[Paper Review] A criterion for irreducibility of parabolic baby Verma modules of reductive Lie algebras

Yiyang Li, Bin Shu+1 more|arXiv (Cornell University)|Apr 19, 2014

Advanced Algebra and Geometry6 references3 citations

TL;DR

This paper establishes a sufficient condition for the irreducibility of parabolic baby Verma modules in reductive Lie algebras of types $A_n$, $B_n$, $C_n$, and $D_n$ over an algebraically closed field of prime characteristic $p$, when the $p$-character $\chi$ is of standard Levi form. The key result shows that such modules are irreducible when the $p$-regular weight $\lambda_0$ lies in specific alcoves, partially answering a question by Friedlander and Parshall on irreducibility of induced modules.

ABSTRACT

Let $G$ be a connected, reductive algebraic group over an algebraically closed field $k$ of prime characteristic $p$ and $\mathfrak{g}=Lie(G)$. In this paper, we study representations of $\mathfrak{g}$ with a $p$-character $χ$ of standard Levi form. When $\mathfrak{g}$ is of type $A_n, B_n, C_n$ or $D_n$, a sufficient condition for the irreducibility of standard parabolic baby Verma $\mathfrak{g}$-modules is obtained. This partially answers a question raised by Friedlander and Parshall in [Friedlander E. M. and Parshall B. J., Deformations of Lie algebra representations, Amer. J. Math. 112 (1990), 375-395]. Moreover, as an application, in the special case that $\mathfrak{g}$ is of type $A_n$ or $B_n$, and $χ$ lies in the sub-regular nilpotent orbit, we recover a result of Jantzen in [Jantzen J. C., Subregular nilpotent representations of $sl_n$ and $so_{2n+1}$, Math. Proc. Cambridge Philos. Soc. 126 (1999), 223-257].

Motivation & Objective

• To address a long-standing open question by Friedlander and Parshall on necessary and sufficient conditions for irreducibility of induced modules from parabolic subalgebras.
• To study representations of reductive Lie algebras with $p$-characters of standard Levi form, particularly in the context of modular representation theory.
• To provide a sufficient condition for the irreducibility of parabolic baby Verma modules in types $A_n$, $B_n$, $C_n$, and $D_n$.
• To recover and generalize a result of Jantzen on subregular nilpotent orbits in types $A_n$ and $B_n$.

Proposed method

• The authors define parabolic baby Verma modules as induced modules from irreducible representations of Levi subalgebras $\mathfrak{g}_J$ with trivial action on the nilradical $\mathfrak{u}_J^+$.
• They use the decomposition $\lambda = \lambda_0 + p\lambda_1$ with $\lambda_0 \in X_1'(T)$, the set of representatives for $X(T)/pX(T)$, to analyze weights in the first dominant alcove $C_0$.
• The proof relies on the action of the affine Weyl group and the use of the dot action to relate weights in the same block.
• They apply results from support variety theory and the structure of $U_\chi(\mathfrak{g})$-modules to analyze irreducibility.
• For types $A_n$ and $B_n$, they verify that the condition $\lambda_0 \in C_0$ with $\lambda_0 + \rho$ satisfying certain bounds on root sum implies irreducibility of the induced module.
• They use explicit computation of Weyl group orbits and weight shifts to verify that the induced modules remain irreducible under the given conditions.

Experimental results

Research questions

• RQ1 Under what conditions is a parabolic baby Verma module irreducible when the $p$-character $\chi$ is of standard Levi form?
• RQ2 Can a sufficient condition for irreducibility of such modules be established in types $A_n$, $B_n$, $C_n$, and $D_n$?
• RQ3 Does the condition on $\lambda_0$ lying in the first dominant alcove $C_0$ with bounded root sum ensure irreducibility of the induced module?
• RQ4 Can the result recover Jantzen’s theorem on subregular nilpotent orbits in $\mathfrak{sl}_n$ and $\mathfrak{so}_{2n+1}$?
• RQ5 Is the irreducibility preserved under the action of the affine Weyl group in the block of a $p$-regular weight?

Key findings

• For reductive Lie algebras of type $A_n$, $B_n$, $C_n$, and $D_n$, a sufficient condition for the irreducibility of parabolic baby Verma modules is established when the $p$-regular weight $\lambda_0$ lies in the first dominant alcove $C_0$ and satisfies $0 \leq \langle \lambda_0 + \rho, \alpha^\vee \rangle < p$ for all positive roots $\alpha$.
• In type $A_n$, the induced module $\widehat{\mathcal{Z}}_P(\lambda)$ is irreducible if $\lambda_0 + \rho = (r_1, \dots, r_n)$ with $0 \leq \sum r_i \leq p$, and the dimension of the module is $r_{n-i} p^{N-1}$, matching known results from [6, Theorem 2.6].
• In type $B_n$, the induced module $\widehat{\mathcal{Z}}_P(\lambda)$ is irreducible when $\lambda_1 + \rho = (r_1, \dots, r_n)$ with $0 \leq \sum_{i=1}^{n-1} 2r_i + r_n \leq p$, and the dimension is $r_i p^{N-1}$ for $1 \leq i \leq n-1$ and $r_{2n-i} p^{N-1}$ for $n+1 \leq i \leq 2n-1$, consistent with [6, Proposition 3.13].
• The authors recover Jantzen’s result on subregular nilpotent orbits in $\mathfrak{sl}_n$ and $\mathfrak{so}_{2n+1}$ as a special case when $\chi$ lies in the subregular nilpotent orbit and $\lambda_0$ satisfies the alcove condition.
• The proof confirms that the induced module remains irreducible under the action of the affine Weyl group, provided $\lambda_0$ lies in $C_0$ and the weight is $p$-regular.
• The condition on $\lambda_0$ being in $C_0$ ensures that all Weyl group translates $\lambda_i$ also lie in $C_0$, which is essential for irreducibility via the structure of the block.