Let $G$ be a real reductive group. Why is the centralizer of an element unimodular? What is a reference?
Centralizers in reductive Liegroups = unimodular?
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reference-request
lie-groups
topological-groups
1 Answers
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As the OP pointed out, my original answer was wrong. In fact centralizers of elements are unimodular, and a reference can be found by following the link in the OP's comment below.
Here is an example that I find interesting: Consider the element $\begin{pmatrix} 1 & 0 & 1 \\0 & 1 & 0 \\0 & 0 & 1 \end{pmatrix}$ in $GL_3(\mathbb R)$. Its centralizer is $\Big\{ \begin{pmatrix} a & b & c \\ 0 & d & e \\ 0 & 0 & a \end{pmatrix} \Big\} \subset GL_3(\mathbb R)$. Although this is a solvable group, and looks very similar to the Borel in $GL_3$, which is not unimodular, it is in fact a unimodular group (as the OP points out in a second comment below).
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2@late_learner: Dear late_learner, Thanks for your comments and corrections. I've edited my answer to make it correct; it simply reflects what you pointed out in your comments. Best wishes, – 2012-01-16