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Let $\mathfrak{p}\subseteq \mathfrak{gl}_n(\mathbb{C})$ be a parabolic algebra of a parabolic group $P\subseteq GL_n(\mathbb{C})$.

What is the difference among the nilradical of $\mathfrak{p}$, the nilpotent ideal of $\mathfrak{p}$, and the Lie algebra of the intersection of the kernels of all homomorphisms from $P$ to the multiplicative group in $\mathbb{C}$?

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The nilradical of $\mathfrak{p}$ is the largest nilpotent ideal of $\mathfrak{p}$. The phrase "the nilpotent ideal of $\mathfrak{p}$" is ambiguous because in general there are many such ideals. However there is always a largest such ideal, called the nilradical.

The intersection of the kernels of all homomorphisms from $P$ to $\mathbb{C}^\times$ is a subgroup of $P$ which contains the derived subgroup $P'$. In particular it is not a Lie algebra, so it cannot be meaningfully compared to the other objects in your question.

Furthermore, the Lie algebra of this kernel is in general strictly larger than the nilradical of $\mathfrak{p}$. They coincide precisely when $P$ is a minimal parabolic.

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  • $\begingroup$ Thank you. Can you clarify something? Do you mean that there is no such thing as the Lie algebra of the intersection of the kernels of all homomorphisms from $P$ to $\mathbb{C}^{\times}$ or do you mean the intersection of the kernels of all homomorphisms from $P$ to $\mathbb{C}^{\times}$ cannot be a Lie group? Another question: can you provide an example that the Lie algebra of this kernel is in general strictly larger than the nilradical of $\mathfrak{p}$? Thank you. $\endgroup$ – user137969 Mar 25 '14 at 21:18
  • $\begingroup$ The intersection of the kernels is a Lie group, so it has a Lie algebra. I just mean you have to make a distinction between Lie groups and Lie algebras. For your second question you can take any proper maximal parabolic subgroup of $GL_3(\mathbb{C})$ such as $P = \{M \in GL_3(\mathbb{C}) : M_{31} = M_{32} = 0\}$. $\endgroup$ – Konstantin Ardakov Mar 25 '14 at 21:42
  • $\begingroup$ So for your example using $GL_3(\mathbb{C})$ and $P$, the nilradical of $\mathfrak{p}$ is the set of all matrices with $0$ in all coordinates except in coordinates $M_{13}$ and $M_{23}$, right? But what is the Lie algebra of the intersection of the kernels of all homomorphisms from $P$ to $\mathbb{C}^{\times}$? Thanks again! $\endgroup$ – user137969 Mar 25 '14 at 21:50

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