An sl3 weight with cohomology in degree one

Given: The Axiom of Choice, $\mathfrak{g}={\mathfrak{s}\mathfrak{l}}_3(\mathbb{C})$ with standard Cartan and Borel, simple roots ${\alpha }_1,{\alpha }_2$, fundamental weights ${\omega }_1,{\omega }_2$, $\rho ={\omega }_1+{\omega }_2$, the simple reflection $s_1$, the longest element $w_0$ of the Weyl group, and $\lambda =s_1\cdot \rho =\rho -2{\alpha }_1=-3{\omega }_1+3{\omega }_2$.

In $A_2$ the simple roots are ${\alpha }_1,{\alpha }_2$, the positive roots are ${\alpha }_1,{\alpha }_2,{\alpha }_1+{\alpha }_2$, and $\rho ={\omega }_1+{\omega }_2$ satisfies $⟨\rho ,{\alpha }_i^{\vee }⟩=1$; the simple reflection acts by $s_1(\nu )=\nu -⟨\nu ,{\alpha }_1^{\vee }⟩{\alpha }_1$, and the dot action is $w\cdot \nu =w(\nu +\rho )-\rho$ with $\mathrm{\ell }(s_1)=1$ (Classical complex matrix Lie algebras, Root systems of the classical complex Lie algebras, Fundamental weights for a chosen simple root system, The Weyl vector in fundamental coordinates, The Weyl vector, Root reflections and the Weyl group action).

Borel-Weil-Bott: if $\nu +\rho$ is regular with Weyl element $v$, then $H^{\mathrm{\ell }(v)}(X,{\mathcal{L}}_{\nu })\cong L(v\cdot \nu {)}^{\ast }$ and all other cohomology vanishes (The Borel-Weil-Bott theorem).

The Weyl dimension formula gives $\dim L(\rho )={\prod }_{\beta \in {\mathrm{\Phi }}^{+}}(2\rho ,\beta )\mathrm{/}(\rho ,\beta )=2^{\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }}=2^3=8$, since $(2\rho ,\beta )=2(\rho ,\beta )$ and $(\rho ,\beta )>0$ for the three positive roots; in particular $H^1\cong L(\rho {)}^{\ast }$ is eight-dimensional (The Weyl dimension formula, Root systems of the classical complex Lie algebras, The Weyl vector in fundamental coordinates).

Euler-characteristic form of Borel-Weil-Bott: ${\sum }_i(-1{)}^i\mathrm{c}\mathrm{h}{H}^i(X,{\mathcal{L}}_{\nu })=(-1{)}^{\mathrm{\ell }(v)}\mathrm{c}\mathrm{h}L(v\cdot \nu {)}^{\ast }$ when $\nu +\rho$ is regular with Weyl element $v$ (The Borel-Weil-Bott Euler character is a signed dual Weyl character).

For a dominant integral weight $\nu$ the dual $L(\nu {)}^{\ast }$ is irreducible of highest weight $-w_0\nu$; since $w_0$ maps ${\mathrm{\Phi }}^{+}$ onto ${\mathrm{\Phi }}^{-}$, it sends $2\rho ={\sum }_{\alpha >0}\alpha$ to $-2\rho$, so $w_0\rho =-\rho$ and $-w_0\rho =\rho$, hence $L(\rho {)}^{\ast }$ is irreducible of highest weight $\rho$ and therefore $L(\rho {)}^{\ast }\cong L(\rho )$ by the classification of finite-dimensional irreducibles, with $\mathrm{c}\mathrm{h}L(\rho {)}^{\ast }=\mathrm{c}\mathrm{h}L(\rho )$ (Highest weight of the dual representation, Length and longest Weyl-group element, Highest-weight classification).