 Weyl reflection lift in an sl2 representation

In a finite-dimensional sl2 Lie algebra representation, raising and lowering operators are nilpotent operators, so their exponentials are finite sums. In the defining two-dimensional representation, the displayed product is $\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$. It sends $h$ to $-h$ by conjugation, lifting the nontrivial Weyl group reflection. Its square acts on the irreducible highest-weight module $L(n)={\mathrm{Sym}}^n{\mathbb{C}}^2$ by $(-1{)}^n$. Thus a Weyl reflection can have a representative of order four on a representation.

// Target: linear-operator-theory.bigb