My understanding is that "formally" there are many irreps which are not highest weight (e.g. unitary principal series, Whitakker modules), however "informally" it is not true (highest weight vector is always nearby one need just to open eyes:).

The point is how do we think of representation space "V" - if we think it abstractly as some abstract vector space - okay - can be no highest weight vector. However if we think of $V$ as some subset of functions of some variables (x_1,...,x_n), which satisfy some condition (e.g. square integrable or whatever), then we can "open eyes" and find the highest weight vector - the subtlety is that it may not obey the "condition", so it does not belong $V$, but still it corresponds to some function $f(x_1,...,x_n)$.

**Example.**

Take $sl_2$ and its standard representation by differential operators (sorry for some misprints):

e = d/dx

h = xd/dx - l x

f = x^2d/dx - l^2x

1)
Consider C[x] - then it is highest weight module with "1" as an highest weight vector.

2) Consider quasi-polynoms space {f(x) = p(x)exp(m x) }
This is actually a Whittaker module wiht character "m", and function exp(m x) is Whittaker vector i.e. it is eigen for "e".

3) Consider square integrable functions with appropriate scalar product and condition on $l$. You can get unitary principal series representation in this way.

So to my taste all these representations are essentially equivalent and they have highest weight vector "1", but "formally" it is of course not true.

for ${\mathfrak sl}(2,\mathbb C)$". A similar construction can be done in general, but it seems like a pain. $\endgroup$4more comments