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Journal of Lie Theory 33 (2023), No. 3, 887--918
Copyright Heldermann Verlag 2023

Representations of the Special Lie Superalgebra with p-Character of Height One

Feifei Duan
(1) School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, Hebei, P.R. China
(2) Hebei Key Laboratory of Computational Mathematics and Applications, Shijiazhuang, Hebei, P.R. China


\def\bk{\mathbf{k}} \def\ggg{{\frak g}} \def\gl{\mathfrak{gl}(n)} \def\uuu{U_{\chi}(\frak g)} Let $\bk$ be an algebraically closed field of prime characteristic and $S(n)$ be the special Lie superalgebra of Cartan type over $\bk$. Define $\bar{S}(n)=S(n)\oplus\bk\mbox{-}\{\xi_1D_1 \}$. So $\bar{S}(n)_0\cong\gl$. Let $\ggg=S(n)$ or $\bar{S}(n)$. We investigate in this paper the representations of $\ggg$ when $\chi$ is restricted or $\mathrm{ht}(\chi)=1$. The main results are listed below.\\ (1) When $\mathrm{ht}(\chi)=1$, the irreducible representations of $U_{\chi}(\ggg)$ are considered. Precisely, the composition factors of the Kac modules are confirmed and the dimensions of simple modules are given.\\ (2) When $\chi=0$ or $\mathrm{ht}(\chi)=1$, the structures of indecomposable projective modules are studied and the Cartan invariants of $\uuu$ are given.\\ (3) When $\chi=0$ or $\mathrm{ht}(\chi)=1$, we show that the representation category over $U_{\chi}(\ggg)$ has only one block (reckoning parities in).

Keywords: Special Lie superalgebra, irreducible representations, projective representations, Cartan invariants, block.

MSC: 17B10,17B50,17B35.

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