Borel-Weil Theorem for Algebraic Supergroups

Abstract

We study the structure of an algebraic supergroup $\mathbb{G}$ and establish the Borel-Weil theorem for $\mathbb{G}$ to give a systematic construction of all simple supermodules over an arbitrary field. Especially when $\mathbb{G}$ has a distinguished parabolic super-subgroup, we show that the set of all simple supermodules of $\mathbb{G}$ is parameterized by the set of all dominant weights for the even part of $\mathbb{G}$, prove a super-analogue of the Kempf vanishing theorem, and give a description of Euler characteristics.

arXiv comment34 pages: The title have been changed. Some comments have been added. There are also some minor changes

References in corpus (7)

• A new proof of the Mullineux conjecture
• Chevalley Supergroups
• Modular representations of the ortho-symplectic supergroups
• Semisimple algebraic tensor categories
• Algebraic supergroups with Lie superalgebras of classical type
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Cited by in corpus (5)

• Affine algebraic super-groups with integral
• Classification of irreducible representations of affine group superschemes and the division superalgebras of their endomorphisms
• Frobenius kernels of algebraic supergroups and Steinberg's tensor product theorem
• Color-Flavor Transformation Revisited
• Mixed Tensor Products, Capelli Berezinians, and Newton's Formula for $\mathfrak{gl}(m|n)$