Existence and uniqueness of cuspidal support

A smallest block statistic with nonzero invariants. The set $\mathcal{S}:=\{\eta :\eta \text{ an ordered partition of }\{1,\dots ,n\}\text{ with }{M}^{U_{\eta }}\mathrm{\ne }0\}$ is nonempty, because the one-block partition ${\eta }_0=(\{1,\dots ,n\})$ has $U_{{\eta }_0}=\{I_n\}$ and hence $M^{U_{{\eta }_0}}=M\mathrm{\ne }0$ by [L1] and [L7]. Since there are finitely many ordered partitions, the positive integers $d(\eta )={\sum }_i\mathrm{\mid }{S}_i{\mathrm{\mid }}^2$ for $\eta \in \mathcal{S}$ have a least value, attained at some $\gamma \in \mathcal{S}$; fix such a $\gamma$. If $L_{\eta }\le L_{\gamma }$, then every block of $\eta$ is contained in a block of $\gamma$: for any two indices in one $\eta$-block their transposition matrix lies in $L_{\eta }$, hence in $L_{\gamma }$, so they lie in one $\gamma$-block by [L3]. Thus $\eta$ refines $\gamma$. Splitting a block of size $a+b$ into nonempty blocks of sizes $a,b$ decreases $d$ by $2ab>0$, so $d(\eta )\le d(\gamma )$, with equality only when $L_{\eta }=L_{\gamma }$. For such $\eta \in \mathcal{S}$ the minimality of $d(\gamma )$ forces equality, and therefore $L_{\eta }=L_{\gamma }$.

Transport of Harish-Chandra induction. Let $\tau \in S_n$, put $w:=w_{\tau }=P_{\tau }$, and let $W$ be a complex $L_{\gamma }$-module. For $f\in R_{L_{\gamma }}^G(W)$ define $f^{\mathrm{\prime }}(g):=f(gw)$. If $p^{\mathrm{\prime }}\in P_{\tau (\gamma )}=w{P}_{\gamma }{w}^{-1}$, then $p:=w^{-1}{p}^{\mathrm{\prime }}w\in P_{\gamma }$, and the covariance of $f$ gives $f^{\mathrm{\prime }}(g{p}^{\mathrm{\prime }})=f(g{p}^{\mathrm{\prime }}w)=f(gwp)={\pi }_{\gamma }(p{)}^{-1}f(gw)={\pi }_{\tau (\gamma )}(p^{\mathrm{\prime }}{)}^{-1}{\cdot }_{W^{\tau }}{f}^{\mathrm{\prime }}(g)\mathrm{.}$ Here ${\pi }_{\tau (\gamma )}(p^{\mathrm{\prime }})=w{\pi }_{\gamma }(p)w^{-1}$ by the transported Levi decomposition of [L3], and its action on $W^{\tau }$ is the action of ${\pi }_{\gamma }(p)$ on $W$ by [L4]. Thus $f^{\mathrm{\prime }}\in R_{L_{\tau (\gamma )}}^G(W^{\tau })$. The assignment $f\mapsto f^{\mathrm{\prime }}$ is a $G$-linear bijection: its inverse is $h\mapsto (g\mapsto h(g{w}^{-1}))$, and left translation commutes with the fixed right translation $g\mapsto gw$. Hence $R_{L_{\gamma }}^G(W)\cong R_{L_{\tau (\gamma )}}^G(W^{\tau })\mathrm{.}$ Moreover every $G$-module $Y$ is isomorphic to its $\tau$-transport $Y^{\tau }$: if ${\rho }_Y$ denotes its action, ${\rho }_Y(w^{-1}):Y\to Y^{\tau }$ intertwines the actions, since ${\rho }_Y(w^{-1}){\rho }_Y(g)={\rho }_Y(w^{-1}gw){\rho }_Y(w^{-1})$. In particular the two Harish-Chandra inductions have the same simple constituents, with $M^{\tau }\cong M$. If a cuspidal pair $(L_{{\gamma }^{\mathrm{\prime }}},N^{\mathrm{\prime }})$ satisfies $L_{{\gamma }^{\mathrm{\prime }}}=L_{\tau (\gamma )}$ and $N^{\mathrm{\prime }}\cong N^{\tau }$, parabolic independence [L5] handles any different order of the same blocks; consequently $M\in \mathrm{Irr}(G\mathrm{\mid }(L_{\gamma },N))$ if and only if $M^{\tau }\in \mathrm{Irr}(G\mathrm{\mid }(L_{{\gamma }^{\mathrm{\prime }}},N^{\mathrm{\prime }}))$.

The mixed Hom-space and its Mackey expansion. Suppose now that $M$ lies in the series of both cuspidal pairs, that is $R_{L_{\gamma }}^G(N)\twoheadrightarrow M$ and $R_{L_{{\gamma }^{\mathrm{\prime }}}}^G(N^{\mathrm{\prime }})\twoheadrightarrow M$ by [L4]. Since $R_{L_{{\gamma }^{\mathrm{\prime }}}}^G(N^{\mathrm{\prime }})$ is completely reducible by [L7] and $M$ is one of its quotients, $M$ is a direct summand of $R_{L_{{\gamma }^{\mathrm{\prime }}}}^G(N^{\mathrm{\prime }})$, so the composite of the surjection $R_{L_{\gamma }}^G(N)\twoheadrightarrow M$ with an inclusion $M\hookrightarrow R_{L_{{\gamma }^{\mathrm{\prime }}}}^G(N^{\mathrm{\prime }})$ is a nonzero $G$-linear map and ${\mathrm{Hom}}_G(R_{L_{\gamma }}^G(N),R_{L_{{\gamma }^{\mathrm{\prime }}}}^G(N^{\mathrm{\prime }}))\mathrm{\ne }0$. By adjunction [L2] this space is isomorphic to ${\mathrm{Hom}}_{L_{\gamma }}(N,{}^{\ast }\text{\hspace{0.17em}⁣}{R}_{L_{\gamma }}^G(R_{L_{{\gamma }^{\mathrm{\prime }}}}^G(N^{\mathrm{\prime }})))$, and the Mackey formula [L6] applied to the $L_{{\gamma }^{\mathrm{\prime }}}$-module $N^{\mathrm{\prime }}$ rewrites the inner restriction as a direct sum, so that $0\mathrm{\ne }{⨁}_{\rho \in \mathcal{D}}{\mathrm{Hom}}_{L_{\gamma }}(N,R_{C_{\rho }}^{L_{\gamma }}(({}^{\rho }{N}^{\mathrm{\prime }}{)}^{D_{\rho }})),$ with $C_{\rho }=L_{\gamma }\cap w_{\rho }{L}_{{\gamma }^{\mathrm{\prime }}}{w}_{\rho }^{-1}$, $D_{\rho }=U_{\gamma }\cap w_{\rho }{L}_{{\gamma }^{\mathrm{\prime }}}{w}_{\rho }^{-1}$, $A_{\rho }=L_{\gamma }\cap w_{\rho }{U}_{{\gamma }^{\mathrm{\prime }}}{w}_{\rho }^{-1}$ with intersected parabolics in $w_{\rho }{L}_{{\gamma }^{\mathrm{\prime }}}{w}_{\rho }^{-1}$ and $L_{\gamma }$, respectively, as in [L6]; hence there is $\rho \in \mathcal{D}$ with ${\mathrm{Hom}}_{L_{\gamma }}(N,R_{C_{\rho }}^{L_{\gamma }}(Y_{\rho }))\mathrm{\ne }0$ for $Y_{\rho }:=({}^{\rho }{N}^{\mathrm{\prime }}{)}^{D_{\rho }}$, a complex $C_{\rho }$-module.

$M^{U_{\gamma }}$ is cuspidal. Let $\delta$ be a proper refinement of $\gamma$, listed block by block in the order of $\gamma$, as [L4] allows. Then $L_{\delta }\le L_{\gamma }$ and $U_{\delta }=U_{\gamma }⋊(U_{\delta }\cap L_{\gamma })$ with $M^{U_{\delta }}=(M^{U_{\gamma }}{)}^{U_{\delta }\cap L_{\gamma }}$ by [L4] and [L5]. If $\delta \in \mathcal{S}$ then $L_{\delta }=L_{\gamma }$ by step 1.1; but a proper refinement $\delta$ of $\gamma$ has $L_{\delta }\mathrm{\ne }{L}_{\gamma }$, since the two set partitions differ and so some two indices lie in a common block of exactly one of them, whence the transposition matrix $w_{(kl)}$ lies in exactly one of $L_{\delta },L_{\gamma }$ by [L3]. Hence $\delta \mathrm{\notin }\mathcal{S}$, that is $M^{U_{\delta }}=0$, and therefore $(M^{U_{\gamma }}{)}^{U_{\delta }\cap L_{\gamma }}=0$. For an arbitrary proper refinement ${\delta }^{\mathrm{\prime }}$ of $\gamma$, let $\delta$ be the rearrangement of ${\delta }^{\mathrm{\prime }}$ that lists blocks block by block in the order of $\gamma$; the defining entry conditions for $P_{{\delta }^{\mathrm{\prime }}}\cap L_{\gamma }$ and $P_{\delta }\cap L_{\gamma }$ on a $\gamma$-block diagonal matrix involve only the relative order of the blocks of ${\delta }^{\mathrm{\prime }}$ inside each block of $\gamma$, which the rearrangement preserves, so $P_{{\delta }^{\mathrm{\prime }}}\cap L_{\gamma }=P_{\delta }\cap L_{\gamma }$ and hence $U_{{\delta }^{\mathrm{\prime }}}\cap L_{\gamma }=U_{\delta }\cap L_{\gamma }$ by [L3], [L4]. Therefore ${}^{\ast }\text{\hspace{0.17em}⁣}{R}_{L_{{\delta }^{\mathrm{\prime }}}}^{L_{\gamma }}(M^{U_{\gamma }})=(M^{U_{\gamma }}{)}^{U_{{\delta }^{\mathrm{\prime }}}\cap L_{\gamma }}=0$ for every proper refinement ${\delta }^{\mathrm{\prime }}$ of $\gamma$, and $M^{U_{\gamma }}$ is a cuspidal $L_{\gamma }$-module by [L4].

A simple cuspidal submodule. The module $M^{U_{\gamma }}$ is nonzero, finite-dimensional and complex, so it is completely reducible and hence a direct sum of simple submodules by [L7]; fix a simple $L_{\gamma }$-submodule $N\subseteq M^{U_{\gamma }}$ with $N\mathrm{\ne }0$. For every proper refinement $\delta$ of $\gamma$ the inclusion $N\hookrightarrow M^{U_{\gamma }}$ gives an inclusion ${}^{\ast }\text{\hspace{0.17em}⁣}{R}_{L_{\delta }}^{L_{\gamma }}(N)\hookrightarrow {}^{\ast }\text{\hspace{0.17em}⁣}{R}_{L_{\delta }}^{L_{\gamma }}(M^{U_{\gamma }})=0$ by exactness [L8] and step 2.1, so ${}^{\ast }\text{\hspace{0.17em}⁣}{R}_{L_{\delta }}^{L_{\gamma }}(N)=0$ and the simple module $N$ is cuspidal; hence $(L_{\gamma },N)$ is a cuspidal pair by [L4].

$M$ lies in the series. The inclusion $N\subseteq M^{U_{\gamma }}={}^{\ast }\text{\hspace{0.17em}⁣}{R}_{L_{\gamma }}^G(M)$ is a nonzero $L_{\gamma }$-linear map from $N$ to ${}^{\ast }\text{\hspace{0.17em}⁣}{R}_{L_{\gamma }}^G(M)$, so its image under the adjunction bijection of [L2] is a nonzero $G$-linear map $R_{L_{\gamma }}^G(N)\to M$. The image of that map is a nonzero submodule of the simple module $M$ and therefore equals $M$, so the map is surjective and $M\in \mathrm{Irr}(G\mathrm{\mid }(L_{\gamma },N))$ by [L4].

Terms over a proper Levi vanish. Let $C\le L_{\gamma }$ be a proper coordinate Levi, say $C=L_{\eta }$ with $\eta$ a proper refinement of $\gamma$, let $A=U_{\eta }\cap L_{\gamma }$ be the radical of the coordinate parabolic $P_{\eta }\cap L_{\gamma }=L_{\eta }⋉(U_{\eta }\cap L_{\gamma })$ of $L_{\gamma }$ by [L4], and let $Y$ be a complex $C$-module. Then ${\mathrm{Hom}}_{L_{\gamma }}(R_C^{L_{\gamma }}(Y),N)\cong {\mathrm{Hom}}_C(Y,{}^{\ast }\text{\hspace{0.17em}⁣}{R}_C^{L_{\gamma }}(N))={\mathrm{Hom}}_C(Y,N^A)=0$ by adjunction [L2] applied to the parabolic $P_{\eta }\cap L_{\gamma }$ of $L_{\gamma }$ and by the assumed cuspidality of $N$ from [L4]. Since $R_C^{L_{\gamma }}(Y)$ is completely reducible by [L7] and $N$ is simple, both ${\mathrm{Hom}}_{L_{\gamma }}(N,R_C^{L_{\gamma }}(Y))$ and ${\mathrm{Hom}}_{L_{\gamma }}(R_C^{L_{\gamma }}(Y),N)$ have dimension equal to the multiplicity of $N$ in $R_C^{L_{\gamma }}(Y)$, by Schur's lemma [L7]; hence ${\mathrm{Hom}}_{L_{\gamma }}(N,R_C^{L_{\gamma }}(Y))=0$ as well.

The contributing double coset has Levi $L_{\gamma }$. In the nonzero Hom-space of step 1.3 the group $C_{\rho }=L_{\gamma }\cap w_{\rho }{L}_{{\gamma }^{\mathrm{\prime }}}{w}_{\rho }^{-1}$ is the coordinate Levi $L_{\eta }$ of $L_{\gamma }$ for the ordered partition $\eta$ whose blocks are the nonempty intersections of a block of $\gamma$ with a block of $\rho ({\gamma }^{\mathrm{\prime }})$ and which lists its blocks block by block in the order of $\gamma$: a matrix is block diagonal for both $\gamma$ and $\rho ({\gamma }^{\mathrm{\prime }})$ exactly when it is block diagonal for the common refinement $\eta$ by [L3], and $L_{\eta }\le L_{\gamma }$ with $L_{\eta }\le L_{\rho ({\gamma }^{\mathrm{\prime }})}$ by [L4]. Similarly the parabolic $L_{\gamma }\cap w_{\rho }{P}_{{\gamma }^{\mathrm{\prime }}}{w}_{\rho }^{-1}=C_{\rho }⋉A_{\rho }$ of [L6] equals $P_{\eta }\cap L_{\gamma }=L_{\eta }⋉(U_{\eta }\cap L_{\gamma })$, so that $A_{\rho }=U_{\eta }\cap L_{\gamma }$: a matrix $g$ lies in $L_{\gamma }\cap w_{\rho }{P}_{{\gamma }^{\mathrm{\prime }}}{w}_{\rho }^{-1}$ exactly when it is $\gamma$-block diagonal and satisfies the entry conditions ${\mathrm{blk}}_{\rho ({\gamma }^{\mathrm{\prime }})}(k)\le {\mathrm{blk}}_{\rho ({\gamma }^{\mathrm{\prime }})}(l)$ for all pairs with $g_{kl}\mathrm{\ne }0$, and on a $\gamma$-block diagonal matrix the conditions of [L3] for $P_{\eta }$ reduce to the same requirements, because the entries of $g$ outside a single block of $\gamma$ vanish and the blocks of $\rho ({\gamma }^{\mathrm{\prime }})$ meeting a block of $\gamma$ occur in $\eta$ in their order of $\rho ({\gamma }^{\mathrm{\prime }})$. Since $Y_{\rho }=({}^{\rho }{N}^{\mathrm{\prime }}{)}^{D_{\rho }}$ is a complex $C_{\rho }$-module and the induction in the summand of step 1.3 is along $L_{\gamma }\cap w_{\rho }{P}_{{\gamma }^{\mathrm{\prime }}}{w}_{\rho }^{-1}=C_{\rho }⋉A_{\rho }$ by [L6], step 4.2 with $C=C_{\rho }$ shows that $C_{\rho }\mathrm{\ne }{L}_{\gamma }$ is impossible; hence $C_{\rho }=L_{\gamma }$, that is $L_{\gamma }\le w_{\rho }{L}_{{\gamma }^{\mathrm{\prime }}}{w}_{\rho }^{-1}$ and $d(\gamma )\le d({\gamma }^{\mathrm{\prime }})$.

The reverse inequality. The argument of steps 1.3, 4.2 and 5.1 applied to the same $M$ with the two pairs interchanged uses ${\mathrm{Hom}}_G(R_{L_{{\gamma }^{\mathrm{\prime }}}}^G(N^{\mathrm{\prime }}),R_{L_{\gamma }}^G(N))\mathrm{\ne }0$, which holds because $M$ is a quotient of $R_{L_{{\gamma }^{\mathrm{\prime }}}}^G(N^{\mathrm{\prime }})$ and a direct summand of the completely reducible module $R_{L_{\gamma }}^G(N)$ by [L7]; expanding $0\mathrm{\ne }{\mathrm{Hom}}_{L_{{\gamma }^{\mathrm{\prime }}}}(N^{\mathrm{\prime }},{}^{\ast }\text{\hspace{0.17em}⁣}{R}_{L_{{\gamma }^{\mathrm{\prime }}}}^G(R_{L_{\gamma }}^G(N)))$ by [L2] and the Mackey formula [L6], and discarding the summands over proper coordinate Levis of $L_{{\gamma }^{\mathrm{\prime }}}$ by step 4.2 with $N^{\mathrm{\prime }}$ in place of $N$ (which is cuspidal by [L4]), yields $d({\gamma }^{\mathrm{\prime }})\le d(\gamma )$. Hence $d(\gamma )=d({\gamma }^{\mathrm{\prime }})$, and the inclusion of step 5.1 is an equality: $L_{\gamma }=w_{\rho }{L}_{{\gamma }^{\mathrm{\prime }}}{w}_{\rho }^{-1}$ for the $\rho \in \mathcal{D}$ of step 1.3. With $\tau :={\rho }^{-1}\in S_n$ and $w_{\tau }=w_{\rho }^{-1}$ by [L9] this reads $w_{\tau }{L}_{\gamma }{w}_{\tau }^{-1}=L_{{\gamma }^{\mathrm{\prime }}}$, that is $L_{\tau (\gamma )}=L_{{\gamma }^{\mathrm{\prime }}}$ by [L3], so that ${\gamma }^{\mathrm{\prime }}$ and $\tau (\gamma )$ have the same blocks.

The modules are transported. In the summand of step 1.3 with $C_{\rho }=L_{\gamma }$ the group $w_{\rho }{L}_{{\gamma }^{\mathrm{\prime }}}{w}_{\rho }^{-1}$ equals $L_{\gamma }$ by step 6.1, so $D_{\rho }=U_{\gamma }\cap w_{\rho }{L}_{{\gamma }^{\mathrm{\prime }}}{w}_{\rho }^{-1}=U_{\gamma }\cap L_{\gamma }=\{I_n\}$, $A_{\rho }=L_{\gamma }\cap w_{\rho }{U}_{{\gamma }^{\mathrm{\prime }}}{w}_{\rho }^{-1}\subseteq w_{\rho }{L}_{{\gamma }^{\mathrm{\prime }}}{w}_{\rho }^{-1}\cap w_{\rho }{U}_{{\gamma }^{\mathrm{\prime }}}{w}_{\rho }^{-1}=w_{\rho }(L_{{\gamma }^{\mathrm{\prime }}}\cap U_{{\gamma }^{\mathrm{\prime }}})w_{\rho }^{-1}=\{I_n\}$, the parabolic $L_{\gamma }\cap w_{\rho }{P}_{{\gamma }^{\mathrm{\prime }}}{w}_{\rho }^{-1}=C_{\rho }⋉A_{\rho }$ is $L_{\gamma }$, the functor $R_{C_{\rho }}^{L_{\gamma }}=R_{L_{\gamma }}^{L_{\gamma }}$ is the identity of [L1], and $Y_{\rho }=({}^{\rho }{N}^{\mathrm{\prime }}{)}^{D_{\rho }}={}^{\rho }{N}^{\mathrm{\prime }}$ is the transport of $N^{\mathrm{\prime }}$ to $L_{\gamma }$ by [L4]. Step 1.3 therefore gives ${\mathrm{Hom}}_{L_{\gamma }}(N,(N^{\mathrm{\prime }}{)}^{\rho })\mathrm{\ne }0$; as $N$ and $(N^{\mathrm{\prime }}{)}^{\rho }$ are simple $L_{\gamma }$-modules, $N\cong (N^{\mathrm{\prime }}{)}^{\rho }$ by Schur's lemma [L7]. Transporting this isomorphism by $\tau ={\rho }^{-1}$ and using $(Y^{\rho }{)}^{{\rho }^{-1}}\cong Y$ of [L4] gives $N^{\tau }\cong N^{\mathrm{\prime }}$, so that the pairs are conjugate in the sense of claim 2.

The partition of the irreducibles. By step 4.1 every simple complex $G$-module $M$ lies in the series of at least one cuspidal pair, and by step 7.1 any two cuspidal pairs whose series both contain $M$ are conjugate in the sense of claim 2; conversely step 1.2 shows that conjugate pairs have their series matched by the transport bijection, so that pairs in one conjugacy class carry the same irreducibles. Hence the assignment to $M$ of the conjugacy class of a cuspidal pair $(L_{\gamma },N)$ with $M\in \mathrm{Irr}(G\mathrm{\mid }(L_{\gamma },N))$ is well defined, series attached to pairwise non-conjugate cuspidal pairs are pairwise disjoint, and the unions of the series over one conjugacy class partition $\mathrm{Irr}(\mathbb{C}G)$. This proves all three claims. ∎