Transitivity and parabolic independence of Harish-Chandra induction

The refinement factorisation. Let $v\in U_{\delta }$. Its blocks of rows and columns for $\gamma$ are unions of blocks for $\delta$ by [L3], so we may write $v=AD$, where $D$ is the $\gamma$-block diagonal matrix obtained from $v$ by replacing every entry lying in an off-diagonal $\gamma$-block $(k,l)$ (that is ${\mathrm{blk}}_{\gamma }(k)\mathrm{\ne }{\mathrm{blk}}_{\gamma }(l)$) by $0$, and $A:=v{D}^{-1}$. Each diagonal $\gamma$-block of $v$ is itself a $\delta$-block upper unitriangular matrix (after ordering coordinates by their $\delta$-blocks, its diagonal blocks are identities and its blocks below them vanish), so it is invertible and $D$ is invertible; $D$ is block diagonal for $\gamma$ and lies in $U_{\delta }$, hence $D\in L_{\gamma }\cap U_{\delta }$, while $A=v{D}^{-1}$ is the product of two $\gamma$-block upper triangular matrices (by [L2] applied to $v$ and to the block diagonal matrix $D^{-1}$), so $A$ is $\gamma$-block upper triangular with identity diagonal blocks, that is $A\in U_{\gamma }$. Every $v\in U_{\delta }$ therefore lies in $U_{\gamma }\cdot (L_{\gamma }\cap U_{\delta })$, and since $U_{\gamma }\le U_{\delta }$ and $L_{\gamma }\cap U_{\delta }\le U_{\delta }$ the product is contained in $U_{\delta }$, so $U_{\delta }=U_{\gamma }(L_{\gamma }\cap U_{\delta })$; the intersection $U_{\gamma }\cap (L_{\gamma }\cap U_{\delta })=L_{\gamma }\cap U_{\gamma }$ is trivial by [L2], the factorisation is unique because $D$ is the diagonal part of $v$, and $U_{\gamma }⊴U_{\delta }$ because $U_{\delta }\le P_{\gamma }$ by [L3] and $U_{\gamma }⊴P_{\gamma }$ by [L2]; hence $U_{\delta }=U_{\gamma }⋊(L_{\gamma }\cap U_{\delta })$.

The induction hypothesis and products. Prove claim 2 for ${\mathrm{GL}}_m({\mathbb{F}}_q)$ by strong induction on $m$. For $m=1$, and also whenever $L=G$, the only coordinate parabolic with Levi $L$ is $G$, so both functors are identities. Assume the result for degrees less than $m$. For $H=H_1\times H_2$ and $P_i=L_i⋉U_i\le H_i$, the bijection $H\mathrm{/}(U_1\times U_2)\to (H_1\mathrm{/}{U}_1)\times (H_2\mathrm{/}{U}_2)$ is equivariant for left $H$ and right $L_1\times L_2$. Linearising gives the exterior tensor product of the two induction bimodules of [L1]. A natural isomorphism between induction functors for two parabolics in $H_i$, evaluated on the regular $\mathbb{C}{L}_i$-module, is an isomorphism of $(H_i,L_i)$-bimodules: naturality for right multiplication by each element of $L_i$ proves right equivariance. Tensoring these bimodule isomorphisms and then tensoring over $\mathbb{C}[L_1\times L_2]$ with any module proves independence of induction in $H$ for arbitrary modules, not only exterior products. Adjunction [L4], applied factorwise, gives natural isomorphisms between the represented Hom functors for the two restrictions in $H$. Evaluating these isomorphisms and their inverses at identity maps gives inverse module maps, proving independence of restriction in $H$. Iterating this construction shows that the induction hypothesis applies to a proper coordinate Levi of ${\mathrm{GL}}_m$, since each of its factors has degree strictly smaller than $m$, although the sum of their degrees remains $m$.

The intersected parabolics and the pairing formula. Let $P=L⋉U=P_{\gamma }$ and $Q=L⋉V=P_{{\gamma }^{\mathrm{\prime }}}$ be as in claim 2, let $X=\lambda$ be an irreducible complex $L$-module, and let $\mathcal{D}$, $w_{\rho }$, $C_{\rho }$, $D_{\rho }$, $A_{\rho }$ be as in [L5]. Then ${\mathrm{Hom}}_G(R_L^{G,P}\lambda ,R_L^{G,Q}\lambda )\cong {\mathrm{Hom}}_L(\lambda ,{}^{\ast }\text{\hspace{0.17em}⁣}{R}_L^{G,P}(R_L^{G,Q}\lambda ))$ by adjunction [L4], and expanding the restriction by the Mackey formula [L5] turns this into a direct sum, so that $\dim {\mathrm{Hom}}_G(R_L^{G,P}\lambda ,R_L^{G,Q}\lambda )={\sum }_{\rho \in \mathcal{D}}{T}_{\rho }(P,Q),T_{\rho }(P,Q):=\dim {\mathrm{Hom}}_L(\lambda ,R_{C_{\rho }}^{L,L\cap w_{\rho }Q{w}_{\rho }^{-1}}(({}^{\rho }\lambda {)}^{D_{\rho }})),$ where $({}^{\rho }\lambda {)}^{D_{\rho }}={}^{\ast }\text{\hspace{0.17em}⁣}{R}_{C_{\rho }}^{w_{\rho }L{w}_{\rho }^{-1},P\cap w_{\rho }L{w}_{\rho }^{-1}}({}^{\rho }\lambda )$ is the $C_{\rho }$-module obtained by restricting from $w_{\rho }L{w}_{\rho }^{-1}$ along the parabolic $P\cap w_{\rho }L{w}_{\rho }^{-1}=C_{\rho }⋉D_{\rho }$ of [L5], and the outer induction is taken along the parabolic $L\cap w_{\rho }Q{w}_{\rho }^{-1}=C_{\rho }⋉A_{\rho }$; they are parabolics of $w_{\rho }L{w}_{\rho }^{-1}$ and $L$, respectively, with common Levi $C_{\rho }$ by [L5]. The same formula with $Q$ replaced by $P$ computes $\dim {\mathrm{Hom}}_G(R_L^{G,P}\lambda ,R_L^{G,P}\lambda )$.

The terms with $C_{\rho }=L$. Suppose $C_{\rho }=L$, that is $w_{\rho }\in N_G(L)$, equivalently $\rho$ lies in the normaliser $N_{S_n}(W_L)$ of the parabolic Weyl group. Then $w_{\rho }L{w}_{\rho }^{-1}=L$ and [L5] gives $D_{\rho }=U\cap L=\{I_n\}$ and $A_{\rho }=L\cap w_{\rho }V{w}_{\rho }^{-1}=w_{\rho }(L\cap V)w_{\rho }^{-1}=\{I_n\}$; the parabolic $L\cap w_{\rho }Q{w}_{\rho }^{-1}$ is $L$ itself, so $R_{C_{\rho }}^{L,L}=R_L^{L,L}$ is the identity functor and $({}^{\rho }\lambda {)}^{D_{\rho }}$ is the module ${}^{\rho }\lambda$ with underlying space $\lambda$ and $L$-action $l\cdot x=(w_{\rho }^{-1}l{w}_{\rho })\cdot x$. Hence $T_{\rho }(P,Q)=\dim {\mathrm{Hom}}_L(\lambda ,{}^{\rho }\lambda )$, which is $1$ if $\lambda \cong {}^{\rho }\lambda$ and $0$ otherwise; in particular this term depends only on $L$, $\lambda$ and $\rho$, and not on the parabolics $P$ and $Q$.

The bijection between coset sets. Let $L=L_{\gamma }$, $K=L_{\delta }$ and $H:=L\cap U_{\delta }$. By step 1.1, $U_{\delta }=U_{\gamma }⋊H$, and $K$ normalises $H$ because $K⋉H=L\cap P_{\delta }$ is a parabolic of $L$ by [L2] and [L3]. Form the balanced product $G\mathrm{/}{U}_{\gamma }{\times }_{L}L\mathrm{/}H$ of the right $L$-set $G\mathrm{/}{U}_{\gamma }$ and the left $L$-set $L\mathrm{/}H$ of [L8]. The map $\mathrm{\Phi }:[g{U}_{\gamma },lH]⟼gl{U}_{\delta }$ is well defined: changing $g$ to $gu$ for $u\in U_{\gamma }$ does not change the image since $l^{-1}ul\in U_{\gamma }\subseteq U_{\delta }$ by [L2]; changing $l$ to $lh$ for $h\in H$ does not change it since $h\in U_{\delta }$; and the balancing relation $(g{l}_0{U}_{\gamma },lH)\sim (g{U}_{\gamma },l_{0}lH)$ has the same image by associativity. It is surjective since $g{U}_{\delta }=\mathrm{\Phi }([g{U}_{\gamma },H])$. To prove injectivity, suppose $gl{U}_{\delta }=g^{\mathrm{\prime }}{l}^{\mathrm{\prime }}{U}_{\delta }$, so $gl=g^{\mathrm{\prime }}{l}^{\mathrm{\prime }}{u}_{\delta }$ for some $u_{\delta }\in U_{\delta }$. Write $u_{\delta }=uh$ with $u\in U_{\gamma }$, $h\in H$ by step 1.1. Then $g=g^{\mathrm{\prime }}{l}^{\mathrm{\prime }}uh{l}^{-1}$, and $l^{\mathrm{\prime }}u(l^{\mathrm{\prime }}{)}^{-1}\in U_{\gamma }$ by [L2], so $g{U}_{\gamma }=g^{\mathrm{\prime }}{l}^{\mathrm{\prime }}h{l}^{-1}{U}_{\gamma }$. Balancing by $l_0=l^{\mathrm{\prime }}h{l}^{-1}\in L$ gives $[g{U}_{\gamma },lH]=[g^{\mathrm{\prime }}{U}_{\gamma },l^{\mathrm{\prime }}hH]=[g^{\mathrm{\prime }}{U}_{\gamma },l^{\mathrm{\prime }}H],$ proving injectivity. The map is equivariant for left multiplication by $G$. It is also equivariant for the right $K$-actions $[g{U}_{\gamma },lH]\cdot k:=[g{U}_{\gamma },lkH]$ and $(gl{U}_{\delta })\cdot k:=glk{U}_{\delta }$: the first is well defined since $K$ normalises $H$, and both images are $glk{U}_{\delta }$.

The remaining terms. Suppose $C_{\rho }\mathrm{\ne }L$. Then $C_{\rho }$ is a proper Levi subgroup of $L$, and $L$ is a product of general linear groups over ${\mathbb{F}}_q$; by step 1.2 the induction hypothesis applies in $L$, so the functors $R_{C_{\rho }}^{L,L\cap w_{\rho }Q{w}_{\rho }^{-1}}$ and $R_{C_{\rho }}^{L,L\cap w_{\rho }P{w}_{\rho }^{-1}}$ are isomorphic, both parabolics having Levi $C_{\rho }$ by [L5]. Applying the isomorphism to the module $({}^{\rho }\lambda {)}^{D_{\rho }}$ shows that the term $T_{\rho }(P,Q)$ of step 1.3 equals the corresponding term $T_{\rho }(P,P)$ in which $R_L^{G,Q}$ is replaced by $R_L^{G,P}$; the module $({}^{\rho }\lambda {)}^{D_{\rho }}$ is unchanged in this replacement because $D_{\rho }$ refers to the first parabolic $P$ in both cases. Likewise, the induction hypothesis applied in the group $w_{\rho }L{w}_{\rho }^{-1}$, whose factors, like those of the proper Levi $L$, each have degree smaller than $n$, shows that the functors ${}^{\ast }\text{\hspace{0.17em}⁣}{R}_{C_{\rho }}^{w_{\rho }L{w}_{\rho }^{-1},P\cap w_{\rho }L{w}_{\rho }^{-1}}$ and ${}^{\ast }\text{\hspace{0.17em}⁣}{R}_{C_{\rho }}^{w_{\rho }L{w}_{\rho }^{-1},Q\cap w_{\rho }L{w}_{\rho }^{-1}}$ from $w_{\rho }L{w}_{\rho }^{-1}$ to $C_{\rho }$ are isomorphic, both parabolics $P\cap w_{\rho }L{w}_{\rho }^{-1}=C_{\rho }⋉D_{\rho }$ and $Q\cap w_{\rho }L{w}_{\rho }^{-1}=C_{\rho }⋉D_{\rho }^{\mathrm{\prime }}$ having Levi $C_{\rho }$ by [L5]; hence $T_{\rho }(P,P)=T_{\rho }(Q,Q)$, and the two applications together give $T_{\rho }(P,Q)=T_{\rho }(P,P)=T_{\rho }(Q,Q)$ for every $\rho \in \mathcal{D}$.

Transitivity of Harish-Chandra induction. Let $W$ be a complex $K$-module. Inflating through the surjection $P_{\delta }\to K$ and inducing in two steps gives ${\mathrm{Ind}}_{P_{\delta }}^G({\mathrm{Inf}}_K^{P_{\delta }}W)\cong {\mathrm{Ind}}_{P_{\gamma }}^G({\mathrm{Ind}}_{P_{\delta }}^{P_{\gamma }}({\mathrm{Inf}}_K^{P_{\delta }}W))$ by transitivity of ordinary induction [L9], and ${\mathrm{Ind}}_{P_{\delta }}^{P_{\gamma }}({\mathrm{Inf}}_K^{P_{\delta }}W)\cong \mathbb{C}[P_{\gamma }\mathrm{/}{U}_{\delta }]{\otimes }_{\mathbb{C}K}W$ as $P_{\gamma }$-modules. Restricting the bijection $\mathrm{\Phi }$ of step 2.1 to $P_{\gamma }\mathrm{/}{U}_{\delta }\subseteq G\mathrm{/}{U}_{\delta }$ gives a $(P_{\gamma },K)$-equivariant bijection $P_{\gamma }\mathrm{/}{U}_{\delta }\cong P_{\gamma }\mathrm{/}{U}_{\gamma }{\times }_{L}L\mathrm{/}(L\cap U_{\delta })$: well-definedness and equivariance are inherited, and every $(p{U}_{\gamma },l(L\cap U_{\delta }))$ with $p\in P_{\gamma }$, $l\in L$ has $\mathrm{\Phi }(p{U}_{\gamma },l(L\cap U_{\delta }))=pl{U}_{\delta }$ with $pl\in P_{\gamma }L=P_{\gamma }$, since $P_{\gamma }=L{U}_{\gamma }$ by [L2]. Therefore $\mathbb{C}[P_{\gamma }\mathrm{/}{U}_{\delta }]{\otimes }_{\mathbb{C}K}W\cong \mathbb{C}[P_{\gamma }\mathrm{/}{U}_{\gamma }]{\otimes }_{\mathbb{C}L}(\mathbb{C}[L\mathrm{/}(L\cap U_{\delta })]{\otimes }_{\mathbb{C}K}W)$, which is the inflation to $P_{\gamma }$ of $R_K^{L,L\cap P_{\delta }}(W)=R_K^L(W)$ taken with respect to the parabolic $L\cap P_{\delta }=K(L\cap U_{\delta })$ of $L$; the latter is a parabolic with radical $L\cap U_{\delta }$ by step 1.1 and [L2], and its Levi subgroup is $K$. Applying ${\mathrm{Ind}}_{P_{\gamma }}^G$ gives the first displayed isomorphism of claim 1, namely $R_K^{G,P_{\delta }}\cong R_L^{G,P_{\gamma }}\circ R_K^{L,L\cap P_{\delta }}$, with the identification $\mathbb{C}[G\mathrm{/}{U}_{\gamma }]{\otimes }_{\mathbb{C}L}(\mathbb{C}[L\mathrm{/}(L\cap U_{\delta })]{\otimes }_{\mathbb{C}K}W)$ of the two sides.

Equal pairings and equal characters. Fix an irreducible $\lambda$. By steps 1.3, 1.4 and 2.2 every summand of the three sums for $\dim {\mathrm{Hom}}_G(R_L^{G,P}\lambda ,R_L^{G,Q}\lambda )$, $\dim {\mathrm{Hom}}_G(R_L^{G,P}\lambda ,R_L^{G,P}\lambda )$ and $\dim {\mathrm{Hom}}_G(R_L^{G,Q}\lambda ,R_L^{G,Q}\lambda )$ coincides, so the three numbers are equal. Writing ${\chi }_{\lambda }^P,{\chi }_{\lambda }^Q$ for the characters of $R_L^{G,P}\lambda$ and $R_L^{G,Q}\lambda$, positivity of the inner product [L7] gives $0\le ⟨{\chi }_{\lambda }^P-{\chi }_{\lambda }^Q,{\chi }_{\lambda }^P-{\chi }_{\lambda }^Q{⟩}_G=⟨{\chi }_{\lambda }^P,{\chi }_{\lambda }^P{⟩}_G-2⟨{\chi }_{\lambda }^P,{\chi }_{\lambda }^Q{⟩}_G+⟨{\chi }_{\lambda }^Q,{\chi }_{\lambda }^Q{⟩}_G=0$, where the inner products are the dimensions computed above by [L7]; therefore ${\chi }_{\lambda }^P={\chi }_{\lambda }^Q$ by [L7]. Both modules are completely reducible by [L6], and a completely reducible complex $G$-module is determined up to isomorphism by its character because its multiplicities are inner products with irreducible characters by [L6]; hence $R_L^{G,P}\lambda \cong R_L^{G,Q}\lambda$ for every irreducible $\lambda$.

Transitivity of Harish-Chandra restriction. For a complex $G$-module $X$ one has $X^{U_{\delta }}=X^{U_{\gamma }⋊(L\cap U_{\delta })}=(X^{U_{\gamma }}{)}^{L\cap U_{\delta }}$ by step 1.1, since a vector is fixed by every element of $U_{\delta }$ exactly when it is fixed by $U_{\gamma }$ and by $L\cap U_{\delta }$. The $L$-module $X^{U_{\gamma }}={}^{\ast }\text{\hspace{0.17em}⁣}{R}_L^{G,P_{\gamma }}(X)$ restricts under $L\cap P_{\delta }$ to the module whose $L\cap U_{\delta }$-invariants are $X^{U_{\delta }}$, and $L\cap U_{\delta }$ is the unipotent radical of $L\cap P_{\delta }$ by step 3.1, so $(X^{U_{\gamma }}{)}^{L\cap U_{\delta }}={}^{\ast }\text{\hspace{0.17em}⁣}{R}_K^{L,L\cap P_{\delta }}({}^{\ast }\text{\hspace{0.17em}⁣}{R}_L^{G,P_{\gamma }}(X))$ as $K$-modules; this is the second displayed isomorphism of claim 1, and its construction uses no choice beyond the fixed data.

From simple modules to natural functors. Choose representatives $S_1,\dots ,S_t$ for the finitely many simple complex $L$-modules, and choose isomorphisms ${\phi }_i:R_L^{G,P}{S}_i\to R_L^{G,Q}{S}_i$ using step 3.2. There are finitely many simple types because each is a quotient of the finite-dimensional regular module, which is completely reducible by [L6]. For every complex $L$-module $W$, the evaluation map ${⨁}_{i=1}^t{S}_i{\otimes }_{\mathbb{C}}{\mathrm{Hom}}_L(S_i,W)\to W$, $s\otimes f\mapsto f(s)$, is a natural isomorphism. For finite-dimensional $W$ this follows from complete reducibility and ${\mathrm{End}}_L(S_i)=\mathbb{C}$ in [L6]. For arbitrary $W$, each vector generates the finite-dimensional submodule $\mathbb{C}Lw$, giving surjectivity; any element of the kernel involves only finitely many maps from the finite-dimensional $S_i$, whose images lie in a finite-dimensional submodule, giving injectivity by the finite-dimensional case. The tensor model [L1] therefore identifies each induced module naturally with ${⨁}_i{R}_L^{G,P}{S}_i{\otimes }_{\mathbb{C}}{\mathrm{Hom}}_L(S_i,W)$, or its $Q$ version. The maps ${⨁}_i({\phi }_i\otimes \mathrm{i}\mathrm{d})$ define a natural isomorphism between them, since a map $W\to W^{\mathrm{\prime }}$ acts on the Hom factors by postcomposition. Only the finitely many ${\phi }_i$ are chosen; no bases or decompositions of arbitrary $W$ are chosen.

Independence of the restriction. Let $X$ be a complex $G$-module. Composing the natural isomorphisms ${\mathrm{Hom}}_L(W,X^U)\cong {\mathrm{Hom}}_G(R_L^{G,P}W,X)$ of [L4] with the natural isomorphism $R_L^{G,P}\cong R_L^{G,Q}$ of step 4.2 and with the second adjunction ${\mathrm{Hom}}_G(R_L^{G,Q}W,X)\cong {\mathrm{Hom}}_L(W,X^V)$ yields a natural isomorphism of functors of $W$ from ${\mathrm{Hom}}_L(-,X^U)$ to ${\mathrm{Hom}}_L(-,X^V)$, hence an $L$-linear isomorphism ${\varphi }_X:X^U\to X^V$: evaluate the natural Hom-space map at $W=X^U$ on ${\mathrm{i}\mathrm{d}}_{X^U}$, and evaluate its inverse at $W=X^V$ on ${\mathrm{i}\mathrm{d}}_{X^V}$. Naturality with respect to these maps makes their two composites the respective identities, and the construction is natural in $X$ because the adjunction isomorphisms are natural in both variables. Thus ${}^{\ast }\text{\hspace{0.17em}⁣}{R}_L^{G,P}\cong {}^{\ast }\text{\hspace{0.17em}⁣}{R}_L^{G,Q}$ as functors, which completes both displayed isomorphisms of claim 2.

Every construction in the preceding steps used only the fixed data, finitely many double-coset representatives, finitely many irreducible modules and the inner-product averages, so no form of the axiom of choice was used; and the functors $R_L^{G,P_{\gamma }}\cong R_L^{G,P_{{\gamma }^{\mathrm{\prime }}}}$ and ${}^{\ast }\text{\hspace{0.17em}⁣}{R}_L^{G,P_{\gamma }}\cong {}^{\ast }\text{\hspace{0.17em}⁣}{R}_L^{G,P_{{\gamma }^{\mathrm{\prime }}}}$ of claim 2 hold with all parabolics the coordinate parabolics $P_{\gamma },P_{{\gamma }^{\mathrm{\prime }}}$ and Levi $L=L_{\gamma }=L_{{\gamma }^{\mathrm{\prime }}}$, in particular for the standard parabolic $P_{\alpha }$ and any reordering of the blocks of $\alpha$. This proves both claims. ∎