1 Introduction
1.1 Invariant theory
Given a connected reductive group G and a finite-dimensional representation V of G, the quotient stack
$\mathfrak {X}=V/G$
and the Geometric Invariant Theory (GIT) quotient
$\mathcal {X}=V/\!\!/ G$
are fundamental objects in invariant theory [Reference Mumford, Fogarty and KirwanMFK94; Reference Dieudonné and CarrellDC70; Reference Popov and VinbergPV94], with significant applications to geometry and representation theory, e.g. [Reference NakajimaNak94; Reference ReinekeRei08; Reference Maulik and OkounkovMO19] for connections with Lie theory.
Recent research has continued to explore the geometry of the quotient
$V/G$
. For instance, in [Reference Špenko and den BerghŠV17; Reference Špenko and den BerghŠV21], the authors study semiorthogonal decompositions of the coherent derived category
$\mathcal {D}(V/G)$
to obtain noncommutative resolutions of
$V/\!\!/ G$
[Reference Van den BerghVan04; Reference LuntsLun10; Reference Kuznetsov and LuntsKL15], using the framework of window categories developed by Halpern-Leistner [Reference Halpern-LeistnerHal15]. Their approach focuses on quasi-symmetric representations of reductive groups, which are the representations for which the sum of weights on any line through the origin in the character lattice is zero.
In this paper, we consider the case where V is a weakly symmetric representation of G (Definition 2.3). This encompasses the case of self-dual representations. We prove a decomposition of the (infinite-dimensional) cohomology vector space
$\mathrm {H}^*(V/G)$
into a finite collection of finite-dimensional, cohomologically graded vector spaces
$\mathcal {P}_{\lambda }$
indexed by cocharacters
$\lambda \in X_*(T)$
of a maximal torus T of G (Theorem 1.1). This decomposition, which employs parabolic induction (§2), is known as cohomological integrality and can be seen as a cohomological version of a generalised Springer decomposition [Reference LusztigLus84] in this context. We then define the refined Donaldson–Thomas invariants of
$V/G$
as the Betti numbers of these vector spaces
$\mathcal {P}_{\lambda }$
(Definition 1.4) following [Reference Meinhardt and ReinekeMR19; Reference MeinhardtMei15; Reference Davison and MeinhardtDM20a] and initiate their study (Proposition 1.5). These invariants are expected to encode rich representation-theoretic information about the action of G on V and topological information about the GIT quotient
$V/\!\!/ G$
. In particular, we expect that the vector space
$\mathcal {P}_{\lambda }$
computes the intersection cohomology of some GIT quotient, thereby providing an efficient algorithmic and inductive way to determine the intersection Betti numbers of a large class of GIT quotients. We study this question in [Reference HennecartHen24] and in forthcoming work.
Stacks of the form
$V/G$
and their moduli spaces
$V/\!\!/ G$
serve as local models for smooth stacks
$\mathfrak {X}$
with good moduli spaces
$\mathcal {X}$
by the Luna étale slice theorem [Reference Alper, Hall and RydhAHR20]. Therefore, our results provide a foundation for exploring the geometry and topology of morphisms
$\mathfrak {X}\rightarrow \mathcal {X}$
from a weakly symmetric smooth stack to its good moduli space [Reference AlperAlp13]. While this paper develops the core algebraic results, the applications to the geometry of a class of significant stacks in algebraic geometry and representation theory are investigated in [Reference HennecartHen24].
1.2 Cohomological integrality
Cohomological integrality results, originally developed by Kontsevich and Soibelman in the context of cohomological Hall algebras and wall-crossing for moduli spaces [Reference Kontsevich and SoibelmanKS08; Reference Kontsevich and SoibelmanKS11], involve factorisations of generating series into plethystic exponentials. For symmetric quivers without potential, Efimov [Reference EfimovEfi12] proved a cohomological integrality isomorphism, answering a conjecture by Kontsevich and Soibelman [Reference Kontsevich and SoibelmanKS11], which identifies the cohomological Hall algebra (CoHA) to a supercommutative algebra over a graded vector space with finite-dimensional graded components. Efimov’s isomorphism can be seen as a categorification of integrality formulas, in the sense that an equality between two generating series is turned into an isomorphism between two graded vector spaces whose graded characters recover the series.
Efimov’s integrality theorem forms the basis of all subsequent cohomological integrality results for CoHAs [Reference Meinhardt and ReinekeMR19; Reference Davison and MeinhardtDM20a; Reference Kapranov and VasserotKV22; Reference Davison, Hennecart and Schlegel MejiaDHM22; Reference Davison, Hennecart and Schlegel MejiaDHM23]. These results provide deep structural insights and link the topology of the stack to the good moduli space. In this context, cohomological integrality is often expressed through Poincaré-Birkhoff-Witt (PBW) theorems, analogous to those in the theory of Lie algebras, and leads to refined cohomological invariants of interest in algebraic geometry, forming the basis of cohomological Donaldson–Thomas theory [Reference SzendroiSze14; Reference Davison and MeinhardtDM20a; Reference DavisonDav23].
In contrast to the traditional approach involving cohomological Hall algebras, our work deals directly with parabolic induction morphisms (§2) for more general smooth weakly symmetric stacks arising from weakly symmetric representations of reductive groups. We define the cohomological integrality morphism explicitly in Theorem 1.1. It does not involve the symmetric power of a vector space as in the quiver case, but some combinatorial replacement of it for any Weyl group, defined from the lattice of cocharacters of a maximal torus.
This paper can be viewed as a generalization of Efimov’s integrality isomorphism [Reference EfimovEfi12] to all weakly symmetric stacks of the form
$V/G$
, not just those arising from quivers. It lays the groundwork for extending cohomological Donaldson–Thomas theory to more general stacks and moduli spaces beyond those parameterising objects in an Abelian category. The cohomological integrality we present here is a purely algebraic result, with a purely algebraic proof. We explain in [Reference HennecartHen24] the geometric implications of this result.
There are also motivic versions of integrality results, such as those studied in [Reference BuBu24] following [Reference Kontsevich and SoibelmanKS08] and in the context of motivic Donaldson–Thomas theory [Reference Kontsevich and SoibelmanKS10; Reference MeinhardtMei17; Reference Bussi, Joyce and MeinhardtBJM19], which are different from ours.
1.3 Main results
In this paper, we work over an algebraically closed field of characteristic zero. Moreover, cohomology is considered with
$\mathbf {Q}$
-coefficients.
1.3.1 Parabolic induction
We refer to §2.1 for the precise definitions. Let G be a connected reductive group and V a representation of G. We let
$T\subseteq G$
be a maximal torus. For any cocharacter
$\lambda \colon \mathbf {G}_{\mathrm {m}}\rightarrow T$
, we let
$G^{\lambda }\subseteq G$
be the corresponding Levi subgroup of G [Reference Digne and MichelDM20b, Proposition 1.22] and
$V^{\lambda }$
the representation of
$G^{\lambda }$
given by the
$\lambda $
-fixed subspace of V. If
$\lambda =0$
is the trivial cocharacter, we write
$V=V^0$
and
$G=G^0$
.
There is an induction map
$$ \begin{align} \mathrm{Ind}_{\lambda}\colon \mathrm{H}^{*+d_{\lambda}}(V^{\lambda}/G^{\lambda})\rightarrow\mathrm{H}^{*+d_{0}}(V/G)\,, \end{align} $$
where
$d_{\lambda }=\dim V^{\lambda }-\dim G^{\lambda }$
(§2.2). If V is symmetric (i.e. V and
$V^*$
have the same sets of weights, Definition 2.3), then the induction map preserves shifted cohomological degrees (Lemma 2.7), that is
$\mathrm {Ind}_{\lambda }(\mathrm {H}^{i+d_{\lambda }}(V^{\lambda }/G^{\lambda }))\subseteq \mathrm {H}^{i+d_0}(V/G)$
for any
$i\in \mathbf {Z}$
(this is the reason for the notation
$\mathrm {H}^{*+d_{\lambda }}$
in (1.1), which emphasises that the natural grading is the cohomological grading shifted by
$d_{\lambda }$
).
We define an order on the set
$X_*(T)$
of cocharacters of T as follows:
$$ \begin{align} \lambda\preceq\mu \stackrel{\text{def.}}{\iff} \left\{ \begin{aligned} &V^{\lambda}\subseteq V^{\mu}\\ & \mathfrak{g}^{\lambda}\subseteq\mathfrak{g}^{\mu}\,. \end{aligned} \right. \end{align} $$
We let 
, where
$\sim $
is the equivalence relation given by
$\lambda \sim \mu \iff (\lambda \preceq \mu \text { and }\mu \preceq \lambda )$
. In other words, two cocharacters are equivalent if their fixed-point sets inside V and
$\mathfrak {g}$
(or equivalently inside V and G) both coincide. The set
$\mathscr {P}_V$
is finite. If V is symmetric, the induction map
$\mathrm {Ind}_{\lambda }$
only depends on the class
$\overline {\lambda }\in \mathscr {P}_V$
up to a non-zero scalar (Lemma 2.16).
We let
$W=N_G(T)/T$
be the Weyl group of G. The natural action of W on
$X_*(T)$
descends to a W-action on
$\mathscr {P}_V$
(§2.2). The image of the induction map
$\mathrm {Ind}_{\lambda }$
only depends on the class
$\widetilde {\lambda }\in \mathscr {P}_V/W$
of
$\lambda $
(Lemma 2.17).
1.3.2 Cohomological integrality
Let G be a reductive group with maximal torus T and V be a finite-dimensional weakly symmetric representation of G (Definition 2.3). For any
$\overline {\lambda }\in \mathscr {P}_V$
, we let
$\lambda \in X_*(T)$
be an arbitrary lift. For
$\widetilde {\lambda }\in \mathscr {P}_V/W$
, we let
$\lambda \in X_*(T)$
be an arbitrary lift.
For
$\lambda \in X_*(T)$
, we let
$W_{\lambda }=\{w\in W\mid V^{w\cdot \lambda }=V^{\lambda }\text { and }\mathfrak {g}^{w\cdot \lambda }=\mathfrak {g}^{\lambda }\}$
. This is a subgroup of W (Lemma 4.2).
We let
$G_{\lambda }\subseteq G$
be the intersection of the kernel of the action map
$G^{\lambda }\rightarrow \mathrm {GL}(V^{\lambda })$
with the center of
$G^{\lambda }$
and 
. Note that the group
$G_{\lambda }$
is not necessarily connected, but this can essentially be ignored (Lemma 4.1) because we consider cohomology with
$\mathbf {Q}$
-coefficients.
For
$\alpha \in X^*(T)$
, we let 
be the
$\alpha $
-weight space of V. We define
$\mathfrak {g}_{\alpha }$
similarly.
We let
We may call
$k_{\lambda }$
the induction kernel, see the shuffle-like formula in Proposition 2.11. We define
$\varepsilon _{V,\lambda }\colon W_{\lambda }\rightarrow \mathbf {C}^*$
to be the character such that for any
$w\in W_{\lambda }$
,
$w(k_{\lambda })=\varepsilon _{V,\lambda }(w)^{-1}k_{\lambda }$
(Proposition 4.6).
Theorem 1.1 (Proposition 4.16 + Corollary 4.24).
There exist
$W_{\lambda }$
-stable cohomologically graded vector spaces
$\mathcal {P}_{\lambda }\subseteq \mathrm {H}^{*+d_{\lambda }}(V^{\lambda }/G^{\lambda })$
,
$\lambda \in X_*(T)$
, such that the map
$$\begin{align*}\bigoplus_{\widetilde{\lambda}\in\mathscr{P}_{V}/W}(\mathcal{P}_{\lambda}\otimes \mathrm{H}^*(\mathrm{pt}/G_{\lambda}))^{\varepsilon_{V,\lambda}}\xrightarrow{\sum_{\widetilde{\lambda}\in\mathscr{P}_V/W}\mathrm{Ind}_{\lambda}} \mathrm{H}^*(V/G) \end{align*}$$
induced by the induction morphisms (1.1) is a graded isomorphism, where
$\mathrm {H}^*(\mathrm {pt}/G_{\lambda })\cong \mathrm {Sym}(\mathfrak {g}_{\lambda }^*)$
,
$\mathfrak {g}_{\lambda }^*$
sits in degree
$2$
and
$(\mathcal {P}_{\lambda }\otimes \mathrm {H}^*(\mathrm {pt}/G_{\lambda }))^{\varepsilon _{V,\lambda }}$
denotes the
$\varepsilon _{V,\lambda }$
-isotypic component for the natural
$W_{\lambda }$
-action. Moreover, the vector spaces
$\mathcal {P}_{\lambda }$
are finite-dimensional and the inclusion
$\mathcal {P}_{\lambda }\subseteq \mathrm {H}^{*+d_{\lambda }}(V^{\lambda }/G^{\lambda })$
factors through the inclusion
$\mathrm {H}^{*+d_{\lambda }}(V^{\lambda }/(G^{\lambda }/G_{\lambda }))\rightarrow \mathrm {H}^{*+d_{\lambda }}(V^{\lambda }/G^{\lambda })$
. In addition, the vector spaces
$\mathcal {P}_{\mu }$
for
$\mu \preceq \lambda $
satisfy the cohomological integrality isomorphism for
$(V^{\lambda },G^{\lambda })$
.
This type of theorems is sometimes referred to as cohomological integrality. We can reconstruct the infinite dimensional vector space
$\mathrm {H}^*(V/G)$
using a finite number of finite-dimensional vector spaces
$\mathcal {P}_{\lambda }$
,
$\widetilde {\lambda }\in \mathscr {P}_V/W$
, allowing us to extract fundamental enumerative invariants associated to the stack
$V/G$
(Definition 1.4).
For a cocharacter
$\lambda \in X_*(T)$
, there is a natural W-equivariant map
$\mathscr {P}_{V^{\lambda }}\rightarrow \mathscr {P}_V$
, which can be defined as follows. Given a cocharacter
$\mu \in X_*(T)$
, there exists a cocharacter
$\nu \in X_*(T)$
such that
$V^{\lambda }\cap V^{\mu }=V^{\nu }$
and
$G^{\lambda }\cap G^{\mu }=G^{\nu }$
(Lemma 2.8). We have
$\nu \preceq \lambda $
and moreover,
$\overline {\nu }=\overline {\mu }$
in
$\mathscr {P}_{V^{\lambda }}$
. The map sends
$\overline {\mu }=\overline {\nu }\in \mathscr {P}_{V^{\lambda }}$
to
$\overline {\nu }\in \mathscr {P}_V$
. Then, the last sentence of Theorem 1.1 means that the vector spaces
$\mathcal {P}_{\nu }$
satisfy the cohomological integrality isomorphism for
$(V^{\lambda },G^{\lambda })$
.
Remark 1.2. Let
$Q=(Q_0,Q_1)$
be a symmetric quiver with set of vertices
$Q_0$
and set of arrows
$Q_1$
. Let
$\mathbf {d}\in \mathbf {N}^{Q_0}$
be a dimension vector. If
$V_{\mathbf {d}}$
is the representation space of
$\mathbf {d}$
-dimensional representations of Q which is acted on by 
, then the isomorphism given by Theorem 1.1 essentially recovers [Reference EfimovEfi12, Theorem 1.1]. In loc.cit, it was convenient for the author to twist the cohomological Hall algebra multiplication to make it supercommutative, as in [Reference Kontsevich and SoibelmanKS11]. This twist does not change the images of the induction/multiplication morphisms which means that it is possible to have cohomological integrality isomorphisms even without this twist. This remark inspired the present work, in which the twist is replaced by the character
$\varepsilon _{V,\lambda }$
of the group
$W_{\lambda }$
(Proposition 4.6). When V is symmetric (and not only weakly symmetric), the character
$\varepsilon _{V,\lambda }$
takes values in
$\{\pm 1\}$
.
Remark 1.3. In this paper, we consider cohomology with
$\mathbf {Q}$
-coefficients, which is crucial for the cohomological integrality map (Theorem 1.1) to be an isomorphism. It is crucial to work with
$\mathbf {Q}$
-coefficients from §2.5 until the end of the paper. However, one can define the induction morphisms (§2) for any ring of coefficients. It then becomes an interesting question to study the analogous question of cohomological integrality in positive characteristic, especially when the cohomology of the classifying space
$\mathrm {H}^*(\mathrm {pt}/G)$
has torsion.
Definition 1.4. We define the cohomologically refined Donaldson–Thomas invariants of
$V/G$
as the dimensions 
of the cohomological degree i piece of the vector space
$\mathcal {P}_{\lambda }$
for
$\lambda \in X_*(T)$
. We may call the Euler characteristics 
of
$\mathcal {P}_{\lambda }$
the Donaldson–Thomas invariants of
$V/G$
.
1.3.3 Vanishing bound
The vector spaces
$\mathcal {P}_{\lambda }$
given by Theorem 1.1 are finite dimensional (Proposition 4.16), and we can give a rough bound on the cohomological degrees for which they possibly do not vanish. We let 
, where
$d_{\lambda }=\dim V^{\lambda }-\dim G^{\lambda }$
. The natural grading on
$\mathcal {H}_{\lambda }$
is given by
$\mathcal {H}_{\lambda }^k=\mathrm {H}^{k+d_{\lambda }}(V^{\lambda }/G^{\lambda })$
. It induces a cohomological grading on
$\mathcal {P}_{\lambda }$
.
Proposition 1.5. The cohomologically graded vector space
$\mathcal {P}_{\lambda }$
given by Theorem 1.1 are concentrated in cohomological degrees
$[\dim G^{\lambda }-\dim V^{\lambda }, \dim G^{\lambda }]$
, that is
$\mathcal {P}_{\lambda }^k=0$
unless
$k\in [\dim G^{\lambda }-\dim V^{\lambda }, \dim G^{\lambda }]$
.
1.4 Notations and conventions
-
• If V is a representation of a finite group W and
$\chi $
an irreducible character of W, we let
$V^{\chi }$
be the
$\chi $
-isotypic component of V. -
• If X is a complex algebraic variety acted upon by an algebraic group G, we let
$X/G=[X/G]$
be the quotient stack; that is, we omit the brackets. -
• The multiplicative group is denoted by
$\mathbf {G}_{\mathrm {m}}$
. The Abelian group of characters of an algebraic torus T is
$\mathrm {Hom}_{\mathbf {Z}}(T,\mathbf {G}_{\mathrm {m}})$
. The set of cocharacters of T is the Abelian group
$\mathrm {Hom}_{\mathbf {Z}}(\mathbf {G}_{\mathrm {m}},T)$
. We have natural embedding and
$X^*(T)\subseteq \mathfrak {t}^*$
respectively given by
$\lambda \mapsto \frac {\mathrm {d\lambda }}{\mathrm {d}t}_{|t=1}$
and
$\alpha \mapsto \mathrm {d}\alpha (1)$
, the differential of a character
$\alpha $
at the neutral element
$1\in T$
. Moreover, these inclusions are compatible with the natural pairings
$\langle -,-\rangle \colon X_*(T)\times X^*(T)\rightarrow \mathbf {Z}$
and
$\mathfrak {t}\times \mathfrak {t}^*\rightarrow \mathbf {C}$
. -
• If A is an Abelian group and
$S\subseteq A$
is a subset of A, we let be the opposite of S. We will only consider the case
$A=X^*(T)$
. -
• Let
$H\subseteq G$
be algebraic groups and X an H-variety. We let where H acts on
$X\times G$
by
$h\cdot (x,g)=(h\cdot x,gh^{-1})$
. The formula gives a G-action on
$X\times ^HG$
. -
• If R is a domain (e.g.
$R=\mathrm {H}^*_T(\mathrm {pt})$
for some algebraic torus T), we denote by
$\mathrm {Frac}(R)$
its fraction field. -
• If G is a possibly disconnected algebraic group, then we denote by
$G^{\circ }$
its neutral component. -
• Given a scheme X, we denote by
$\mathcal {O}_X$
its sheaf of regular functions. If X is a complex scheme, then set ring of regular functions on X. -
• If S is a set (for example, a vector space) with the action of a group G, we denote by
$S^G$
the fixed subset.
and 
be the opposite of S. We will only consider the case 
where H acts on 
gives a G-action on 
set ring of regular functions on X.
2 Parabolic induction for representations of reductive groups
2.1 The induction diagram
Let G be a reductive group,
$T\subseteq G$
a maximal torus of G and V a finite-dimensional representation of G. For a cocharacter
$\lambda \colon {\mathbf {G}}_{\mathrm {m}}\rightarrow T$
, we let 
be the centraliser of
$\lambda $
, a Levi subgroup of G [Reference Digne and MichelDM20b, Proposition 1.22] and 
be the fixed locus, a representation of
$G^{\lambda }$
. We also let 
, a parabolic subgroup of G and 
, a representation of
$P_{\lambda }$
.
A weight of V is a character
$\alpha \colon T\rightarrow \mathbf {G}_{\mathrm {m}}$
such that there exists
$v\in V\smallsetminus \{0\}$
such that for any
$t\in T$
,
$t\cdot v=\alpha (t)v$
.
We let
$\mathcal {W}'(V)\subseteq X^*(T)$
be the set of weights of the representation V. We have a direct sum decomposition
$$\begin{align*}V\cong\bigoplus_{\alpha\in\mathcal{W}'(V)}V_{\alpha} \end{align*}$$
where
$V_{\alpha }$
consists of
$0$
and nonzero vectors of weight
$\alpha $
, where 
.
We let
$\mathcal {W}(V)$
be the collection of weights of V, counted with multiplicities:
$\alpha \in \mathcal {W}'(V)$
appears
$\dim V_{\alpha }$
times in
$\mathcal {W}(V)$
.
In particular, we have
$\mathcal {W}'(\mathfrak {g})$
and
$\mathcal {W}(\mathfrak {g})$
where
$\mathfrak {g}$
is seen as the adjoint representation of G. We have the natural pairing
$$\begin{align*}\langle-,-\rangle\colon X_*(T)\times X^*(T)\rightarrow\mathbf{Z} \end{align*}$$
between characters and cocharacters. If
$(\lambda ,\alpha )\in X_*(T)\times X^*(T)$
, then
$\alpha \circ \lambda \in \mathrm {End}(\mathbf {G}_{\mathrm {m}})$
, and so it is given by raising to some power, which is
$\langle \lambda ,\alpha \rangle \in \mathbf {Z}$
. We let 
and we define similarly
$\mathcal {W}^{\lambda =0}(V)$
,
$\mathcal {W}^{\lambda \geqslant 0}(V)$
,
$\mathcal {W}^{\lambda <0}(V)$
and
$\mathcal {W}^{\lambda \leqslant 0}(V)$
and their
$'$
versions, by forgetting the multiplicities.
It is immediate that
$V^{\lambda \geqslant 0}=\bigoplus _{\alpha \in \mathcal {W}^{'\lambda \geqslant 0}(V)}V_{\alpha }$
.
We consider the (commutative) induction diagram
where 
is the affine GIT quotient of V by G and the map
$\imath _{\lambda }$
is induced by the equivariant closed immersion
$(V^{\lambda },G^{\lambda })\rightarrow (V,G)$
.
Lemma 2.1. The map
$q_{\lambda }$
is a vector bundle stack of rank 
and the map
$p_{\lambda }$
is a proper and representable morphism of stacks.
Proof. The map
$p_{\lambda }$
can be presented as the map of quotient stacks
$(V^{\lambda \geqslant 0}\times ^{P_{\lambda }} G)/G\rightarrow V/G$
coming from the G-equivariant map
which can be factored as a closed immersion followed by a projective map
$$\begin{align*}\begin{matrix} V^{\lambda\geqslant 0}\times^{P_{\lambda}}G&\rightarrow&V\times G/P_{\lambda}&\rightarrow&V\\ (v,g)&\mapsto&(g\cdot v,gP_{\lambda})&&\\ &&(v,gP_{\lambda})&\mapsto&v \end{matrix} \end{align*}$$
and is therefore projective. Therefore,
$p_{\lambda }$
is representable and projective.
The fact that
$q_{\lambda }$
is smooth, surjective of relative dimension
$\#\mathcal {W}^{\lambda>0}(V)-\mathcal {W}^{\lambda >0}(\mathfrak {g})$
is easily seen. That it is in addition a vector bundle stack is a rather classical fact (which is not used in this paper). It comes from the facts that
$V^{\lambda \geqslant 0}\rightarrow V^{\lambda }$
is a vector bundle (by the Białinicky-Birula decomposition [Reference Białynicki-BirulaBia73], but this can be seen directly in the case of linear actions) and the kernel of the map
$P_{\lambda }\rightarrow G^{\lambda }$
is the unipotent radical of
$P_{\lambda }$
.
Lemma 2.2. The map
$\imath _{\lambda }$
is a finite map.
Proof. The arguments are parallel to that of the proof [Reference Meinhardt and ReinekeMR19, Lemma 2.1]. Namely, we consider the commutative diagram
where 
(see §1.4), so that we have an equivalence of stacks
$V^{\lambda \geqslant 0}/P_{\lambda }\simeq \widetilde {V}^{\lambda \geqslant 0}/G$
. We now explain the definitions of the maps in this diagram.
The map
$q_{\lambda }$
is given by the equivariant map
$(V^{\lambda \geqslant 0},P_{\lambda })\rightarrow (V^{\lambda },G^{\lambda })$
,
$(v,g)\mapsto(\lim \limits _{t\rightarrow 0} \lambda (t)\cdot v,\lim \limits _{t\rightarrow 0}\lambda (t)g\lambda (t)^{-1})$
. Therefore, the pullback of functions by
$q_{\lambda }$
induces a morphism between rings of invariant functions
$$\begin{align*}\overline{q}_{\lambda}^*\colon\mathbf{C}[V^{\lambda}]^{G^{\lambda}}\rightarrow \mathbf{C}[\widetilde{V}^{\lambda\geqslant 0}]^{G} \end{align*}$$
whose dual is the morphism between affine schemes
$$\begin{align*}\overline{q}_{\lambda}\colon \mathrm{Spec}(\mathbf{C}[\widetilde{V}^{\lambda\geqslant 0}]^{G})\rightarrow\mathrm{Spec}(\mathbf{C}[V^{\lambda}]^{G^{\lambda}})=V^{\lambda}/\!\!/ G^{\lambda}\,. \end{align*}$$
The map
$\jmath _{\lambda }$
is given by the equivariant morphism
$(V^{\lambda },G^{\lambda })\rightarrow (V^{\lambda \geqslant 0},P_{\lambda })$
,
$(v,g)\mapsto (v,g)$
which induces the morphism of algebras
$$\begin{align*}\overline{\jmath}_{\lambda}^*\colon \mathbf{C}[\widetilde{V}^{\lambda\geqslant 0}]^{G}\rightarrow\mathbf{C}[V^{\lambda}]^{G^{\lambda}} \end{align*}$$
and dually the morphism of schemes
$$\begin{align*}\overline{\jmath}_{\lambda}\colon\mathrm{Spec}(\mathbf{C}[V^{\lambda}]^{G^{\lambda}})\rightarrow\mathrm{Spec}(\mathbf{C}[\widetilde{V}^{\lambda\geqslant 0}]^{G})\,. \end{align*}$$
Since
$q_{\lambda }\circ \jmath _{\lambda }=\mathrm {id}$
, we have
$\overline {q}_{\lambda }\circ \overline {\jmath }_{\lambda }=\mathrm {id}$
. On the ring of functions, we have
$\overline {\jmath }_{\lambda }^*\circ \overline {q}_{\lambda }^*=\mathrm {id}$
and so
$\overline {\jmath }_{\lambda }^*$
is surjective. This tells us that
$\overline {\jmath }_{\lambda }$
is a closed immersion. Moreover,
$\imath _{\lambda }=\overline {p}_{\lambda }\circ \overline {\jmath }_{\lambda }$
and so it suffices to prove that
$\overline {p}_{\lambda }$
is a finite map. Since it is a morphism of finite type complex schemes, it suffices to prove that it is an integral morphism [Sta24, Lemma 01WJ].
Since
$\widetilde {p}_{\lambda }\colon \widetilde {V}^{\lambda \geqslant 0}\rightarrow V$
is a projective morphism (Proof of Lemma 2.1),
$(\widetilde {p}_{\lambda })_*\mathcal {O}_{\widetilde {V}^{\lambda \geqslant 0}}$
is a coherent
$\mathcal {O}_{V}$
-module. Therefore, the map
$\mathbf {C}[V]\rightarrow \mathbf {C}[\widetilde {V}^{\lambda \geqslant 0}]$
is finite and hence integral [Sta24, Lemma 01WJ]. We deduce that the map
$$\begin{align*}\overline{p}_{\lambda}^*\colon\mathbf{C}[V]^G\rightarrow \mathbf{C}[\widetilde{V}^{\lambda\geqslant 0}]^G \end{align*}$$
is integral, and hence finite, as follows. Let
$a\in \mathbf{C}[\widetilde{V}^{\lambda\geqslant 0}]^G $
. It is a solution of a monic polynomial with coefficients in
$\mathbf {C}[V]$
. By applying the Reynolds operator to the equation, we obtain a monic equation with coefficients in
$\mathbf {C}[V]^G$
of which a is a zero. This concludes the proof.
We define the equivalence relation
$\equiv $
on
$X_*(T)$
by
$\alpha \equiv \beta \iff \exists a,b\in \mathbf {Z}_{>0},\quad a\alpha =b\beta $
. The quotient set is denoted by 
. If
$S\subseteq X_*(T)$
, we denote by
$\overline {S}$
its image in
$\overline {X_*(T)}$
. Similarly, if S is a collection of characters (with multiplicities allowed), we denote by
$\overline {S}$
the corresponding collection of elements of
$\overline {X_*(T)}$
(with multiplicities taken into account).
Definition 2.3. We say that a representation V of a reductive group G is
-
(1) symmetric if V and its dual
$V^*$
have the same sets of weights counted with multiplicities:
$\mathcal {W}(V)=\mathcal {W}(V^*)$
. -
(2) weakly symmetric if
$\overline {\mathcal {W}(V)}=\overline {\mathcal {W}(V^*)}$
.
Since a representation of a reductive group is determined by its character, a representation V of G is symmetric if and only if it is self-dual, that is
$V\cong V^*$
as representations of G.
Example 2.4. The following are symmetric representations:
-
(1) The trivial representation
$\{0\}$
of a reductive group G, -
(2) The adjoint representation
$\mathfrak {g}=\mathrm {Lie}(G)$
of a reductive group G, -
(3) The cotangent representation
$\mathrm {T}^*V$
of any representation V of a reductive group G, -
(4) The representation space of
$\mathbf {d}\in \mathbf {C}^{Q_0}$
-dimensional representations of a symmetric quiver
$Q=(Q_0,Q_1)$
, for
$G=\prod _{i\in Q_0}\mathrm {GL}_{\mathbf {d}_i}$
. -
(5) Any representations of products of simple groups of types
$B_n, C_n, D_n (n \text { even}), E_7, E_8, F_4, G_2$
. Indeed, for such a group G,
$w_0=-1\in W$
and for any
$w\in W$
,
$\dim V_{\alpha }=\dim V_{w\cdot \alpha }$
[Reference BourbakiBB07, Planches I–IX], [Reference HumphreysHum90, Exercise and Table 1 page 59].
Example 2.5. The two-dimensional representation of
$\mathbf {C}^*$
given by
$t\cdot (u,v)=(tu,t^{-2}v)$
is weakly symmetric but not symmetric.
For
$\lambda \in X_*(T)$
, we define 
. The quantity
$r_{\lambda }$
is defined in Lemma 2.1. For later use, we give the following easy lemma.
Lemma 2.6. Let V be a weakly symmetric representation of G. Then,
$$\begin{align*}d_{\lambda}+2r_{\lambda}=d\,. \end{align*}$$
Proof. We have
$$\begin{align*}\mathcal{W}(V)=\mathcal{W}^{\lambda>0}(V)\sqcup\mathcal{W}^{\lambda=0}(V)\sqcup\mathcal{W}^{\lambda<0}(V) \end{align*}$$
and by weak symmetry (Definition 2.3),
$\#\mathcal {W}^{\lambda>0}(V)=\#\mathcal {W}^{\lambda <0}(V)$
. Therefore,
$\#\mathcal {W}(V)=2\#\mathcal {W}^{\lambda>0}(V)+\#\mathcal {W}^{\lambda =0}(V)$
. These equalities are also valid for
$V=\mathfrak {g}$
under the adjoint action of G (which is a symmetric representation, Example 2.4). We obtain the lemma by combining them together.
2.2 Parabolic induction
For
$\lambda \in X_*(T)$
, we let 
be the intersection complex monodromic mixed Hodge module of the smooth stack
$V^{\lambda }/G^{\lambda }$
. We refer to [Reference Davison and MeinhardtDM20a] for the necessary background regarding monodromic mixed Hodge modules, and in particular regarding the square-root
$\mathscr {L}^{1/2}$
of the Tate twist
$\mathscr {L}$
. The Tate twist 
is a pure weight
$0$
complex of mixed Hodge structures, concentrated in degree
$2$
. Alternatively, for the purposes of this paper, the reader may prefer to consider instead constructible sheaves. The Tate twist
$-\otimes \mathscr {L}$
is then replaced by the shift
$[-2]$
.
By smoothness of
$q_{\lambda }$
(Lemma 2.1), we have
$q_{\lambda }^*\cong q_{\lambda }^!\otimes \mathscr {L}^{r_{\lambda }}$
and thus we obtain
$$\begin{align*}q_{\lambda}^*\mathbb{D}\mathbf{Q}_{V^{\lambda}/G^{\lambda}}^{\mathrm{vir}}\cong\mathbb{D}\mathbf{Q}_{V^{\lambda\geqslant 0}/P_{\lambda}}\otimes\mathscr{L}^{r_{\lambda}+d_{\lambda}/2} \end{align*}$$
which by adjunction provides us with the map
$$ \begin{align} \mathbb{D}\mathbf{Q}_{V^{\lambda}/G^{\lambda}}^{\mathrm{vir}}\rightarrow(q_{\lambda})_*\mathbb{D}\mathbf{Q}_{V^{\lambda\geqslant 0}/P_{\lambda}}\otimes\mathscr{L}^{r_{\lambda}+d_{\lambda}/2}\,. \end{align} $$
Moreover, the map
$p_{\lambda }$
is proper and representable (Lemma 2.1), and so by dualising the adjunction map
$$\begin{align*}\mathbf{Q}_{V/G}\rightarrow (p_{\lambda})_*\mathbf{Q}_{V^{\lambda\geqslant 0}/P_{\lambda}}\,, \end{align*}$$
and using
$\mathbb {D} (p_{\lambda })_*=(p_{\lambda })_!\mathbb {D}$
and
$(p_{\lambda })_*=(p_{\lambda })_!$
, we obtain
$$ \begin{align} (p_{\lambda})_*\mathbb{D}\mathbf{Q}_{V^{\lambda\geqslant 0}/P_{\lambda}}\rightarrow\mathbb{D}\mathbf{Q}_{V/G}\,. \end{align} $$
We eventually obtain the map
$$ \begin{align} \mathrm{Ind}_{\lambda}\colon (\imath_{\lambda})_*(\pi_{\lambda})_*\mathbb{D}\mathbf{Q}_{V^{\lambda}/G^{\lambda}}^{\mathrm{vir}}\rightarrow \pi_*\mathbb{D}\mathbf{Q}_{V/G}\otimes\mathscr{L}^{r_{\lambda}+d_{\lambda}/2} \end{align} $$
by composing
$(\imath _{\lambda })_*(\pi _{\lambda })_*$
applied to (2.2) with
$\pi _*$
applied to (2.3) twisted by
$\mathscr {L}^{r_{\lambda }+d_{\lambda }/2}$
. We used the commutativity of (2.1) to identify
$\pi _*(p_{\lambda })_*\mathbb {D}\mathbf {Q}_{V^{\lambda \geqslant 0}/P_{\lambda }}$
and
$(\imath _{\lambda })_*(\pi _{\lambda })_*(q_{\lambda })_*\mathbb {D}\mathbf {Q}_{V^{\lambda \geqslant 0}/P_{\lambda }}$
.
Lemma 2.7. Let V be a weakly symmetric representation of G. Then, the induction morphism
$\mathrm {Ind}_{\lambda }\colon \mathrm {H}^{*+d_{\lambda }}(V^{\lambda }/G^{\lambda })\rightarrow \mathrm {H}^{*+d}(V/G)$
preserves shifted cohomological degrees, that is
$\mathrm {Ind}_{\lambda }(\mathrm {H}^{i+d_{\lambda }}(V^{\lambda }/G^{\lambda }))\subseteq \mathrm {H}^{i+d}(V/G)$
for any
$i\in \mathbf {Z}$
.
Proof. The induction at the level of cohomology vector spaces is given by applying the derived global section functor to the morphism of complexes (2.4) (the sheafified induction). Moreover, by Lemma 2.6,
$d/2=r_{\lambda }+d_{\lambda }/2$
. Therefore, in the weakly symmetric case, the right-hand side of (2.4) is precisely
$\pi _*\mathbb {D}\mathbf {Q}_{V/G}^{\mathrm {vir}}$
. Moreover,
$\mathbb {D}\mathbf {Q}_{V/G}^{\mathrm {vir}}\cong \mathbf {Q}_{V/G}^{\mathrm {vir}}$
and
$\mathbb {D}\mathbf {Q}_{V^{\lambda }/G^{\lambda }}^{\mathrm {vir}}\cong \mathbf {Q}_{V^{\lambda }/G^{\lambda }}^{\mathrm {vir}}$
since
$V/G$
and
$V^{\lambda }/G^{\lambda }$
are smooth and the chosen shifts/Tate twists (by the dimension) make the considered constant sheaves Verdier self-dual.
2.3 Augmentation
Let
$\lambda \in X_*(T)$
. We defined a representation
$V^{\lambda }$
of
$G^{\lambda }$
in §2.1. The induction morphisms, which we have defined for
$(V,G)$
, can also be defined in the exact same way for
$(V^{\lambda },G^{\lambda })$
. We can see T as a maximal torus of
$G^{\lambda }$
. If
$\mu \in X_*(T)$
, we have
$(V^{\lambda })^{\mu }=V^{\lambda }\cap V^{\mu }$
,
$(G^{\lambda })^{\mu }=G^{\lambda }\cap G^{\mu }$
,
$(V^{\lambda })^{\mu \geqslant 0}=V^{\lambda }\cap V^{\mu \geqslant 0}$
and 
.
In this case, the induction diagram (2.1) becomes
The definition of the induction map associated with this datum following §2.2 is
$$\begin{align*}\mathrm{Ind}_{\mu,\lambda}\colon \mathrm{H}^*((V^{\lambda})^{\mu}/(G^{\lambda})^{\mu})\rightarrow \mathrm{H}^*(V^{\lambda}/G^{\lambda})\,. \end{align*}$$
If
$\mu \preceq \lambda $
(1.2), then
$(V^{\lambda })^{\mu }=V^{\mu }$
and
$(G^{\lambda })^{\mu }=G^{\mu }$
, so that the induction
$\mathrm {Ind}_{\mu ,\lambda }$
is
$$\begin{align*}\mathrm{Ind}_{\mu,\lambda}\colon \mathrm{H}^*(V^{\mu}/G^{\mu})\rightarrow\mathrm{H}^*(V^{\lambda}/G^{\lambda})\,. \end{align*}$$
Lemma 2.8. For any
$\lambda ,\mu \in X_*(T)$
, one can find
$\nu \in X_*(T)$
such that
$V^{\nu }=V^{\mu }\cap V^{\lambda }$
,
$G^{\nu }=G^{\mu }\cap G^{\lambda }$
,
$(V^{\lambda })^{\nu \geqslant 0}=(V^{\lambda })^{\mu \geqslant 0}$
and
$P_{\mu ,\lambda }=P_{\nu ,\lambda }$
.
Proof. The cocharacters
$\lambda ,\mu $
define a morphism
$$\begin{align*}\lambda\times\mu\colon \mathbf{G}_{\mathrm{m}}^{\times2}\rightarrow T\,. \end{align*}$$
We obtain
$\nu $
by composing a general one-parameter subgroup
$\zeta \colon \mathbf {G}_{\mathrm {m}}\rightarrow \mathbf {G}_{\mathrm {m}}^{\times 2}$
,
$t\mapsto (t^a,t^b)$
with
$\lambda \times \mu $
. This guarantees the first two equalities of the Lemma. To guarantee the last two, it suffices to take a general one-parameter subgroup with the constraint
$b>0$
by replacing
$\zeta $
by
$\zeta ^{-1}$
if necessary.
2.4 Associativity
Proposition 2.9. Let
$\lambda ,\mu ,\nu \in X_*(T)$
be such that
$V^{\lambda }\subseteq V^{\mu }\subseteq V^{\nu }$
,
$G^{\lambda }\subseteq G^{\mu }\subseteq G^{\nu }$
and
$(V^{\nu })^{\mu \geqslant 0}\subseteq (V^{\nu })^{\lambda \geqslant 0}$
,
$P_{\mu ,\nu }\subseteq P_{\lambda ,\nu }$
. Then,
$$\begin{align*}\mathrm{Ind}_{\lambda,\nu}=\mathrm{Ind}_{\mu,\nu}\circ\mathrm{Ind}_{\lambda,\mu}\,. \end{align*}$$
Proof. The equality classically follows by base-change in the commutative diagram
2.5 Explicit formula for the parabolic induction
If
$\mathfrak {t}_{\mathbf {Q}}$
is a rational form of the Lie algebra of T, we have
$\mathrm {H}^*_T(\mathrm {pt})\cong \mathrm {Sym}(\mathfrak {t}^*)$
. Any character
$\alpha \in X_*(T)$
may be identified with an element of
$\mathfrak {t}^*$
and hence
$\mathrm {H}^*_T(\mathrm {pt})$
via
$\alpha \mapsto \mathrm {d}\alpha (1)$
(see §1.4). By abuse, we will still denote by
$\alpha $
the corresponding element of
$\mathrm {H}^*_T(\mathrm {pt})$
.
For
$\lambda \in X_*(T)$
, we define the induction kernel
We let 
be the Weyl group of
$G^{\lambda }$
.
Lemma 2.10. For any
$\lambda \in X_*(T)$
,
$k_{\lambda }$
is
$W^{\lambda }$
-invariant.
Proof. It suffices to prove that
$W^{\lambda }$
sends
$\mathcal {W}^{\lambda <0}(V)\subseteq \mathcal {W}(V)$
to itself, and similarly for
$\mathcal {W}^{\lambda <0}(\mathfrak {g})$
. This comes from the invariance of the pairing between characters and cocharacters: for any
$w\in W$
,
$\langle w\cdot \lambda ,w\cdot \alpha \rangle =\langle \lambda ,\alpha \rangle $
and if
$w\in W^{\lambda }$
,
$w\cdot \lambda =\lambda $
. Therefore, for any
$\alpha \in X^*(T)$
,
$\langle \lambda ,\alpha \rangle <0$
if and only if
$\langle \lambda ,w\cdot \alpha \rangle <0$
.
Proposition 2.11. Let
$\lambda \colon \mathbf {G}_{\mathrm {m}}\rightarrow T$
be a cocharacter and
$f\in \mathrm {H}^*(V^{\lambda }/G^{\lambda })\cong \mathrm {H}^*(\mathrm {pt}/T)^{W^{\lambda }}$
. We have
$$\begin{align*}\mathrm{Ind}_{\lambda}(f)=\sum_{w\in W/W^{\lambda}}w\cdot(fk_{\lambda})=\frac{1}{\# W^{\lambda}}\sum_{w\in W}w\cdot(fk_{\lambda})\,. \end{align*}$$
Proof. The second equality follows from the
$W^{\lambda }$
-invariance of both f (since
$f\in \mathrm {H}^*(V^{\lambda }/G^{\lambda })=\mathrm {H}^*(V^{\lambda }/T)^{W^{\lambda }}$
) and
$k_{\lambda }$
(Lemma 2.10). The first equality is a computation of Euler classes exactly as in the proof of [Reference Kontsevich and SoibelmanKS11, Theorem 2] in the case of quiver representations.
More generally, let
$\lambda ,\mu \in X_*(T)$
. We let
Proposition 2.12 (Augmentation of Proposition 2.11).
For
$f\in \mathrm {H}^*((V^{\mu })^{\lambda }/(G^{\mu })^{\lambda })\cong \mathrm {H}^*(\mathrm {pt}/T)^{W^{\lambda }\cap W^{\mu }}$
, we have
$$\begin{align*}\mathrm{Ind}_{\lambda,\mu}(f)=\sum_{w\in W^{\mu}/(W^{\lambda}\cap W^{\mu})}w(fk_{\lambda,\mu})\,. \end{align*}$$
Proof. This follows immediately from Proposition 2.11 applied to
$V^{\mu }$
as a representation of
$G^{\mu }$
and the cocharacter
$\lambda $
of T, seen as a maximal torus of
$G^{\mu }$
, and the identification of the Weyl group of
$(G^{\mu })^{\lambda }=G^{\lambda }\cap G^{\mu }$
with
$W^{\lambda }\cap W^{\mu }$
.
2.6 Parabolic induction for weakly symmetric representations
Proposition 2.13. Let V be a (weakly) symmetric representation of a reductive group G and
$\lambda \in X_*(T)$
a cocharacter of the maximal torus of T. Then, the representation
$V^{\lambda }$
of
$G^{\lambda }$
is also (weakly) symmetric.
Proof. It is immediate to check that
$\mathcal {W}'(V^{\lambda })=\{\alpha \in \mathcal {W}'(V)\mid \langle \lambda ,\alpha \rangle =0\}$
and for any
$\alpha \in \mathcal {W}'(V^{\lambda })$
, we can identify the weight spaces
$V^{\lambda }_{\alpha }=V_{\alpha }$
. Therefore, the (weak) symmetry of V implies that of
$V^{\lambda }$
.
Recall the order relation on
$X_*(T)$
defined in the introduction (1.2)
$$\begin{align*}\lambda\preceq \mu\overset{\text{def.}}{\iff} \left\{ \begin{aligned}V^{\lambda}\subseteq V^{\mu}\\ \mathfrak{g}^{\lambda}\subseteq\mathfrak{g}^{\mu}\,. \end{aligned} \right. \end{align*}$$
for any
$\lambda ,\mu \in X_*(T)$
.
We obtain an equivalence relation on the set
$X_*(T)$
$$\begin{align*}\lambda\sim \mu\overset{\text{def.}}{\iff} \lambda\preceq\mu \text{ and }\mu\preceq\lambda\,. \end{align*}$$
We let 
be the quotient of
$X_*(T)$
by this equivalence relation. If
$\lambda \in X_*(T)$
,
$\overline {\lambda }$
denotes its class in
$\mathscr {P}_V$
. It has an induced order relation still denoted by
$\preceq $
:
$$\begin{align*}\overline{\lambda}\preceq\overline{\mu}\iff \lambda\preceq\mu\,. \end{align*}$$
It is immediate that this is well-defined, i.e. the definition does not depend on the representatives
$\lambda ,\mu $
of
$\overline {\lambda },\overline {\mu }$
.
Lemma 2.14. Let V be a weakly symmetric representation of a reductive group G. If
$\lambda ,\mu \in X_*(T)$
are such that
$\lambda \sim \mu $
, then
-
(1)
$\#\mathcal {W}^{\lambda>0}(V)=\#\mathcal {W}^{\mu >0}(V)=\#\mathcal {W}^{\lambda <0}(V)=\#\mathcal {W}^{\mu <0}(V)$
, -
(2)
$\#\mathcal {W}^{\lambda>0}(\mathfrak {g})=\#\mathcal {W}^{\mu >0}(\mathfrak {g})=\#\mathcal {W}^{\lambda <0}(\mathfrak {g})=\#\mathcal {W}^{\mu <0}(\mathfrak {g})$
.
Proof. We prove the statement for V. The statement for
$\mathfrak {g}$
is obtained by specialising
$V=\mathfrak {g}$
, since
$\mathfrak {g}$
under the adjoint action of G is a symmetric representation of G (by the existence of invariant pairings, nondegenerate up to the center of
$\mathfrak {g}$
, Example 2.4).
Since V is weakly symmetric, we have
$\#\mathcal {W}^{\lambda>0}(V)=\#\mathcal {W}^{\lambda <0}(V)$
and moreover,
$\#\mathcal {W}(V)=\#\mathcal {W}^{\lambda>0}(V)+\#\mathcal {W}^{\lambda <0}(V)+\mathcal {W}(V^{\lambda })$
. Since we have the same equalities with
$\lambda $
replaced by
$\mu $
and
$\mathcal {W}^{\lambda =0}(V)=\mathcal {W}^{\mu =0}(V)$
, we obtain
$\#\mathcal {W}^{\lambda>0}(V)=\#\mathcal {W}^{\mu >0}(V)$
. The other equalities follow.
Lemma 2.15. If
$\lambda ,\mu \in X_*(T)$
are such that
$\lambda \sim \mu $
, then
$k_{\lambda }=\xi k_{\mu }$
for some
$\xi \in \mathbf {Q} ^*$
.
Proof. If
$\lambda \sim \mu $
, we have
$\mathcal {W}^{\lambda>0}(V)\sqcup \mathcal {W}^{\lambda <0}(V)=\mathcal {W}^{\mu>0}(V)\sqcup \mathcal {W}^{\mu <0}(V)$
and
$\mathcal {W}^{\lambda>0}(\mathfrak {g})\sqcup \mathcal {W}^{\lambda <0}(\mathfrak {g})=\mathcal {W}^{\mu>0}(\mathfrak {g})\sqcup \mathcal {W}^{\mu <0}(\mathfrak {g})$
and since V and
$\mathfrak {g}$
are weakly symmetric representations of G,
$\overline {\mathcal {W}^{\lambda>0}(V)}=-\overline {\mathcal {W}^{\lambda <0}(V)}$
,
$\overline {\mathcal {W}^{\mu>0}(V)}=-\overline {\mathcal {W}^{\mu <0}(V)}$
and similarly for
$\mathfrak {g}$
. Therefore, for any
$\alpha \in \mathcal {W}^{\lambda <0}(V)$
(resp.
$\mathcal {W}^{\lambda <0}(\mathfrak {g})$
), there exists
$\xi _{\alpha }\in \mathbf {Q}^*$
(resp.
$\zeta _{\alpha }\in \mathbf {Q}^*$
) such that
$$\begin{align*}\begin{matrix} \mathcal{W}^{\lambda<0}(V)&\rightarrow&\mathcal{W}^{\mu<0}(V)\\ \alpha&\mapsto &\xi_{\alpha}\alpha \end{matrix} \end{align*}$$
and
$$\begin{align*}\begin{matrix} \mathcal{W}^{\lambda<0}(\mathfrak{g})&\rightarrow&\mathcal{W}^{\mu<0}(\mathfrak{g})\\ \alpha&\mapsto &\zeta_{\alpha}\alpha \end{matrix} \end{align*}$$
are bijections. Then,
$$\begin{align*}\frac{\prod_{\alpha\in\mathcal{W}^{\lambda<0}(V)}\xi_{\alpha}}{\prod_{\alpha\in\mathcal{W}^{\lambda<0}(\mathfrak{g})}\zeta_{\alpha}} k_{\lambda}=k_{\mu}\,. \end{align*}$$
We may therefore set 
.
Lemma 2.16. We assume that V is a weakly symmetric representation of G. Then, for any
$\lambda , \mu \in X_*(T)$
such that
$\overline {\lambda }=\overline {\mu }$
, the induction maps
$\mathrm {Ind}_{\lambda }$
and
$\mathrm {Ind}_{\mu }$
differ by a non-zero scalar. In particular, the images of
$\mathrm {Ind}_{\lambda }$
and
$\mathrm {Ind}_{\mu }$
inside
$\mathrm {H}^{*+d}(V/G)$
coincide.
Proof. We use the explicit formula for the induction (Proposition 2.11). Let
$f\in \mathrm {H}^*(V^{\lambda }/G^{\lambda })=\mathrm {H}^*(V^{\mu }/G^{\mu })$
. We have
$$\begin{align*}\mathrm{Ind}_{\lambda}(f)=\sum_{w\in W/W^{\lambda}}w\cdot(fk_{\lambda}),\quad \mathrm{Ind}_{\mu}(f)=\sum_{w\in W/W^{\mu}}w\cdot(fk_{\mu})\,. \end{align*}$$
Moreover,
$G^{\lambda }=G^{\mu }$
,
$W^{\lambda }=W^{\mu }$
and by Lemma 2.15,
$k_{\lambda }=\xi k_{\mu }$
. This concludes the proof.
The W-action on
$X_*(T)$
induces a W-action on
$\mathscr {P}_V$
. For
$\lambda \in X_*(T)$
, we let
$\widetilde {\lambda }\in \mathscr {P}_V/W$
be its W-orbit.
Lemma 2.17. Let
$\lambda ,\mu \in X_*(T)$
be such that
$\widetilde {\lambda }=\widetilde {\mu }$
. Then, the images of
$\mathrm {Ind}_{\lambda }$
and
$\mathrm {Ind}_{\mu }$
coincide.
Proof. By Lemma 2.16, we may assume that for some
$w\in W$
,
$\mu =w\cdot \lambda $
. Then,
$\mathcal {W}^{\mu <0}(V)=w\cdot \mathcal {W}^{\lambda <0}(V)$
and
$\mathcal {W}^{\mu <0}(\mathfrak {g})=w\cdot \mathcal {W}^{\lambda <0}(\mathfrak {g})$
. Therefore,
$w(k_{\lambda })=k_{\mu }$
. Using the explicit formula for the induction product (Proposition 2.11), we obtain
$$\begin{align*}\mathrm{Ind}_{\mu}(f)=\mathrm{Ind}_{\lambda}(w^{-1}(f))\,, \end{align*}$$
and therefore, since
$w^{-1}$
induces an isomorphism
$\mathrm {H}^*(V^{\mu }/G^{\mu })=\mathrm {H}^*(V^{w\cdot \lambda }/G^{w\cdot \lambda })\cong \mathrm {H}^*(V^{\lambda }/G^{\lambda })$
, the images of
$\mathrm {Ind}_{\lambda }$
and
$\mathrm {Ind}_{\mu }$
coincide.
3 Action of tautological classes on the cohomology of the stack
Let G be a reductive group and V a representation of G. We let
$\pi \colon V/G\rightarrow V/\!\!/ G$
be the good moduli space map. We let
$\kappa \colon V/\!\!/ G\rightarrow \mathrm {pt}$
be the projection to the point.
Then, for any line bundle
$\mathcal {L}$
over
$V/G$
, we have an operation
$\pi _*\mathbf {Q}_{V/G}\rightarrow \pi _*\mathbf {Q}_{V/G}[-2]$
of multiplication by the first Chern class
$c_1(\mathcal {L})$
. More precisely, we have the following lemma.
Lemma 3.1. There is a morphism of graded rings
$$\begin{align*}\mathrm{H}^*(\mathrm{pt}/G)\xrightarrow{a} \mathrm{RHom}(\pi_*\mathbf{Q}_{V/G},\pi_*\mathbf{Q}_{V/G}) \end{align*}$$
such that the composition
$\kappa _*\circ a$
where
$\kappa _*\colon \mathrm {RHom}(\pi _*\mathbf {Q}_{V/G},\pi _*\mathbf {Q}_{V/G})\rightarrow \mathrm {RHom}(\kappa _*\pi _*\mathbf {Q}_{V/G}, \kappa _*\pi _*\mathbf {Q}_{V/G})\cong \mathrm {RHom}(\mathrm {H}^*(\mathrm {pt}/G),\mathrm {H}^*(\mathrm {pt}/G))$
, sends
$a\in \mathrm {H}^*(\mathrm {pt}/G)$
to the endomorphism of
$\mathrm {H}^*(\mathrm {pt}/G)$
given by the cup product with a.
Proof. We have a canonical adjunction isomorphism
$$\begin{align*}\mathrm{RHom}(\pi_*\mathbf{Q}_{V/G},\pi_*\mathbf{Q}_{V/G})\cong\mathrm{RHom}(\pi^*\pi_*\mathbf{Q}_{V/G},\mathbf{Q}_{V/G})\,. \end{align*}$$
By precomposing with the counit map
$\pi ^*\pi _*\mathbf {Q}_{V/G}\rightarrow \mathbf {Q}_{V/G}$
, we obtain the morphism
$$\begin{align*}\mathrm{RHom}(\mathbf{Q}_{V/G},\mathbf{Q}_{V/G})\cong\mathrm{H}^*(V/G)\cong\mathrm{H}^*(\mathrm{pt}/G)\rightarrow \mathrm{RHom}(\pi_*\mathbf{Q}_{V/G}\,,\pi_*\mathbf{Q}_{V/G}). \end{align*}$$
The last statement of the theorem follows from the functoriality of adjunctions.
We now state and prove an elementary lemma regarding Cartan subalgebras of reductive Lie algebras.
Lemma 3.2. Let
$\mathfrak {g}$
be a reductive Lie algebra,
$\mathfrak {h}\subseteq \mathfrak {g}$
a Lie ideal and
$\mathfrak {t}$
a Cartan subalgebra. Then,
$\mathfrak {h}\cap \mathfrak {t}$
is a Cartan subalgebra of
$\mathfrak {h}$
.
Proof. Let
$\pi \colon \mathfrak {g}\rightarrow \mathfrak {g}/\mathfrak {h}$
be the projection. Since
$\mathfrak {g}$
is reductive, we have a (noncanonical) decomposition
$\mathfrak {g}\cong \mathfrak {h}\oplus (\mathfrak {g}/\mathfrak {h})$
as Lie algebras. If
$\mathfrak {t}'$
is a Cartan subalgebra of
$\mathfrak {h}$
and
$\mathfrak {t}"$
is a Cartan subalgebra of
$\mathfrak {g}/\mathfrak {h}$
, then
$\mathfrak {t}'\oplus \mathfrak {t}"$
is a Cartan subalgebra of
$\mathfrak {g}$
. Moreover,
$\pi (\mathfrak {t})\cong \mathfrak {t}/\mathfrak {t}\cap \mathfrak {h}$
is an Abelian subalgebra of
$\mathfrak {g}/\mathfrak {h}$
and
$\mathfrak {t}\cap \mathfrak {h}$
is an Abelian subalgebra of
$\mathfrak {h}$
. By maximality of
$\dim \mathfrak {t}'$
, we have
$$\begin{align*}\dim \pi(\mathfrak{t})=\dim \mathfrak{t}-\dim(\mathfrak{t}\cap\mathfrak{h})\geqslant \dim \mathfrak{t}-\dim \mathfrak{t}'=\dim\mathfrak{t}"\,. \end{align*}$$
By maximality of
$\dim \mathfrak {t}"$
, we have equality in this chain of inequalities. This means that
$\dim (\mathfrak {t}\cap \mathfrak {h})=\dim \mathfrak {t}'$
and therefore,
$\mathfrak {t}\cap \mathfrak {h}$
is a Cartan subalgebra of
$\mathfrak {h}$
.
We have the group version of Lemma 3.2.
Lemma 3.3. Let G be a reductive group,
$H\subseteq G$
a normal subgroup and
$T\subseteq G$
a maximal torus. Then, the neutral component of
$T\cap H$
is a maximal torus of H.
Proof. This follows from Lemma 3.2 by taking the Lie algebras of all groups appearing.
Example 3.4. In the group situation, it is necessary to consider the neutral component of
$T\cap H$
as shown by the elementary case of
$H=\{\pm 1\}\subseteq \mathrm {SL}_2(\mathbf {C})$
, and where
$T\subseteq \mathrm {SL}_2(\mathbf {C})$
is the standard torus of diagonal matrices.
Proposition 3.5. Let G be a reductive group. For any normal subgroup
$H\subseteq G$
acting trivially on V, we have (non-canonical) actions of
$\mathrm {H}^*(\mathrm {pt}/H)$
on
$\mathrm {H}^*(\mathrm {pt}/G)\cong \mathrm {H}^*(V/G)$
and
$\pi _*\mathbf {Q}_{V/G}$
.
Proof. By choosing a decomposition
$\mathrm {Lie}(G)\cong \mathrm {Lie}(H)\oplus \mathrm {Lie}(G/H)$
, we obtain a decomposition
$\mathrm {H}^*(\mathrm {pt}/G)\cong \mathrm {H}^*(\mathrm {pt}/(G/H))\otimes \mathrm {H}^*(\mathrm {pt}/H)$
, giving an action of
$\mathrm {H}^*(\mathrm {pt}/H)$
on
$\mathrm {H}^*(\mathrm {pt}/G)$
by cup-product on the second summand. At the sheaf level, the action of the lemma is obtained by pre-composing the morphism a of Lemma 3.1 with the inclusion
$\mathrm {H}^*(\mathrm {pt}/H)\rightarrow \mathrm {H}^*(\mathrm {pt}/G)$
.
Lemma 3.6. Let G be an algebraic group and
$H\subseteq G$
a normal subgroup. Then, we have a natural inclusion
$\mathrm {H}^*(\mathrm {pt}/(G/H))\rightarrow \mathrm {H}^*(\mathrm {pt}/G)$
.
Proof. The map
$\mathrm {H}^*(\mathrm {pt}/(G/H))\rightarrow \mathrm {H}^*(\mathrm {pt}/G)$
is the pullback in cohomology for the morphism of stacks
$\mathrm {pt}/G\rightarrow \mathrm {pt}/(G/H)$
induced by the surjective morphism of groups
$G\rightarrow G/H$
.
Proposition 3.7. Let G be a reductive algebraic group, and V a finite-dimensional representation of G. Let H be a normal subgroup of G acting trivially on V. Then, the morphism
$$\begin{align*}\mathrm{H}^*(\mathrm{pt}/H)\otimes\mathrm{H}^*(V/(G/H))\rightarrow\mathrm{H}^*(V/G) \end{align*}$$
obtained by combining the natural pullback map
$\mathrm {H}^*(V/(G/H))\rightarrow \mathrm {H}^*(V/G)$
coming from the morphism of stacks
$V/G\rightarrow V/(G/H)$
(Lemma 3.6) with the
$\mathrm {H}^*(\mathrm {pt}/H)$
-action on
$\mathrm {H}^*(V/G)$
(Proposition 3.5) is an isomorphism.
Proof. We let
$\mathfrak {g}=\mathrm {Lie}(G)$
and
$\mathfrak {h}=\mathrm {Lie}(H)$
. We let
$\mathfrak {t}\subseteq \mathfrak {g}$
be a Cartan subalgebra. We have
$\mathfrak {g}\cong \mathfrak {h}\oplus (\mathfrak {g}/\mathfrak {h})$
and therefore a decomposition of the Weyl group
$W_{\mathfrak {g}}\cong W_{\mathfrak {h}}\times W_{\mathfrak {g}/\mathfrak {h}}$
. We let
$\pi \colon \mathfrak {g}\rightarrow \mathfrak {g}/\mathfrak {h}$
be the projection. The morphism of the proposition can then be identified with the isomorphism
$$\begin{align*}\mathrm{Sym}((\mathfrak{t}\cap\mathfrak{h})^*)^{W_{\mathfrak{h}}}\otimes\mathrm{Sym}(\pi(\mathfrak{t})^*)^{W_{\mathfrak{g}/\mathfrak{h}}}\rightarrow\mathrm{Sym}(\mathfrak{t}^*)^{W}\,, \end{align*}$$
see also Lemma 3.2.
4 Cohomological integrality for weakly symmetric representations of reductive groups
We fix a weakly symmetric representation V (Definition 2.3) of a connected reductive group G and a maximal torus
$T\subseteq G$
.
4.1 Weyl groups
Let
$\lambda \colon \mathbf {G}_{\mathrm {m}}\rightarrow T$
. We let
$G^{\lambda }\subseteq G$
be the corresponding Levi subgroup (§2.1) and 
the intersection of the kernel of the action of
$G^{\lambda }$
on
$V^{\lambda }$
and the center of
$G^{\lambda }$
. We let
$W^{\lambda }=N_{G^{\lambda }}(T)$
be the Weyl group of
$G^{\lambda }$
and 
. The group
$W_{\lambda }$
may be thought of as a kind of relative Weyl group.
Lemma 4.1. Let
$G_{\lambda }^{\circ }$
be the neutral component of
$G_{\lambda }$
. Let
$k=\dim G_{\lambda }$
. Then,
$G_{\lambda }^{\circ }\cong (\mathbf {C}^*)^k$
for some
$k\geqslant 0$
and the natural pullback map
$\mathrm {H}^*(\mathrm {pt}/G_{\lambda })\rightarrow \mathrm {H}^*(\mathrm {pt}/G_{\lambda }^{\circ })$
is an isomorphism.
Proof. The group
$G_{\lambda }$
is a subgroup of the center of
$G^{\lambda }$
, and so it is a diagonalisable algebraic group. Therefore, its neutral component is a torus. We have a splitting
$G\cong G_{\lambda }^{\circ }\times \pi _0(G_{\lambda })$
[Reference HumphreysHum12, Theorem p.104] and so, by the Künneth formula,
$$\begin{align*}\mathrm{H}^*(\mathrm{pt}/G_{\lambda})\cong\mathrm{H}^*(\mathrm{pt}/G_{\lambda}^{\circ})\otimes\mathrm{H}^*(\mathrm{pt}/\pi_0(G_{\lambda}))\,. \end{align*}$$
Since
$\pi _0(G_{\lambda })$
is a finite group,
$\mathrm {H}^*(\mathrm {pt}/\pi _0(G_{\lambda }))\cong \mathrm {H}^*(\mathrm {pt})^{\pi _0(G_{\lambda })}\cong \mathbf {Q}$
, which concludes.
Lemma 4.2. The group
$W_{\lambda }$
is a subgroup of W and it contains
$W^{\lambda }$
as a normal subgroup.
Proof. We let
$\dot {w}\in N_G(T)$
be a lift of
$w\in W$
. Then,
$V^{w\cdot \lambda }=\dot {w}V^{\lambda }$
. The first statement of the lemma follows then immediately from the fact that if
$w,w'\in W$
, and
$\dot {w}, \dot {w'}\in N_G(T)$
are respective lifts, then,
$\dot {w}\dot {w'}$
is a lift of
$ww'$
and
$\dot {w}^{-1}$
a lift of
$w^{-1}$
.
We prove that
$W^{\lambda }\subseteq W_{\lambda }$
. Let
$w\in W^{\lambda }$
and
$\dot {w}\in N_{G^{\lambda }}(T)$
be a lift of w. Then,
$\dot {w}G^{\lambda }\dot {w}^{-1}=G^{w\cdot \lambda }=G^{\lambda }$
and
$\dot {w}V^{\lambda }=V^{w\cdot \lambda }=V^{\lambda }$
and so
$w\in W_{\lambda }$
. This proves the inclusion.
We now show that
$W^{\lambda }$
is a normal subgroup of
$W_{\lambda }$
. It suffices to check that if
$w\in W_{\lambda }$
and
$w'\in W^{\lambda }$
, and
$\dot {w}\in N_G(T)$
,
$\dot {w'}\in N_{G^{\lambda }}(T)$
are any lifts, then
$\dot {w}\dot {w'}\dot {w}^{-1}\in G^{\lambda }$
. We have
$\dot {w}\dot {w'}\dot {w}^{-1}\in \dot {w}G^{\lambda }\dot {w}^{-1}=G^{w\cdot \lambda }=G^{\lambda }$
. This concludes the proof.
Corollary 4.3. The group
$W_{\lambda }$
acts on
$\mathrm {H}^*(\mathrm {pt}/T)^{W^{\lambda }}\cong \mathrm {H}^*(\mathrm {pt}/G^{\lambda })$
.
Proof. The Weyl group W of G acts on
$\mathrm {H}^*(\mathrm {pt}/T)$
. We just have to check that the induced action of the subgroup
$W_{\lambda }$
preserves
$W^{\lambda }$
-invariants. This is a consequence of Lemma 4.2.
Lemma 4.4. The group
$W_{\lambda }$
acts naturally on
$\mathrm {H}^*(\mathrm {pt}/G_{\lambda })$
. The subgroup
$W^{\lambda }\subseteq W_{\lambda }$
acts trivially on
$\mathrm {H}^*(\mathrm {pt}/G_{\lambda })$
.
Proof. By definition,
$G_{\lambda }$
is preserved by
$W_{\lambda }$
, as
$G_{\lambda }$
only depends on
$\overline {\lambda }\in \mathscr {P}_V$
and
$W_{\lambda }$
is the stabiliser of
$\overline {\lambda }$
. The second statement follows from the fact that
$G_{\lambda }$
is contained in the center of
$G^{\lambda }$
.
We let 
. Since
$G_{\lambda }$
is a normal subgroup of
$G^{\lambda }$
,
$\overline {G^{\lambda }}$
is a reductive group. Since
$G_{\lambda }$
is contained in the center of
$G^{\lambda }$
, the Weyl group of
$\overline {G^{\lambda }}$
is isomorphic to
$W^{\lambda }$
.
Lemma 4.5. The group
$W_{\lambda }$
acts naturally on
$\mathrm {H}^*(V^{\lambda }/\overline {G^{\lambda }})$
. Moreover, the
$W_{\lambda }$
-action on
$\mathrm {H}^*(\mathrm {pt}/G^{\lambda })\cong \mathrm {H}^*(V^{\lambda }/G^{\lambda })$
obtained in Corollary 4.3 and the
$W_{\lambda }$
-action on
$\mathrm {H}^*(V^{\lambda }/\overline {G^{\lambda }})\otimes \mathrm {H}^*(\mathrm {pt}/G_{\lambda })$
obtained by the tensor product of the actions given in Lemmas 4.4 and 4.5 coincide via the isomorphism given in Proposition 3.7.
Proof. We have
$\mathrm {H}^*(V^{\lambda }/\overline {G^{\lambda }})\cong \mathrm {H}^*(\mathrm {pt}/(T/G_{\lambda }^{\circ }))^{W^{\lambda }}$
. The restriction to
$W_{\lambda }$
of the W action on T preserves
$G_{\lambda }$
since
$G_{\lambda }$
is in the center of
$G^{\lambda }$
. Therefore, we obtain a
$W_{\lambda }$
-action on
$T/G_{\lambda }^{\circ }$
. It induces a
$W_{\lambda }$
-action on
$\mathrm {H}^*(\mathrm {pt}/(T/G_{\lambda }^{\circ }))$
. The fact that
$W_{\lambda }$
normalises
$W^{\lambda }$
(Lemma 4.2) implies that
$\mathrm {H}^*(\mathrm {pt}/(T/G_{\lambda }^{\circ }))^{W^{\lambda }}$
is preserved by the
$W_{\lambda }$
-action.
The tensor product decomposition of the action follows immediately from this and Lemma 4.4.
Proposition 4.6. There exists a character
$\varepsilon _{V,\lambda }\colon W_{\lambda }\rightarrow \mathbf {Q}^*$
such that for any
$w\in W_{\lambda }$
,
$w(k_{\lambda })=\varepsilon _{V,\lambda }(w)^{-1}k_{\lambda }\in \mathrm {H}^*_T(\mathrm {pt})$
.
Proof. It suffices to prove the existence and unicity of the rational number
$\varepsilon _{V,\lambda }(w)\in \mathbf {Q}^*$
satisfying the equality of the proposition, for any
$w\in W_{\lambda }$
. The multiplicativity then follows from the unicity of this sign.
For
$w\in W_{\lambda }$
, the cocharacters
$\lambda ,w^{-1}\cdot \lambda $
of T are such that
$\overline {\lambda }=\overline {w^{-1}\cdot \lambda }$
. Therefore, by Lemma 2.15, there exists
$\xi \in \mathbf {Q}^*$
such that
$w(k_{\lambda })=k_{w^{-1}\cdot \lambda }=\xi k_{\lambda }$
. We set 
. This proves the existence. The unicity of
$\xi $
is straightforward.
Proposition 4.7. Let
$\lambda \in X_*(T)$
,
$w\in W_{\lambda }$
, and
$f\in \mathrm {H}^*(V^{\lambda }/G^{\lambda })$
. Then,
$\mathrm {Ind}_{\lambda }(w\cdot f)=\varepsilon _{V,\lambda }(w)\mathrm {Ind}_{\lambda }(f)$
.
Proof. By Proposition 2.11, we have
$$\begin{align*}\begin{aligned} \mathrm{Ind}_{\lambda}(w\cdot f)&=\frac{1}{\# W^{\lambda}}\sum_{w'\in W}w'(wf)w'(k_{\lambda})\\ &=\frac{1}{\# W^{\lambda}}\sum_{w"\in W}(w"f)w"(w^{-1}k_{\lambda})\\ &=\frac{1}{\# W^{\lambda}}\sum_{w"\in W}(w"f)w"(\varepsilon_{V,\lambda}(w)k_{\lambda}) \quad\text{ by Proposition}~4.6 \end{aligned} \end{align*}$$
which concludes.
We now state an elementary result from the representation theory of finite groups for later use.
Proposition 4.8. Let G be a finite group
$\chi \colon G\rightarrow \mathbf {C}^*$
a character of G and V a finite-dimensional representation of G. Then, the formula
$$\begin{align*}p_{\chi}(v)=\frac{1}{\# G}\sum_{g\in G}\chi(g)^{-1}g\cdot v \end{align*}$$
is the projector onto the
$\chi $
-isotypic component
$V^{\chi }$
of V.
Proof. This is [Reference SerreSer71, §2.6, Théorème 8].
4.2 Proof of the cohomological integrality theorem
We let V be a weakly symmetric representation of a reductive group G. In the rest of this section, we prove Theorem 1.1.
For
$\lambda \in X_*(T)$
, we define the
$\mathbf {Z}$
-graded vector space
When
$\lambda =0$
is the trivial cocharacter, we write 
. More generally, for any pair
$\lambda ,\mu $
, we define 
. By §2.3, we have induction maps
$\mathrm {Ind}_{\lambda }\colon \mathcal {H}_{\lambda }\rightarrow \mathcal {H}$
for
$\lambda \in X_*(T)$
and more generally,
$\mathrm {Ind}_{\mu ,\lambda }\colon \mathcal {H}_{\lambda ,\mu }\rightarrow \mathcal {H}_{\lambda }$
for
$\lambda ,\mu \in X_*(T)$
.
For
$\lambda \in X_*(T)$
, we choose a
$W_{\lambda }$
-invariant splitting
$\mathfrak {g}^{\lambda }=\mathfrak {g}_{\lambda }\oplus (\mathfrak {g}^{\lambda }/\mathfrak {g}_{\lambda })$
, where 
is the Lie algebra of
$G_{\lambda }$
(§4.1). It induces a splitting of the maximal torus
$\mathfrak {t}=\mathfrak {g}_{\lambda }\oplus (\mathfrak {t}/\mathfrak {g}_{\lambda })$
(Lemma 3.2). We let
so that
$\mathcal {H}_{\lambda }\cong \mathcal {H}_{\lambda }^{\mathrm {prim}}\otimes \mathrm {H}^*(\mathrm {pt}/G_{\lambda })$
(Proposition 3.7). We let
so that
$\mathcal {A}_{\lambda }\cong \mathcal {A}_{\lambda }^{\mathrm {prim}}\otimes \mathrm {H}^*(\mathrm {pt}/G_{\lambda })$
,
$\mathcal {H}_{\lambda }=\mathcal {A}_{\lambda }^{W^{\lambda }}$
and
$\mathcal {H}_{\lambda }^{\mathrm {prim}}\cong (\mathcal {A}_{\lambda }^{\mathrm {prim}})^{W^{\lambda }}$
.
For each
$\lambda \in X_*(T)$
, we let
$J_{\lambda }$
be the smallest
$W^{\lambda }$
-stable
$\mathcal {A}_{\lambda }^{\mathrm {prim}}$
-submodule of the localisation
containing
$$\begin{align*}k_{\mu,\lambda}=\frac{\prod_{\alpha\in\mathcal{W}^{\mu<0}(V^{\lambda})}\alpha}{\prod_{\alpha\in\mathcal{W}^{\mu<0}(\mathfrak{g}^{\lambda})}\alpha} \end{align*}$$
for all
$\mu \in X_*(T)$
such that
$\lambda \not \preceq \mu $
for the order relation
$\preceq $
defined in §2.6.
We let
and
(for this last isomorphism, we use the fact that
$W^{\lambda }$
acts trivially on
$\mathrm {H}^*(\mathrm {pt}/G_{\lambda })$
as
$G_{\lambda }$
is in the center of
$G^{\lambda }$
, Lemma 4.4).
Lemma 4.9. We can characterise
$J_{\lambda }$
as the smallest
$W^{\lambda }$
-stable
$\mathcal {A}_{\lambda }^{\mathrm {prim}}$
-submodule of
$\mathcal {A}^{\mathrm {prim},\mathrm {loc}}$
containing
$$\begin{align*}k_{\mu,\lambda}=\frac{\prod_{\alpha\in\mathcal{W}^{\mu<0}(V^{\lambda})}\alpha}{\prod_{\alpha\in\mathcal{W}^{\mu<0}(\mathfrak{g}^{\lambda})}\alpha} \end{align*}$$
for all
$\mu \in X_*(T)$
such that
$\mu \prec \lambda $
.
Proof. If
$\mu \prec \lambda $
, then we have
$\dim V^{\mu }\leqslant \dim V^{\lambda }$
and
$\dim \mathfrak {g}^{\mu }\leqslant \dim \mathfrak {g}^{\lambda }$
with at least one strict inequality (where the order relation
$\preceq $
is defined in §2.6). If by contradiction we have
$\lambda \preceq \mu $
, then we have
$$\begin{align*}\dim V^{\lambda}\leqslant \dim V^{\mu}\,, \quad \dim\mathfrak{g}^{\lambda}\leqslant\dim \mathfrak{g}^{\mu}\,. \end{align*}$$
Therefore, if
$\mu \prec \lambda $
, then
$\lambda \not \preceq \mu $
and by definition,
$k_{\mu ,\lambda }\in J_{\lambda }$
.
Conversely, if
$\lambda \not \preceq \mu $
, we let
$\nu \in X_*(T)$
be such that
$V^{\nu }=V^{\lambda }\cap V^{\mu }$
,
$\mathfrak {g}^{\nu }=\mathfrak {g}^{\lambda }\cap \mathfrak {g}^{\mu }$
,
$(V^{\lambda })^{\nu \geqslant 0}=(V^{\lambda })^{\mu \geqslant 0}$
and
$(\mathfrak {g}^{\lambda })^{\nu \geqslant 0}=(\mathfrak {g}^{\lambda })^{\mu \geqslant 0}$
(Lemma 2.8). We have
$V^{\nu }\subseteq V^{\lambda }$
and
$\mathfrak {g}^{\nu }\subseteq \mathfrak {g}^{\lambda }$
with at least one strict inclusion. Therefore,
$\nu \prec \lambda $
and moreover,
$k_{\nu ,\lambda }=k_{\mu ,\lambda }$
. This proves the lemma.
Lemma 4.10. We have
$J_{\lambda }^{W^{\lambda }}\subseteq \mathcal {H}_{\lambda }^{\mathrm {prim}}$
. Moreover,
$J_{\lambda }\cap \mathcal {H}_{\lambda }^{\mathrm {prim}}=J_{\lambda }^{W^{\lambda }}$
.
Proof. This comes from the fact that the induction product §2.2 is well-defined on the cohomology, without requiring localisation, even though the explicit formula (Proposition 2.11) involves rational fractions. More precisely,
$J_{\lambda }^{W^{\lambda }}$
is linearly generated by the elements
$$\begin{align*}\sum_{w\in W^{\lambda}}w\cdot(fk_{\mu,\lambda}) \end{align*}$$
for
$f\in \mathcal {A}_{\lambda }^{\mathrm {prim}}$
and
$\mu \prec \lambda $
, by Lemma 4.9. These are averages over
$W^{\lambda }$
of elements of
$J_{\lambda }$
. We can rewrite this sum as
$$\begin{align*}\sum_{\overline{w'}\in W^{\lambda}/W^{\mu}}w'\cdot\left[\left(\sum_{w\in W^{\mu}}(w\cdot f)\right)k_{\mu,\lambda}\right] \end{align*}$$
since
$k_{\mu ,\lambda }$
is
$W^{\mu }$
-invariant (Lemma 2.10). This is (Proposition 2.11) the formula for
Since
$\sum _{w\in W^{\mu }}(w\cdot f)$
is polynomial, its induction I is also polynomial. Therefore, it belongs to
$\mathcal {H}_{\lambda }$
. Moreover,
$I\in J_{\lambda }\subseteq \mathcal {A}_{\lambda }^{\mathrm {prim}}[\prod _{\alpha \in \mathcal {W}(\mathfrak {g}^{\lambda })\smallsetminus \{0\}}\alpha ^{-1}]$
by
$W^{\lambda }$
-invariance of
$J_{\lambda }$
. Therefore,
$I\in \mathcal {H}_{\lambda }\cap \mathcal {A}_{\lambda }^{\mathrm {prim}}[\prod _{\alpha \in \mathcal {W}(\mathfrak {g}^{\lambda })\smallsetminus \{0\}}\alpha ^{-1}]=\mathcal {H}_{\lambda }^{\mathrm {prim}}$
.
For the second claim of the lemma, we have obviously
$J_{\lambda }^{W^{\lambda }}\subseteq J_{\lambda }$
. Combined with the first part of the lemma, this gives the reverse inclusion
$J_{\lambda }^{W^{\lambda }}\subseteq J_{\lambda }\cap \mathcal {H}_{\lambda }^{\mathrm {prim}}$
. For the direct inclusion, if
$f\in J_{\lambda }$
is also in
$\mathcal {H}_{\lambda }^{\mathrm {prim}}=(\mathcal {A}_{\lambda }^{\mathrm {prim}})^{W^{\lambda }}$
, it clearly is
$W^{\lambda }$
-invariant:
$f\in J_{\lambda }^{W^{\lambda }}$
.
Lemma 4.11. The submodule
$J_{\lambda }$
is
$W_{\lambda }$
-invariant.
Proof. Let
$w\in W_{\lambda }$
. We let 
. Then,
$J_{\lambda }^w$
is
$W^{\lambda }$
-stable by Lemma 4.2. It is an
$\mathcal {A}_{\lambda }^{\mathrm {prim}}$
-submodule of
$\mathcal {A}_{\lambda }^{\mathrm {prim}}[\prod _{\alpha \in \mathcal {W}(\mathfrak {g}^{\lambda })\smallsetminus \{0\}}(w\cdot \alpha )^{-1}]=\mathcal {A}_{\lambda }^{\mathrm {prim}}[\prod _{\alpha \in \mathcal {W}(\mathfrak {g}^{\lambda })\smallsetminus \{0\}}\alpha ^{-1}]$
. Moreover, it contains
$$\begin{align*}k_{w\cdot\mu,w\cdot\lambda} \end{align*}$$
for
$\lambda \not \preceq \mu $
. Now, we use that by Lemma 2.15, for
$w\in W_{\lambda }$
,
$\{\pm k_{\mu ,\lambda }\colon \lambda \not \preceq \mu \}=\{\pm k_{w\cdot \mu ,w\cdot \lambda }\mid \lambda \not \preceq \mu \}$
since for
$w\in W_{\lambda }$
,
$\overline {w\cdot \lambda }=\overline {\lambda }$
. By minimality of
$J_{\lambda }$
, we deduce that
$J_{\lambda }\subseteq J_{\lambda }^w$
. By symmetry of the argument, we conclude
$J_{\lambda }^w=J_{\lambda }$
.
Corollary 4.12.
$J_{\lambda }^{W^{\lambda }}$
is
$W_{\lambda }$
-invariant.
We let
$\mathcal {P}_{\lambda }\subseteq \mathcal {H}_{\lambda }^{\mathrm {prim}}$
be a
$W_{\lambda }$
-stable direct sum complement of
$J_{\lambda }^{W^{\lambda }}$
(using Corollary 4.12):
$$\begin{align*}\mathcal{H}_{\lambda}^{\mathrm{prim}}=J_{\lambda}^{W^{\lambda}}\oplus \mathcal{P}_{\lambda}\,. \end{align*}$$
Lemma 4.13. For any
$\lambda \in X_*(T)$
, the image of
$$\begin{align*}\bigoplus_{\lambda\not\preceq\mu}\mathcal{H}_{\mu,\lambda}\xrightarrow{\bigoplus\mathrm{Ind}_{\mu,\lambda}}\mathcal{H}_{\lambda} \end{align*}$$
is
$J_{\lambda }^{W^{\lambda }}\otimes \mathrm {H}^*(\mathrm {pt}/G_{\lambda })$
.
Similarly, the image of
$$\begin{align*}\bigoplus_{\mu\prec\lambda}\mathcal{H}_{\mu,\lambda}\xrightarrow{\bigoplus\mathrm{Ind}_{\mu,\lambda}}\mathcal{H}_{\lambda} \end{align*}$$
is
$J_{\lambda }^{W^{\lambda }}\otimes \mathrm {H}^*(\mathrm {pt}/G_{\lambda })$
.
Proof. We only prove the second part of the lemma, as the second part can be proved in the exact same way by using Lemma 4.9.
This is a calculation very similar to that in the proof of Lemma 4.11. Namely, if
$f\in \mathcal {H}_{\mu ,\lambda }$
and
$g\in \mathrm {H}^*(\mathrm {pt}/G_{\lambda })$
,
$\mathrm {Ind}_{\mu ,\lambda }(fg)=\mathrm {Ind}_{\mu ,\lambda }(f)g$
by
$W^{\lambda }$
-invariance of g (Lemma 4.4). Moreover,
$fg\in \mathcal {H}_{\mu ,\lambda }$
. Therefore, a spanning subset of
$J_{\lambda }^{W^{\lambda }}\otimes \mathrm {H}^*(\mathrm {pt}/G_{\lambda })$
can be obtained in this way by considering all
$\mu \prec \lambda $
. Conversely, if
$f\in \mathcal {H}_{\mu ,\lambda }\subseteq \mathcal {A}_{\mu ,\lambda }=\mathcal {A}_{\lambda }=\mathrm {H}^*(\mathrm {pt}/T)$
for
$\mu \prec \lambda $
, we write
$f=\sum _{i=1}^Nf_i\otimes g_i$
with
$f_i\in \mathcal {A}_{\lambda }^{\mathrm {prim}}$
and
$g_i\in \mathrm {H}^*(\mathrm {pt}/G_{\lambda })$
so that
$$\begin{align*}\mathrm{Ind}_{\mu,\lambda}(f)=\sum_{i=1}^N\mathrm{Ind}_{\mu,\lambda}(f_i)\otimes g_i \end{align*}$$
indeed belongs to
$J_{\lambda }^{W^{\lambda }}\otimes \mathrm {H}^*(\mathrm {pt}/G_{\lambda })$
.
Lemma 4.14. For any
$\lambda \in X_*(T)$
, the map
$$\begin{align*}\bigoplus_{\substack{\mu\in X_*(T)\\\mu\preceq\lambda}}\mathcal{P}_{\mu}\otimes\mathrm{H}^*(\mathrm{pt}/G_{\mu})\xrightarrow{\bigoplus_{\mu\preceq\lambda}\mathrm{Ind}_{\mu,\lambda}} \mathcal{H}_{\lambda} \end{align*}$$
is surjective.
Proof. We proceed by induction on
$\lambda \in X_*(T)$
. If
$\lambda $
is minimal, then
$J_{\lambda }=\{0\}$
(by Lemma 4.9) and so
$\mathcal {P}_{\lambda }=\mathrm {H}^*(V^{\lambda }/(G^{\lambda }/G_{\lambda }))$
. By Proposition 3.7, the map
$$\begin{align*}\mathcal{P}_{\lambda}\otimes\mathrm{H}^*(\mathrm{pt}/G_{\lambda})\rightarrow\mathrm{H}^*(V^{\lambda}/G^{\lambda}) \end{align*}$$
of the lemma is indeed surjective (as it even is an isomorphism).
Now, let
$\lambda \in X_*(T)$
. We assume that the map of the lemma is surjective for any
$\lambda '\prec \lambda $
. By associativity of the induction (Proposition 2.9), the image of the map of the lemma coincides with the image of the composition
$$\begin{align*}\left(\bigoplus_{\mu\prec\lambda}\bigoplus_{\mu'\preceq\mu}\mathcal{P}_{\mu'}\otimes\mathrm{H}^*(\mathrm{pt}/G_{\mu'})\right)\oplus \left(\mathcal{P}_{\lambda}\otimes\mathrm{H}^*(\mathrm{pt}/G_{\lambda})\right)\xrightarrow{I\oplus\mathrm{id}} \left(\bigoplus_{\mu\prec\lambda}\mathcal{H}_{\mu}\right)\oplus\left(\mathcal{P}_{\lambda}\otimes\mathrm{H}^*(\mathrm{pt}/G_{\lambda})\right)\rightarrow\mathcal{H}_{\lambda} \end{align*}$$
where the morphism I is
$\bigoplus _{\mu \prec \lambda }\bigoplus _{\mu '\preceq \mu }\mathrm {Ind}_{\mu ',\mu }$
. By induction hypothesis,
$$\begin{align*}\left(\bigoplus_{\mu\prec\lambda}\bigoplus_{\mu'\preceq\mu}\mathcal{P}_{\mu'}\otimes\mathrm{H}^*(\mathrm{pt}/G_{\mu'})\right)\rightarrow\left(\bigoplus_{\mu\prec\lambda}\mathcal{H}_{\mu}\right) \end{align*}$$
is surjective and by Lemma 4.13, the image of
$$\begin{align*}\left(\bigoplus_{\mu\prec\lambda}\mathcal{H}_{\mu}\right)\rightarrow\mathcal{H}_{\lambda} \end{align*}$$
is
$J_{\lambda }^{W^{\lambda }}\otimes \mathrm {H}^*(\mathrm {pt}/G_{\lambda })$
, of which
$\mathcal {P}_{\lambda }\otimes \mathrm {H}^*(\mathrm {pt}/G_{\lambda })$
is a direct sum complement by definition of
$\mathcal {P}_{\lambda }$
. This proves the surjectivity.
For
$y\in \mathfrak {t}$
, we let
$\mathfrak {g}^{y}=\{\xi \in \mathfrak {g}\mid [y,\xi ]=0\}$
be the centraliser of y in
$\mathfrak {g}$
and
$V^{y}=\{v\in V\mid y\cdot v=0\}$
where y acts on V via the infinitesimal action.
Lemma 4.15. For any
$y\in \mathfrak {t}$
, there exists
$\lambda \in X_*(T)$
such that
$\mathfrak {g}^{\lambda }=\mathfrak {g}^{y}$
and
$V^{\lambda }=V^{y}$
.
Proof. We may assume that
$y\neq 0$
. Via the natural embeddings
$X_*(T)\subseteq \mathfrak {t}_{\mathbf {Q}}$
and
$X^*(T)\subseteq \mathfrak {t}_{\mathbf {Q}}^*$
, the natural pairing
$X_*(T)\times X^*(T)\rightarrow \mathbf {Z}$
corresponds to the natural pairing
$\langle -,-\rangle \colon \mathfrak {t}\times \mathfrak {t}^*\rightarrow \mathbf {C}$
(see also §1.4). We let
and
The rational vector space
$\cap _{\alpha \in \mathcal {W}_1}\alpha ^{\perp }\subseteq \mathfrak {t}_{\mathbf {Q}}$
is non-zero, as it is defined by the rational equations
$\langle -,\alpha \rangle =0$
and its complexification contains y. The complement of the finite union of rational hyperplanes 
is therefore also non-empty. We let
$y'\in U_{y}$
be a rational point (
$y\in \mathfrak {t}_{\mathbf {Q}}$
). For some
$N\geqslant 1$
,
$Ny'\in X_*(T)\subseteq \mathfrak {t}$
. We let 
. It satisfies the property of the lemma.
Proposition 4.16. For any
$\lambda \in X_*(T)$
,
$\mathcal {P}_{\lambda }$
is a finite-dimensional vector space.
Proof. By possibly replacing V by
$V^{\lambda }$
and G by
$G^{\lambda }$
, we may assume that
$\lambda =0$
. It suffices to prove that
$J_{\lambda }^{W^{\lambda }}\subseteq \mathcal {H}_{\lambda }^{\mathrm {prim}}$
has finite codimension. If we replace V by
$V\times \mathfrak {g}$
, then the corresponding module
$J_{\lambda }\subseteq \mathcal {A}^{\mathrm {prim}}_{\lambda }[\prod _{\alpha \in \mathcal {W}(\mathfrak {g})\smallsetminus \{0\}}\alpha ^{-1}]$
is smaller than the one for V (by the very definition of
$J_{\lambda }$
at the beginning of §4.2, since when replacing V by
$V\times \mathfrak {g}$
, the new
$k_{\lambda }$
is equal to the numerator of the old
$k_{\lambda }$
), and therefore so is the corresponding module of invariants
$J_{\lambda }^{W^{\lambda }}$
, and therefore its codimension is bigger. Moreover, in this case,
$J_{\lambda }\subseteq \mathcal {A}_{\lambda }^{\mathrm {prim}}$
, i.e. there is no need to localise.
We have an inclusion
$$\begin{align*}\mathcal{H}_{\lambda}^{\mathrm{prim}}/J_{\lambda}^{W^{\lambda}}\rightarrow\mathcal{A}_{\lambda}^{\mathrm{prim}}/J_{\lambda}\,, \end{align*}$$
because
$\mathcal {H}_{\lambda }^{\mathrm {prim}}\cap J_{\lambda }=J_{\lambda }^{W^{\lambda }}$
(second part of Lemma 4.10). Therefore, it suffices to prove that the ideal
$J_{\lambda }\subseteq \mathcal {A}_{\lambda }^{\mathrm {prim}}$
has finite codimension. We see
$\mathcal {A}_{\lambda }^{\mathrm {prim}}$
as the ring of regular functions on the affine algebraic variety
$\mathfrak {t}/\mathfrak {g}_{\lambda }$
. We prove that the closed subscheme defined by
$J_{\lambda }$
is supported on
$\{0\}$
. If the ideal
$J_{\lambda }$
vanishes on
$y\in (\mathfrak {t}/\mathfrak {g}_{\lambda })_{\overline {\mathbf {Q}}}\smallsetminus \{0\}$
(where
$\overline {\mathbf {Q}}$
is the field of algebraic numbers) we let
$y'\in \mathfrak {t}$
be a lift of y and
$\nu $
be given by Lemma 4.15. It satisfies
$\mathcal {W}^{\nu =0}(V)=\{\alpha \in \mathcal {W}(V)\mid \langle y,\alpha \rangle =0\}\subseteq \mathcal {W}(V)$
and
$\mathcal {W}^{\nu =0}(\mathfrak {g})=\{\alpha \in \mathcal {W}(\mathfrak {g})\mid \langle y,\alpha \rangle =0\} \subseteq \mathcal {W}(\mathfrak {g})$
. Moreover, at least one of these inequalities is strict. Indeed, we have
$y'\not \in \mathfrak {g}_{\lambda }$
, which means that either
$y'$
is not in the center of
$\mathrm {Lie}(G^{\lambda })$
and so
$\nu $
defines a strict Levi
$G^{\nu }=G^y$
of
$G^{\lambda }$
or
$y'$
is not in the Lie algebra of the kernel of the action
$G^{\lambda }\rightarrow \mathrm {GL}(V^{\lambda })$
in which case
$V^{\nu }=V^{y}\subsetneq V^{\lambda }$
. Therefore,
$\nu \prec \lambda =0$
, i.e
$\overline {\nu }$
is nontrivial. By definition, we have
$\prod _{\alpha \in \mathcal {W}^{\nu <0}(V^{\lambda })}\alpha \in J_{\lambda }$
.
We also have
$\prod _{\alpha \in \mathcal {W}^{\nu <0}(V^{\lambda })}\langle y,\alpha \rangle \neq 0$
: indeed, by choice of
$\nu $
, for any
$\alpha \in \mathcal {W}^{\nu <0}(V^{\lambda })$
,
$\langle \nu ,\alpha \rangle =0$
if and only if
$\langle y,\alpha \rangle =0$
. This is a contradiction since we found an element
$\prod _{\alpha \in \mathcal {W}^{\nu <0}(V^{\lambda })}\alpha \neq 0$
of
$J_{\lambda }$
not vanishing on y. Therefore,
$J_{\lambda }$
defines a subscheme of
$\mathfrak {t}/\mathfrak {g}_{\lambda }$
supported on
$0$
.
The rest of this section is devoted to the proof of the injectivity of the cohomological integrality map in Theorem 1.1.
For
$\lambda \in X_*(T)$
, we let 
. This is a
$\mathcal {A}_{\lambda }$
-submodule of
$\mathcal {A}_{\lambda }^{\mathrm {loc}}$
.
Recall the inclusion
$\mathcal {H}_{\lambda }=\mathcal {A}_{\lambda }^{W^{\lambda }}\rightarrow \mathcal {A}_{\lambda }$
.
Lemma 4.17. The elements
$\alpha \in \mathcal {W}^{\lambda <0}(V)\cup \mathcal {W}^{\lambda <0}(\mathfrak {g})$
(or more generally,
$\alpha \in X^*(T)$
such that
$\langle \lambda ,\alpha \rangle \neq 0$
), seen as elements of
$\mathcal {A}_{\lambda }$
, are not zero divisors in the quotient
$\mathcal {A}_{\lambda }$
-module
$\mathcal {A}_{\lambda }^{\mathrm {loc}}/\widetilde {J}_{\lambda }$
.
Proof. Let
$\alpha \in \mathcal {W}^{\lambda <0}(V)\cup \mathcal {W}^{\lambda <0}(\mathfrak {g})$
. We see
$\alpha \in \mathfrak {t}^*$
via the natural inclusion
$X^*(T)\subseteq \mathfrak {t}^*$
and
$\langle \lambda ,\alpha \rangle \neq 0$
implies that
$\alpha $
does not vanish on
$\mathfrak {g}_{\lambda }$
since it takes a nonzero value on
$\left (\frac {\mathrm {d}}{\mathrm {d}t}\lambda (t)\right )_{t=1}\in \mathfrak {g}_{\lambda }\subseteq \mathfrak {t}$
. Therefore, the projection
$\alpha '\in \mathfrak {g}_{\lambda }^*$
of
$\alpha $
with respect to the chosen (or any) decomposition
$\mathfrak {t}^*\cong (\mathfrak {t}/\mathfrak {g}_{\lambda })^*\oplus \mathfrak {g}_{\lambda }^*$
is nonzero.
We choose a basis 
of
$\mathfrak {g}_{\lambda }^*$
such that
$x_1=\alpha '$
and we order the monomials by the lexicographic order so that 
. Let
$g\in \mathcal {A}_{\lambda }^{\mathrm {loc}}\cong \mathcal {A}_{\lambda }^{\mathrm {prim},\mathrm {loc}}\otimes \mathrm {H}^*(\mathrm {pt}/G_{\lambda })$
. It may be written in a unique way (see Lemma 4.1)
with
$g_{\nu }\in \mathcal {A}_{\lambda }^{\mathrm {prim},\mathrm {loc}}$
. We have an equivalence between the following two statements:
-
(1)
$g\not \in \widetilde {J}_{\lambda }$
, -
(2)
$g_{\nu }\not \in J_{\lambda }$
for some
$\nu \in \mathbf {N}^{\dim \mathfrak {g}_{\lambda }}$
.
The implication
$(1)\Rightarrow (2)$
is immediate as if all
$g_{\nu }$
belong to
$J_{\lambda }$
, then g would be in
$\widetilde {J}_{\lambda }$
. The reverse implication
$(2)\Rightarrow (1)$
is also true by the unicity of the decomposition (4.1) of g.
Assume that
$g\not \in \widetilde {J}_{\lambda }$
. We need to show that for
$\alpha \in \mathcal {W}^{\lambda <0}(V)\cup \mathcal {W}^{\lambda <0}(\mathfrak {g})$
,
$\alpha g\not \in \widetilde {J}_{\lambda }$
. We let
$\nu =\max \{\nu '\in \mathbf {N}^{\dim \mathfrak {g}_{\lambda }}\colon g_{\nu '}\neq 0\}$
(the maximum is taken with respect to the lexicographic order). We may assume that
$g_{\nu }\not \in J_{\lambda }$
. We have
$$\begin{align*}\begin{aligned} \alpha g&=(\alpha-\alpha')g+\alpha'g\\ &=(\alpha-\alpha')g+x_1g \end{aligned} \end{align*}$$
and
$\alpha -\alpha '\in (\mathfrak {t}/\mathfrak {g}_{\lambda })^*\subseteq \mathrm {Sym}((\mathfrak {t}/\mathfrak {g}_{\lambda })^*)$
. Therefore, the term of highest degree in the variables 
in
$\alpha g$
is
$g_{\nu }x_1x^{\nu }$
and
$g_{\nu }\not \in J_{\lambda }$
. By the equivalence above,
$\alpha g\not \in \widetilde {J}_{\lambda }$
. This concludes the proof.
We let 
, 
and 
. We have the localisation maps
$$\begin{align*}L\colon \mathcal{A}_{\lambda}\rightarrow\mathcal{A}_{\lambda}',\quad L\colon\mathcal{A}_{\lambda}^{\mathrm{loc}}\rightarrow\mathcal{A}_{\lambda}^{\prime\mathrm{loc}}, \quad L\colon \mathcal{H}_{\lambda}\rightarrow\mathcal{H}_{\lambda}'\,. \end{align*}$$
We let 
be the localised
$\mathcal {A}_{\lambda }'$
-module inside
$\mathcal {A}_{\lambda }^{\prime \mathrm {loc}}$
(i.e. the
$\mathcal {A}_{\lambda }'$
-submodule of
$\mathcal {A}_{\lambda }^{\prime \mathrm {loc}}$
generated by
$J_{\lambda }$
). We emphasised here the fact that
$\widetilde {J}^{\prime }_{\lambda }$
is not a
$\mathcal {A}^{\prime \mathrm {loc}}_{\lambda }$
-submodule.
Lemma 4.18. The localisation maps
$$\begin{align*}L\colon \mathcal{A}_{\lambda}^{\mathrm{loc}}/\widetilde{J}_{\lambda}\rightarrow \mathcal{A}_{\lambda}^{\prime\mathrm{loc}}/\widetilde{J}_{\lambda}', \quad L\colon \mathcal{H}_{\lambda}/(\widetilde{J}_{\lambda})^{W^{\lambda}}\rightarrow \mathcal{H}_{\lambda}'/(\widetilde{J}_{\lambda}')^{W^{\lambda}} \end{align*}$$
are injective.
Proof. This follows from Lemma 4.17 as we localise by a set of nonzero divisors of the quotient
$\mathcal {A}_{\lambda }$
-module
$\mathcal {A}_{\lambda }^{\mathrm {loc}}/\widetilde {J}_{\lambda }$
in
$\mathcal {A}_{\lambda }$
. More precisely, the first localisation morphism
$L\colon \mathcal {A}_{\lambda }^{\mathrm {loc}}/\widetilde {J}_{\lambda }\rightarrow \mathcal {A}_{\lambda }^{\prime \mathrm {loc}}/\widetilde {J}_{\lambda }'$
is the localisation with respect to
$\prod _{\alpha \in \mathcal {W}^{\lambda <0}(V)\cup \mathcal {W}^{\lambda <0}(\mathfrak {g})}\alpha $
for the
$\mathcal {A}_{\lambda }$
-module
$\mathcal {A}_{\lambda }^{\mathrm {loc}}/\widetilde {J}_{\lambda }$
, and the localisation element is not a zero-divisor of
$\mathcal {A}_{\lambda }^{\mathrm {loc}}/\widetilde {J}_{\lambda }$
. Therefore, the first localisation morphism is injective.
Now, the two localisation morphisms of the lemma fit into the commutative diagram
The vertical map a is induced by the inclusion
$\mathcal {H}_{\lambda }=\mathcal {H}_{\lambda }^{W^{\lambda }}\subseteq \mathcal {A}_{\lambda }\subseteq \mathcal {A}_{\lambda }^{\mathrm {loc}}$
. The map a is injective since
$\mathcal {H}_{\lambda }\cap \widetilde {J}_{\lambda }=(\widetilde {J}_{\lambda })^{W^{\lambda }}$
, which follows from the arguments of the proof of Lemma 4.10. Therefore, the second localisation map (the top line of (4.2)) must be injective.
Lemma 4.19. Let
$\lambda , \mu \in X_{*}(T)$
and
$\nu $
as in Lemma 2.8. Then,
$$\begin{align*}k_{\mu}=k_{\nu,\lambda}\frac{\prod_{\alpha\in\mathcal{W}^{\mu<0}(V)\cap\mathcal{W}^{\lambda\neq0}(V)}\alpha}{\prod_{\alpha\in\mathcal{W}^{\mu<0}(\mathfrak{g})\cap\mathcal{W}^{\lambda\neq0}(\mathfrak{g})}\alpha}\,. \end{align*}$$
Proof. By the properties satisfied by the cocharacter
$\nu $
given by Lemma 2.8, we have
$\mathcal {W}^{\mu <0}(V)=\mathcal {W}^{\nu <0}(V)\cap \mathcal {W}^{\lambda =0}(V)\sqcup \mathcal {W}^{\mu <0}(V)\cap \mathcal {W}^{\lambda \neq 0}(V)$
and similarly for V and
$\mathfrak {g}$
exchanged. Therefore,
$$\begin{align*}\begin{aligned} k_{\mu}&=\frac{\prod_{\alpha\in\mathcal{W}^{\mu<0}(V)}\alpha}{\prod_{\alpha\in\mathcal{W}^{\mu<0}(\mathfrak{g})}\alpha}\\ &=\frac{\prod_{\alpha\in\mathcal{W}^{\nu<0}(V)\cap\mathcal{W}^{\lambda=0}(V)}\alpha}{\prod_{\alpha\in\mathcal{W}^{\nu<0}(\mathfrak{g})\cap\mathcal{W}^{\lambda=0}(\mathfrak{g})}\alpha}\frac{\prod_{\alpha\in\mathcal{W}^{\mu<0}(V)\cap\mathcal{W}^{\lambda\neq0}(V)}\alpha}{\prod_{\alpha\in\mathcal{W}^{\mu<0}(\mathfrak{g})\cap\mathcal{W}^{\lambda\neq0}(\mathfrak{g})}\alpha}\\ &=k_{\nu,\lambda}\frac{\prod_{\alpha\in\mathcal{W}^{\mu<0}(V)\cap\mathcal{W}^{\lambda\neq0}(V)}\alpha}{\prod_{\alpha\in\mathcal{W}^{\mu<0}(\mathfrak{g})\cap\mathcal{W}^{\lambda\neq0}(\mathfrak{g})}\alpha}\,. \end{aligned} \end{align*}$$
We define
This is an invertible element in
$\mathcal {A}_{\lambda }'$
.
Lemma 4.20. Let
$\lambda , \mu \in X_{*}(T)$
and w in W be such that
$\lambda \not \preceq w^{-1}\cdot \mu $
. Then,
$w\cdot k_{\mu }\in \widetilde {J}^{\prime }_{\lambda }$
.
Proof. Of course, the presence of W is superfluous since
$w k_{\mu }=k_{w^{-1}\cdot \mu }$
for any
$w\in W$
and it suffices to prove that for any
$\lambda ,\mu \in X_*(T)$
such that
$\lambda \not \preceq \mu $
,
$k_{\mu }\in \widetilde {J}^{\prime }_{\lambda }$
.
We let
$\nu \in X_*(T)$
be such that
$V^{\nu }=V^{\lambda }\cap V^{\mu }$
,
$\mathfrak {g}^{\nu }=\mathfrak {g}^{\lambda }\cap \mathfrak {g}^{\mu }$
,
$(V^{\lambda })^{\nu \geqslant 0}=(V^{\lambda })^{\mu \geqslant 0}$
and
$(\mathfrak {g}^{\lambda })^{\nu \geqslant 0}=(\mathfrak {g}^{\lambda })^{\mu \geqslant 0}$
(Lemma 2.8). By assumption,
$\nu \prec \lambda $
.
We can write, by Lemma 4.19,
$$\begin{align*}k_{\mu}=k_{\nu,\lambda}\underbrace{\frac{\prod_{\alpha\in\mathcal{W}^{\mu<0}(V)\cap\mathcal{W}^{\lambda\neq0}(V)}\alpha}{\prod_{\alpha\in\mathcal{W}^{\mu<0}(\mathfrak{g})\cap\mathcal{W}^{\lambda\neq0}(\mathfrak{g})}\alpha}}_{=k^{\prime}_{\mu,\lambda}\in\mathcal{A}^{\prime}_{\lambda}}\,. \end{align*}$$
This element is in
$\widetilde {J}^{\prime }_{\lambda }$
since
$k_{\nu ,\lambda }$
is in
${J}_{\lambda }$
, by Lemma 4.9.
Lemma 4.21. Let
$\lambda \in X_*(T)$
be such that for some
$w\in W$
,
$\lambda \preceq w\cdot \lambda $
. Then,
$\overline {w\cdot \lambda }=\overline {\lambda }$
.
Proof. The lemma follows from the inclusions
$V^{\lambda }\subseteq V^{w\cdot \lambda }$
and
$\mathfrak {g}^{\lambda }\subseteq \mathfrak {g}^{w\cdot \lambda }$
and the equalities of dimensions
$\dim V^{\lambda }=\dim V^{w\cdot \lambda }$
and
$\dim \mathfrak {g}^{\lambda }=\dim \mathfrak {g}^{w\cdot \lambda }$
.
For
$\lambda \in X_*(T)$
, we have an inclusion
$$\begin{align*}\mathrm{Res}_{\lambda}\colon \mathcal{H}\cong\mathcal{A}^{W}\cong\mathcal{A}_{\lambda}^{W}\rightarrow\mathcal{H}_{\lambda}\cong \mathcal{A}_{\lambda}^{W^{\lambda}}\,. \end{align*}$$
since
$\mathcal {A}\cong \mathcal {A}_{\lambda }=\mathrm {H}^*(\mathrm {pt}/T)$
and
$W^{\lambda }\subseteq W$
(Lemma 4.2).
Corollary 4.22. For any
$f_{\mu }\in \mathcal {H}_{\mu }$
,
$\widetilde {\mu }\in \mathscr {P}_V/W$
such that
$f_{\mu }=0$
if
$\lambda \prec w\cdot \mu $
for some
$w\in W$
, we have
$$\begin{align*}L\mathrm{Res}_{\lambda}\left(\sum_{\widetilde{\mu}\in\mathscr{P}_V/W}\mathrm{Ind}_{\mu}(f_{\mu})\right)\equiv k_{\lambda}\cdot\sum_{w\in W_{\lambda}/W^{\lambda}}\varepsilon_{V,\lambda}(w)^{-1}(w\cdot f_{\lambda})\quad \pmod{\widetilde{J}^{\prime}_{\lambda}}\,. \end{align*}$$
Proof. We calculate:
$$\begin{align*}\begin{aligned} &L\mathrm{Res}_{\lambda}\left(\sum_{\widetilde{\mu}\in\mathscr{P}_V/W}\mathrm{Ind}_{\mu}(f_{\mu})\right)\\ &\quad =L\mathrm{Res}_{\lambda}\left( \sum_{\widetilde{\mu}\in\mathscr{P}_V/W}\frac{1}{\# W^{\mu}}\sum_{w\in W}(w\cdot f_{\mu})(w\cdot k_{\mu})\right)\quad\text{by Proposition}~2.11\\ &\quad \equiv\frac{1}{\# W^{\lambda}}\sum_{\substack{w\in W\\ w\overline{\lambda}=\overline{\lambda}}}(w\cdot f_{\lambda})(w\cdot k_{\lambda})\quad \pmod{\widetilde{J}_{\lambda}'}\quad\text{by Lemmas}~4.20 \text{ and } 2.21\\ &\quad = \frac{1}{\# W^{\lambda}}k_{\lambda}\cdot\sum_{w\in W_{\lambda}}\varepsilon_{V,\lambda}(w)^{-1}(w\cdot f_{\lambda})\quad\text{by definition of the character} \varepsilon_{V,\lambda}, \text{ Proposition}~4.6\,. \end{aligned}\\[-18pt] \end{align*}$$
Proposition 4.23. The map
$$\begin{align*}\bigoplus_{\widetilde{\mu}\in\mathscr{P}_V/W}(\mathcal{P}_{\mu}\otimes\mathrm{H}^*(\mathrm{pt}/G_{\mu}))^{\varepsilon_{V,\mu}}\xrightarrow{\bigoplus\mathrm{Ind}_{\mu}}\mathrm{H}^*(V/G) \end{align*}$$
induced by the parabolic induction morphisms is injective.
Proof. Let
$f_{\mu }\in (\mathcal {P}_{\mu }\otimes \mathrm {H}^*(\mathrm {pt}/G_{\mu }))^{\varepsilon _{V,\mu }}$
,
$\widetilde {\mu }\in \mathscr {P}_V/W$
. We let 
. We assume that not all of the
$f_{\mu }$
’s are zero, and we have to prove that
$U\neq 0$
. We let
$\widetilde {\lambda }\in \mathscr {P}_V/W$
be such that
$f_{\lambda }\neq 0$
and for any
$\widetilde {\mu }\in \mathscr {P}_V/W$
such that
$f_{\mu }\neq 0$
, for any
$w\in W$
,
$\lambda \not \preceq w\cdot \mu $
(otherwise, we may replace
$\widetilde {\lambda }$
by
$\widetilde {\mu }$
, etc.).
By Corollary 4.22, we have
$$\begin{align*}\begin{aligned} L\mathrm{Res}_{\lambda}\left(\sum_{\widetilde{\mu}\in\mathscr{P}_V/W}\mathrm{Ind}_{\mu}(f_{\mu})\right)&\equiv k_{\lambda}\cdot\sum_{w\in W_{\lambda}/W^{\lambda}}\varepsilon_{V,\lambda}(w)^{-1}(w\cdot f_{\lambda})\quad \pmod{\widetilde{J}^{\prime}_{\lambda}}\\ &\equiv k_{\lambda}\# (W_{\lambda}/W^{\lambda}) f_{\lambda}\quad \pmod{\widetilde{J}^{\prime}_{\lambda}}\quad\text{by Proposition } 4.8\,. \end{aligned} \end{align*}$$
and moreover,
$f_{\lambda }\not \in \widetilde {J}^{\prime }_{\lambda }$
since
$f_{\lambda }\in \mathcal {P}_{\lambda }\otimes \mathrm {H}^*(\mathrm {pt}/G_{\lambda })$
, and
$k_{\lambda }$
is invertible in
$\mathcal {A}_{\lambda }'$
. We obtain that
$\mathrm {Res}_{\lambda }(U)\neq 0$
. Therefore,
$U\neq 0$
, proving injectivity.
We are now ready to prove Theorem 1.1.
Corollary 4.24. The map
$$\begin{align*}\bigoplus_{\widetilde{\mu}\in\mathscr{P}_V/W}(\mathcal{P}_{\mu}\otimes\mathrm{H}^*(\mathrm{pt}/G_{\mu}))^{\varepsilon_{V,\mu}}\xrightarrow{\bigoplus_{\widetilde{\mu}\in\mathscr{P}_V/W}\mathrm{Ind}_{\mu}}\mathrm{H}^*(V/G) \end{align*}$$
is an isomorphism.
Proof. The injectivity is Proposition 4.23. The surjectivity comes from Lemmas 4.14 and 2.17. Indeed, by Lemma 4.14, any element
$f\in \mathrm {H}^*(V/G)$
has the form
$f=\sum _{\mu \preceq 0}\mathrm {Ind}_{\mu }(g_{\mu })$
for some
$g_{\mu }\in \mathcal {P}_{\mu }\otimes \mathrm {H}^*(\mathrm {pt}/G_{\mu })$
. By Lemma 2.17, we may rewrite
$f=\sum _{\widetilde {\mu }\in \mathscr {P}_{V}/W}\mathrm {Ind}_{\mu }(\widetilde {g}_{\mu })$
. We have the formula
$$\begin{align*}\mathrm{Ind}_{\mu}(\widetilde{g}_{\mu})=\frac{1}{\# W^{\mu}}\sum_{\overline{w'}\in W/W_{\mu}}w'\cdot\left(\sum_{w\in W_{\mu}}\varepsilon_{V,\mu}(w)^{-1}w\cdot \widetilde{g}_{\mu} \right)(w'\cdot k_{\mu}) \end{align*}$$
for
$\widetilde {g}_{\mu }\in \mathcal {H}_{\mu }$
, deduced from Propositions 2.11 and 4.6 which implies, using Proposition 4.8, that
$\mathrm {Ind}_{\mu }(\widetilde {g}_{\mu })$
vanishes if f is not in the
$\varepsilon _{V,\mu }$
-isotypic component of
$\mathcal {P}_{\mu }\otimes \mathrm {H}^*(\mathrm {pt}/G_{\mu })$
for the
$W_{\mu }$
-action. Therefore, we may assume that
$\widetilde {g}_{\mu }$
is
$\varepsilon _{V,\lambda }$
-invariant. This concludes the proof of the corollary.
4.3 The BPS subspaces
In analogy to the case of quivers [Reference EfimovEfi12; Reference Meinhardt and ReinekeMR19] and also [Reference Davison and MeinhardtDM20a] for quivers with potential, we may call the subspaces
$\mathcal {P}_{\lambda }\subseteq \mathrm {H}^*(V^{\lambda }/G^{\lambda })$
the BPS subspaces. Although we lack both qualitative and quantitative properties of these spaces, there are cases when we can prove they vanish.
Proposition 4.25. Let
$\lambda \in X_*(T)$
. If
$\ker (G^{\lambda }\rightarrow \mathrm {GL}(V^{\lambda }))^{\circ }\not \subseteq Z(G^{\lambda })$
, then
$\mathcal {P}_{\lambda }=\{0\}$
.
Proof. By possibly replacing V by
$V^{\lambda }$
and G by
$G^{\lambda }$
, we may assume that
$\lambda $
is the trivial cocharacter of T. If
$\ker (G\rightarrow \mathrm {GL}(V))^{\circ }\not \subseteq Z(G)$
, then there exists a cocharacter
$\mu \colon \mathbf {G}_{\mathrm {m}}\rightarrow \ker (G\rightarrow \mathrm {GL}(V))\subseteq T$
which does not factor through
$Z(G)$
. We have
$\overline {\mu }\in \mathscr {P}_V\smallsetminus \{\overline {0}\}$
. Then, the induction diagram reads
and the induction map
$\mathrm {Ind}_{\mu }\colon \mathrm {H}^*(V/G^{\mu })=\mathrm {H}^*(\mathrm {pt}/T)^{W^{\mu }}\rightarrow \mathrm {H}^*(V/G)=\mathrm {H}^*(\mathrm {pt}/T)$
is the map
$$\begin{align*}f\mapsto \sum_{w\in W/W^{\mu}}w\cdot\left(\frac{f}{\prod_{\alpha\in\mathcal{W}^{\mu<0}(\mathfrak{g})}\alpha}\right)\,. \end{align*}$$
It is surjective. Hence,
$\mathcal {P}_0=\{0\}$
.
5 Vanishing bounds for the BPS spaces
We recall that
$\mathcal {H}_{\lambda }=\mathrm {H}^*(V^{\lambda }/G^{\lambda })$
for
$\lambda \in X_*(T)$
. We recall the chosen
$\mathbf {Z}$
-grading on
$\mathcal {H}_{\lambda }$
:
$\mathcal {H}_{\lambda }^k=\mathrm {H}^{k+d_{\lambda }}(V^{\lambda }/G^{\lambda })$
for
$k\in \mathbf {Z}$
. This is just a shift of the cohomological grading, chosen so that induction morphisms preserve this grading (Lemma 2.7). In this section, we prove Proposition 1.5.
Proof of the lower bound in Proposition 1.5.
First, we have an inclusion
$\mathcal {P}_{\lambda }^k\subseteq \mathcal {H}_{\lambda }^k=\mathrm {H}^{k+d_{\lambda }}(V^{\lambda }/G^{\lambda })$
, and the latter vector space is zero for
$k<-d_{\lambda }=\dim G^{\lambda }-\dim V^{\lambda }$
.
The proof of the upper bound uses [Reference EfimovEfi12, Lemma 3.6] and is analogous to the proof of [Reference EfimovEfi12, Theorem 3.5].
Lemma 5.1 [Reference EfimovEfi12, Lemma 3.6].
Let k be a field and let 
be an algebra of polynomials with
$\deg (z_i)=2$
(to match the cohomological degrees). Suppose that 
are pairwise linearly independent linear forms in
$z_i$
. Let 
be non-empty set of polynomials of the form
$$\begin{align*}P_i=\prod_{j=1}^s\ell_j^{d_{i,s}} \end{align*}$$
for some
$d_{i,j}\in \mathbf {Z}_{\geqslant 0}$
. Let 
for
$1\leqslant j\leqslant s$
. Then, the following are equivalent:
-
(i)
for
$d\geqslant \sum _{j=1}^sd_j-n+1$
, -
(ii) the ideal
has finite codimension, -
(iii) For any sequence
of numbers in such that
$d_{i,p_i}>0$
for
$1\leqslant i\leqslant r$
, the linear forms generate the space
$B^1$
.
for 
has finite codimension,
of numbers in 
such that 
Proof of the upper bound in Proposition 1.5.
As we already noticed in the proof of Proposition 4.16, if we replace V by
$V\times \mathfrak {g}$
, then we have inclusions
$\mathcal {P}_{V,\lambda }\subseteq \mathcal {P}_{V\times \mathfrak {g},\lambda }$
, and therefore, it suffices to give the upper bound for
$\mathcal {P}_{V\times \mathfrak {g},\lambda }$
. We set 
, the (polynomial) algebra of functions on
$\mathfrak {g}/\mathfrak {g}_{\lambda }$
. The proof of Proposition 4.16 shows that the ideal
$J_{\lambda }\subseteq \mathrm {Sym}[(\mathfrak {t}/\mathfrak {g}_{\lambda })^*]$
has finite codimension, that is satisfies the second point of Lemma 5.1. In our case, the quantity
$\sum _{j=1}^sd_j$
is equal to
$\frac {\dim V^{\lambda }-\dim V^T}{2}$
. Therefore, an upper bound for
$\sum _{j=1}^sd_j-n+1$
is
$\frac {\dim V^{\lambda }}{2}$
.
Now, Lemma 5.1 gives
$(\mathcal {A}_{\lambda }^{\mathrm {prim}})^{2d}\subseteq J_{\lambda }$
for
$d\geqslant \frac {\dim V^{\lambda }}{2}$
(for the unshifted cohomological gradings), where
$(\mathcal {A}_{\lambda }^{\mathrm {prim}})^e$
denotes degree e polynomials inside
$\mathcal {A}_{\lambda }^{\mathrm {prim}}$
. By taking into account the cohomological shift by
$\dim G^{\lambda }-\dim V^{\lambda }$
, we obtain the upper bound:
$\mathcal {P}_{\lambda }^k=0$
for
$k>\dim V^{\lambda }+\dim G^{\lambda }-\dim V^{\lambda }=\dim G^{\lambda }$
.
6 Examples
In this section, we give some explicit examples of cohomological integrality isomorphisms for some choices of pairs
$(G,V)$
of a reductive group G and a (weakly) symmetric representation V of G. We give each time the explicit formula for the cohomological integrality isomorphisms produced by Theorem 1.1. In each case, it is possible to verify by hand that they are indeed isomorphisms.
6.1
$(\mathbf {C}^*)^2\curvearrowright \mathbf {C}^2\oplus (\mathbf {C}^2)^{\vee }$
We let 
act on 
by
We can identify the set of equivalence classes of cocharacters (§2.6)
$\mathscr {P}_{V}=\{\overline {\lambda _0},\overline {\lambda _1},\overline {\lambda _2},\overline {\lambda _3}\}$
where
$$\begin{align*}\begin{matrix} \lambda_0&\colon&\mathbf{G}_{\mathrm{m}}&\rightarrow&(\mathbf{C}^*)^2\\ &&v&\mapsto&(1,1)\,, \end{matrix} \end{align*}$$
$$\begin{align*}\begin{matrix} \lambda_1&\colon&\mathbf{G}_{\mathrm{m}}&\rightarrow&(\mathbf{C}^*)^2\\ &&v&\mapsto&(v^{-1},1)\,, \end{matrix} \end{align*}$$
$$\begin{align*}\begin{matrix} \lambda_2&\colon&\mathbf{G}_{\mathrm{m}}&\rightarrow&(\mathbf{C}^*)^2\\ &&v&\mapsto&(1,v^{-1})\,, \end{matrix} \end{align*}$$
$$\begin{align*}\begin{matrix} \lambda_3&\colon&\mathbf{G}_{\mathrm{m}}&\rightarrow&(\mathbf{C}^*)^2\\ &&v&\mapsto&(v^{-1},v^{-1})\,. \end{matrix} \end{align*}$$
We have
$V^{\lambda _0}=V$
,
$V^{\lambda _1}=(0\oplus \mathbf {C})\oplus (0\oplus \mathbf {C}^{\vee })$
,
$V^{\lambda _2}=(\mathbf {C}\oplus 0)\oplus (\mathbf {C}^{\vee }\oplus 0)$
and
$V^{\lambda _3}=\{0\}$
. We have
$G^{\lambda _0}=G^{\lambda _1}=G^{\lambda _2}=G^{\lambda _3}=(\mathbf {C}^*)^2$
. We can describe the induction maps as follows:
$$\begin{align*}\begin{matrix} \mathrm{Ind}_{\lambda_1,\lambda_0}&\colon& \overbrace{\mathbf{Q}[x_1,x_2]}^{=\mathrm{H}^*(V^{\lambda_1}/G^{\lambda_1})}&\rightarrow&\overbrace{\mathbf{Q}[x_1,x_2]}^{=\mathrm{H}^*(V/G)}\\&&f&\mapsto&x_1f \end{matrix} \end{align*}$$
$$\begin{align*}\begin{matrix} \mathrm{Ind}_{\lambda_2,\lambda_0}&\colon& \overbrace{\mathbf{Q}[x_1,x_2]}^{=\mathrm{H}^*(V^{\lambda_2}/G^{\lambda_2})}&\rightarrow&\overbrace{\mathbf{Q}[x_1,x_2]}^{=\mathrm{H}^*(V/G)}\\&&f&\mapsto&x_2f \end{matrix} \end{align*}$$
$$\begin{align*}\begin{matrix} \mathrm{Ind}_{\lambda_3,\lambda_0}&\colon& \overbrace{\mathbf{Q}[x_1,x_2]}^{=\mathrm{H}^*(V^{\lambda_3}/G^{\lambda_3})}&\rightarrow&\overbrace{\mathbf{Q}[x_1,x_2]}^{\mathrm{H}^*(V/G)}\\&&f&\mapsto&x_1x_2f \end{matrix} \end{align*}$$
$$\begin{align*}\begin{matrix} \mathrm{Ind}_{\lambda_3,\lambda_1}&\colon& \overbrace{\mathbf{Q}[x_1,x_2]}^{=\mathrm{H}^*(V^{\lambda_3}/G^{\lambda_3})}&\rightarrow&\overbrace{\mathbf{Q}[x_1,x_2]}^{=\mathrm{H}^*(V^{\lambda_1}/G^{\lambda_1})}\\&&f&\mapsto&x_2f \end{matrix} \end{align*}$$
$$\begin{align*}\begin{matrix} \mathrm{Ind}_{\lambda_3,\lambda_2}&\colon& \overbrace{\mathbf{Q}[x_1,x_2]}^{=\mathrm{H}^*(V^{\lambda_3}/G^{\lambda_3})}&\rightarrow&\overbrace{\mathbf{Q}[x_1,x_2]}^{=\mathrm{H}^*(V^{\lambda_2}/G^{\lambda_2})}\\ &&f&\mapsto&x_1f \end{matrix} \end{align*}$$
We have
$$\begin{align*}\mathcal{P}_{\lambda_0}=\mathbf{Q}\subseteq \mathbf{Q}[x_1,x_2],\quad \mathcal{P}_{\lambda_1}=\mathbf{Q}\subseteq \mathbf{Q}[x_1,x_2],\quad \mathcal{P}_{\lambda_2}=\mathbf{Q}\subseteq \mathbf{Q}[x_1,x_2],\quad \mathcal{P}_{\lambda_3}=\mathbf{Q}\subseteq \mathbf{Q}[x_1,x_2]\,. \end{align*}$$
We have
$$\begin{align*}G_{\lambda_0}=1\subseteq(\mathbf{C}^*)^2,\quad G_{\lambda_1}=\mathbf{C}^*\times 1\subseteq(\mathbf{C}^*)^2,\quad G_{\lambda_2}=1\times\mathbf{C}^*\subseteq(\mathbf{C}^*)^2,\quad G_{\lambda_3}=(\mathbf{C}^*)^2\,. \end{align*}$$
The cohomological integrality isomorphism is
$$\begin{align*}\begin{matrix} \overbrace{\mathbf{Q}}^{=\mathcal{P}_{\lambda_0}}\oplus \overbrace{\mathbf{Q}[x_1]}^{=\mathcal{P}_{\lambda_1}\otimes\mathrm{H}^*(\mathrm{pt}/G_{\lambda_1})}\oplus\overbrace{\mathbf{Q}[x_2]}^{=\mathcal{P}_{\lambda_2}\otimes\mathrm{H}^*(\mathrm{pt}/G_{\lambda_2})}\oplus\overbrace{\mathbf{Q}[x_1,x_2]}^{=\mathcal{P}_{\lambda_3}\otimes\mathrm{H}^*(\mathrm{pt}/G_{\lambda_3})}&\rightarrow&\overbrace{\mathbf{Q}[x_1,x_2]}^{=\mathrm{H}^*(V/G)}\\ (a_0,f(x_1),g(x_2),h(x_1,x_2))&\mapsto&a_0+x_1f(x_1)+x_2g(x_2)+x_1x_2h(x_1,x_2)\,. \end{matrix} \end{align*}$$
6.2
$\mathrm {GL}_2(\mathbf {C})\curvearrowright \mathbf {C}^2\oplus (\mathbf {C}^2)^{\vee }$
We let 
act on 
via
where the superscript t indicates the transpose. We let 
be the standard torus. The Weyl group is
$W=\mathfrak {S}_2$
. We have
$\mathscr {P}_V/W=\{\widetilde {\lambda _0},\widetilde {\lambda _1},\widetilde {\lambda _2}, \widetilde {\lambda }_3\}$
where
$$\begin{align*}\begin{matrix} \lambda_0&\colon&\mathbf{G}_{\mathrm{m}}&\rightarrow&(\mathbf{C}^*)^2\\ &&v&\mapsto&(1,1) \end{matrix} \end{align*}$$
$$\begin{align*}\begin{matrix} \lambda_1&\colon&\mathbf{G}_{\mathrm{m}}&\rightarrow&(\mathbf{C}^*)^2\\ &&v&\mapsto&(v^{-1},1) \end{matrix} \end{align*}$$
$$\begin{align*}\begin{matrix} \lambda_2&\colon&\mathbf{G}_{\mathrm{m}}&\rightarrow&(\mathbf{C}^*)^2\\ &&v&\mapsto&(v^{-1},v^{-2}) \end{matrix} \end{align*}$$
$$\begin{align*}\begin{matrix} \lambda_3&\colon&\mathbf{G}_{\mathrm{m}}&\rightarrow&(\mathbf{C}^*)^2\\ &&v&\mapsto&(v^{-1},v^{-1}) \end{matrix} \end{align*}$$
Therefore, we have
$V^{\lambda _0}=V$
,
$V^{\lambda _1}=(0\oplus \mathbf {C})\oplus (0\oplus \mathbf {C}^{\vee })$
,
$V^{\lambda _2}=\{0\}$
, and
$V^{\lambda _3}=\{0\}$
. We have
$G^{\lambda _0}=\mathrm {GL}_2(\mathbf {C}), G^{\lambda _1}=T$
,
$G^{\lambda _2}=T$
and
$G^{\lambda _3}=G$
. We can describe the induction maps as follows:
$$\begin{align*}\begin{matrix} \mathrm{Ind}_{\lambda_1,\lambda_0}&\colon& \overbrace{\mathbf{Q}[x_1,x_2]}^{=\mathrm{H}^*(V^{\lambda_1}/G^{\lambda_1})}&\rightarrow&\overbrace{\mathbf{Q}[x_1,x_2]^{\mathfrak{S}_2}}^{=\mathrm{H}^*(V/G)}=\mathbf{Q}[x_1+x_2,x_1x_2]\\ &&f&\mapsto&\frac{x_1}{x_1-x_2}f(x_1,x_2)+\frac{x_2}{x_2-x_1}f(x_2,x_1) \end{matrix} \end{align*}$$
$$\begin{align*}\begin{matrix} \mathrm{Ind}_{\lambda_2,\lambda_0}&\colon& \overbrace{\mathbf{Q}[x_1,x_2]}^{=\mathrm{H}^*(V^{\lambda_2}/G^{\lambda_2})}&\rightarrow&\overbrace{\mathbf{Q}[x_1,x_2]^{\mathfrak{S}_2}}^{=\mathrm{H}^*(V/G)}=\mathbf{Q}[x_1+x_2,x_1x_2]\\ &&f&\mapsto&\frac{x_1x_2}{x_1-x_2}f(x_1,x_2)+\frac{x_1x_2}{x_2-x_1}f(x_2,x_1) \end{matrix} \end{align*}$$
$$\begin{align*}\begin{matrix} \mathrm{Ind}_{\lambda_2,\lambda_1}&\colon& \overbrace{\mathbf{Q}[x_1,x_2]}^{=\mathrm{H}^*(V^{\lambda_2}/G^{\lambda_2})}&\rightarrow&\overbrace{\mathbf{Q}[x_1,x_2]}^{\mathrm{H}^*(V^{\lambda_1}/G^{\lambda_1})}\\ &&f&\mapsto&x_2f \end{matrix} \end{align*}$$
$$\begin{align*}\begin{matrix} \mathrm{Ind}_{\lambda_2,\lambda_3}&\colon& \overbrace{\mathbf{Q}[x_1,x_2]}^{=\mathrm{H}^*(V^{\lambda_2}/G^{\lambda_2})}&\rightarrow&\overbrace{\mathbf{Q}[x_1+x_2,x_1x_2]}^{\mathrm{H}^*(V^{\lambda_3}/G^{\lambda_3})}\\ &&f&\mapsto&\frac{f(x_1,x_2)-f(x_2,x_1)}{x_1-x_2} \end{matrix} \end{align*}$$
The induction
$\mathrm {Ind}_{\lambda _1,\lambda _0}$
is surjective, since
$$\begin{align*}x_1+x_2=\mathrm{Ind}_{\lambda_1,\lambda_0}(x_1) \end{align*}$$
and
$$\begin{align*}x_1x_2=\mathrm{Ind}_{\lambda_1,\lambda_0}\left(\frac{1}{2}x_1x_2\right)\,. \end{align*}$$
The induction
$\mathrm {Ind}_{\lambda _2,\lambda _3}$
is surjective as any symmetric polynomial f is the image by
$\mathrm {Ind}_{\lambda _2,\lambda _3}$
of
$(x_1-x_2)f$
.
A direct sum complement of the image of
$\mathrm {Ind}_{\lambda _2,\lambda _1}$
is
$\mathbf {Q}[x_1]=\mathcal {P}_{\lambda _1}\otimes \mathbf {Q}[x_1]$
by definition of
$\mathcal {P}_{\lambda _1}$
.
Therefore, we have
$\mathcal {P}_{\lambda _0}=0$
,
$\mathcal {P}_{\lambda _1}=\mathbf {Q}$
,
$\mathcal {P}_{\lambda _2}=\mathbf {Q}$
and
$\mathcal {P}_{\lambda _3}=0$
. We have
$G_{\lambda _0}=1, G_{\lambda _1}=\mathbf {C}^*\times 1$
,
$G_{\lambda _2}=T$
and
$G_{\lambda _3}=G$
. The cohomological integrality isomorphism reads
$$\begin{align*}\begin{matrix} \overbrace{\mathbf{Q}[x_1]}^{=\mathcal{P}_{\lambda_1}\otimes\mathrm{H}^*(\mathrm{pt}/G_{\lambda_1})}\oplus\overbrace{\mathbf{Q}[x_1,x_2]^{\mathrm{sgn}}}^{=(\mathcal{P}_{\lambda_2}\otimes\mathrm{H}^*(\mathrm{pt}/G_{\lambda_2}))^{\varepsilon_{V,\lambda_2}}}&\rightarrow&\overbrace{\mathbf{Q}[x_1,x_2]^{\mathfrak{S}_2}}^{=\mathrm{H}^*(V/G)}\\ (f(x_1),g(x_1,x_2))&\mapsto&\frac{x_1f(x_1)-x_2f(x_2)}{x_1-x_2}+2x_1x_2\frac{g(x_1,x_2)}{x_1-x_2} \end{matrix} \end{align*}$$
where
$\mathbf {Q}[x_1,x_2]^{\mathrm {sgn}}$
is the sign-isotypic component, since the kernel
$\frac {x_1x_2}{x_1-x_2}$
changes sign when we exchange the variables
$x_1,x_2$
, and so the character
$\varepsilon _{V,\lambda _2}$
of
$W_{\lambda _2}=\mathfrak {S}_2$
given by Proposition 4.6 is the sign.
6.3
$\mathrm {GL}_2(\mathbf {C})\curvearrowright (\mathbf {C}^2\oplus (\mathbf {C}^2)^{\vee })^g$
We generalise the previous situation by considering the diagonal action of
$\mathrm {GL}_2(\mathbf {C})$
on
$(\mathbf {C}^2\oplus (\mathbf {C}^2)^{\vee })^g$
(
$g\geqslant 0$
). The set
$\mathscr {P}_{V}/W$
is the same as in §6.2. The fixed point sets
$V^{\lambda _0}, V^{\lambda _1}$
, and
$V^{\lambda _2}$
can be computed similarly and the Levi subgroups
$G^{\lambda _0}, G^{\lambda _1}$
and
$G^{\lambda _2}$
are the ones given in §6.2. We can describe the induction maps as follows, analogously to §6.2.
$$\begin{align*}\begin{matrix} \mathrm{Ind}_{\lambda_1,\lambda_0}&\colon& \mathbf{Q}[x_1,x_2]&\rightarrow&\mathbf{Q}[x_1,x_2]^{\mathfrak{S}_2}=\mathbf{Q}[x_1+x_2,x_1x_2]\\&&f&\mapsto&\frac{x_1^g}{x_1-x_2}f(x_1,x_2)+\frac{x_2^g}{x_2-x_1}f(x_2,x_1) \end{matrix} \end{align*}$$
$$\begin{align*}\begin{matrix} \mathrm{Ind}_{\lambda_2,\lambda_0}&\colon& \mathbf{Q}[x_1,x_2]&\rightarrow&\mathbf{Q}[x_1,x_2]^{\mathfrak{S}_2}=\mathbf{Q}[x_1+x_2,x_1x_2]\\&&f&\mapsto&\frac{x_1^gx_2^g}{x_1-x_2}f(x_1,x_2)+\frac{x_1^gx_2^g}{x_2-x_1}f(x_2,x_1) \end{matrix} \end{align*}$$
$$\begin{align*}\begin{matrix} \mathrm{Ind}_{\lambda_2,\lambda_1}&\colon& \mathbf{Q}[x_1,x_2]&\rightarrow&\mathbf{Q}[x_1,x_2]\\&&f&\mapsto&x_2^gf \end{matrix} \end{align*}$$
$$\begin{align*}\begin{matrix} \mathrm{Ind}_{\lambda_2,\lambda_3}&\colon&\mathbf{Q}[x_1,x_2]&\rightarrow&\mathbf{Q}[x_1+x_2,x_1x_2]\\&&f&\mapsto&\frac{f(x_1,x_2)-f(x_2,x_1)}{x_1-x_2} \end{matrix} \end{align*}$$
We have
$\mathcal {P}_{\lambda _0}=\bigoplus _{j=0}^{g-2}\mathbf {Q}(x_1+x_2)^{j}$
,
$\mathcal {P}_{\lambda _1}=\bigoplus _{j=0}^{g-1}\mathbf {Q} x_2^j$
,
$\mathcal {P}_{\lambda _2}=\mathbf {Q}$
and
$\mathcal {P}_{\lambda _3}=0$
and the cohomological integrality isomorphism reads
$$\begin{align*}\begin{matrix} \mathcal{P}_{\lambda_{0}}\oplus (\mathcal{P}_{\lambda_1}\otimes\mathbf{Q}[x_1])\oplus\mathbf{Q}[x_1,x_2]^{\mathrm{sgn}}&\rightarrow&\mathbf{Q}[x_1,x_2]^{\mathfrak{S}_2}\\ (f(x_1,x_2),g(x_1,x_2),h(x_1,x_2))&\mapsto&f(x_1,x_2)+\frac{x_1^gg(x_1,x_2)-x_2^gg(x_2,x_1)}{x_1-x_2}+2x_1^gx_2^g\frac{h(x_1,x_2)}{x_1-x_2}\,. \end{matrix} \end{align*}$$
6.4
$\mathrm {SL}_2(\mathbf {C})\curvearrowright \mathbf {C}^2$
We let 
act on
$\mathbf {C}^2$
via the natural action. It is a symmetric representation. We have
$W=\mathfrak {S}_2$
. We let
$T\subseteq \mathrm {SL}_2(\mathbf {C})$
be the maximal torus of diagonal matrices. We have
$\mathscr {P}_V/W=\{\widetilde {\lambda _0},\widetilde {\lambda _1}\}$
where
$$\begin{align*}\begin{matrix} \lambda_0&\colon&\mathbf{G}_{\mathrm{m}}&\rightarrow&T\\ &&v&\mapsto&(1,1) \end{matrix} \end{align*}$$
$$\begin{align*}\begin{matrix} \lambda_1&\colon&\mathbf{G}_{\mathrm{m}}&\rightarrow&T\\ &&v&\mapsto&(v^{-1},v)\,. \end{matrix} \end{align*}$$
We have
$V^{\lambda _0}=V$
,
$V^{\lambda _1}=\{0\}$
. We can describe the induction map as follows.
$$\begin{align*}\begin{matrix} \mathrm{Ind}_{\lambda_1,\lambda_0}&\colon& \mathbf{Q}[x]&\rightarrow&\mathbf{Q}[x^2]\\&&f&\mapsto&\frac{1}{2}(f(x)+f(-x))\,. \end{matrix} \end{align*}$$
The factor
$\frac {1}{2}$
comes from the fact that the T-weight of
$\mathfrak {g}^{\lambda _1>0}=\begin {pmatrix} 0&*\\ 0&0 \end {pmatrix} $
is
$2$
. We have
$\mathcal {P}_{\lambda _1}=\mathbf {Q}$
and
$\mathcal {P}_{\lambda _0}=\{0\}$
and the cohomological integrality isomorphism is
$$\begin{align*}\begin{matrix} \mathbf{Q}[x^2]&\rightarrow&\mathbf{Q}[x^2]\\ f&\mapsto&f\,. \end{matrix} \end{align*}$$
6.5
$\mathrm {SL}_2(\mathbf {C})\curvearrowright \mathbf {C}^d$
Let
$V_d$
be the d-dimensional irreducible representation of
$\mathrm {SL}_2(\mathbf {C})$
. Up to a scaling factor
$C_{d}$
, the induction morphism is
$$\begin{align*}\begin{matrix} \mathrm{Ind}_{\lambda_1,\lambda_0}&\colon& \mathbf{Q}[x]&\rightarrow&\mathbf{Q}[x^2]\\ &&f&\mapsto&C_{d}x^{\lfloor\frac{d}{2}\rfloor-1}f(x)+C_{d}(-x)^{\lfloor\frac{d}{2}\rfloor-1}f(-x)\,. \end{matrix} \end{align*}$$
We have
$\mathcal {P}_{\lambda _1}=\mathbf {Q}$
and
$\mathcal {P}_{\lambda _0}=\mathbf {Q}[x^2]_{\deg <2(\lfloor \frac {d}{2}\rfloor -1)}$
where
$C_{d}\in \mathbf {Q}$
is some constant depending only on d (note that
$\deg (x)=2$
in the cohomological grading).
The cohomological integrality isomorphism is
$$\begin{align*}\begin{matrix} \mathcal{P}_{\lambda_0}\oplus\mathbf{Q}[x]^{\varepsilon_{d}}&\rightarrow&\mathbf{Q}[x^2]\\ (f,g)&\mapsto&f+2C_{d}x^{\lfloor\frac{d}{2}\rfloor-1}g \end{matrix} \end{align*}$$
where
$\mathbf {Q}[x]^{\varepsilon _{d}}=\mathbf {Q}[x^2]$
if
$\lfloor \frac {d}{2}\rfloor -1$
is even, and
$\mathbf {Q}[x]^{\varepsilon _{d}}=x\mathbf {Q}[x^2]$
if
$\lfloor \frac {d}{2}\rfloor -1$
is odd.
6.6
$\mathrm {SL}_2(\mathbf {C})\curvearrowright (\mathfrak {sl}_2(\mathbf {C}))^g$
Let
$g\geqslant 0$
. We consider the diagonal adjoint action of
$\mathrm {SL}_2(\mathbf {C})$
on 
. Again,
$\mathscr {P}_V=\{\overline {\lambda _0},\overline {\lambda _1}\}$
with
$\lambda _0,\lambda _1$
as in §6.4. The induction map reads
$$\begin{align*}\begin{matrix} \mathrm{Ind}_{\lambda_1,\lambda_0}&\colon&\mathbf{Q}[x]&\rightarrow&\mathbf{Q}[x^2]\\ &&f&\mapsto&(2x)^{g-1}f(x)+(-2x)^{g-1}f(-x)\,. \end{matrix} \end{align*}$$
We have
$V^{\lambda _1}\cong \mathbf {C}^g$
with the trivial action of T. Therefore,
$\mathcal {P}_{\lambda _1}=\mathbf {Q}$
and
$\mathcal {P}_{\lambda _0}=\mathbf {Q}[x^2]_{\deg <2(g-1)}$
. The cohomological integrality isomorphism is
$$\begin{align*}\begin{matrix} \mathbf{Q}[x]^{\varepsilon_g}\oplus\mathcal{P}_0&\rightarrow&\mathbf{Q}[x^2]\\ (f,g)&\mapsto&g(x)+2^gx^{g-1}f(x) \end{matrix} \end{align*}$$
where
$\mathbf {Q}[x]^{\varepsilon _g}=\mathbf {Q}[x^2]$
if
$g-1$
is even and
$\mathbf {Q}[x]^{\varepsilon _g}=x\mathbf {Q}[x^2]$
is
$g-1$
is odd.
Remark 6.2. In this example,
$G_{\lambda _0}=\{\pm 1\}$
is a finite group. We see that it does not affect the cohomological integrality isomorphism.
6.7
$G\curvearrowright \{0\}$
In this section, we present the cohomological integrality isomorphism for the trivial representation of a reductive group G. Then,
$\mathscr {P}_{\{0\}}/W$
is in bijection with the set of conjugacy classes of Levi subgroups of G. If
$\lambda \in X_*(T)$
is a general cocharacter, then the induction morphism
$$\begin{align*}\mathrm{Ind}_{\lambda}\colon \mathrm{H}^*(\mathrm{pt}/T)\rightarrow\mathrm{H}^*(\mathrm{pt}/G) \end{align*}$$
is surjective. Indeed, we have
$$\begin{align*}\mathrm{Ind}_{\lambda}(f)=\sum_{w\in W}w\left(\frac{f}{\prod_{\substack{\alpha\in\mathcal{W}(\mathfrak{g})\\\langle\lambda,\alpha\rangle<0}}\alpha}\right). \end{align*}$$
Therefore,
$f=\mathrm {Ind}_{\lambda }\left (\frac {1}{\# W}f\prod _{\substack {\alpha \in \mathcal {W}(\mathfrak {g})\\\langle \lambda ,\alpha \rangle <0}}\alpha \right )$
.
This implies that
$\mathcal {P}_{\lambda }=\{0\}$
if
$G^{\lambda }$
is not the maximal torus of G and
$\mathcal {P}_{\lambda }=\mathbf {Q}$
if
$G^{\lambda }=T$
. In particular, the cohomological integrality reads (for a choice of generic
$\lambda \in X_*(T)$
):
$$\begin{align*}\begin{matrix} \mathbf{Q}[\mathfrak{t}]^{\varepsilon_W}&\rightarrow&\mathbf{Q}[\mathfrak{t}]^W\\ f&\mapsto& \frac{f}{\prod_{\substack{\alpha\in\mathcal{W}(\mathfrak{g})\\\langle\lambda,\alpha\rangle<0}}\alpha}\,, \end{matrix} \end{align*}$$
where
$\varepsilon _W(w)=(-1)^{\ell (w)}$
is the sign character of the Coxeter group W;
$\ell (w)$
denotes the length of
$w\in W$
. This morphism is indeed an isomorphism since
$\prod _{\substack {\alpha \in \mathcal {W}(\mathfrak {g})\\\langle \lambda ,\alpha \rangle <0}}\alpha $
is a generator of the
$\mathbf {Q}[\mathfrak {t}]^W$
-module of anti-invariant polynomials.
6.8
$G\curvearrowright \mathfrak {g}$
Let G be a reductive group and
$\mathfrak {g}$
its Lie algebra with the adjoint action. Then,
$\mathscr {P}_{\mathfrak {g}}$
is in one-to-one correspondence with the set of standard Levi subgroups of G and
$\mathscr {P}_{\mathfrak {g}}/W$
is in bijection with conjugacy classes of Levi subgroups of
$\mathfrak {g}$
. In this case,
$\mathbf {Q}[\mathfrak {t}]^W$
is a polynomial algebra and for
$\lambda \preceq \mu \in X_*(T)$
, the induction map is
$$\begin{align*}\begin{matrix} \mathrm{Ind}_{\lambda,\mu}&\colon&\mathbf{Q}[\mathfrak{t}]^{W^{\lambda}}&\rightarrow &\mathbf{Q}[\mathfrak{t}]^{W^{\mu}}\\ &&f&\mapsto&\sum_{w\in W^{\mu}/W^{\lambda}}w\cdot f\,. \end{matrix} \end{align*}$$
Therefore, the induction maps are surjective. This implies that
$\mathcal {P}_{\lambda }=0$
except if
$\lambda $
is a generic cocharacter (i.e.
$\mathfrak {g}^{\lambda }$
is a torus), in which case
$\mathcal {P}_{\lambda }=\mathbf {Q}$
. The cohomological integrality isomorphism is then
$$\begin{align*}\begin{matrix} \mathbf{Q}[\mathfrak{t}]^W&\rightarrow&\mathbf{Q}[\mathfrak{t}]^W\\ f&\mapsto&(\#W)\cdot f\,. \end{matrix} \end{align*}$$
6.9 Topology of the algebra of invariants
Computing the polynomial invariants of binary forms is a long-standing problem in invariant theory, solved for binary forms with small degrees, and still challenging in higher degrees.
The strong version of the sheafified upgrade of our cohomological integrality theorem, which is the subject of a forthcoming work, would give an algorithm to compute the intersection cohomology of the GIT quotients
$V/\!\!/ G$
for symmetric representations V of a reductive group G.
Conjecture 6.3 (Strong integrality conjecture).
Let
$\lambda \in X_*(T)$
. Then, there is a canonical identification
$\mathcal {P}_{\lambda }\cong \mathrm {IH}^*(V^{\lambda }/\!\!/ G^{\lambda })$
if a general closed orbit of
$G^{\lambda }/G_{\lambda }$
inside
$V^{\lambda }$
has finite stabiliser in
$G^{\lambda }/G_{\lambda }$
, and
$\mathcal {P}_{\lambda }=0$
otherwise.
In particular, for
$G=\mathrm {SL}_2(\mathbf {C})$
, it gives a computation of the intersection cohomology of
$V_{d}/\!\!/ \mathrm {SL}_2$
, where
$V_d$
is the d-dimensional irreducible representation of
$\mathrm {SL}_2(\mathbf {C})$
. In this section, we verify this strong integrality conjecture for some representations of
$\mathrm {SL}_2(\mathbf {C})$
. More precisely, we check that for small values of d, when a general closed orbit in
$V_d$
has finite stabilisers, then
$\dim \mathrm {IH}(V_d/\!\!/ \mathrm {SL}_2(\mathbf {C}))=\dim \mathcal {P}_0$
, where
$\mathcal {P}_0$
is the vector space given by Theorem 1.1 for the trivial cocharacter
$\lambda =0=\lambda _0$
.
In our situation, motivated by integrality results, we also want to determine the stable locus inside
$V_{d}$
, that is, the open subset of closed
$\mathrm {SL}_2(\mathbf {C})$
-orbits with finite stabiliser.
Remark 6.4. While this paper was under refereeing process, the work [Bu+25] appears. The authors solve Conjecture 6.3 under an orthogonality assumption for the representation V of G: they assume that there is a nondegenerate symmetric G-invariant bilinear form on V. This condition is stronger than symmetricity (and a fortiori, weak symmetricity) we assume in the present paper, as demonstrated for example by the natural representation
$\mathbf {C}^{2n}$
of the symplectic group
$\mathrm {Sp}(2n)$
. This example can be treated by a direct approach. In combination with [Reference KnopKno07], it is possible to prove Conjecture 6.3 for any symmetric representation V of G such that
$\mathbf {C}[V]^G=\mathbf {C}$
, that is,
$V/\!\!/ G=\mathrm {pt}$
.
6.9.1 Invariants of linear binary forms
Let
$V_2=\mathbf {C}^2$
be the natural representation of
$\mathrm {SL}_2(\mathbf {C})$
. Then, we calculate easily
$V_2/\!\!/\mathrm {SL}_2(\mathbf {C})=\mathrm {pt}$
since the action of
$\mathrm {SL}_2(\mathbf {C})$
on
$V_2$
has two orbits,
$\{0\}$
and
$V_2\smallsetminus \{0\}$
, of those only
$\{0\}$
is closed. The stabiliser of the closed orbit
$\{0\}$
is
$\mathrm {SL}_2(\mathbf {C})$
, which is not finite. Therefore, the stable locus is empty. One expects that
$\mathcal {P}_{\lambda _0}=0$
. This is confirmed by §6.4.
6.9.2 Invariants of binary quadratic forms
We let
$V_3$
be the
$3$
-dimensional irreducible representation of
$\mathrm {SL}_2(\mathbf {C})$
. There is a single polynomial invariant for binary quadratic forms
$ax^2+bxy+cy^2$
, which is the discriminant
$$\begin{align*}D=b^2-4ac\,, \end{align*}$$
see [Reference SchurSch13, Satz 1.9, 2.8]. Therefore, we have
$V_2/\!\!/\mathrm {SL}_2(\mathbf {C})\cong \mathbf {C}$
. Since
$\dim V_3/\mathrm {SL}_2(\mathbf {C})=0$
and
$\dim \mathbf {C}=1$
, the stable locus is empty. Therefore, we expect that
$\mathcal {P}_{\lambda _0}=0$
, and this is confirmed by §6.5.
6.9.3 Invariants of binary cubic forms
We let
$V_4$
be the
$4$
-dimensional irreducible representation of
$\mathrm {SL}_2(\mathbf {C})$
. There is one polynomial invariant for binary cubic forms
$ax^3+3bx^2y+3cxy^2+dy^3$
, which is the discriminant
$$\begin{align*}D=3b^2c^2+6abcd-4b^3d-4c^3a-a^2d^2\,, \end{align*}$$
see [Reference SchurSch13, Satz 2.8]. Therefore,
$V_4/\!\!/\mathrm {SL}_2(\mathbf {C})\cong \mathbf {C}$
. Moreover, since
$\dim V_4/\mathrm {SL}_2(\mathbf {C})=1=\dim \mathbf {C}$
, the stable locus is non-empty. By
$\mathbf {C}^*$
-equivariance, the stable locus is the open locus of binary cubic forms of non-zero discriminant.
We notice that
$\dim \mathrm {IH}(\mathbf {C})=1=\dim \mathcal {P}_{\lambda _0}=1$
where
$\mathcal {P}_{\lambda _0}$
is given in §6.5 for
$d=4$
.
6.9.4 Invariants of binary quartic forms
Let
$V_5$
be the irreducible
$5$
-dimensional representation of
$\mathrm {SL}_2(\mathbf {C})$
. The ring of invariants of binary quartic forms has two algebraically independent generators
$i,j$
[Reference SchurSch13, Satz 2.9]. Therefore,
$V_5/\!\!/\mathrm {SL}_2(\mathbf {C})\cong \mathbf {C}^2$
. Moreover,
$\dim V_5/\mathrm {SL}_2(\mathbf {C})=2=\dim \mathbf {C}^2$
and therefore, the stable locus is non-empty. In this case, we expect that
$P_{\lambda _0}$
is of dimension
$1=\dim \mathrm {IH}^2(\mathbf {C}^2)$
, and this is again confirmed by §6.5 for
$d=5$
.
Acknowledgements
At various stages of the preparation of this work, the author was supported by the Royal Society, by the National Science Foundation under Grant No. DMS-1928930 and by the Alfred P. Sloan Foundation under grant G-2021-16778, while the author was in residence at the Simons Laufer Mathematical Sciences Institute (formerly MSRI) in Berkeley, California, during the Spring 2024 semester. The author thanks the SLMath Sciences Institute and the University of Edinburgh for the excellent working conditions. The author is grateful to Ben Davison for the support provided, in particular via postdoctoral fellowships, to Woonam Lim for pointing out a mistake in the example section, to Dragoş Frăţilă for interesting comments, to Sebastian Schlegel Mejia for suggesting improvements and corrections, to Dimitri Wyss for interesting related discussions, and to Andrei Neguţ for suggesting me to investigate weakenings of the symmetricity condition, which led to the current version involving weakly symmetric representations (Definition 2.3). I am grateful to Daniel Juteau for discussions on perverse sheaves. I would also like to thank the anonymous referee for allowing me to improve this paper.
Competing interests
The author has no competing interest to declare.
Data availability statement
No data are associated with this study.
