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LSpice
LSpice
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irreducible Irreducible non-reductive subgroup in GL(n) over a characteristic 0 field

Let $V$ be a nan $n$-dimensional vector space over a characteristic $0$ field $k$ (or better, let $V=\mathbb{A}^n_k$). I wonder whether the following is true: $$\begin{align*} &\text{Absolutely irreducible subgroup $H$ of GL$(n)$=End$(V)$, where the latter two groups are regarded as algebraic}\\ &\text{groups, are reductive.} \end{align*}$$ By

Absolutely irreducible subgroups $H$ of $\operatorname{GL}(n)=\operatorname{Aut}(V)$, where the latter two groups are regarded as algebraic groups, are reductive.

By absolute irreducibility I mean the subgroup $H_K$ acts irreducibly on $V_K$ for any field extension $k\to K$, i.e., there is no parabolic subgroup of GL$(n)$$\operatorname{GL}(n)$ containing $H$. If it is true, I would like a reference to a proof; if it is not true, I wonder where I can find a counter-example.

My opinion on this is that I need to see whether the set of irreducible groups can inject into the set of reductive groups (or the set of non-reductive ones can inject into the set of non-irreducible ones). On the one hand, there is a correspondence of parabolic subgroups and cocharacters of GL$(n)$$\operatorname{GL}(n)$. However, on the other hand, I don't know a classification of reductive subgroups or that of non-reductive ones.

Any help will be appreciated.

irreducible non-reductive subgroup in GL(n) over a characteristic 0 field

Let $V$ be a n-dimensional vector space over a characteristic $0$ field $k$ (or better, let $V=\mathbb{A}^n_k$). I wonder whether the following is true: $$\begin{align*} &\text{Absolutely irreducible subgroup $H$ of GL$(n)$=End$(V)$, where the latter two groups are regarded as algebraic}\\ &\text{groups, are reductive.} \end{align*}$$ By absolute irreducibility I mean the subgroup $H_K$ acts irreducibly on $V_K$ for any field extension $k\to K$, i.e., there is no parabolic subgroup of GL$(n)$ containing $H$. If it is true, I would like a reference to a proof; if it is not true, I wonder where I can find a counter-example.

My opinion on this is that I need to see whether the set of irreducible groups can inject into the set of reductive groups (or the set of non-reductive ones can inject into the set of non-irreducible ones). On the one hand, there is a correspondence of parabolic subgroups and cocharacters of GL$(n)$. However, on the other hand, I don't know a classification of reductive subgroups or that of non-reductive ones.

Any help will be appreciated.

Irreducible non-reductive subgroup in GL(n) over a characteristic 0 field

Let $V$ be an $n$-dimensional vector space over a characteristic $0$ field $k$ (or better, let $V=\mathbb{A}^n_k$). I wonder whether the following is true:

Absolutely irreducible subgroups $H$ of $\operatorname{GL}(n)=\operatorname{Aut}(V)$, where the latter two groups are regarded as algebraic groups, are reductive.

By absolute irreducibility I mean the subgroup $H_K$ acts irreducibly on $V_K$ for any field extension $k\to K$, i.e., there is no parabolic subgroup of $\operatorname{GL}(n)$ containing $H$. If it is true, I would like a reference to a proof; if it is not true, I wonder where I can find a counter-example.

My opinion on this is that I need to see whether the set of irreducible groups can inject into the set of reductive groups (or the set of non-reductive ones can inject into the set of non-irreducible ones). On the one hand, there is a correspondence of parabolic subgroups and cocharacters of $\operatorname{GL}(n)$. However, on the other hand, I don't know a classification of reductive subgroups or that of non-reductive ones.

Any help will be appreciated.

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mhahthhh
mhahthhh
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irreducible non-reductive subgroup in GL(n) over a characteristic 0 field

Let $V$ be a n-dimensional vector space over a characteristic $0$ field $k$ (or better, let $V=\mathbb{A}^n_k$). I wonder whether the following is true: $$\begin{align*} &\text{Absolutely irreducible subgroup $H$ of GL$(n)$=End$(V)$, where the latter two groups are regarded as algebraic}\\ &\text{groups, are reductive.} \end{align*}$$ By absolute irreducibility I mean the subgroup $H_K$ acts irreducibly on $V_K$ for any field extension $k\to K$, i.e., there is no parabolic subgroup of GL$(n)$ containing $H$. If it is true, I would like a reference to a proof; if it is not true, I wonder where I can find a counter-example.

My opinion on this is that I need to see whether the set of irreducible groups can inject into the set of reductive groups (or the set of non-reductive ones can inject into the set of non-irreducible ones). On the one hand, there is a correspondence of parabolic subgroups and cocharacters of GL$(n)$. However, on the other hand, I don't know a classification of reductive subgroups or that of non-reductive ones.

Any help will be appreciated.