Very specific question. We work over $\mathbb{C}$, although really just want alg. closed of char. 0.


Suppose that $G$ is an algebraic group and $V$ is a finite-dimensional $G$-module, meaning that we have a comodule map
$$
a:V\to\mathbb{C}[G]\otimes V.
$$
Let $v\in V$ be a non-zero vector; then we may choose a basis $v\_1=v, v\_2,\dots,v\_n$ of $V$, and write
$$
a(v\_1)=\sum\limits\_if\_i\otimes v\_i.
$$
Now it is clear that the ideal $(f\_1-1,f\_2,\dots,f\_n)$ cuts out exactly the subgroup of $G$ given by the stabilizer of $v$ in $G$.


My question is as follows: how could I go about computing the cohomology of the Koszul complex defined by the elements $f\_1-1,f\_2,\dots,f\_n$ in $\mathbb{C}[G]$?


**Edit:** I should add that I have an idea of what the answer 'should' be: let $\mathfrak{g}$ denote the Lie algebra of $G$, and write $H$ for the stabilizer of $v$ in $G$. Write $W\subseteq V$ for the subspace $\mathfrak{g}\cdot v$. Then this is a representation of $H$, so $V/W$ in particular is also a representation of $H$. I expect that the cohomology of the Koszul complex is isomorphic to the ring:
$$
\mathbb{C}[H]\otimes \bigwedge(V/W)
$$
Here I'm viewing the Koszul complex as an operator on the supercommutative superalgebra $\mathbb{C}[G]\otimes\bigwedge V$, where $f\_i$ 'corresponds to' the basis vector $v\_i\in V$.



Koszul complex of equations defining a stabilizer
See arXiv:1411.7994, Proposition 1.28 for a description of the cohomology dg-algebra of a Koszul complex in the case when the zero locus has codimension smaller than the number of equations.


