So this is asking a basic and/or stupid question (my apology and appreciation) about Soergel modules that comes out of exercises by me who knows little about the subject.


Let $W$ be a finite Weyl group with standard representation $V$ ($=X^\*(T)\otimes\_{\mathbb{Z}}\mathbb{R}$). Let $R=\mathbb{R}[V]$ and $I\_W$ be the ideal generated by strictly positive degree $W$-invariant elements in $R$ so that $C:=R/I\_W$ is the coinvariant algebra. Soergel modules can and will be viewed as $C$-modules. Let $\mathbb{L}\in C$ be the product of (differentials of) positive roots, the highest degree element. Let $\partial\_w, w\in W$ be the BGG-Demazure operators ($\partial\_s.v:=\frac{v-s.v}{\alpha\_s}$). Is it by any chance that



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> The indecomposable Soergel module indexed by $w\in W$ is isomorphic to a shift of $\langle\partial\_w\mathbb{L}\rangle\_C$ (the ideal in $C$ generated by $\partial\_w\mathbb{L}$) as $C$-modules?
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Description of Soergel modules
The cyclicity of the Soergel modules (the ideals you write are principal) is equivalent to rational smoothness of the corresponding Schubert varieties. Or more generally, this can be written as the following condition on Kazhdan-Lusztig polynomials $P\_{xy}(q)=q^{l(x)-l(y)}$, which works e.g. for dihedral groups and any $x,y$. See for example "4." on p. 3 of [Soergel's paper](https://www.ams.org/journals/jams/1990-03-02/S0894-0347-1990-1029692-5/). In general it's more involved.


