Exercise 8, § 1

Tag 01JN, permanent. Cite it as [Bourbaki, Tag 01JN].

Lie Groups and Lie Algebras, VIII, 221

Show that the $\mathfrak{s}$ -module $V(m)$ can be realized as the space of homogeneous polynomials $f(u,v)$ of degree $m$ in two variables, the operators $X_+$ , $X_-$ and $H$ being given by:

X_+f=u\frac{\partial f}{\partial v},\qquad X_-f=-v\frac{\partial f}{\partial u},\qquad Hf=u\frac{\partial f}{\partial u}-v\frac{\partial f}{\partial v}.

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