The adjoint representation and highest root

The vector $E_{1n}$ has weight ${\epsilon }_1-{\epsilon }_n$: $[H,E_{1n}]=(H_{11}-H_{nn})E_{1n}$ by [L2].

$E_{1n}$ is killed by every positive root vector: for $i<j$ we have $[E_{ij},E_{1n}]={\delta }_{j1}{E}_{in}-{\delta }_{in}{E}_{1j}$, and ${\delta }_{j1}=0$ because $i<j$ with $i\ge 1$, while ${\delta }_{in}=0$ because $i<j\le n$ forces $i<n$; hence $[E_{ij},E_{1n}]=0$.

The adjoint submodule generated by $E_{1n}$ is all of ${\mathfrak{s}\mathfrak{l}}_n$. It contains $H=[E_{n1},E_{1n}]=E_{nn}-E_{11}$; then $[E_{nk},H]$ is a nonzero scalar multiple of $E_{nk}$ for every $k<n$. It also contains $E_{jn}=[E_{j1},E_{1n}]$ for $1<j<n$, as well as the original $E_{1n}$. Finally, $[E_{nk},E_{jn}]={\delta }_{kj}{E}_{nn}-E_{jk}$ supplies every off-diagonal $E_{jk}$ with $j,k<n$ and every diagonal difference $E_{nn}-E_{jj}$. These matrices span ${\mathfrak{s}\mathfrak{l}}_n$.

In the Euclidean model of the roots take the traceless vector $v=(n-1,n-3,\dots ,1-n)$: $(v,{\epsilon }_i-{\epsilon }_j)=2(j-i)$, so it is regular and selects exactly $i<j$. Each positive root has the expansion ${\epsilon }_i-{\epsilon }_j={\alpha }_i+\cdots +{\alpha }_{j-1}$. The $n-1$ vectors ${\alpha }_j$ are independent: comparing successive coordinates in $\sum c_j({\epsilon }_j-{\epsilon }_{j+1})=0$ gives every $c_j=0$. If $j>i+1$ the root splits at $i+1$ into two positive roots; an adjacent root cannot so split because the two nonempty interval expansions would have to sum to its single coefficient one. Thus these adjacent differences are exactly the simple roots.

For every positive root ${\epsilon }_i-{\epsilon }_j$ with $i<j$, one has $({\epsilon }_1-{\epsilon }_n)-({\epsilon }_i-{\epsilon }_j)=({\alpha }_1+\cdots +{\alpha }_{i-1})+({\alpha }_j+\cdots +{\alpha }_{n-1}),$ a nonnegative integral combination of simple roots. Thus ${\epsilon }_1-{\epsilon }_n$ is the highest root. By steps 1.1–1.3, $E_{1n}$ has that weight, is killed by all positive root spaces, and generates the adjoint module, so it is a highest weight vector and the adjoint module has highest weight ${\epsilon }_1-{\epsilon }_n$. (Height and highest root, Highest-weight vectors and modules, step 1.1, step 1.2, step 1.3, step 1.4, L4) ∎