Quasi-reductive supergroups with small even parts

Technical Summary: Quasi-Reductive Supergroups with Small Even Parts

1. Problem Statement
The paper addresses the classification of affine algebraic supergroups $G$ over an algebraically closed field $k$ (of arbitrary characteristic) where the largest even (super)subgroup, denoted $G_{ev}$, is isomorphic to one of the classical groups of rank one: $GL_2$, $SL_2$, or $PSL_2$.

This problem arises naturally in the structural theory of quasi-reductive supergroups. Specifically, if $G$ is a quasi-reductive supergroup and $T$ is a maximal torus in $G_{ev}$, the centralizer of a subtorus $T_\alpha = \ker(\alpha)$ (where $\alpha$ is a root) is itself a quasi-reductive supergroup. The even part of such a centralizer is often a reductive group of semisimple rank one (isomorphic to $SL_2$, $PSL_2$, or $GL_2$). Therefore, a complete description of these "small even part" supergroups is a prerequisite for understanding the centralizers of tori in general quasi-reductive supergroups.

2. Methodology
The author employs the Harish-Chandra pair method, which establishes an equivalence of categories between algebraic supergroups and Harish-Chandra pairs $(G, V)$.

• The Pair: A pair consists of an algebraic group $G$ (here $GL_2, SL_2,$ or $PSL_2$) and a $G$-module $V$ (representing the odd part $G_{\bar{1}}$), equipped with a symmetric bilinear map $[\cdot, \cdot]: V \times V \to \text{Lie}(G)$ satisfying specific equivariance and Jacobi-like identities ($[[v,v],v]=0$).
• Strategy:
  1. Classify all possible $G$-module structures for $V = G_{\bar{1}}$ compatible with the Lie bracket constraints.
  2. Analyze the representation theory of the underlying even groups ($SL_2$ and $GL_2$), particularly focusing on Weyl modules $V(n)$, simple modules $L(n)$, and their behavior in positive characteristic $p$.
  3. Utilize the Loewy length (the length of the composition series) of the module $V$ to inductively classify the structures.
  4. Distinguish between split and non-split extensions, and identify cases where the unipotent radical $R_u(G)$ is purely odd.

3. Key Contributions and Results

The paper provides a complete classification of these supergroups up to isomorphism, divided by the isomorphism type of $G_{ev}$.

A. Case $G_{ev} \simeq SL_2$
The classification depends on the Loewy length $l$ of $G_{\bar{1}}$ and the characteristic $p$.

• Split Case: If $G_{\bar{1}}$ is a trivial module or the bracket is zero, $G$ is a split supergroup ($G \simeq SL_2 \ltimes G_{odd}$).
• Non-Split Cases:
  1. $G_{\bar{1}} \simeq L(1)$: The supergroup is isomorphic to the orthosymplectic supergroup $SpO(2|1)$.
  2. $G_{\bar{1}} \simeq L(0) \oplus L(2)$: The supergroup is denoted $H(0 \oplus 2)$. Here $L(2) \simeq \mathfrak{sl}_2$ (adjoint action) and $L(0)$ is trivial. The derived subgroup is a semi-direct product of $SL_2$ and a unipotent radical.
  3. $G_{\bar{1}} \simeq V(3)$ (Weyl module of highest weight 3): This case occurs only if $p=3$. The resulting supergroup is denoted $H(3/1)$, which is almost-simple.
• Main Theorem (3.18): Any quasi-reductive supergroup with $G_{ev} \simeq SL_2$ is an extension of one of the above by a purely odd unipotent radical.

B. Case $G_{ev} \simeq PSL_2$
This case is reduced to the $SL_2$ case by considering the action of the center $-I_2$.

• The only non-trivial cases correspond to $H(0 \oplus 2)/\mu_2$.
• The supergroup $H(3/1)$ does not descend to $PSL_2$ because the module $V(3)$ does not admit a trivial action by $-I_2$ in the required way for the quotient.

C. Case $G_{ev} \simeq GL_2$
This is the most complex case. The classification relies on the decomposition of $G_{\bar{1}}$ into weight spaces $M_k$ with respect to the center $Z \simeq \mathbb{G}_m$ of $GL_2$. The Lie bracket $[M_k, M_{-k}]$ determines the structure.

• Case $[G_{\bar{1}}, G_{\bar{1}}] = \mathfrak{z}$ (Center): Corresponds to Proposition 5.2.
• Case $[G_{\bar{1}}, G_{\bar{1}}] = \mathfrak{sl}_2$:
  • If $k=0$ (trivial center action on the bracket), $G_{\bar{1}} \simeq \mathfrak{gl}_2$. This yields a family of supergroups $Q(2; a, c)$, which are deformations of the Queer supergroup $Q(2)$. The isomorphism class depends on the ratio of parameters $a, c$.
  • If $k \neq 0$ and $p|k$, the structure involves modules like $L((t, t))$ and $L((t+1, t-1))$. This leads to families $S(t)$ and $L(t)$.
• Case $[G_{\bar{1}}, G_{\bar{1}}] = \mathfrak{gl}_2$:
  • If $p \nmid k$, the odd part is $V(\lambda) \oplus V(\lambda)^*$ where $\lambda = (t, t-1)$. This yields the family $H(t)$.
  • Isomorphism Result: $H(t) \simeq H(t')$ if and only if $t=t'$ or $t+t'=1$. Remarkably, all Lie superalgebras $\mathfrak{h}(t)$ are isomorphic to $\mathfrak{sl}(2|1)$, but the supergroups are distinct for different $t$.
  • If $p|k$, the structure involves non-trivial kernels and leads to the families $S(t)$ and $L(t)$ mentioned above.

4. Applications to Centralizers of Tori
The paper applies these classification results to describe the structure of centralizers $Cent_G(S)$ for tori $S$ in a quasi-reductive supergroup $G$.

• Reductivity of Centralizers: The author proves that if $G$ is a quasi-reductive and reductive supergroup, then its Cartan supersubgroup $T$ (the centralizer of a maximal torus) is also reductive.
• Structure of $G_\alpha$: For a root $\alpha$, the centralizer $G_\alpha = Cent_G(T_\alpha)$ is shown to be a quasi-reductive supergroup. The paper explicitly describes its solvable radical $R(G_\alpha)$ and the reductive quotient $G_\alpha/R(G_\alpha)$ using the classification from Sections 3–5.
• Hypothesis 6.6: The author conjectures that centralizers of tori in any quasi-reductive supergroup are reductive, extending a known result from classical algebraic groups.

5. Significance

• Structural Foundation: This work fills a critical gap in the representation theory of supergroups by providing the "building blocks" (rank one even parts) necessary for analyzing general quasi-reductive supergroups.
• Characteristic Dependence: The results highlight the profound impact of the field characteristic $p$. New families of supergroups (like $H(3/1)$) emerge only in specific characteristics ($p=3$), and the classification of $GL_2$-based supergroups is heavily dependent on whether $p$ divides the weight indices.
• Deformation Theory: The identification of $Q(2; a, c)$ as deformations of the Queer supergroup $Q(2)$ provides new examples of non-classical supergroup structures.
• Methodological Rigor: The use of Harish-Chandra pairs combined with detailed modular representation theory (Loewy filtrations, block decompositions) offers a robust framework for future classifications of higher-rank supergroups.

In summary, Zubkov provides a comprehensive taxonomy of supergroups with small even parts, resolving the classification for $SL_2, PSL_2,$ and $GL_2$ even parts, and demonstrating their utility in understanding the local structure (centralizers) of larger quasi-reductive supergroups.