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Why are they so important?

I see them appear in Lie theory, algebraic geometry, etc.

Can somebody elaborate?

For example, can someone explain why they are such natural groups to consider in algebraic geometry?

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up vote 2 down vote accepted

They are solvable groups and there representation theory is by far easier than the general representation theory (Mackey-Clifford-etc.)

In principle, it is sufficient to classify the representations of unipotent groups and reductive groups, in order to classify all the representation in full.

Every lie or algebraic group is a semi direct product of a reductive one and a unipotent one.

Also the reductive subgroups of a reductive group determine to a large extent the representation theory via parabolic induction. This involves necessary understanding the parabolic subgroups, hence the unipotent subgroups.

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So the algebraic group $\mathrm{GL}_n$ can be written as the semi-product of which reductive and which unipotent one? (This must have something to do with some well-known matrix decomposition.) – Harry Mar 19 '12 at 18:49
@Harry: $GL_n$ is reductive (its only normal subgroups are central). – Alastair Litterick Mar 19 '12 at 21:15
A nontrivial (but still easy) example of the decomposition is given by the Borel subgroup (maximal closed solvable subgroup) of a reductive linear group; in $GL_n$, we can write $B = UT$ where $B$ is the (solvable) subgroup of upper-triangular matrices, $U$ is the subgroup of strictly-upper-triangular matrices (i.e. 1's on the diagonal; this is unipotent and normal in $B$) and $T$ is a maximal torus of $G$ (diagonal matrices), which is reductive. – Alastair Litterick Mar 19 '12 at 21:30

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