Coincidence of the two Springer correspondences for the dual of a classical Lie algebra in characteristic 2 (Xue's Remark 5.19 expectation)

Let $k$ be an algebraically closed field of characteristic $2$ and let $G_{ad}$ be an adjoint algebraic group of type $B$, $C$ or $D$ over $k$, with Lie algebra $\mathfrak{g}_{ad}$ and dual space $\mathfrak{g}_{ad}^{*}$. Two a priori different Springer correspondences are known for $\mathfrak{g}_{ad}^{*}$: the correspondence constructed directly by Xue (2009) by sheaf-theoretic methods on the simply connected group of the same type and transported across the isogeny class, and the correspondence obtained by applying the Fourier-Deligne transform to the Springer correspondence for $\mathfrak{g}_{ad}$ constructed in the companion paper. Xue stated (Remark 5.19) the expectation that these two bijections from irreducible representations of the Weyl group to pairs of nilpotent coadjoint orbits and irreducible equivariant local systems coincide up to tensoring by the sign representation, and gave no proof. The problem is to prove or disprove this expectation, or to determine the precise relation between the two constructions, for all types $B_n$, $C_n$, $D_n$.

Background

Let $k$ be an algebraically closed field of characteristic $2$ and let $G$ be a connected reductive algebraic group over $k$ of classical type, i.e. symplectic type $C_n$ or orthogonal types $B_n$ and $D_n$, or a group isogenous to one of these (two groups are isogenous if their root data differ only by the intermediate lattice between the character lattice and its dual; the simply connected and adjoint groups are the two extremes of an isogeny class). Write $\mathfrak{g}$ for the Lie algebra of $G$ and $\mathfrak{g}^{*}$ for its dual vector space. The group $G$ acts on $\mathfrak{g}^{*}$ by the coadjoint action, $g.\xi(x)=\xi(\operatorname{Ad}(g^{-1})x)$ for $g\in G$, $\xi\in\mathfrak{g}^{*}$, $x\in\mathfrak{g}$. Following Kac and Weisfeiler, an element $\xi\in\mathfrak{g}^{*}$ is called nilpotent if it annihilates the Lie algebra of some Borel subgroup (a maximal connected solvable subgroup) of $G$; its orbit under the coadjoint action is a nilpotent coadjoint orbit.

The Springer correspondence is a fundamental bridge between the geometry of nilpotent orbits and the representation theory of Weyl groups. The Weyl group $W=N_{G}(T)/T$ is the finite reflection group determined by a maximal torus $T$ of $G$. In its sheaf-theoretic form, the correspondence is a bijection from the set $\operatorname{Irr}(W)$ of isomorphism classes of irreducible representations of $W$ to the set of pairs $(\mathcal{O},\mathcal{F})$ where $\mathcal{O}$ is a nilpotent orbit and $\mathcal{F}$ is an irreducible $G$-equivariant $\overline{\mathbb{Q}}_{\ell}$-local system (a locally constant sheaf for a prime $\ell\neq\operatorname{char}k$) on $\mathcal{O}$, up to isomorphism; such local systems are classified by the irreducible characters of the component group $A_{G}(\xi)=Z_{G}(\xi)/Z_{G}^{0}(\xi)$, the group of connected components of the centralizer of a point $\xi\in\mathcal{O}$. Springer constructed the correspondence for $\mathfrak{g}$ when the characteristic is large, and Lusztig generalized it (for the group, in arbitrary characteristic) using intersection cohomology complexes, which are the simple objects of the category of perverse sheaves (an abelian category of constructible $\ell$-adic complexes adapted to this theory).

When the characteristic is good, $\mathfrak{g}$ carries a $G$-invariant nondegenerate bilinear form identifying $\mathfrak{g}$ with $\mathfrak{g}^{*}$, and the adjoint and coadjoint theories coincide. In characteristic $2$ for types $B$, $C$, $D$ the invariant trace form is degenerate, so $\mathfrak{g}$ and $\mathfrak{g}^{*}$ are not $G$-equivariantly isomorphic, and nilpotent orbits in $\mathfrak{g}^{*}$ form a genuinely different family: Xue classified them for symplectic and orthogonal groups over algebraically closed and finite fields, computed the component groups of the centralizers, and constructed a Springer correspondence $\gamma_{dir}:\operatorname{Irr}(W)\to\mathcal{S}_{\mathfrak{g}^{*}}$ directly, by a method parallel to Lusztig's construction for the group, where $\mathcal{S}_{\mathfrak{g}^{*}}$ denotes the set of pairs $(\mathcal{O},\mathcal{F})$ as above with $\mathcal{O}$ a nilpotent coadjoint orbit. Her construction proceeds through the variety of pairs $(\xi,gB)$ with $g^{-1}.\xi$ in the annihilator of the Borel subalgebra and a principal $W$-bundle over the regular semisimple elements of $\mathfrak{g}^{*}$, and it relies on the theorem of Kac-Weisfeiler that regular semisimple elements of $\mathfrak{g}^{*}$ exist and form an open dense subset, which is guaranteed when $G$ is simply connected and can fail otherwise (this is Lemma 5.2 and Remark 5.3 of the source paper).

Independently, for the adjoint group $G_{ad}$ of type $B$, $C$ or $D$, a Springer correspondence for the Lie algebra $\mathfrak{g}_{ad}$ itself was constructed in the companion paper (assuming $G_{ad}$ adjoint for the analogous reason). The Fourier-Deligne transform is a standard equivalence on the derived category of $\ell$-adic constructible sheaves from a vector space $V$ to its dual $V^{*}$, defined by the kernel $\psi(\langle v,\xi\rangle)$ for a nontrivial additive character $\psi$; it carries $G_{ad}$-equivariant perverse sheaves on $\mathfrak{g}_{ad}$ supported on the nilpotent cone to $G_{ad}$-equivariant perverse sheaves on $\mathfrak{g}_{ad}^{*}$, and hence converts the Springer correspondence for $\mathfrak{g}_{ad}$ into a second Springer correspondence $\gamma_{FT}:\operatorname{Irr}(W)\to\mathcal{S}_{\mathfrak{g}^{*}}$ for the dual.

These two routes to a Springer correspondence on $\mathfrak{g}^{*}$ are logically independent: the direct one is built for the simply connected group and transported across the isogeny class (later work of Xue established that nilpotent coadjoint orbits and component groups of centralizers are naturally identified for groups in the same isogeny class), while the Fourier one is built on the adjoint group. In Remark 5.19 of the published paper, Xue states the expectation that the two correspondences coincide up to tensoring by the sign representation $\operatorname{sgn}$ of $W$ (the one-dimensional character on which each simple reflection acts as $-1$), i.e. $\gamma_{FT}(E)=\gamma_{dir}(E\otimes\operatorname{sgn})$ for every $E\in\operatorname{Irr}(W)$. The sign twist is the natural guess: in type $A$, where the correspondence is classical, the zero and regular nilpotent orbits carry the sign and trivial representations respectively, and these are exactly the two extremes interchanged by the Fourier transform. No proof of the expectation is known in bad characteristic, where the classical good-characteristic arguments break down because there is no $G$-invariant identification of $\mathfrak{g}$ with $\mathfrak{g}^{*}$.

Problem Statement

Let $k$ be an algebraically closed field of characteristic $2$, let $G_{ad}$ be an adjoint connected algebraic group of type $B_{n}$, $C_{n}$ or $D_{n}$ over $k$, let $W$ be its Weyl group, and let $\mathfrak{g}_{ad}$ and $\mathfrak{g}_{ad}^{*}$ be its Lie algebra and the dual vector space. Let $\mathcal{S}_{\mathfrak{g}^{*}}$ denote the set of pairs $(\mathcal{O},\mathcal{F})$, up to isomorphism, where $\mathcal{O}$ is a nilpotent coadjoint orbit in $\mathfrak{g}_{ad}^{*}$ and $\mathcal{F}$ is an irreducible $G_{ad}$-equivariant $\overline{\mathbb{Q}}_{\ell}$-local system on $\mathcal{O}$ ($\ell\neq 2$). Consider the following two bijections from $\operatorname{Irr}(W)$ to $\mathcal{S}_{\mathfrak{g}^{*}}$:

(1) $\gamma_{dir}$, the Springer correspondence for $\mathfrak{g}^{*}$ constructed directly by Xue for the simply connected group of the same type, transported to $G_{ad}$ through the natural identification of nilpotent coadjoint orbits and component groups of centralizers across the isogeny class;

(2) $\gamma_{FT}$, the Springer correspondence for $\mathfrak{g}_{ad}^{*}$ obtained by applying the Fourier-Deligne transform to the Springer correspondence for $\mathfrak{g}_{ad}$ (constructed by Xue for the adjoint group).

Prove or disprove that, for every type $B_{n}$, $C_{n}$, $D_{n}$ and every irreducible representation $E\in\operatorname{Irr}(W)$, one has $\gamma_{FT}(E)=\gamma_{dir}(E\otimes\operatorname{sgn})$, where $\operatorname{sgn}$ is the sign representation of $W$. A proof of this coincidence must establish it for all types $B_{n}$, $C_{n}$, $D_{n}$ and all $E\in\operatorname{Irr}(W)$; a single type and a single $E\in\operatorname{Irr}(W)$ with $\gamma_{FT}(E)\neq\gamma_{dir}(E\otimes\operatorname{sgn})$ refutes it. If the coincidence fails, the task then requires determining, for each type $B_{n}$, $C_{n}$, $D_{n}$, the precise relation between $\gamma_{FT}$ and $\gamma_{dir}$ on $\operatorname{Irr}(W)$; this determination is a requirement in its own right, so a bare counterexample refutes the coincidence but does not by itself resolve the task. In particular, make explicit how the identification of nilpotent coadjoint orbits and centralizer component groups between the simply connected and adjoint groups enters the comparison, so that the statement is well posed for the adjoint group on which the Fourier construction naturally lives.

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