Modular representations of strange classical Lie superalgebras and the first super Kac-Weisfeiler conjecture

TL;DR

This paper explores modular representations of certain queer and periplectic Lie superalgebras over fields of characteristic p>2, verifying a conjecture on the maximal dimensions of irreducible modules for these structures.

Contribution

It initiates the study of modular representations of periplectic Lie superalgebras and verifies the first super Kac-Weisfeiler conjecture for these cases.

Findings

Verification of the super Kac-Weisfeiler conjecture for periplectic Lie superalgebras.

First analysis of modular representations for queer and periplectic Lie superalgebras.

Extension of known results to new classes of Lie superalgebras.

Abstract

Suppose $\mathfrak{g}=\mathfrak{g}_{\bar 0}+\mathfrak{g}_{\bar 1} is a Lie superalgebra of queer type or periplectic type over an algebraically closed field $\textbf{k}$ of characteristic $p>2$. In this article, we initiate preliminarily to investigate modular representations of periplectic Lie superalgebras and then verify the first super Kac-Weisfeiler conjecture on the maximal dimensions of irreducible modules for $\mathfrak{g}$ proposed by the second-named author in [Shu] where the conjecture is targeted at all finite-dimensional restricted Lie superalgebras over $\bk$, and already proved to be true for basic classical Lie superalgebras and completely solvable restricted Lie superalgebras.