Irreducible modules for pseudo-reductive groups

Bate, Michael; Stewart, David I.

doi:10.4171/jems/1153

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Cited by 4 publications

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“…) of k i , where v p (λ i ) is the highest power of p [BS22,Thm. 5.8] gives the dimension of L C (λ) as [κ : k] where κ is the compositum of the k i (λ i ) in k. We need a slight upgrade of this result.…”

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A construction of pseudo-reductive groups with non-reduced root system

Bate¹,

Röhrle²,

Sercombe³

et al. 2022

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confidence: 99%

A construction of pseudo-reductive groups with non-reduced root system

Bate¹,

Röhrle²,

Sercombe³

et al. 2022

Preprint

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A Construction of Pseudo-reductive Groups with Non-reduced Root Systems

Bate,

Röhrle,

Sercombe

et al. 2024

Transformation Groups

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Complete reducibility for Lie subalgebras and semisimplification

Bate,

Böhm,

Martin

et al. 2023

European Journal of Mathematics

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Let G be a connected reductive linear algebraic group over a field k. Using ideas from geometric invariant theory, we study the notion of G-complete reducibility over k for a Lie subalgebra $${\mathfrak {h}}$$ h of the Lie algebra $${\mathfrak {g}} = \hbox {Lie} (G)$$ g = Lie ( G ) of G and prove some results when $${\mathfrak {h}}$$ h is solvable or $$\textrm{char}\hspace{0.55542pt}(k)= 0$$ char ( k ) = 0 . We introduce the concept of a k-semisimplification$${\mathfrak {h}}'$$ h ′ of $${\mathfrak {h}}$$ h ; $${\mathfrak {h}}'$$ h ′ is a Lie subalgebra of $${\mathfrak {g}}$$ g associated to $${\mathfrak {h}}$$ h which is G-completely reducible over k. This is the Lie algebra counterpart of the analogous notion for subgroups studied earlier by the first, third and fourth authors. As in the subgroup case, we show that $${\mathfrak {h}}'$$ h ′ is unique up to $$\textrm{Ad}\hspace{0.55542pt}(G(k))$$ Ad ( G ( k ) ) -conjugacy in $${\mathfrak {g}}$$ g . Moreover, we prove that the two concepts are compatible: for H a closed subgroup of G and $$H'$$ H ′ a k-semisimplification of H, the Lie algebra $$\textrm{Lie}\hspace{0.55542pt}(H')$$ Lie ( H ′ ) is a k-semisimplification of $$\textrm{Lie}\hspace{0.55542pt}(H)$$ Lie ( H ) .

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Irreducible modules for pseudo-reductive groups

Cited by 4 publications

References 6 publications

A construction of pseudo-reductive groups with non-reduced root system

A construction of pseudo-reductive groups with non-reduced root system

A Construction of Pseudo-reductive Groups with Non-reduced Root Systems

Complete reducibility for Lie subalgebras and semisimplification

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