Standard intertwining operators for the finite principal series

Let $n\ge 1$, let $q$ be a prime power, put $G={\mathrm{GL}}_n({\mathbb{F}}_q)$ with upper triangular Borel $B$, diagonal torus $T$ and unipotent radical $U$, let $W=S_n$ be the Weyl group with inversion length $\mathrm{\ell }$ and canonical permutation matrix $\dot{w}$ for $w\in W$ (Permutation Weyl group and inversion length), let $\chi \in \widehat{T}$ be a character with Weyl stabiliser $W_{\chi }=\{w\in W:w\cdot \chi =\chi \}$ (Diagonal torus characters and the Weyl action), and let $e_{\chi }=\frac{1}{\mathrm{\mid }B\mathrm{\mid }}{\sum }_{b\in B}\tilde{\chi }(b{)}^{-1}b$ be the idempotent of $\tilde{\chi }$ in the corner $e_{\chi }\mathbb{C}[G]e_{\chi }$, whose elements $e_{\chi }\dot{w}{e}_{\chi }$, $w\in W_{\chi }$, form a $\mathbb{C}$-basis, with $\mathbb{C}[G]e_{\chi }\cong I(\chi )$ as left $\mathbb{C}[G]$-modules (Principal series endomorphisms as the chi-idempotent corner, The principal series module for finite GL_n).

For $w\in W_{\chi }$ put ${\mathrm{\Theta }}_w:=q^{\mathrm{\ell }(w)}{e}_{\chi }\dot{w}{e}_{\chi }\in e_{\chi }\mathbb{C}[G]e_{\chi },$ the normalization by $q^{\mathrm{\ell }(w)}$ being the one of the Bruhat double-coset basis of the finite Hecke algebra (The Bruhat double-coset basis of the finite Hecke algebra), where $\mathrm{\mid }B\dot{w}B\mathrm{\mid }\mathrm{/}\mathrm{\mid }B\mathrm{\mid }=q^{\mathrm{\ell }(w)}$ (Cardinality of a finite Bruhat cell). The standard intertwining operator attached to $w\in W_{\chi }$ is the endomorphism $B_w:=R_{{\mathrm{\Theta }}_{w^{-1}}}\in {\mathrm{End}}_{\mathbb{C}[G]}(\mathbb{C}[G]e_{\chi })\cong {\mathrm{End}}_G(I(\chi )),$ where $R_a(x{e}_{\chi }):=x{e}_{\chi }a$ is right multiplication by $a$, the operators being transported to ${\mathrm{End}}_G(I(\chi ))$ along the isomorphism of Principal series endomorphisms as the chi-idempotent corner. The index $w^{-1}$ is forced by the reversal of composition in the identification of the endomorphism algebra with the opposite of the corner: $R_a{R}_b=R_{ba}$. In particular ${\mathrm{\Theta }}_1=q^0{e}_{\chi }=e_{\chi }$ and $B_1=\mathrm{i}\mathrm{d}$, and since ${\mathrm{\Theta }}_w$, $w\in W_{\chi }$, is a $\mathbb{C}$-basis of the corner, the operators $B_w$, $w\in W_{\chi }$, form a $\mathbb{C}$-basis of ${\mathrm{End}}_G(I(\chi ))$.

Representative independence. Let $w\in W_{\chi }$ and $g=b_1\dot{w}{b}_2$ with $b_1,b_2\in B$. Because $e_{\chi }b=\tilde{\chi }(b)e_{\chi }$ for $b\in B$ one has $e_{\chi }g{e}_{\chi }=\tilde{\chi }(b_1)\tilde{\chi }(b_2)e_{\chi }\dot{w}{e}_{\chi },$ so the compensated element $\tilde{\chi }(b_1{)}^{-1}\tilde{\chi }(b_2{)}^{-1}{e}_{\chi }g{e}_{\chi }$ equals $e_{\chi }\dot{w}{e}_{\chi }$. If $g=b_1^{\mathrm{\prime }}\dot{w}{b}_2^{\mathrm{\prime }}$ is a second decomposition, the two compensated elements agree: comparing the decompositions gives $b_1^{-1}{b}_1^{\mathrm{\prime }}=\dot{w}{b}_2{b}_2^{\mathrm{\prime }-1}{\dot{w}}^{-1}\in B$, and $\tilde{\chi }(b_1^{-1}{b}_1^{\mathrm{\prime }})=\tilde{\chi }(b_2{b}_2^{\mathrm{\prime }-1})$, so $\tilde{\chi }(b_1)\tilde{\chi }(b_2)=\tilde{\chi }(b_1^{\mathrm{\prime }})\tilde{\chi }(b_2^{\mathrm{\prime }})$. For a monomial representative $n_w=t\dot{w}$ with $t\in T$, the torus factor is $\tilde{\chi }(t)=\chi (t)$, and $\chi (t{)}^{-1}{e}_{\chi }{n}_w{e}_{\chi }=e_{\chi }\dot{w}{e}_{\chi },$ the special case $b_1=t$, $b_2=1$ displayed in the normalization. Thus the compensated corner element, and hence $B_w$, is independent of the choice of double-coset representatives; the uncompensated element $e_{\chi }g{e}_{\chi }$ generally is not. This is the one-dimensional case of the simultaneous representative and ${\gamma }_n$ convention of Dudas-Michel, Section 11.3.

Remarks on indexing. For $\chi =1$ this is the usual spherical basis with inverse index in the endomorphism model. Raw basis elements for characters that have not been sorted out by the Weyl stabiliser use the ambient length $\mathrm{\ell }(w)$ of the permutation matrix; the Hecke-algebra basis after Weyl sorting is specified later in this page. All constructions use finite sums, the explicit permutation matrices $\dot{w}$ and the fixed idempotent $e_{\chi }$, so no choice principle is used.