Unipotent double-coset biset splitting

The transported blocks and the groups $M,V,X$. Put $\kappa (k):={\mathrm{blk}}_{\beta }({\tau }^{-1}(k))$ for $1\le k\le n$, so that $k\in \tau (J_j)$ exactly when $\kappa (k)=j$; by [L2] one has $(wA{w}^{-1}{)}_{kl}=A_{{\tau }^{-1}(k),{\tau }^{-1}(l)}$ for every matrix $A$, so applying the criteria of [L1] to the composition $\beta$ gives $X=\{g:g_{kl}=0\text{ whenever }\kappa (k)>\kappa (l)\},M=\{g:g_{kl}=0\text{ whenever }\kappa (k)\mathrm{\ne }\kappa (l)\},V=\{g\in X:g_{kk}=1\text{ for all }k,g_{kl}=0\text{ whenever }k\mathrm{\ne }l\text{ and }\kappa (k)\ge \kappa (l)\},$ and then $M,V\subseteq X$, $M\cap V=\{I_n\}$ (both contain $w{I}_n{w}^{-1}=I_n$, and an element $wp{w}^{-1}=wq{w}^{-1}$ with $p\in L_{\beta }$, $q\in U_{\beta }$ satisfies $p=q\in L_{\beta }\cap U_{\beta }=\{I_n\}$), $X=MV$ (as $P_{\beta }=L_{\beta }{U}_{\beta }$), $X,M,V$ are subgroups of $G$ and $V⊴X$, because products and inverses are carried by $w(pq)w^{-1}=(wp{w}^{-1})(wq{w}^{-1})$ and $(wq{w}^{-1}{)}^{-1}=w{q}^{-1}{w}^{-1}$ by [L3], and for $x=wp{w}^{-1}\in X$ and $v=wu{w}^{-1}\in V$ one has $xv{x}^{-1}=w(pu{p}^{-1})w^{-1}\in V$ since $pu{p}^{-1}\in U_{\beta }$ by [L1].

The subgroups $C,A,D$. The sets $C=L_{\alpha }\cap M$, $A=L_{\alpha }\cap V$ and $D=U_{\alpha }\cap M$ are subgroups of $G$ by [L4], and $C\subseteq L_{\alpha }$, $C\subseteq M$, $A\subseteq V$, $D\subseteq U_{\alpha }$ hold by definition; moreover $A\cap C\subseteq V\cap M=\{I_n\}$ by step 1.1.

The projection onto $M$. Since $X=MV$ with $M\cap V=\{I_n\}$ and $V⊴X$ by step 1.1, every $g\in X$ has a unique factorisation $g=mu$ with $m\in M$ and $u\in V$: if $m_1{u}_1=m_2{u}_2$ with $m_i\in M$, $u_i\in V$, then $m_2^{-1}{m}_1=u_2{u}_1^{-1}\in M\cap V=\{I_n\}$, so $m_1=m_2$ and $u_1=u_2$. Define $\pi :X\to M$ by $\pi (g):=m$ for $g=mu$; then $\pi (g{)}^{-1}g\in V$ for all $g\in X$, and $\pi$ is a group homomorphism with kernel $V$, because for $g=mu$ and $g^{\mathrm{\prime }}=m^{\mathrm{\prime }}{u}^{\mathrm{\prime }}$ one has $g{g}^{\mathrm{\prime }}=m{m}^{\mathrm{\prime }}(m^{\mathrm{\prime }-1}u{m}^{\mathrm{\prime }}{u}^{\mathrm{\prime }})$ with $m^{\mathrm{\prime }-1}u{m}^{\mathrm{\prime }}\in V$ and $u^{\mathrm{\prime }}\in V$ by [L3] and step 1.1, so the factorisation of $g{g}^{\mathrm{\prime }}$ is $(m{m}^{\mathrm{\prime }})\cdot (m^{\mathrm{\prime }-1}u{m}^{\mathrm{\prime }}{u}^{\mathrm{\prime }})$ and $\pi (g{g}^{\mathrm{\prime }})=m{m}^{\mathrm{\prime }}=\pi (g)\pi (g^{\mathrm{\prime }})$, while $\pi (g)=I_n$ holds exactly when $g\in V$.

The components of $P_{\alpha }\cap X$ lie in $X$. Let $g\in P_{\alpha }\cap X$ and let $g=l_0{u}_0$ be its factorisation from [L1] with $l_0\in L_{\alpha }$ and $u_0\in U_{\alpha }$; put $l:=l_0$ and $u:=g{l}_0^{-1}=l_0{u}_0{l}_0^{-1}\in U_{\alpha }$, so that $g=ul$ with $u\in U_{\alpha }$ and $l\in L_{\alpha }$. Then $l$ is the $\alpha$-block diagonal part of $g$: if $k,j$ lie in the same $\alpha$-block $I_t$, then, since $l$ is block diagonal and $u$ has identity diagonal blocks, $g_{kj}=(ul{)}_{kj}={\sum }_m{u}_{km}{l}_{mj}={\sum }_{m\in I_t}{u}_{km}{l}_{mj}=l_{kj}$; entries of $l$ between different $\alpha$-blocks are zero. Hence $l\in X$: if $\kappa (k)>\kappa (j)$ then either ${\mathrm{blk}}_{\alpha }(k)\mathrm{\ne }{\mathrm{blk}}_{\alpha }(j)$ and $l_{kj}=0$, or ${\mathrm{blk}}_{\alpha }(k)={\mathrm{blk}}_{\alpha }(j)$ and $l_{kj}=g_{kj}=0$ because $g\in X$ by the criterion of step 1.1. Finally $u=g{l}^{-1}\in X$, because $X$ is a subgroup containing $g$ and $l$ by step 1.1.

$\mathrm{\Phi }$ is well defined. First, changing representatives of the two cosets does not change the image. If $a\in A=L_{\alpha }\cap V$, then $m^{-1}am\in V$ because $M$ normalises $V$ by step 1.1; hence $U_{\alpha }lamw{U}_{\beta }=U_{\alpha }lm(m^{-1}am)w{U}_{\beta }=U_{\alpha }lmw{U}_{\beta }$, because $w^{-1}(m^{-1}am)w\in U_{\beta }$. If $d\in D=U_{\alpha }\cap M$, then $ld{l}^{-1}\in U_{\alpha }$ by [L1], so $U_{\alpha }ldmw{U}_{\beta }=U_{\alpha }lmw{U}_{\beta }$. Second, for $c\in C$ the two sides of the generating balancing relation have the same image, since $(lc)m=l(cm)$ by [L3]: $U_{\alpha }(lc)mw{U}_{\beta }=U_{\alpha }l(cm)w{U}_{\beta }\mathrm{.}$ Finally, $\mathrm{\Phi }$ takes values in the specified $(U_{\alpha },U_{\beta })$-double cosets inside $P_{\alpha }w{P}_{\beta }$: writing $m=w{m}_0{w}^{-1}$ with $m_0\in L_{\beta }$ gives $lmw=lw{m}_0\in P_{\alpha }w{P}_{\beta }$.

Normalisation by $C$. Let $c\in C$. Then $c{L}_{\alpha }{c}^{-1}=L_{\alpha }$ and $c{U}_{\alpha }{c}^{-1}=U_{\alpha }$, because $C\subseteq L_{\alpha }\subseteq P_{\alpha }$ and $U_{\alpha }⊴P_{\alpha }$ by [L1]; and $cM{c}^{-1}=M$, because $C\subseteq M$; and $cV{c}^{-1}=V$, because $C\subseteq X$ and $V⊴X$ by step 1.1. Hence $cA{c}^{-1}=c(L_{\alpha }\cap V)c^{-1}=(c{L}_{\alpha }{c}^{-1})\cap (cV{c}^{-1})=L_{\alpha }\cap V=A$ and $cD{c}^{-1}=c(U_{\alpha }\cap M)c^{-1}=(c{U}_{\alpha }{c}^{-1})\cap (cM{c}^{-1})=U_{\alpha }\cap M=D$, so $C$ normalises $A$ and $D$. Consequently right multiplication by $C$ on $L_{\alpha }\mathrm{/}A$ and left multiplication by $C$ on $D\mathrm{\backslash }M$ are well defined.

The $\kappa$-block diagonal part, and $L_{\alpha }\cap X$. For $g\in X$ let $d$ be the $\kappa$-block diagonal part of $g$, that is $d_{kl}:=g_{kl}$ when $\kappa (k)=\kappa (l)$ and $d_{kl}:=0$ otherwise; then $d=\pi (g)$. Indeed, write $g=wp{w}^{-1}$ with $p\in P_{\beta }$ and $p=\lambda u$ with $\lambda \in L_{\beta }$, $u\in U_{\beta }$ as in [L1]; the conjugation formula of [L2] gives $d=w\lambda w^{-1}$, and $w\lambda w^{-1}\in M=\{g:g_{kl}=0\text{ whenever }\kappa (k)\mathrm{\ne }\kappa (l)\}$ by step 1.1, while $d^{-1}g=w({\lambda }^{-1}p)w^{-1}\in V$ by [L1], so the factorisation $g=d\cdot (d^{-1}g)$ with $d\in M$ and $d^{-1}g\in V$ is the one of step 2.2 and $d=\pi (g)$. Consequently, for $g\in L_{\alpha }\cap X$ the element $\pi (g)$ lies in $L_{\alpha }$: its entries are entries of $g$ inside $\kappa$-blocks and zero elsewhere, and $g$ vanishes off the $\alpha$-block diagonal by [L1]; hence $\pi (g)\in C=L_{\alpha }\cap M$. Moreover $g=a\pi (g)$ with $a:=\pi (g)(\pi (g{)}^{-1}g)\pi (g{)}^{-1}$: here $\pi (g{)}^{-1}g\in V$ by step 2.2, and $a\in V$ because $\pi (g)\in M\subseteq X$ normalises the normal subgroup $V⊴X$ of step 1.1, while $a\in L_{\alpha }$ as a product of elements of $L_{\alpha }$; so $a\in A=L_{\alpha }\cap V$.

$\mathrm{\Phi }$ is surjective. Let $x\in P_{\alpha }w{P}_{\beta }$, say $x=pwq$ with $p\in P_{\alpha }$ and $q\in P_{\beta }$. By [L1] write $p=lu$ with $l\in L_{\alpha }$, $u\in U_{\alpha }$ and $q=m_0{u}_0$ with $m_0\in L_{\beta }$, $u_0\in U_{\beta }$, and put $m:=w{m}_0{w}^{-1}\in M$, so that $w{m}_0=mw$ and $x=lumw{u}_0$. Then $U_{\alpha }x{U}_{\beta }=U_{\alpha }lumw{u}_0{U}_{\beta }=U_{\alpha }lmw{U}_{\beta }=\mathrm{\Phi }([lA,Dm]),$ because $lu{l}^{-1}\in U_{\alpha }$ (as $U_{\alpha }⊴P_{\alpha }$ and $l\in L_{\alpha }\subseteq P_{\alpha }$ by [L1]) makes $U_{\alpha }lu=U_{\alpha }l$, and $u_0{U}_{\beta }=U_{\beta }$ since $u_0\in U_{\beta }$. Hence every $(U_{\alpha },U_{\beta })$-double coset inside $P_{\alpha }w{P}_{\beta }$ lies in the image of $\mathrm{\Phi }$.

Claim 3: the two biset structures agree. For $a\in L_{\alpha }$ and $b\in L_{\beta }$ the formulas $a\cdot (U_{\alpha }x{U}_{\beta }):=U_{\alpha }ax{U}_{\beta }$ and $(U_{\alpha }x{U}_{\beta })\cdot b:=U_{\alpha }xb{U}_{\beta }$ are well defined (as $a{U}_{\alpha }{a}^{-1}=U_{\alpha }$ and $b{U}_{\beta }{b}^{-1}=U_{\beta }$ by [L1]) and satisfy the axioms of [L6], so they are a left action of $L_{\alpha }$ and a right action of $L_{\beta }$. On the balanced product the formulas $a\cdot [lA,Dm]:=[alA,Dm]$ and $[lA,Dm]\cdot b:=[lA,Dmwb{w}^{-1}]$ are well defined, because they carry the generating relation to a relation of the same form, $((al)cA,Dm)\sim (alA,Dcm)$ and $(lcA,Dmwb{w}^{-1})\sim (lA,Dcmwb{w}^{-1})$ for $c\in C$; they are actions because $M$ is a subgroup and $b\mapsto wb{w}^{-1}$ is a homomorphism $L_{\beta }\to M$ (it is conjugation by the fixed element $w$ and $w{L}_{\beta }{w}^{-1}=M$, while $w(b{b}^{\mathrm{\prime }})w^{-1}=(wb{w}^{-1})(w{b}^{\mathrm{\prime }}{w}^{-1})$ and $w{I}_n{w}^{-1}=I_n$ by [L3]). Finally, for $b^w:=wb{w}^{-1}$ one has $b^{w}w=wb$, so $\mathrm{\Phi }(a\cdot [lA,Dm])=U_{\alpha }almw{U}_{\beta }=a\cdot \mathrm{\Phi }([lA,Dm]),\mathrm{\Phi }([lA,Dm]\cdot b)=U_{\alpha }lm{b}^{w}w{U}_{\beta }=U_{\alpha }lmwb{U}_{\beta }=\mathrm{\Phi }([lA,Dm])\cdot b\mathrm{.}$

The $M$-component of $U_{\alpha }\cap X$. For $u\in U_{\alpha }\cap X$ one has $\pi (u)\in D=U_{\alpha }\cap M$. Indeed $\pi (u)$ is the $\kappa$-block diagonal part $d$ of $u$ by step 3.2, so $d\in M$ and $d_{kk}=u_{kk}=1$ for all $k$, while $d_{kl}=0$ whenever $k\mathrm{\ne }l$ and ${\mathrm{blk}}_{\alpha }(k)\ge {\mathrm{blk}}_{\alpha }(l)$: if $\kappa (k)\mathrm{\ne }\kappa (l)$ then $d_{kl}=0$, and if $\kappa (k)=\kappa (l)$ then $d_{kl}=u_{kl}=0$ by the criterion for $U_{\alpha }$ in [L1]. Hence $d\in U_{\alpha }$ and $d\in D$.

$\mathrm{\Phi }$ is injective. Let $l,l^{\mathrm{\prime }}\in L_{\alpha }$ and $m,m^{\mathrm{\prime }}\in M$ with $\mathrm{\Phi }([l^{\mathrm{\prime }}A,D{m}^{\mathrm{\prime }}])=\mathrm{\Phi }([lA,Dm])$, that is $U_{\alpha }{l}^{\mathrm{\prime }}{m}^{\mathrm{\prime }}w{U}_{\beta }=U_{\alpha }lmw{U}_{\beta }$. Then there are $u_0\in U_{\alpha }$ and $u_2\in U_{\beta }$ with $l^{\mathrm{\prime }}{m}^{\mathrm{\prime }}w{u}_2=u_{0}lmw$; putting $v:=w{u}_2{w}^{-1}\in V$ gives $l^{\mathrm{\prime }}{m}^{\mathrm{\prime }}v=u_{0}lm$ and hence $m^{\mathrm{\prime }}=l^{\mathrm{\prime }-1}{u}_{0}lm{v}^{-1}=(l^{\mathrm{\prime }-1}{u}_0{l}^{\mathrm{\prime }})(l^{\mathrm{\prime }-1}l)m{v}^{-1}=u^{\mathrm{\prime }}\lambda m{v}_1,$ where $u^{\mathrm{\prime }}:=l^{\mathrm{\prime }-1}{u}_0{l}^{\mathrm{\prime }}\in U_{\alpha }$, $\lambda :=l^{\mathrm{\prime }-1}l\in L_{\alpha }$ and $v_1:=v^{-1}\in V$. Therefore $m^{\mathrm{\prime }}{m}^{-1}=u^{\mathrm{\prime }}\lambda (m{v}_1{m}^{-1})=u^{\mathrm{\prime }}\lambda v_2$ with $v_2:=m{v}_1{m}^{-1}\in V$, because $m\in M\subseteq X$ normalises $V⊴X$ by step 1.1; so $u^{\mathrm{\prime }}\lambda =(m^{\mathrm{\prime }}{m}^{-1})v_2^{-1}\in M\cdot V\subseteq X$ by step 1.1 and $u^{\mathrm{\prime }}\lambda \in U_{\alpha }{L}_{\alpha }=P_{\alpha }$ by [L1], that is $u^{\mathrm{\prime }}\lambda \in P_{\alpha }\cap X$, and step 2.3 gives $u^{\mathrm{\prime }}\in X$ and $\lambda \in X$. Applying the homomorphism $\pi$ of step 2.2 to $m^{\mathrm{\prime }}{m}^{-1}=u^{\mathrm{\prime }}\lambda v_2$ gives $m^{\mathrm{\prime }}{m}^{-1}=\pi (m^{\mathrm{\prime }}{m}^{-1})=\pi (u^{\mathrm{\prime }})\pi (\lambda )\pi (v_2)=\pi (u^{\mathrm{\prime }})\pi (\lambda ),$ since $m^{\mathrm{\prime }}{m}^{-1}\in M$ satisfies $\pi (m^{\mathrm{\prime }}{m}^{-1})=m^{\mathrm{\prime }}{m}^{-1}$ and $v_2\in V=\ker \pi$ by step 2.2; with $d:=\pi (u^{\mathrm{\prime }})\in D$ by step 4.1 and $c:=\pi (\lambda )\in C$ by step 3.2 this is $m^{\mathrm{\prime }}=dcm$. By step 3.2 applied to $\lambda \in L_{\alpha }\cap X$ we have $\lambda =ac$ with $a\in A$, so $l^{\mathrm{\prime }}=l{\lambda }^{-1}=l{c}^{-1}{a}^{-1}$. Hence $l^{\mathrm{\prime }}A=l{c}^{-1}A$ and $D{m}^{\mathrm{\prime }}=Dcm$, and the pair $(l^{\mathrm{\prime }}A,D{m}^{\mathrm{\prime }})=(l{c}^{-1}A,Dcm)$ is related to $(lA,Dm)$ by the generating relation, because $(l{c}^{-1})c=l$ and $c(Dm)=Dcm$; therefore $[l^{\mathrm{\prime }}A,D{m}^{\mathrm{\prime }}]=[lA,Dm]$, so $\mathrm{\Phi }$ is injective.

Claims 1 and 2. Steps 1.1, 2.1 and 3.1 prove claim 1. Step 2.4 shows that $\mathrm{\Phi }$ is well defined on the balanced product, step 3.3 that it is surjective onto the set of $(U_{\alpha },U_{\beta })$-double cosets inside $P_{\alpha }w{P}_{\beta }$, and step 5.1 that it is injective; hence $\mathrm{\Phi }$ is a bijection, which is claim 2, the set $P_{\alpha }w{P}_{\beta }$ being by [L5] the double coset attached to the $(W_{\alpha },W_{\beta })$-double coset of $\tau$. The entry and projection arguments above used no block-increasing hypothesis.

The extreme cases. For $\alpha =\beta =(n)$ one has $U_{\alpha }=U_{\beta }=\{I_n\}$ and $L_{\alpha }=L_{\beta }=G$ by [L7], so $M=X=G=L_{\alpha }$, $V=\{I_n\}$, $C=G$ and $A=D=\{I_n\}$; the balanced product is the quotient of $G\times G$ by $(gc,m)\sim (g,cm)$ and is in bijection with $G$ via the well-defined map $(l,m)\mapsto lm$ (with inverse $g\mapsto (g,I_n)$), while the double cosets inside $P_{\alpha }w{P}_{\beta }=G$ are the singletons, so claim 2 reads $G\cong G$ via $g\mapsto gw$, each target double coset being a singleton. For $\alpha =\beta =(1^n)$ one has $U_{\alpha }=U_{\beta }=U$ and $L_{\alpha }=L_{\beta }=T$ by [L7], so $M=wT{w}^{-1}=T$ (the conjugation formula of [L2] only permutes the diagonal positions), $V=wU{w}^{-1}$, $C=T\cap T=T$, and $A=T\cap V=\{I_n\}$, $D=U\cap T=\{I_n\}$ because $T\cap U=\{I_n\}$ by [L7] and $V$ is a conjugate of $U$; the balanced product is therefore $(T\mathrm{/}\{I_n\}){\times }_T(\{I_n\}\mathrm{\backslash }T)\cong T$ and claim 2 specialises to a bijection $T\to U\mathrm{\backslash }(BwB)\mathrm{/}U$. For $n=1$ one has $\alpha =\beta =(1)$, $G=B=T={\mathbb{F}}_q^{\times }$ and $U=\{I_1\}$ by [L7], so $M=C=G$, $A=D=\{I_1\}$ and claim 2 is the tautological bijection from $G$ onto the singletons $\{g\}=U_{\alpha }\{g\}{U}_{\beta }$ inside $P_{\alpha }w{P}_{\beta }=G$.

Conclusion. Claims 1, 2 and 3 of the statement are steps 6.1, 6.1 and 3.4, with the extreme cases verified in step 7.1; the proof is complete. ∎