Elliptic Twisted Levi Subgroups

• Elliptic twisted Levi subgroups are E-pseudo-Levi subgroups defined as the connected centralizers associated with degree zero semistable G-bundles on elliptic curves.
• They stratify the moduli space of G-bundles via a refined Jordan–Chevalley decomposition, linking group theory insights with elliptic curve geometry.
• Their classification employs Borel–de Siebenthal theory and affine Dynkin diagrams, yielding explicit examples for both classical and exceptional groups.

Elliptic twisted Levi subgroups, also known as $E$-pseudo-Levi subgroups, arise in the study of degree zero semistable $G$-bundles over elliptic curves, where $G$ is a connected reductive group and $E$ is an irreducible curve of arithmetic genus one. These subgroups and their associated root-theoretic structures play a fundamental role in describing the stratification of the moduli of bundles, generalizing the concept of Levi subgroups in the context of elliptic curves, and refining the classical Jordan–Chevalley decomposition for $G$-bundles on $E$ (Frăţilă et al., 2020).

1. Definition of Elliptic Twisted Levi Subgroups

Let $G$ be a connected reductive group with maximal torus $T \subset G$, character lattice $X^*(T)$, and root system $\Phi \subset X^*(T)$. Over the elliptic curve $E$, the torus of $E$-points is $T_E \cong \operatorname{Hom}(X^*(T), J(E))$, where $J(E) = \mathrm{Pic}^0(E)$ is the Jacobian of $E$. For any $p \in T_E$, define the $E$-root subsystem

$$\Sigma_p := \{ \alpha \in \Phi \mid \alpha_*(p) = 1_{J(E)} \},$$

where $\alpha_*: T_E \to J(E)$ arises from the character $\alpha$. The subsystems $\Sigma_p$ are closed root subsytems, and every such closed subsystem corresponds to some $p$ via Borel–de Siebenthal theory and elliptic arguments.

The associated connected reductive subgroup is

$G(\Sigma) := C_G(Z(\Sigma))^\circ,$

where $$Z(\Sigma) = \bigcap_{\alpha \in \Sigma}\ker(\alpha: T \to \mathbb{G}_m)$$.

Definition: An $E$-pseudo-Levi subgroup of $G$ is any subgroup of the form

$H = \operatorname{Stab}_G(p)^\circ = G(\Sigma_p)$

for some semisimple $p \in G_E$ (moduli stack of degree 0 semistable $G$-bundles) or equivalently $p \in T_E$. $H$ is also characterized as the connected centralizer in $G$ of a degree zero element of the $E$-points of the center $Z(H)_E$ [(Frăţilă et al., 2020), Prop 3.3.3].

2. Stratification of Semistable Moduli Stacks and the Jordan–Chevalley Decomposition

Let $G_E = \mathrm{Bun}_G^{0,ss}(E)$ denote the moduli stack of degree 0 semistable $G$-bundles on $E$, and let $\widehat{G}_E$ be the space of framed bundles at a chosen basepoint $x_0$, so that $\widehat{G}_E \to G_E$ is a $G$-torsor.

The key properties are:

• The substacks $(G_E)_{[H]}$ of bundles whose semisimple part has connected centralizer conjugate to $H$ provide a locally-closed decomposition:

$G_E = \bigsqcup_{[H]} (G_E)_{[H]}$

indexed by $G$-conjugacy classes of $E$-pseudo-Levi subgroups $H$.

• For each such $H$, let $Z(H)_E^{\text{reg}}$ denote the open locus of regular central (degree 0) $H$-bundles with full stabilizer in $G$ exactly $H$, and $H^{\text{uni}}$ the locus of unipotent framed $H$-bundles. The induction map

$$\pi_H : Z(H)_E^{\text{reg}} \times H^{\text{uni}} \to \widehat{G}_E$$

is a $W_{G,H}$-Galois cover onto the framed stratum over $(G_E)_{[H]}$, with $W_{G,H} := N_G(H)/H$ the relative Weyl group.

• Every framed bundle $p \in \widehat{G}_E$ admits a unique factorization (Jordan–Chevalley decomposition):

$p = p_s \cdot p_u,$

where $p_s \in Z(H)_E^{\text{reg}}$, $p_u \in H^{\text{uni}}$, and $H = \operatorname{Stab}_G(p_s)^\circ$ [(Frăţilă et al., 2020), Thms 4.1.1, 4.3.1–4.3.2].

3. Classification via Borel–de Siebenthal and Extended Dynkin Diagrams

Connected reductive subgroups of $G$ containing $T$ correspond to closed root subsystems $\Sigma \subset \Phi$ arising as intersections with the $\mathbb{Z}$-span of subsets of the "extended simple roots" $\widetilde{\Delta} = \Delta \cup \{\alpha_0\}$, where $\alpha_0 = -(\text{highest root})$. This is encoded in the affine Dynkin diagram for $G$:

• Levi subgroups are characterized by deleting exactly one node.
• $E$-pseudo-Levi subgroups arise by deleting two nodes (possibly from different components) and forming the connected centralizer for the residual subdiagram.

For each such deletion, one obtains:

• The closed root subsystem $\Sigma_H$ from the disconnected Dynkin subdiagrams.
• $H = G(\Sigma) = C_G(Z(\Sigma))^\circ$ with corresponding semisimple type.
• The defining cocharacter $p \in T_E$ such that $\alpha_i(p) = 1_{J(E)}$ for the deleted nodes and no further vanishing.
• The relative Weyl group $W_{G,H} = \operatorname{Norm}_W(\Sigma) / W_\Sigma$, typically a finite abelian or symmetric group.

This provides a classification of $E$-pseudo-Levi subgroups up to $W$-conjugacy in terms of the affine Dynkin diagram, deleting pairs of nodes, and centralizer structure in $G$ [(Frăţilă et al., 2020), Thm 3.2.2, Lemma A.1.1, Prop A.2.3].

4. Explicit Examples for Classical and Exceptional Types

The classification yields detailed cases for different types of groups:

Group $G$	Deletion Pattern	Resulting $H$	Relative Weyl Group $W_{G,H}$
$A_n$ ($SL_{n+1}$)	Remove nodes $i,j$	$S(GL_{j-i} \times GL_{n+1-(j-i)})$	$S_2$ if block sizes equal, else $1$
$G_2$	Remove $\{0,1\}$ or $\{0,2\}$	$SL_2 \times SL_2 / \Delta\mu_2$ or $SL_2$	$\mathbb{Z}/2$ or $1$
$C_3$ ($Sp_6$)	Remove two nodes (various)	$Sp_4 \times SL_2$, $SL_2 \times Sp_4$	$1$
		$SL_2^3$	$S_3$
$D_4$ ($Spin_8$)	Remove two opposite legs	$(SL_2)^4 / \mu_2^{\mathrm{diag}}$	$\mathbb{Z}/2 \times \mathbb{Z}/2$

The table entries correspond to explicit removal of nodes from the affine Dynkin diagram, yielding closed root subsystems and corresponding $E$-pseudo-Levi subgroups per the established rules (Frăţilă et al., 2020).

5. Significance in the Theory of $G$-Bundles on Elliptic Curves

The partition of the moduli stack of semistable $G$-bundles into strata labeled by $E$-pseudo-Levi subgroups underpins a refinement of the Jordan–Chevalley decomposition for bundles on elliptic curves. This structural insight links the geometry of $G$-bundles to group-theoretic invariants arising from root system combinatorics and centralizer analysis. For instance, the loci of framed unipotent bundles on an ordinary elliptic curve are equivariantly isomorphic to the unipotent cone in $G$—a result that allows for explicit analysis of moduli in terms of simpler group-theoretic data (Frăţilă et al., 2020).

6. Connections to Representation Theory and Algebraic Geometry

The introduction and classification of elliptic twisted Levi subgroups synthesize earlier machinery from the theory of moduli and group actions with new stratifications and factorization phenomena specific to genus one curves. The partition by connected stabilizers relates to the Tannakian formalism and to the description of moduli via root-theoretic and combinatorial constructions, leveraging the Borel–de Siebenthal algorithm refined by elliptic structures. These results are central to understanding higher-level phenomena such as the behavior of the unipotent cone, stable degenerations of moduli, and explicit isomorphism classes for $G$-bundles with additional structures (Frăţilă et al., 2020).