# Koszul complex of equations defining a stabilizer

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 $$v_i\in V$$.