Degree-one Kostant classes correspond to simple reflections

Given: The setting of Kostant's nilradical cohomology theorem, with simple reflections $s_1,\dots ,s_r$ attached to a base $\mathrm{\Delta }=\{{\alpha }_1,\dots ,{\alpha }_r\}$ of ${\mathrm{\Phi }}^{+}$.

[L1] $H^k({\mathfrak{n}}^{+},V)\cong {⨁}_{\mathrm{\ell }(w)=k}{\mathbb{C}}_{w\cdot \lambda }$ as $\mathfrak{h}$-modules, and consequently $H^1({\mathfrak{n}}^{+},V)={⨁}_{\mathrm{\ell }(w)=1}{\mathbb{C}}_{w\cdot \lambda }$ (Kostant's nilradical cohomology theorem).

[L2] An element of $W$ has length $1$ exactly when it is a simple reflection: the length is the minimum number of simple reflections in an expression, so $\mathrm{\ell }(w)=1$ means $w=s_i$ for some $i$, and conversely each $s_i$ has $\mathrm{\ell }(s_i)=1$ with inversion set $\{{\alpha }_i\}$ (Length and longest Weyl-group element, Finite Weyl positive roots and simple reflections, Simple roots form a signed integral basis, Root reflections and the Weyl group action).

[L3] The classes of the extremal cochains are nonzero and the cochain space in the extremal weight is one-dimensional: $C^{\mathrm{\ell }(w)}({\mathfrak{n}}^{+},V{)}_{w\cdot \lambda }=\mathbb{C}{\gamma }_w$ and $[{\gamma }_w]\mathrm{\ne }0$ (The extremal weight cochain of a Weyl element is closed and unique).

[L4] Distinct simple roots give distinct reflections and distinct dot weights: $s_i\mathrm{\ne }{s}_j$ for $i\mathrm{\ne }j$, and $\lambda +\rho$ has trivial stabilizer, so $s_i\cdot \lambda =s_j\cdot \lambda$ forces $s_i=s_j$ (Positive coroot pairings of a dominant integral weight, Integral, dominant, and strictly dominant weights).

[L5] The degree-one term of the BGG complex is $C_1(\lambda )={⨁}_{\mathrm{\ell }(w)=1}M(w\cdot \lambda )={⨁}_{i=1}^{r}M(s_i\cdot \lambda )$ (The Bruhat graph and the BGG Verma sum in degree k, The classical BGG category O).