Definition 2

Let E be a finite dimensional vector space, and $\rho$ (resp. $\pi$ ) a linear representation of $\mathfrak{s}\mathfrak{l}(2, k)$ (resp. $\mathbf{S}\mathbf{L}(2, k)$) on E. Then $\rho$ and $\pi$ are said to be compatible if, for every nilpotent element $x$ of $\mathfrak{s}\mathfrak{l}(2, k),\pi (e^x) =e^{\rho(x)}$.

In other words, $\rho$ and $\pi$ are compatible if, for every nilpotent element $x$ of $\mathfrak{s}\mathfrak{l}(2, k)$, the restriction of $\rho$ to $kx$ is compatible with the restriction of $\pi$ to the group $1 +kx$ (Chap. VII, §3, no. 1).

If $\rho$ and $\pi$ are compatible, so are the dual representations, the $m$th tensor powers, and the $m$th symmetric powers of $\rho$ and $\pi$, respectively (Chap. VII, §5, no. 4, Lemma 1 (i) and (ii)). Similarly for the representations induced by $\rho$ and $\pi$ on a vector subspace stable under $\rho$ and $\pi ($loc. cit.).

In particular, the representation $\rho_m$ of $\mathfrak{s}\mathfrak{l}(2, k)$ on $V(m)$ (no. 3) is compatible with the $m$th symmetric power $\pi_m$ of the identity representation $\pi_1$ of $\mathbf{S}\mathbf{L}(2, k)$. Putting $e^{(m)}_n=(^m_n)u^{m-n}v^n$ as above, we have

$(m)(m)m-nn$

$\pi_m(s)e_n$ = $(su)(sv)$ (3)

for $s\in \mathbf{S}\mathbf{L}(2, k)$ and $0\leq n\leq m$.