Cuspidal representations and Harish-Chandra series

Refinements of ordered partitions. Let $n\ge 1$ and let $\gamma$ and $\delta$ be ordered partitions of $\{1,\dots ,n\}$ (Ordered partitions and coordinate parabolics). One says that $\delta$ refines $\gamma$ when every block of $\delta$ is contained in a block of $\gamma$, and that $\delta$ is a proper refinement of $\gamma$ when in addition the two set partitions differ, that is, when $\gamma$ does not refine $\delta$. Equivalently, $\gamma$ is obtained from $\delta$ by merging blocks, and at least one block of $\gamma$ is the union of at least two blocks of $\delta$; in particular $L_{\delta }\le L_{\gamma }$, since a matrix that is block diagonal for the finer ordered partition $\delta$ is block diagonal for $\gamma$ (Ordered partitions and coordinate parabolics). Moreover $P_{\delta }\cap L_{\gamma }=L_{\delta }⋉(U_{\delta }\cap L_{\gamma }),$ because every $g\in P_{\delta }\cap L_{\gamma }$ can be written as $g=lu$ with $l\in L_{\delta }$ and $u\in U_{\delta }$ (the standard Levi decomposition of Block Levi decomposition of standard parabolics transported by Ordered partitions and coordinate parabolics), and then $l\in L_{\delta }\le L_{\gamma }$ forces $u=l^{-1}g\in L_{\gamma }$; conversely $L_{\delta }(U_{\delta }\cap L_{\gamma })\subseteq P_{\delta }\cap L_{\gamma }$ since both factors lie in both subgroups, and $L_{\delta }\cap U_{\delta }=\{I_n\}$. By claim 1 of Transitivity and parabolic independence of Harish-Chandra induction the case in which the order of the blocks of $\delta$ is compatible with that of $\gamma$ is the one in which $U_{\gamma }\le U_{\delta }$ and $P_{\delta }\cap L_{\gamma }$ is displayed there as a coordinate parabolic of $L_{\gamma }$; every refinement can be reordered compatibly, that is, its blocks can be listed block by block in the order of $\gamma$. The group $U_{\delta }\cap L_{\gamma }$ is normalised by $L_{\delta }$, because $L_{\delta }\le P_{\delta }$ normalises $U_{\delta }$ (Ordered partitions and coordinate parabolics) and stabilises $L_{\gamma }$; hence for every complex $L_{\gamma }$-module $N$ the invariants $N^{U_{\delta }\cap L_{\gamma }}:=\{x\in N:ux=x\text{ for every }u\in U_{\delta }\cap L_{\gamma }\}$ form a complex $L_{\delta }$-submodule of $N$ (An $R$-linear action of $G$ on a left $R$-module, and a $G$-module over $R$). We write ${}^{\ast }\text{\hspace{0.17em}⁣}{R}_{L_{\delta }}^{L_{\gamma }}(N):=N^{U_{\delta }\cap L_{\gamma }}$ for this $L_{\delta }$-module, the Harish-Chandra restriction of $N$ from $L_{\gamma }$ to $L_{\delta }$ (along the coordinate parabolic $P_{\delta }\cap L_{\gamma }$); by claim 2 of Transitivity and parabolic independence of Harish-Chandra induction it does not depend, up to isomorphism of $L_{\delta }$-modules, on the order in which the blocks of $\delta$ are listed.

Cuspidal modules. Let $\gamma$ be an ordered partition of $\{1,\dots ,n\}$ and let $N$ be a complex $L_{\gamma }$-module. Then $N$ is cuspidal when ${}^{\ast }\text{\hspace{0.17em}⁣}{R}_{L_{\delta }}^{L_{\gamma }}(N)=0\text{for every proper refinement }\delta \text{ of }\gamma ,$ that is, when the invariants of $N$ under the unipotent radical $U_{\delta }\cap L_{\gamma }$ of the coordinate parabolic $P_{\delta }\cap L_{\gamma }$ of $L_{\gamma }$ vanish for every properly smaller coordinate Levi subgroup $L_{\delta }<L_{\gamma }$ obtained from a refinement of $\gamma$. We also say that a complex $L_{\gamma }$-module $N$ is non-cuspidal when it is not cuspidal.

Cuspidal pairs and Harish-Chandra series. A cuspidal pair is a pair $(L_{\gamma },N)$ consisting of a coordinate Levi subgroup $L_{\gamma }\le G$ and an irreducible cuspidal complex $L_{\gamma }$-module $N$. The Harish-Chandra series attached to such a pair is the set $\mathrm{Irr}(G\mid (L_{\gamma },N)):=\{M\in \mathrm{Irr}(\mathbb{C}G):R_{L_{\gamma }}^G(N)\twoheadrightarrow M\}$ of isomorphism classes of irreducible complex $G$-modules that are irreducible quotients of $R_{L_{\gamma }}^G(N)$, the Harish-Chandra induction of Harish-Chandra induction and restriction for finite general linear groups; since $R_{L_{\gamma }}^G(N)$ is a nonzero finite-dimensional complex $G$-module and every finite-dimensional complex module of the finite group $G$ is completely reducible, $M$ belongs to the series if and only if $M$ is isomorphic to a direct summand of $R_{L_{\gamma }}^G(N)$, and the series is nonempty (Maschke's theorem for finite groups over fields whose characteristic does not divide $\mathrm{\mid }G\mathrm{\mid }$).

Transport by a permutation. Let $\sigma \in S_n$ (The symmetric group $\mathrm{Sym}(X)$: the bijections of a set $X$ under composition, $\mathrm{Sym}(X)$ is a group under composition, and it is non-abelian whenever $X$ has at least three distinct elements) and let $w_{\sigma }:=P_{\sigma }$ be the associated permutation matrix (Permutation Weyl group and inversion length), so that conjugation by $w_{\sigma }$ maps the coordinate subgroups of type $\gamma$ onto those of type $\sigma (\gamma )$ (Ordered partitions and coordinate parabolics): for an ordered partition $\gamma =(S_1,\dots ,S_r)$ put $\sigma (\gamma ):=(\sigma (S_1),\dots ,\sigma (S_r))$, so that $L_{\sigma (\gamma )}=w_{\sigma }{L}_{\gamma }{w}_{\sigma }^{-1},P_{\sigma (\gamma )}=w_{\sigma }{P}_{\gamma }{w}_{\sigma }^{-1},U_{\sigma (\gamma )}=w_{\sigma }{U}_{\gamma }{w}_{\sigma }^{-1}\mathrm{.}$ For a complex $L_{\gamma }$-module $N$ let $N^{\sigma }$ be the vector space $N$ with the action of $L_{\sigma (\gamma )}$ given by $m\cdot x:=(w_{\sigma }^{-1}m{w}_{\sigma })\cdot x(m\in L_{\sigma (\gamma )},x\in N),$ which is a complex $L_{\sigma (\gamma )}$-module because $m\mapsto w_{\sigma }^{-1}m{w}_{\sigma }$ is an isomorphism $L_{\sigma (\gamma )}\to L_{\gamma }$ (Matrix multiplication is associative, unital, distributive, and compatible with scalar multiplication, Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes, ${\mathrm{GL}}_n(F)$ is a group under matrix multiplication, including the trivial group ${\mathrm{GL}}_0(F)$); we call $N^{\sigma }$ the transport of $N$ by $\sigma$. For a complex $G$-module $X$ let $X^{\sigma }$ be the vector space $X$ with the $G$-action $g\cdot x:=(w_{\sigma }^{-1}g{w}_{\sigma })\cdot x$; since $g\mapsto w_{\sigma }^{-1}g{w}_{\sigma }$ is an automorphism of $G$, this is again a complex $G$-module. Transport is compatible with composition, $(M^{\rho }{)}^{\sigma }=M^{\sigma \circ \rho }$ and $N^{\mathrm{i}\mathrm{d}}=N$, and a module and its transport by $\sigma$ have the same dimension and the same lattice of submodules, so $N$ is irreducible exactly when $N^{\sigma }$ is.

The definition of cuspidality above tests the quotient $N^{U_{\delta }\cap L_{\gamma }}$ for the standard parabolic $P_{\delta }\cap L_{\gamma }$ of the standard Levi $L_{\gamma }$. Dudas and Michel (Definition 10.2) define cuspidality of a $\mathrm{\Lambda }{G}^F$-module by the vanishing of ${}^{\ast }\text{\hspace{0.17em}⁣}{R}_L^{G^F}$ for every proper $G$-split Levi subgroup $L$ of the ambient group; the independence of the parabolic along which the restriction is taken is their Theorem 10.1, and the case needed above (two coordinate parabolics with the same coordinate Levi) is claim 2 of Transitivity and parabolic independence of Harish-Chandra induction. Taylor (Definition 5.7) uses the same vanishing condition for the standard Levis of a fixed split BN-pair, which is the form in which cuspidality is used by the Harish-Chandra series theorems of this page.

Two extreme cases illustrate the definition. For the one-block ordered partition $\gamma =(\{1,\dots ,n\})$ one has $L_{\gamma }=G$, the proper refinements of $\gamma$ are exactly the ordered partitions of $\{1,\dots ,n\}$ having at least two blocks, and a complex $G$-module is cuspidal in the sense above exactly when its invariants under the unipotent radical of every proper coordinate parabolic of $G$ vanish; for a cuspidal pair $(G,N)$ the series is the singleton $\{N\}$, because $R_{L_{\gamma }}^G$ is the identity functor (Harish-Chandra induction and restriction for finite general linear groups). For the all-singleton ordered partition $\gamma =(\{1\},\dots ,\{n\})$ one has $L_{\gamma }=T$ and there is no proper refinement at all, so every complex $T$-module is cuspidal; the series of the cuspidal pairs $(T,\chi )$ are the principal series of $G$. In both cases cuspidality is the standard one for the corresponding standard Levi, and the list of proper refinements does not depend on the order in which the blocks of $\gamma$ are listed.