Let $G=GL_n$ and $B$ its Borel subgroup consisting of all upper triangular matrices. Let $U^-$ be the unipotent subgroup of $G$ consisting of all unipotent lower triangular matrices. How to show that $B$ embeds to $G/U^-$ and this embedding is dense? Thank you very much.
2026-03-29 10:37:17.1774780637
How to show that $B$ embeds to $G/U^-$ densely?
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The embedding of $B$ is simply given by mapping an element of $B$ to its coset in $G/U^-$. Looking at the Lie algebras, you see immediately that this embedding is a local diffemorphism around $e$. I am not sure what's the simplest way of proving that the image is dense, it should be the "big cell" in the Bruhat decompositon.