }_{b\in B}b$ of the group algebra, and let $H=e_B\mathbb{C}[G]e_B$ be the finite Hecke algebra with unit $e_B$ (The finite Hecke algebra as a convolution corner and its endomorphism interpretation). For $w\in S_n$ let $\dot{w}:=P_w\in G$ be the permutation matrix of $w$ and let $\mathrm{\ell }(w)$ be its inversion length. Define the standard basis element $T_w:=q^{\mathrm{\ell }(w)}{e}_B\dot{w}{e}_B=\frac{1}{\mathrm{\mid }B\mathrm{\mid }}{\sum }_{x\in B\dot{w}B}x\in H,$ where the displayed equality is the computation below and uses $\mathrm{\mid }B\dot{w}B\mathrm{\mid }\mathrm{/}\mathrm{\mid }B\mathrm{\mid }=q^{\mathrm{\ell }(w)}$ (Cardinality of a finite Bruhat cell).

The displayed equality. In the group algebra, $e_B\dot{w}{e}_B=\mathrm{\mid }B{\mathrm{\mid }}^{-2}{\sum }_{b,b^{\mathrm{\prime }}\in B}b\dot{w}{b}^{\mathrm{\prime }}$. Each element $x\in B\dot{w}B$ is hit by exactly $\mathrm{\mid }S\mathrm{\mid }$ pairs $(b,b^{\mathrm{\prime }})$, where $S:=B\cap {\dot{w}}^{-1}B\dot{w}$: writing $x=b_0\dot{w}{b}_0^{\mathrm{\prime }}$, the equation $b\dot{w}{b}^{\mathrm{\prime }}=x$ with $b,b^{\mathrm{\prime }}\in B$ is equivalent to $b_0^{\mathrm{\prime }}{b}^{\mathrm{\prime }-1}\in S$, and then $b$ is determined. Hence $e_B\dot{w}{e}_B=\mathrm{\mid }S\mathrm{\mid }\mathrm{\mid }B{\mathrm{\mid }}^{-2}{\sum }_{x\in B\dot{w}B}x$. The same parametrisation $B\times B\to B\dot{w}B$, $(b,b^{\mathrm{\prime }})\mapsto b\dot{w}{b}^{\mathrm{\prime }}$, shows $\mathrm{\mid }B{\mathrm{\mid }}^2=\mathrm{\mid }S\mathrm{\mid }\cdot \mathrm{\mid }B\dot{w}B\mathrm{\mid }$, equivalently $\mathrm{\mid }B\mathrm{\mid }\mathrm{/}\mathrm{\mid }S\mathrm{\mid }=\mathrm{\mid }B\dot{w}B\mathrm{\mid }\mathrm{/}\mathrm{\mid }B\mathrm{\mid }=q^{\mathrm{\ell }(w)}$; substituting gives $q^{\mathrm{\ell }(w)}{e}_B\dot{w}{e}_B=\mathrm{\mid }B{\mathrm{\mid }}^{-1}{\sum }_{x\in B\dot{w}B}x$, as displayed.

Basic properties.

1. $T_1=e_B$ is the unit of $H$: for $w=1$ the cell is $B$, the sum $\mathrm{\mid }B{\mathrm{\mid }}^{-1}{\sum }_{x\in B}x$ equals $e_B$, and $e_B$ is the unit of the corner.
2. By the Bruhat decomposition $G={⨆}_{w\in S_n}B\dot{w}B$ (Bruhat decomposition of GL_n over a finite field) the elements $T_w=q^{\mathrm{\ell }(w)}{e}_B\dot{w}{e}_B$, $w\in S_n$, are linearly independent and form a $\mathbb{C}$-basis of $H$. Indeed the cells $B\dot{w}B$ are pairwise disjoint and cover $G$, so the sums $\mathrm{\mid }B{\mathrm{\mid }}^{-1}{\sum }_{x\in B\dot{w}B}x$ are linearly independent elements of $\mathbb{C}[G]$, and they span $H$ because $H=e_B\mathbb{C}[G]e_B$ is spanned by the elements $e_{B}g{e}_B$ with $g\in G$, while $e_B(bx{b}^{\mathrm{\prime }})e_B=(e_{B}b)x(b^{\mathrm{\prime }}{e}_B)=e_{B}x{e}_B$ for $b,b^{\mathrm{\prime }}\in B$ shows that $e_{B}g{e}_B$ depends only on the double coset $BgB$; hence $e_{B}g{e}_B=q^{-\mathrm{\ell }(w)}{T}_w$ for the unique $w$ with $g\in B\dot{w}B$. In particular ${\dim }_{\mathbb{C}}H=n!$.
3. Under the identification of $H$ with the convolution algebra of $B$-bi-invariant functions on $G$, the element $T_w$ is the normalized characteristic function of the double coset $B\dot{w}B$, taking the value $\mathrm{\mid }B{\mathrm{\mid }}^{-1}$ on that cell: an element $h\in \mathbb{C}[G]$ is $B$-bi-invariant precisely when $e_{B}h{e}_B=h$, so $H$ is the space of functions constant on double cosets, and the displayed formula exhibits $T_w$ as $\mathrm{\mid }B{\mathrm{\mid }}^{-1}$ times the sum of the basis elements of the cell.

Normalization. This is the Dudas-Michel normalization, used for the rest of this page: the multiplication rules established below on this page take the form in which length-additive products of the standard basis elements are single basis elements, and the rank-one quadratic relation in this normalization is recorded by the page's rank-one relation result, reading $T_s^2=(q-1)T_s+q{T}_1$ for a simple reflection $s$ with corresponding basis element $T_s$. No choice principle is used: the permutation matrices $\dot{w}$ are explicit representatives of the double cosets.