2.1. Induction and restriction from normal subgroups

We fix a field $\mathbb {k}$ . For any finite group K, we will denote by $\mathrm {Rep}_{\mathbb {k}}(K)$ the category of finite-dimensional representations of K over $\mathbb {k}$ .

If H is a finite group and $N \subseteq H$ is a subgroup, we have an exact restriction functor

$$\begin{align*}\mathrm{Res}^H_N : \mathrm{Rep}_{\mathbb{k}}(H) \to \mathrm{Rep}_{\mathbb{k}}(N). \end{align*}$$

We also have an exact induction functor

$$\begin{align*}\mathrm{Ind}_N^H := \mathbb{k} H \otimes_{\mathbb{k} N} (-) : \mathrm{Rep}_{\mathbb{k}}(N) \to \mathrm{Rep}_{\mathbb{k}}(H), \end{align*}$$

which is both left and right adjoint to $\mathrm {Res}^H_N$ , see [Reference Curtis and Reiner6, (10.8) and (10.21)].

In practice, we will assume that N is normal. In this case, we have the following statements about the behavior of these functors with respect to semisimplicity.

2.2. Simple subobjects of induced representations

We continue to consider a normal subgroup N of a finite group H and a field $\mathbb {k}$ . Our goal is to obtain a parameterization of the sets appearing in Corollary 2.2.

If $V \in \mathrm {Rep}_{\mathbb {k}}(N)$ and $g \in H,$ we have the N-module ${}^g V$ whose underlying vector space is V, and the action $\cdot ^g$ of N is defined by $h \cdot ^g v = (g^{-1} h g) \cdot v$ for $v \in V$ and $h \in N$ . Up to isomorphism, this representation only depends on the coset $gN$ . Therefore, the subgroup $\mathrm {Stab}_H(V)=\{h \in H \mid {}^h V \cong V\}$ contains N. It is a standard observation that, in this setting, if $S \subset H$ is a subset of representatives for the cosets in $H/N$ , then we have a canonical isomorphism

(1) $$ \begin{align} \mathrm{Res}^H_N(\mathrm{Ind}_N^H(V)) \cong \bigoplus_{s \in S} {}^s V. \end{align} $$

More specifically, as vector spaces, we have $\mathrm {Ind}_N^H(V) = \bigoplus _{s \in S} \mathbb {k}{\cdot}{s} \otimes V$ , and $\mathbb {k}{\cdot}{s} \otimes V$ is an N-stable subspace which identifies with ${}^s V$ . More generally, if $K \subseteq H$ is a subgroup containing N and $V \in \mathrm {Rep}_{\mathbb {k}}(K)$ , and if $S \subset H$ is a subset of representatives for the cosets in $H/K$ , then we have a canonical isomorphism

(2) $$ \begin{align} \mathrm{Res}^H_N(\mathrm{Ind}_K^H(V)) \cong \bigoplus_{s \in S} {}^s (\mathrm{Res}^K_N(V)). \end{align} $$

The following lemma will serve as an ingredient in the proof of Proposition 2.4 below.

In the next statement, we use the notion of crossed product of a group over an algebra, which generalizes twisted group algebras to the setting where the base field (or ring) can have a nontrivial action of the group (see [Reference Karpilovsky10, Chapter 3, §3] for details).

Proof. (1) As explained in [Reference Karpilovsky10, Chapter 3, Lemma 12.1], the algebra morphism

(3) $$ \begin{align} \mathrm{End}_K(\mathrm{Ind}_N^K(V)) \to \mathrm{End}_H(\mathrm{Ind}_N^H(V)) \end{align} $$

induced by the functor $\mathrm {Ind}_K^H$ is an isomorphism. Then the claim follows from [Reference Karpilovsky10, Chapter 3, Theorem 4.2(ii)].

(2) Using the isomorphism (3) and Lemma 2.3, to prove the proposition, it suffices to construct a canonical bijection between the sets of isomorphism classes of simple subrepresentations of $\mathrm {Ind}_N^K(V)$ and of simple modules over $\mathrm {End}_K(\mathrm {Ind}_N^K(V))$ .

By [Reference Curtis and Reiner6, Theorem 11.17], there exists a functorial isomorphism between the lattices of left ideals in $\mathrm {End}_K(\mathrm {Ind}_N^K(V))$ and of K-subrepresentations of $\mathrm {Ind}_N^K(V)$ . This bijection necessarily induces a bijection between the sets of isomorphism classes of simple $\mathrm {End}_K(\mathrm {Ind}_N^K(V))$ -submodules of $\mathrm {End}_K(\mathrm {Ind}_N^K(V))$ and the set of simple K-subrepresentations of $\mathrm {Ind}_N^K(V)$ , so that to conclude it suffices to show that any simple $\mathrm {End}_K(\mathrm {Ind}_N^K(V))$ -module injects in the free rank- $1$ -module. This property follows from the fact that $\mathrm {End}_K(\mathrm {Ind}_N^K(V))$ is a symmetric algebra, see [Reference Karpilovsky10, Chapter 3, Theorem 12.3].

Remark 2.5. The assertion in Proposition 2.4(1) means in particular that the algebra $\mathrm {End}_H(\mathrm {Ind}_N^H(V))$ admits a canonical $K/N$ -grading, with degree- $1$ component the division algebra $\mathrm {End}_N(V)$ . This grading can be described explicitly as follows. Recall from (3) that we have an algebra isomorphism

$$\begin{align*}\mathrm{End}_H(\mathrm{Ind}_N^H(V)) \cong \mathrm{End}_K(\mathrm{Ind}_N^K(V)); \end{align*}$$

moreover, by adjunction, the right-hand side identifies with $\mathrm {Hom}_N(V,\mathrm {Res}^K_N(\mathrm {Ind}_N^K(V)))$ (as a $\mathbb {k}$ -vector space). Now, by (1), $\mathrm {Res}^K_N(\mathrm {Ind}_N^K(V))$ has a canonical decomposition as a direct sum parameterized by $K/N$ , with each factor (noncanonically) isomorphic to V. We deduce a canonical decomposition of $\mathrm {End}_H(\mathrm {Ind}_N^H(V))$ as a direct sum of subspaces parameterized by $K/N$ , as desired. (All of these subspaces are noncanonically isomorphic to $\mathrm {End}_N(V)$ .)

Assuming that V is absolutely irreducible, so that $\mathrm {End}_N(V)=\mathbb {k}$ , this description simplifies and reduces to a twisted group algebra for $K/N$ over $\mathbb {k}$ in the sense of [Reference Curtis and Reiner6, Example 8.33(ii)] or [Reference Karpilovsky10, Chapter 3, §6].

3.1. Parabolic and Levi subgroups

As usual, given a complex algebraic group H, we will denote by $H^\circ $ its neutral connected component, and by $\mathrm {Z}(H)$ its center. Given a subset $K \subset H$ , we denote by $\mathrm {Z}_H(K)=\{h \in H \mid \forall k \in K, hkh^{-1} = k\}$ the centralizer of K in H, and by $\mathrm {N}_H(K)=\{h \in H \mid hKh^{-1} = K\}$ its normalizer.

Let G be a (possibly disconnected) complex reductive algebraic group. Following [Reference Dillery and Schwein7, §2.3], we say that a subgroup P of G is parabolic if the quotient variety $G/P$ is proper, or equivalently, if $P^\circ $ is a parabolic subgroup (in the usual sense) of the connected reductive algebraic group $G^\circ $ . See [Reference Dillery and Schwein7], §2.3 for a combinatorial description of conjugacy classes of such subgroups. If P is a parabolic subgroup of G, with unipotent radical U, then a Levi factor of P is a subgroup L such that $P=L \ltimes U$ . Given P, such a subgroup always exists (see again [Reference Dillery and Schwein7, §2.3] for details). For simplicity, we will use the terminology “Levi subgroup” of G for “Levi factor of a parabolic subgroup” of G.

A Levi subgroup L of G will be called special Footnote ³ if we have

$$\begin{align*}L=\mathrm{Z}_G(\mathrm{Z}(L^{\circ})^{\circ}). \end{align*}$$

Note that in case G is connected, all Levi subgroups are special. In general, as explained in [Reference Dillery and Schwein7, §2.6], the assignments $L \mapsto L^\circ $ and $M \mapsto \mathrm {Z}_G(\mathrm {Z}(M)^{\circ })$ induce (G-equivariant) bijections between the sets of special Levi subgroups of G and Levi subgroups of $G^\circ $ . If P is a parabolic subgroup of G, we will say that P is special if one (equivalently, all) of its Levi factors is special.

We will say that a parabolic subgroup P of G is maximal if $P^\circ = G^\circ $ . By the comments above, there exists exactly one parabolic subgroup which is both special and maximal, but this subgroup is not necessarily G (since the latter is not necessarily a special parabolic subgroup). This subgroup is G in case G is semisimple, or if G is itself a special Levi subgroup in a larger reductive algebraic group H.

We record the following property for later use.

3.2. Equivariant sheaves on nilpotent cones

From now on, we fix a field $\mathbb {k}$ of coefficients for our (complexes of) sheaves and continue with our possibly disconnected complex reductive algebraic group G. For a complex algebraic group H, we will denote by $\mathfrak {h}$ its Lie algebra, and by $\mathcal {N}_H$ the nilpotent cone of H, that is, the closed subvariety of $\mathfrak {h}$ consisting of nilpotent elements. Note that we have $\mathcal {N}_H=\mathcal {N}_{H^\circ }$ . For $x \in \mathfrak {h,}$ we will denote by $\mathrm {Z}_H(x)$ the centralizer of x in H, and set $\mathrm {A}_H(x) := \mathrm {Z}_H(x)/\mathrm {Z}_H(x)^\circ $ .

The main player in this article will be the derived category $D^{\mathrm {b}}_G(\mathcal {N}_G,\mathbb {k})$ of constructible sheaves on the quotient stack $G \backslash \mathcal {N}_G$ , with coefficients in $\mathbb {k}$ . This category can also be described in terms of the Bernstein–Lunts construction, see [Reference Achar1, Chapter 6]. We have a canonical “forgetful” functor

$$\begin{align*}\mathrm{For}^G : D^{\mathrm{b}}_G(\mathcal{N}_G,\mathbb{k}) \to D^{\mathrm{b}}_{\mathrm{c}}(\mathcal{N}_G,\mathbb{k}), \end{align*}$$

where the right-hand side is the constructible derived category of sheaves on $\mathcal {N}_G$ . The category $D^{\mathrm {b}}_G(\mathcal {N}_G,\mathbb {k})$ is equipped with a natural t-structure such that $\mathrm {For}^G$ is t-exact with respect to the perverse t-structure, see, for example, [Reference Achar1, Theorem 6.4.10]. This t-structure is also called the perverse t-structure, and its heart will be denoted $\mathrm {Perv}_G(\mathcal {N}_G,\mathbb {k})$ . In fact, by [Reference Achar1, Theorem 6.4.10] and its proof, this heart consists precisely of the objects of $D^{\mathrm {b}}_G(\mathcal {N}_G,\mathbb {k})$ whose image in $D^{\mathrm {b}}_{\mathrm {c}}(\mathcal {N}_G,\mathbb {k})$ is perverse, and it can equivalently be described as the category of G-equivariant perverse sheaves on $\mathcal {N}_G$ in the sense of [Reference Achar1, Definition 6.2.3]. In case G is connected, the restriction of the functor $\mathrm {For}^G$ to $\mathrm {Perv}_G(\mathcal {N}_G,\mathbb {k})$ is fully faithful (so that a G-equivariant perverse sheaf is a perverse sheaf with a certain property), and its essential image is stable under subquotients, but in general, these properties are not true (in particular, a G-equivariant perverse sheaf should be considered as a perverse sheaf with a certain additional structure).

As in the introduction, we will denote by $\mathfrak {R}_{G,\mathbb {k}}$ the set of isomorphism classes of simple objects of $\mathrm {Perv}_G(\mathcal {N}_G,\mathbb {k})$ . By the general theory of perverse sheaves, every object in $\mathrm {Perv}_G(\mathcal {N}_G,\mathbb {k})$ has finite length, and $\mathfrak {R}_{G,\mathbb {k}}$ identifies with the quotient of the set of pairs $(\mathcal {O},\mathcal {E}),$ where $\mathcal {O} \subset \mathcal {N}_G$ is a G-orbit and $\mathcal {E}$ is a simple G-equivariant $\mathbb {k}$ -local system on $\mathcal {O}$ , by the equivalence relation given by $(\mathcal {O},\mathcal {E}) \sim (\mathcal {O}',\mathcal {E}')$ if $\mathcal {O}=\mathcal {O}'$ and $\mathcal {E} \cong \mathcal {E}'$ . This bijection sends the equivalence class of a pair $(\mathcal {O},\mathcal {E})$ to the isomorphism class of the intersection cohomology complex $\mathcal {IC}(\mathcal {O},\mathcal {E})$ . We will freely identify these two sets in the rest of the article. Recall also that given a G-orbit $\mathcal {O} \subset \mathcal {N}_G$ and a point $x \in \mathcal {O}$ , restriction to x induces an equivalence between the category of G-equivariant $\mathbb {k}$ -local systems on $\mathcal {O}$ and $\mathrm {Rep}_{\mathbb {k}}(\mathrm {A}_G(x))$ , see [Reference Achar1, Proposition 6.2.13].

Let us now study the relation between the categories $D^{\mathrm {b}}_G(\mathcal {N}_G,\mathbb {k})$ and $D^{\mathrm {b}}_{G^\circ }(\mathcal {N}_{G^\circ },\mathbb {k})$ . As explained above, G and $G^\circ $ share the same nilpotent cone, so that we have a canonical t-exact forgetful functor

$$\begin{align*}\mathrm{For}^G_{G^\circ} : D^{\mathrm{b}}_G(\mathcal{N}_G,\mathbb{k}) \to D^{\mathrm{b}}_{G^\circ}(\mathcal{N}_{G},\mathbb{k}). \end{align*}$$

In the language of stacks, this functor is the ( $*$ - or $!$ -, equivalently) pullback under the canonical finite morphism $G^\circ \backslash \mathcal {N}_G \to G \backslash \mathcal {N}_G$ . It admits a left and right adjoint

$$\begin{align*}\gamma_{G^\circ}^G : D^{\mathrm{b}}_{G^\circ}(\mathcal{N}_{G},\mathbb{k}) \to D^{\mathrm{b}}_{G}(\mathcal{N}_{G},\mathbb{k}), \end{align*}$$

given by ( $*$ - or $!$ -, equivalently) pushforward under this map.

Proof. (1) It suffices to prove that for any G-orbit $\mathcal {O} \subset \mathcal {N}_G$ and any simple G-equivariant $\mathbb {k}$ -local system $\mathcal {E}$ on $\mathcal {O}$ , the perverse sheaf $\mathrm {For}^G_{G^\circ }(\mathcal {IC}(\mathcal {O},\mathcal {E}))$ is semisimple. Consider the decomposition $\mathcal {O} = \mathcal {O}_1 \sqcup \cdots \sqcup \mathcal {O}_r$ of $\mathcal {O}$ into $G^\circ $ -orbits, and choose for any i a point $x_i \in \mathcal {O}_i$ . Then we have

(4) $$ \begin{align} \mathrm{For}^G_{G^\circ}(\mathcal{IC}(\mathcal{O},\mathcal{E})) = \bigoplus_{i=1}^r \mathcal{IC}(\mathcal{O}_i,\mathcal{E}_{|\mathcal{O}_i}). \end{align} $$

Moreover, if $i \in \{1, \dots , r\}$ , denoting by $V_i$ the representation of $\mathrm {A}_G(x_i)$ corresponding to the G-equivariant local system $\mathcal {E}$ on $\mathcal {O} = G \cdot x_i$ , the restriction $\mathcal {E}_{|\mathcal {O}_i}$ is the $G^\circ $ -equivariant local system on $\mathcal {O}_i = G^\circ \cdot x_i$ corresponding to the restriction of $V_i$ to the normal subgroup $\mathrm {A}_{G^\circ }(x_i) $ . (To make sense of this, we use the fact that $\mathrm {Z}_{G}(x_i)^\circ = \mathrm {Z}_{G^\circ }(x_i)^\circ $ .) Hence, the desired claim follows from Lemma 2.1(1).

(2) It suffices to prove that for any $G^\circ $ -orbit $\mathcal {O} \subset \mathcal {N}_G$ and any simple $G^\circ $ -equivariant $\mathbb {k}$ -local system $\mathcal {E}$ on $\mathcal {O}$ , the complex $\gamma _{G^\circ }^G(\mathcal {IC}(\mathcal {O},\mathcal {E}))$ is perverse, and semisimple when $\mathrm {char}(\mathbb {k})$ does not divide $|G/G^\circ |$ . Fix such data, and choose $x \in \mathcal {O}$ . Concretely, setting $\mathcal {O}':=G \cdot x$ , this complex is obtained by taking pushforward under the canonical morphism $f : G \times ^{G^\circ } \overline {\mathcal {O}} \to \overline {\mathcal {O}'}$ of the intersection cohomology complex $\mathcal {IC}(G \times ^{G^\circ } \mathcal {O}, \mathcal {E}'),$ where $\mathcal {E}'$ is the G-equivariant local system on $G \times ^{G^\circ } \mathcal {O}$ corresponding to $\mathcal {E}$ under the canonical equivalence between G-equivariant local systems on $G \times ^{G^\circ } \mathcal {O}$ and $G^\circ $ -equivariant local systems on $\mathcal {O}$ . Now this map is finite, and its restriction to $G \times ^{G^\circ } \mathcal {O} = f^{-1}(\mathcal {O}')$ is a Galois covering with Galois group $\mathrm {Z}_{G}(x)/\mathrm {Z}_{G^\circ }(x)$ . Hence, as, for example, in [Reference Achar1, Proposition 3.8.10], $f_* \mathcal {IC}(G \times ^{G^\circ } \mathcal {O}, \mathcal {E}')$ is the intersection cohomology complex of the local system on $\mathcal {O}'$ obtained from $\mathcal {E}'$ by pushforward along the covering $G \times ^{G^\circ } \mathcal {O} \to \mathcal {O}'$ . In particular, this complex is perverse. In terms of representations of finite groups, this pushforward corresponds to the representation of $\mathrm {A}_G(x)$ obtained from the representation of $\mathrm {A}_{G^\circ }(x)$ associated with $\mathcal {E}$ by induction. Hence, the semisimplicity claim follows from Lemma 2.1(2). (Here, $\mathrm {A}_G(x) / \mathrm {A}_{G^\circ }(x)$ is a subgroup of $G/G^\circ $ , hence its order is prime to $\mathrm {char}(\mathbb {k})$ .)

3.3. Induction and restriction functors

Let P be a parabolic subgroup of G, and $L \subset P$ be a Levi factor. Then L is also naturally a quotient of P, and we consider the associated correspondence of stacks

(5) $$ \begin{align} L \backslash \mathcal{N}_L \xleftarrow{q} P \backslash \mathcal{N}_P \xrightarrow{i} G \backslash \mathcal{N}_G. \end{align} $$

We define the associated induction and restriction functors as follows:

(6) $$ \begin{align} \mathrm{Ind}_{L\subseteq P}^G&:=i_! q^* : D^{\mathrm{b}}_L(\mathcal{N}_L,\mathbb{k})\to D^{\mathrm{b}}_G(\mathcal{N}_G,\mathbb{k}), \end{align} $$

(7) $$ \begin{align} \mathrm{Res}_{L\subseteq P}^G&:= q_! i^*: D^{\mathrm{b}}_G(\mathcal{N}_G,\mathbb{k}) \to D^{\mathrm{b}}_L(\mathcal{N}_L,\mathbb{k}), \end{align} $$

(8) $$ \begin{align} {{\,}^{\prime} \mathrm{Res}}_{L\subseteq P}^G&:= q_* i^!: D^{\mathrm{b}}_G (\mathcal{N}_G,\mathbb{k}) \to D^{\mathrm{b}}_L(\mathcal{N}_L,\mathbb{k}). \end{align} $$

Proof. The proofs are similar to those in the connected case (see [Reference Achar, Henderson, Juteau and Riche2, §2.1]) or for characteristic- $0$ coefficients (see Dillery and Schwein [Reference Dillery and Schwein7]). Namely, (1) follows from the facts that the map $\mathcal {N}_P \to \mathcal {N}_G$ is a closed immersion, while the map $\mathcal {N}_P \to \mathcal {N}_L$ is smooth. (2) follows from an application of the base change theorem. Finally, for (3), as in [Reference Dillery and Schwein7, Corollary 2.4.4], we have

(9) $$ \begin{align} \mathrm{For}^L_{L^\circ} \circ \mathrm{Res}_{L\subseteq P}^G \cong \mathrm{Res}^{G^\circ}_{L^\circ \subseteq P^\circ} \circ \mathrm{For}^G_{G^\circ}. \end{align} $$

Therefore, t-exactness of $\mathrm {Res}_{L\subseteq P}^G$ follows from that of $\mathrm {Res}^{G^\circ }_{L^\circ \subseteq P^\circ }$ , for which we refer to [Reference Achar, Henderson, Juteau and Riche2, §2.1]. One checks similarly that ${{\,}^{\prime } \mathrm {Res}}_{L\subseteq P}^G$ is t-exact. Then $\mathrm {Ind}_{L\subseteq P}^G$ is t-exact because it admits t-exact left and right adjoints.

3.4. Cuspidal local systems and cuspidal triples

We will denote by

$$\begin{align*}\mathfrak{R}_{G,\mathbb{k}}^{\mathrm{cusp}} \subset \mathfrak{R}_{G,\mathbb{k}} \end{align*}$$

the subset of (isomorphism classes of) cuspidal objects.

If $\mathcal {F} \in \mathfrak {R}_{G,\mathbb {k}}$ , using (9) and the fact that the functors $\mathrm {For}^G_{G^\circ }$ do not kill any nonzero object, one sees that $\mathcal {F}$ is cuspidal if and only if each composition factor of $\mathrm {For}^G_{G^\circ }(\mathcal {F})$ is cuspidal in the sense above for the group $G^\circ $ , or, in other words, in the sense of Achar et al. [Reference Achar, Henderson, Juteau and Riche2]. (In fact, in view of Lemma 3.3(1), one can replace “composition factor” in this sentence by “simple direct summand.”) Using a similar property for the functors ${{\,}^{\prime } \mathrm {Res}}_{L\subseteq P}^G$ , and [Reference Achar, Henderson, Juteau and Riche2, Proposition 2.1], one sees that $\mathcal {F}$ is cuspidal if and only if it is annihilated by ${{\,}^{\prime } \mathrm {Res}}_{L\subseteq P}^G$ for any non-maximal special parabolic subgroup P with Levi factor L. Then, arguing as in [Reference Achar, Henderson, Juteau and Riche2, Proposition 2.1], one sees that this condition is equivalent to the following ones:

• • $\mathcal {F}$ is not isomorphic to a quotient of an object of the form $\mathrm {Ind}_{L \subseteq P}^G(\mathcal {G})$ with P a non-maximal special parabolic subgroup P with Levi factor L and $\mathcal {G} \in \mathrm {Perv}_L(\mathcal {N}_L,\mathbb {k})$ .

• • $\mathcal {F}$ is not isomorphic to a subobject of an object of the form $\mathrm {Ind}_{L \subseteq P}^G(\mathcal {G})$ with P a non-maximal special parabolic subgroup P with Levi factor L and $\mathcal {G} \in \mathrm {Perv}_L(\mathcal {N}_L,\mathbb {k})$ .

We introduce an equivalence relation on the set of cuspidal triples as follows. If $(L,\mathcal {O},\mathcal {E})$ and $(L',\mathcal {O}',\mathcal {E}')$ are cuspidal triples, we write $(L,\mathcal {O},\mathcal {E}) \sim (L',\mathcal {O}',\mathcal {E}')$ if there exists $g \in G$ such that $L'=gLg^{-1}$ , $\mathcal {O}' = g \cdot \mathcal {O}$ , and $\mathcal {E}'$ is isomorphic to the image of $\mathcal {E}$ under the pushforward functor associated with the isomorphism $\mathcal {O} \xrightarrow {\sim } \mathcal {O}'$ induced by g. We will denote by $\mathfrak {L}_{G,\mathbb {k}}$ the quotient of the set of cuspidal triples by this equivalence relation.

3.5. Fourier transform and applications

The next tool we will need in our study is the Fourier transform. This section is parallel to [Reference Achar, Henderson, Juteau and Riche2, §2.4] or [Reference Dillery and Schwein7, §3.4].

Here, we work as in [Reference Achar, Henderson, Juteau and Riche2, §2.4] with Fourier–Sato transform. (One can also work with the Fourier–Laumon transform as in [Reference Dillery and Schwein7, §3.4], since we will only use some formal properties that the two constructions enjoy.) First, we note that there exists a G-invariant non-degenerate symmetric bilinear form on $\mathfrak {g}$ . In fact, we have a G-equivariant decomposition $\mathfrak {g} = [\mathfrak {g},\mathfrak {g}] \oplus \mathfrak {z}(\mathfrak {g}),$ where $\mathfrak {z}(\mathfrak {g})$ is the center of $\mathfrak {g}$ . If we choose G-invariant non-degenerate symmetric bilinear forms on $[\mathfrak {g},\mathfrak {g}]$ and $\mathfrak {z}(\mathfrak {g})$ , we can assemble them to obtain the desired form on $\mathfrak {g}$ . Here, on $[\mathfrak {g},\mathfrak {g}],$ one can, for example, take the Killing form. To show existence of a G-invariant nondegenerate symmetric bilinear form on $\mathfrak {z}(\mathfrak {g}),$ one can proceed as follows. This Lie algebra is the Lie algebra of the torus $\mathrm {Z}(G^\circ )^\circ $ . Hence, we have $\mathfrak {z}(\mathfrak {g}) = \mathbb {C} \otimes _{\mathbb {Z}} \mathrm {X}_*(\mathrm {Z}(G^\circ )^\circ )$ . The action of G, which factors through an action of the finite group $G/G^\circ $ , is induced by an action on the lattice $\mathrm {X}_*(\mathrm {Z}(G^\circ )^\circ )$ ; it therefore admits a real form, hence an invariant non-degenerate bilinear form.

After fixing such a form, the Fourier transform provides a t-exact autoequivalence $\mathbb {T}_G$ of the derived category of G-equivariant conical sheaves on $\mathfrak {g}$ . (The meaning of “conical” will not be important here. All that we will use is that these objects form a full subcategory of the G-equivariant derived category of $\mathfrak {g}$ , and that the pushforward to $\mathfrak {g}$ of any G-equivariant perverse sheaf on $\mathcal {N}_G$ is conical.)

Lemma 3.11. For any $(\mathcal {O},\mathcal {E}) \in \mathfrak {R}_{G,\mathbb {k}}^{\mathrm {cusp}}$ , there exists a pair $(\mathcal {O}',\mathcal {E}') \in \mathfrak {R}_{G,\mathbb {k}}^{\mathrm {cusp}}$ such that

$$\begin{align*}\mathbb{T}_G(\mathcal{IC}(\mathcal{O},\mathcal{E})) \cong \mathcal{IC}(\mathcal{O}' + \mathfrak{z}(\mathfrak{g}), \mathcal{E}' \boxtimes \underline{\mathbb{k}}_{\mathfrak{z}(\mathfrak{g})}). \end{align*}$$

Note that for any Levi subgroup L of G, the restriction of our non-degenerate G-invariant bilinear form on $\mathfrak {g}$ to $\mathfrak {l}$ is a non-degenerate L-invariant bilinear form on $\mathfrak {l}$ , so that it makes sense to consider the Fourier transform $\mathbb {T}_L$ .

We now fix a special Levi subgroup L of G, an L-orbit $\mathcal {O} \subseteq \mathcal {N}_L$ , and an L-equivariant local system $\mathcal {E}$ on $\mathcal {O}$ . We set

$$\begin{align*}\mathfrak{z}(\mathfrak{l})^\circ := \{ x \in \mathfrak{z}(\mathfrak{l}) \mid \mathrm{Z}_G(x)^\circ = L^\circ\}. \end{align*}$$

(Note that here we have $\mathrm {Z}_G(x)^\circ = \mathrm {Z}_{G^\circ }(x)^\circ $ , so that this definition is in a sense insensitive to disconnectedness.) Let also

$$\begin{align*}\widetilde{Y}_{(L,\mathcal{O})} := G \times^L (\mathcal{O} + \mathfrak{z}(\mathfrak{l})^\circ), \quad Y_{(L,\mathcal{O})} := G \cdot (\mathcal{O} + \mathfrak{z}(\mathfrak{l})^\circ), \end{align*}$$

and consider the stabilizer $\mathrm {N}_G(L,\mathcal {O})$ of $\mathcal {O}$ in $\mathrm {N}_G(L)$ . We have a natural map

$$\begin{align*}\pi_{(L,\mathcal{O})}: \widetilde{Y}_{(L,\mathcal{O})}\to Y_{(L,\mathcal{O})} \end{align*}$$

induced by the G-action on $\mathfrak {g}$ .

Since $\widetilde {Y}_{(L,\mathcal {O})}$ is defined using induction of varieties, there exists an equivalence of categories between the categories of G-equivariant local systems on $\widetilde {Y}_{(L,\mathcal {O})}$ and of L-equivariant local systems on $\mathcal {O} + \mathfrak {z}(\mathfrak {l})^\circ $ . Recall that we have fixed an L-equivariant local system $\mathcal {E}$ on $\mathcal {O}$ ; we will denote by $\widetilde {\mathcal {E}}$ the G-equivariant local system on $\widetilde {Y}_{(L,\mathcal {O})}$ corresponding under this equivalence to $\mathcal {E} \boxtimes \underline {\mathbb {k}}_{\mathfrak {z}(\mathfrak {l})^\circ }$ . By Proposition 3.13, using pushforward, we obtain the G-equivariant local system $(\pi _{(L,\mathcal {O})})_* \widetilde {\mathcal {E}}$ on $Y_{(L,\mathcal {O})}$ .

Let us now fix a parabolic subgroup $P \subset G$ having L as Levi factor. As in [Reference Achar, Henderson, Juteau and Riche2, §2.4] or [Reference Dillery and Schwein7, §2.4], the induction functor $\mathrm {Ind}_{L \subseteq P}^G$ “extends” to a functor from $D^{\mathrm {b}}_L(\mathfrak {l},\mathbb {k})$ to $D^{\mathrm {b}}_G(\mathfrak {g},\mathbb {k})$ , which will be denoted similarly, and which sends conical complexes to conical complexes. (For this construction, one simply replaces the nilpotent cones by the full Lie algebras in the correspondence (5).)

Proposition 3.14. We have a canonical isomorphism

$$\begin{align*}\mathrm{Ind}_{L \subseteq P}^G(\mathcal{IC}(\mathcal{O} + \mathfrak{z}(\mathfrak{l}), \mathcal{E} \boxtimes \underline{\mathbb{k}}_{\mathfrak{z}(\mathfrak{l})})) \cong \mathcal{IC}(Y_{(L,\mathcal{O})}, (\pi_{(L,\mathcal{O})})_* \widetilde{\mathcal{E}}). \end{align*}$$

Proof. As in the proof of [Reference Achar, Henderson, Juteau and Riche2, Proposition 2.17], what we have to prove is that the left-hand side satisfies the usual defining conditions for the right-hand side. For this, we use the “more concrete” description of induction as in [Reference Achar, Henderson, Juteau and Riche2, §2.5]. From this description, it is clear that this complex is supported on the closure of $Y_{(L,\mathcal {O})}$ , and one easily checks that its restriction to $Y_{(L,\mathcal {O})}$ is $(\pi _{(L,\mathcal {O})})_* \widetilde {\mathcal {E}}$ . Next, one needs to check that the restriction, respectively, corestriction, of $\mathrm {Ind}_{L \subseteq P}^G(\mathcal {IC}(\mathcal {O} + \mathfrak {z}(\mathfrak {l}), \mathcal {E} \boxtimes \underline {\mathbb {k}}_{\mathfrak {z}(\mathfrak {l})}))$ to $\overline {Y_{(L,\mathcal {O})}} \smallsetminus Y_{(L,\mathcal {O})}$ is concentrated in perverse degrees $\leq -1$ , respectively, $\geq 1$ . These properties can be checked after applying the functor $\mathrm {For}^G_{G^\circ }$ . Now, if $\Sigma \subset G$ is a subset whose image in $G/G^\circ $ forms coset representatives for $(G/G^\circ )/(P/P^\circ )$ (see Lemma 3.2), one sees that we have

$$\begin{align*}\mathrm{For}^G_{G^\circ} \big( \mathrm{Ind}_{L \subseteq P}^G(\mathcal{IC}(\mathcal{O} + \mathfrak{z}(\mathfrak{l}), \mathcal{E} \boxtimes \underline{\mathbb{k}}_{\mathfrak{z}(\mathfrak{l})})) \big) \cong \bigoplus_{\sigma \in \Sigma} (\mathrm{Ad}_\sigma)_* \mathrm{Ind}_{L^\circ \subseteq P^\circ}^{G^\circ}(\mathcal{IC}(\mathcal{O} + \mathfrak{z}(\mathfrak{l}), \mathcal{E} \boxtimes \underline{\mathbb{k}}_{\mathfrak{z}(\mathfrak{l})})). \end{align*}$$

In the right-hand side, one can write $\mathcal {O}$ as a union of $G^\circ $ -orbits to further decompose this object as a direct sum. Then the direct summands are exactly of the type appearing in [Reference Achar, Henderson, Juteau and Riche2, Proposition 2.17], and one can use this result to conclude.

One can then derive from these results the following important consequence.

Proof. It is enough to prove similar properties for the perverse sheaf $\mathbb {T}_G(\mathrm {Ind}_{L\subseteq P}^G(\mathcal {IC}(\mathcal {O},\mathcal {E})))$ . However, as in [Reference Achar, Henderson, Juteau and Riche2, Lemma 2.9], induction commutes with the Fourier transform in the appropriate sense, so that we have

$$\begin{align*}\mathbb{T}_G(\mathrm{Ind}_{L\subseteq P}^G(\mathcal{IC}(\mathcal{O},\mathcal{E}))) \cong \mathrm{Ind}_{L\subseteq P}^G(\mathbb{T}_L(\mathcal{IC}(\mathcal{O},\mathcal{E}))). \end{align*}$$

Now, by Lemma 3.11, we have

$$\begin{align*}\mathbb{T}_L(\mathcal{IC}(\mathcal{O},\mathcal{E})) \cong \mathcal{IC}(\mathcal{O}' + \mathfrak{z}(\mathfrak{l}), \mathcal{E}' \boxtimes \underline{\mathbb{k}}_{\mathfrak{z}(\mathfrak{l})}) \end{align*}$$

for some pair $(\mathcal {O}',\mathcal {E}') \in \mathfrak {R}_{L,\mathbb {k}}$ . Then we use Proposition 3.14 to deduce an isomorphism

$$\begin{align*}\mathbb{T}_G(\mathrm{Ind}_{L\subseteq P}^G(\mathcal{IC}(\mathcal{O},\mathcal{E}))) \cong \mathcal{IC}(Y_{(L,\mathcal{O}')}, (\pi_{(L,\mathcal{O}')})_* \widetilde{\mathcal{E}'}). \end{align*}$$

The right-hand side clearly does not depend on the choice of P and is invariant under our equivalence relation, which proves the first claim.

The proof of the second claim is identical to that of [Reference Achar, Henderson, Juteau and Riche3, Lemma 2.3]. We recall its main ingredients (presented from a slightly different angle, which follows [Reference Achar, Henderson, Juteau and Riche3, Remark 3.3]) for later reference. Once again, it is enough to prove the similar claim for $\mathbb {T}_G(\mathrm {Ind}_{L\subseteq P}^G(\mathcal {IC}(\mathcal {O},\mathcal {E})))$ . Next, since the $\mathcal {IC}$ functor respects socles and heads, it is enough to prove that the isomorphism classes of simple subobjects and quotients of $(\pi _{(L,\mathcal {O}')})_* \widetilde {\mathcal {E}'}$ coincide. For this, we fix $x \in \mathcal {O}'$ and interpret local systems in terms of representations of finite groups. The local system $\mathcal {E}'$ corresponds to a representation V of $\mathrm {A}_L(x)$ . Now, we have

$$\begin{align*}\mathrm{A}_{\mathrm{N}_G(L)}(x) / \mathrm{A}_L(x) \cong \mathrm{Z}_{\mathrm{N}_G(L)}(x) / \mathrm{Z}_L(x) \cong \mathrm{N}_G(L,\mathcal{O}')/L, \end{align*}$$

and this group is the Galois group of the covering $\pi _{(L,\mathcal {O}')}$ by Proposition 3.13. Hence, any representation of $\mathrm {A}_{\mathrm {N}_G(L)}(x)$ determines a G-equivariant local system on $Y_{(L,\mathcal {O}')}$ , and $(\pi _{(L,\mathcal {O}')})_* \widetilde {\mathcal {E}'}$ corresponds to the representation $\mathrm {Ind}_{\mathrm {A}_L(x)}^{\mathrm {A}_{\mathrm {N}_G(L)}(x)}(V)$ . Hence, the claim follows from Corollary 2.2.

3.6. Induction series

Thanks to Corollary 3.16, it makes sense to define, for a cuspidal triple $(L,\mathcal {O},\mathcal {E})$ , the subset

$$\begin{align*}\mathfrak{R}_{G,\mathbb{k}}^{(L,\mathcal{O},\mathcal{E})} \subseteq \mathfrak{R}_{G,\mathbb{k}} \end{align*}$$

as the set of isomorphism classes of simple subobjects (equivalently, quotients) of the G-equivariant perverse sheaf $\mathrm {Ind}_{L\subseteq P}^G(\mathcal {IC}(\mathcal {O},\mathcal {E}))$ , for any choice of a parabolic subgroup $P \subset G$ with Levi factor L. Moreover, this subset only depends on the image of $(L,\mathcal {O},\mathcal {E})$ in $\mathfrak {L}_{G,\mathbb {k}}$ . It will be called the induction series associated with (the class of) $(L,\mathcal {O},\mathcal {E})$ .

4.1. A Mackey formula

We first prove a “Mackey formula” for compositions of restriction and induction functors, for which the geometric input is given in Dillery and Schwein [Reference Dillery and Schwein7].

For a subgroup $A\subseteq G$ and an element $g\in G$ , we denote by ${}^gA$ the subgroup $gAg^{-1}$ . We fix parabolic subgroups $P,Q$ of G with respective Levi subgroups $L,M$ . By [Reference Dillery and Schwein7, Lemma 2.3.1], there exists a sequence $g_1, \dots , g_s$ of representatives for the double cosets in $Q \backslash G / P$ such that:

• • for any i, $M \cap {{}^{g_i} L}$ contains a maximal torus of G;

• • if $Q g_i P \subseteq \overline {Q g_j P}$ , then $i \leq j$ .

Then, for any i, $M \cap {}^{g_i} P$ is a parabolic subgroup of M with Levi subgroup $M \cap {}^{g_i} L$ , and $Q \cap {}^{g_i} L$ is a parabolic subgroup in ${}^{g_i} L$ with Levi subgroup $M \cap {}^{g_i} L$ .

Theorem 4.1 (Mackey formula).

Let $\mathcal {F}\in \mathrm {Perv}_L(\mathcal {N}_L,\mathbb {k}).$ Then, in $\mathrm {Perv}_M(\mathcal {N}_M,\mathbb {k}),$ we have a filtration

$$\begin{align*}0=\mathcal{F}_0\subseteq \mathcal{F}_1\subseteq \cdots \subseteq \mathcal{F}_s=\mathrm{Res}_{M\subseteq Q}^G\circ \mathrm{Ind}_{L\subseteq P}^G(\mathcal{F}) \end{align*}$$

such that, for any $i \geq 1,$ we have

$$\begin{align*}\mathcal{F}_i/\mathcal{F}_{i-1}\cong \mathrm{Ind}_{M\cap {{}^{g_i}L} \subseteq M\cap {}^{g_i}P}^M\left(\mathrm{Res}_{M\cap {}^{g_i}L\subseteq Q\cap {}^{g_i}L}^{^{g_i}L} \mathrm{Ad}(g_i^{-1})^*\mathcal{F})\right). \end{align*}$$

Proof. The proof is similar to that of [Reference Dillery and Schwein7, Theorem 2.5.1]. Set $\mathfrak {X} = P \backslash \mathcal {N}_P \times _{G \backslash \mathcal {N}_G} Q \backslash \mathcal {N}_Q$ , so that the upper part of the following diagram (in which all other maps are as in (5)) is Cartesian:

Denote by $\pi : \mathfrak {X} \to L \backslash \mathcal {N}_L$ and $\rho : \mathfrak {X} \to M \backslash \mathcal {N}_M$ the composition of maps from this diagram.

There is a natural map from $\mathfrak {X}$ to $P \backslash \mathrm {pt} \times _{G \backslash \mathrm {pt}} Q \backslash \mathrm {pt} \cong Q\backslash G/ P.$ From the (finite) stratification of $Q\backslash G/ P$ by the double cosets $(Q\backslash G /P)_i:=Q\backslash Qg_iP/ P$ , we deduce a stratification of $\mathfrak {X}$ by some substacks $\mathfrak {X}_i$ .

Fix i, and denote by $\pi _i, \rho _i$ the restrictions of $\pi , \rho $ to $\mathfrak {X}_i$ . Let also $\mathcal {N}_i:=\mathcal {N}_{Q\cap ^{g_i}{L}}\times _{\mathcal {N}_{M\cap ^{g_i}{L}}} \mathcal {N}_{M \cap ^{g_i}{P}},$ and $G_i:=(Q\cap ^{g_i}{L})\times _{M\cap ^{g_i}{L}} (M \cap ^{g_i}{P})$ , and consider the commutative diagram

of [Reference Dillery and Schwein7, Theorem 2.5.3]. Here, by definition, the correspondence in the lower line is defining $\mathrm {Ind}_{M\cap {}^{g_i}L\subseteq M\cap {}^{g_i}P}^M \circ \mathrm {Res}_{M\cap {}^{g_i}L\subseteq Q\cap {}^{g_i}L}^{^{g_i}L}$ . Since $f_i$ satisfies the assumptions of [Reference Dillery and Schwein7, Lemma 2.2.3], the functor $f_{i!}f_i^*$ is isomorphic to the identity functor. Therefore, we have

(10) $$ \begin{align} (\rho_{i})_! (\pi_{i})^*\mathcal{F}=\mathrm{Ind}_{M\cap {}^{g_i}L\subseteq M\cap {}^{g_i}P}^M\left(\mathrm{Res}_{M\cap {}^{g_i}L\subseteq Q\cap {}^{g_i}L}^{^{g_i}L} \mathrm{Ad}(g_i^{-1})^*\mathcal{F})\right). \end{align} $$

In particular, $(\rho _{i})_! (\pi _{i})^*\mathcal {F}$ is perverse by Proposition 3.6(3).

For any i, we now set $(Q\backslash G /P)_{\leq i}:=Q\backslash \left ( \bigcup _{j\leq i} Qg_iP \right ) / P$ . Due to the properties of the ordering chosen on the $g_i$ above, we have that, for any i, $(Q\backslash G /P)_{i}$ is an open substack of $(Q\backslash G /P)_{\leq i}$ with closed complement $(Q\backslash G /P)_{\leq i-1}$ . Pulling back these substacks along the map $\mathfrak {X}\rightarrow Q\backslash G/P,$ we get substacks $\mathfrak {X}_{\leq i}, \mathfrak {X}_{\leq i-1}$ . The stack $\mathfrak {X}_{i}$ is then also an open substack of $\mathfrak {X}_{\leq i}$ with closed complement $\mathfrak {X}_{\leq i-1}$ , which gives rise to a distinguished triangle

$$\begin{align*}(\rho_{\leq i-1})_! (\pi_{\leq i-1})^*\mathcal{F} \rightarrow (\rho_{\leq i})_! (\pi_{\leq i})^*\mathcal{F} \rightarrow \rho_{ i!}\pi_{ i}^*\mathcal{F} \xrightarrow{+1}. \end{align*}$$

We set $\mathcal {F}_i:=\rho _{\leq i!}\pi _{\leq i}^*\mathcal {F}$ . Since $\rho _{i!}\pi _{i}^*\mathcal {F}$ is perverse for each i, using induction, one sees that each $\mathcal {F}_i$ is also perverse, and that, for any $i \geq 1,$ the map $\mathcal {F}_{i-1} \to \mathcal {F}_i$ is injective. The triangles above thus become short exact sequences of perverse sheaves, and using (10), the corresponding filtration of $\mathcal {F}_s=\mathrm {Res}_{M\subseteq Q}^G\circ \mathrm {Ind}_{L\subseteq P}^G(\mathcal {F})$ has the desired property.

4.2. Decomposition into induction series

We can now prove the first main result of this article, which says that $\mathfrak {R}_{G,\mathbb {k}}$ is partitioned into induction series.

Theorem 4.3. We have a partition

$$\begin{align*}\mathfrak{R}_{G,\mathbb{k}}=\bigsqcup_{(L,\mathcal{O},\mathcal{E})\in \mathfrak{L}_{G,\mathbb{k}}} \mathfrak{R}_{G,\mathbb{k}}^{(L,\mathcal{O},\mathcal{E})}. \end{align*}$$

Proof. The proof is similar to that of the corresponding statement in case G is connected, given in Achar et al. [Reference Achar, Henderson, Juteau and Riche4].

By a standard argument initially due to Bernstein in the setting of p-adic reductive groups, every simple perverse sheaf $\mathcal {F}\in \mathfrak {R}_{G,\mathbb {k}}$ is part of an induction series. Indeed, consider a special parabolic subgroup P with Levi subgroup L such that $\mathrm {Res}_{L\subseteq P}^G\mathcal {F}\neq 0$ , and which is minimal for this property. Let also $\mathcal {G}$ be a simple quotient of $\mathrm {Res}_{L\subseteq P}^G\mathcal {F}$ . Then, by Proposition 3.6(2) and the choice of P, $\mathrm {Res}_{L\subseteq P}^G\mathcal {F}$ is annihilated by restriction to any Levi subgroup of a non-maximal parabolic subgroup of L, that is, is cuspidal. By exactness (see Proposition 3.6(3)), it follows that $\mathcal {G}$ is cuspidal too. Setting $\mathcal {G}=\mathcal {IC}(\mathcal {O},\mathcal {E})$ , we therefore have a cuspidal triple $(L,\mathcal {O},\mathcal {E})$ . By adjunction (see Proposition 3.6(1)), from the map $\mathrm {Res}_{L\subseteq P}^G\mathcal {F} \twoheadrightarrow \mathcal {G,}$ we get a nonzero map $\mathcal {F}\rightarrow \mathrm {Ind}_{L\subseteq P}^G \mathcal {G}$ . Therefore, $\mathcal {F} \in \mathfrak {R}_{G,\mathbb {k}}^{(L,\mathcal {O},\mathcal {E})}.$

For the disjointness of the induction series, assume $\mathcal {G} \in \mathfrak {R}_{G,\mathbb {k}}^{(L,\mathcal {O}_L,\mathcal {E}_L)}\cap \mathfrak {R}_{G,\mathbb {k}}^{(M,\mathcal {O}_M,\mathcal {E}_M)}$ . Since induction series can be defined either in terms of simple submodules or simple quotients (see Corollary 3.16), $\mathcal {G}$ is a simple quotient of $\mathrm {Ind}_{L\subseteq P}^G(\mathcal {IC}(\mathcal {O}_L,\mathcal {E}_L))$ and also a simple submodule of $\mathrm {Ind}_{M\subseteq Q}^G (\mathcal {IC}(\mathcal {O}_M,\mathcal {E}_M))$ . The existence of the map $\mathrm {Ind}_{L\subseteq P}^G(\mathcal {IC}(\mathcal {O}_L,\mathcal {E}_L))\twoheadrightarrow \mathcal {G} \hookrightarrow \mathrm {Ind}_{M\subseteq Q}^G (\mathcal {IC}(\mathcal {O}_M,\mathcal {E}_M))$ implies that

$$\begin{align*}\mathrm{Hom}_G(\mathrm{Ind}_{L\subseteq P}^G(\mathcal{IC}(\mathcal{O}_L,\mathcal{E}_L)), \mathrm{Ind}_{M\subseteq Q}^G (\mathcal{IC}(\mathcal{O}_M,\mathcal{E}_M)))\neq 0, \end{align*}$$

and then by adjunction that

$$\begin{align*}\mathrm{Hom}_M(\mathrm{Res}_{M\subseteq Q}^G\circ \mathrm{Ind}_{L\subseteq P}^G(\mathcal{IC}(\mathcal{O}_L,\mathcal{E}_L)), \mathcal{IC}(\mathcal{O}_M,\mathcal{E}_M))\neq 0. \end{align*}$$

Let $\mathcal {F}=\mathcal {IC}(\mathcal {O}_L,\mathcal {E}_L)$ , and consider the notation of Theorem 4.1 for the present setting. Then there exists i such that

$$\begin{align*}\mathrm{Hom}_M(\mathrm{Ind}_{M\cap {}^{g_i}L\subseteq M\cap {}^{g_i}P}^M(\mathrm{Res}_{M\cap {}^{g_i}L\subseteq Q\cap {}^{g_i}L}^{^{g_i}L} \mathrm{Ad}(g_i^{-1})^*\mathcal{F})), \mathcal{IC}(\mathcal{O}_M,\mathcal{E}_M))\neq 0. \end{align*}$$

By cuspidality of $\mathcal {F}$ (and thus of $\mathrm {Ad}(g_i^{-1})^*\mathcal {F}$ ), this implies that the parabolic restriction is trivial, so ${}^{g_i}L \subset M$ . We then have

$$\begin{align*}\mathrm{Hom}_M(\mathrm{Ind}_{M\cap {}^{g_i}L\subseteq M\cap {}^{g_i}P}^M( \mathrm{Ad}(g_i^{-1})^*\mathcal{F}), \mathcal{IC}(\mathcal{O}_M,\mathcal{E}_M))\neq 0. \end{align*}$$

By cuspidality of $\mathcal {IC}(\mathcal {O}_M,\mathcal {E}_M),$ this implies that ${}^{g_i} L = M$ , and finally that $(L, \mathcal {O}_L, \mathcal {E}_L)$ and $(M,\mathcal {O}_M,\mathcal {E}_M)$ are equivalent.

5.1. Weyl groups

For any special Levi subgroup L of G and any L-orbit $\mathcal {O} \subseteq \mathcal {N}_L$ , the group $\mathrm {N}_G(L,\mathcal {O})$ acts on $\mathcal {O}$ , which induces an action of $\mathrm {N}_G(L,\mathcal {O})/L$ on the set of isomorphism classes of simple L-equivariant local systems on $\mathcal {O}$ . This action can be described concretely in terms of representations of finite groups as follows. Recall that, after fixing $x \in \mathcal {O}$ , isomorphism classes of simple L-equivariant local systems on $\mathcal {O}$ are in a canonical bijection with isomorphism classes of simple $\mathbb {k}$ -representations of $\mathrm {A}_L(x)$ . Now, as in the proof of Corollary 3.16, we have an identification $\mathrm {N}_G(L)/L \cong \mathrm {A}_{\mathrm {N}_G(L)}(x) / \mathrm {A}_L(x)$ . Using the twist of representations as in §2.2 for the normal subgroup $\mathrm {A}_L(x)$ of $\mathrm {A}_{\mathrm {N}_G(L)}(x)$ , we obtain an action of $\mathrm {A}_{\mathrm {N}_G(L)}(x) / \mathrm {A}_L(x)$ on the set of isomorphism classes of simple $\mathbb {k}$ -representations of $\mathrm {A}_L(x)$ , which corresponds to the action considered above.

For a cuspidal triple $(L,\mathcal {O},\mathcal {E})$ , we will denote by

$$\begin{align*}W_{(L,\mathcal{O},\mathcal{E})} \end{align*}$$

the stabilizer of (the isomorphism class of) $\mathcal {E}$ for the action of $\mathrm {N}_G(L,\mathcal {O}) / L$ on isomorphism classes of simple L-equivariant local systems on $\mathcal {O}$ . This group is a subgroup of $\mathrm {N}_G(L)/L$ . When G is connected, this inclusion is in fact an equality, as follows from [Reference Achar, Henderson, Juteau and Riche3, Proposition 2.6 and Lemma 2.9] and [Reference Achar, Henderson, Juteau and Riche4, Proposition 3.1]. This fact is not true in the disconnectedness case, as shown by the following example, which was communicated to us by Maarten Solleveld.

5.2. Parameterization

Consider once again a cuspidal triple $(L,\mathcal {O},\mathcal {E})$ . Applying Lemma 3.11 to the pair $(\mathcal {O},\mathcal {E}) \in \mathfrak {R}_{L,\mathbb {k}}^{\mathrm {cusp}}$ , we obtain an associated pair $(\mathcal {O}',\mathcal {E}') \in \mathfrak {R}_{L,\mathbb {k}}^{\mathrm {cusp}}$ .

5.3. An example of nontrivial cocycle

In this section, we show through an example that, in general, the algebra in Theorem 5.3 might not be isomorphic to $\mathbb {k}[W_{(L,\mathcal {O},\mathcal {E})}]$ . This example is reproduced from [Reference Aubert, Moussaoui and Solleveld5, Example 3.2], where it was discovered by considerations of the role of the generalized Springer correspondence in the Langlands correspondence. (This example is discussed in Aubert et al. [Reference Aubert, Moussaoui and Solleveld5] to illustrate a slightly different property, related to the discussion in Remark 3.1; the two properties are in fact identical in this case, essentially because the maximal special Levi subgroup of G is $G^\circ $ for this group.)

Example 5.5. Let Q be the subgroup of $\mathrm {SL}_2(\mathbb {C})^5$ defined as

$$ \begin{align*} Q := \left\{ (\pm I_2) \times I_8, \begin{pmatrix} \pm i & 0 \\ 0 & \mp i \end{pmatrix} \times I_4 \times -I_4, \begin{pmatrix} 0 & \pm i \\ \pm i & 0 \end{pmatrix} \times I_2 \times -I_2 \times I_2 \times -I_2,\begin{pmatrix} 0 & \mp 1 \\ \pm 1 & 0 \end{pmatrix} \times I_2 \times -I_4 \times I_2 \right\}. \end{align*} $$

This group is isomorphic to the quaternion group of order 8. We define G as the normalizer $G := \mathrm {N}_{\mathrm {SL}_{10}(\mathbb {C})}(Q)$ . Then we have

$$ \begin{align*} G^\circ & = \mathrm{Z}_{\mathrm{SL}_{10}(\mathbb{C})}(Q) = \left( \mathrm{Z}(\mathrm{GL}_2(\mathbb{C})) \times \mathrm{GL}_2(\mathbb{C})^4 \right) \cap \mathrm{SL}_{10}(\mathbb{C}), \\ G/G^\circ & \cong (\mathbb{Z}/2\mathbb{Z})^2, \\ \mathrm{Z}(G^\circ)& = \left\{ (z_j)_{j=1}^{5} \in \mathrm{Z}(\mathrm{GL}_2(\mathbb{C}))^5 \,\middle|\, \prod_{j=1}^{5} z_j^2 = 1 \right\}. \end{align*} $$

Using the explicit representatives for $G/G^{\circ }$ given in [Reference Aubert, Moussaoui and Solleveld5, Example 3.2], one can check that $\mathrm {Z}_G(\mathrm {Z}(G^{\circ })^\circ )=G^{\circ }$ , so that the maximal special Levi subgroup for G is $G^{\circ }$ .

Let

$$\begin{align*}x = \mathrm{diag} \left( 0_2, \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \right) \in \mathcal{N}_G; \end{align*}$$

we have

$$\begin{align*}\mathrm{A}_{G^\circ}(x) \cong \mathbb{Z}/2\mathbb{Z}, \quad A_G(x) \cong Q. \end{align*}$$

Let us assume $\textrm {char} ( \mathbb {k} ) \neq 2$ , so that $\mathrm {A}_{G^\circ }(x)$ has a nontrivial one-dimensional representation. If $\mathcal {E}$ is the corresponding $G^\circ $ -equivariant local system on $\mathcal {O}=G^\circ \cdot x$ , then $(\mathcal {O},\mathcal {E})$ is a cuspidal pair for $G^\circ $ . We therefore have the cuspidal triple $(G^{\circ }, \mathcal {O}, \mathcal {E})$ , with $W_{(G^{\circ }, \mathcal {O}, \mathcal {E})} = G/G^\circ \cong (\mathbb {Z}/2\mathbb {Z})^2$ .

The perverse sheaf $\mathrm {Ind}_{G^{\circ }\subseteq G^{\circ }}^G (\mathcal {IC}(\mathcal {O},\mathcal {E}))$ identifies with the intersection cohomology complex of the pair $(G \cdot x,\mathcal {F}),$ where $\mathcal {F}$ is the G-equivariant local system associated with the representation of $\mathrm {A}_G(x)$ obtained by induction from the sign character of $\mathrm {A}_{G^\circ }(x)$ . The latter representation is either simple (but not absolutely simple) of dimension $4$ , or the direct sum of two copies of a simple (and absolutely simple) two-dimensional representation. In any case, $\mathfrak {R}_{G,\mathbb {k}}^{(G^{\circ }, \mathcal {O}, \mathcal {E})}$ has only one element, while $(\mathbb {Z}/2\mathbb {Z})^2$ has four isomorphism classes of simple representations, so that the corresponding twisted group algebra of $W_{(G^{\circ }, \mathcal {O}, \mathcal {E})} \cong (\mathbb {Z}/2\mathbb {Z})^2$ must be nontrivial. (In case $\mathrm {Ind}_{G^{\circ }\subseteq G^{\circ }}^G (\mathcal {IC}(\mathcal {O},\mathcal {E}))$ is not simple, this twisted group algebra is isomorphic to $\mathrm {M}_2(\mathbb {k})$ .)

5.4. An example

We conclude this article with an explicit description of our results in the case discussed in Example 5.1. Here, there are two nilpotent orbits: the orbit of $0$ , which satisfies $\mathrm {A}_G(0)=\mathfrak {S}_3$ , and the orbit of $x = \begin {pmatrix} 0 & 1 \\ 0 & 0 \end {pmatrix}$ , where we have $\mathrm {A}_G(x)=(\mathbb {Z}/2\mathbb {Z}) \times \mathfrak {S}_3$ . There are also two conjugacy classes of special Levi subgroups, namely, $T \times \mathfrak {A}_3$ and G. We have $\mathrm {N}_G(T \times \mathfrak {A}_3) / (T \times \mathfrak {A}_3) \cong (\mathbb {Z}/2\mathbb {Z})^2$ . In all cases, the cocycles from Theorem 5.3 will be trivial by the considerations in Remark 5.4.

5.4.1. Characteristics not $2$ or $3$

Let us assume that $\mathrm {char}(\mathbb {k}) \notin \{2,3\}$ . Then $\mathfrak {S}_3$ has three isomorphism classes of simple representations (those of the trivial representation, the sign representation, and a two-dimensional simple representation), giving rise to three simple perverse sheaves denoted $(\mathcal {IC}_i : i \in \{1, 2, 3\}),$ respectively, and $(\mathbb {Z}/2\mathbb {Z}) \times \mathfrak {S}_3$ has six isomorphism classes of simple representations (obtained by tensoring the simple representations of $\mathfrak {S}_3$ either by the trivial representation or by the sign representation). We will denote by $(\mathcal {IC}_i : i \in \{4, 5, 6\})$ the simple perverse sheaves associated with the trivial representation of $\mathbb {Z}/2\mathbb {Z}$ (taking the representations of $\mathfrak {S}_3$ in the same order as above) and by $(\mathcal {IC}_i : i \in \{7, 8, 9\})$ those associated with the sign representation.

Let us assume that $\mathbb {k}$ admits a nontrivial third root of unity. The special Levi subgroup $T \times \mathfrak {A}_3$ has a unique nilpotent orbit $\{0\}$ , but three cuspidal pairs corresponding to the three isomorphism classes of simple $\mathfrak {A}_3$ -representations. The two nontrivial representations are conjugate under the action of the normalizer in G, so only one needs to be considered. For the trivial representation, the associated simple perverse sheaf is the skyscraper sheaf $\delta _0$ with trivial equivariant structure, and denoting by $B \subseteq \mathrm {SL}_2(\mathbb {C})$ the Borel subgroup of upper-triangular matrices, we have

$$\begin{align*}\mathrm{Ind}_{T \times \mathfrak{A}_3 \subset B \times \mathfrak{A}_3}^G(\delta_0) = \mathrm{Ind}_{G^\circ \times \mathfrak{A}_3 \subseteq G^\circ \times \mathfrak{A}_3}^G(\mathrm{Ind}_{T \subseteq B}^{G^\circ}(\delta_0) \boxtimes \mathbb{k}). \end{align*}$$

Here, $\mathrm {Ind}_{T \subseteq B}^{G^\circ }(\delta _0)$ is the Springer sheaf, which is the direct sum of the skyscraper sheaf $\delta _0$ (now seen as a perverse sheaf on $\mathcal {N}_{G^\circ }$ ) and the simple perverse sheaf $\mathcal {IC}(G^\circ \cdot x, \underline {\mathbb {k}})$ . One easily checks using the considerations in the proof of Lemma 3.3(2) that

$$\begin{align*}\mathrm{Ind}_{G^\circ \times \mathfrak{A}_3 \subseteq G^\circ \times \mathfrak{A}_3}^G(\delta_0) \cong \mathcal{IC}_1 \oplus \mathcal{IC}_2, \quad \mathrm{Ind}_{G^\circ \times \mathfrak{A}_3 \subseteq G^\circ \times \mathfrak{A}_3}^G(\mathcal{IC}(G^\circ \cdot x, \underline{\mathbb{k}})) \cong \mathcal{IC}_4 \oplus \mathcal{IC}_5. \end{align*}$$

This induction series therefore has four elements, corresponding to the four isomorphism classes of simple representations of $(\mathbb {Z}/2\mathbb {Z})^2$ .

Similarly, one sees that the induction series associated with a nontrivial character of $\mathfrak {A}_3$ consists of $\mathcal {IC}_3$ and $\mathcal {IC}_6$ , corresponding to the two simple representations of the stabilizer $\mathbb {Z}/2\mathbb {Z}$ .

On the other hand, G has three cuspidal pairs, corresponding to the simple perverse sheaves $\mathcal {IC}_7$ , $\mathcal {IC}_8$ , and $\mathcal {IC}_9$ , which form three other induction series.