What is equivariant chains on a representation sphere?

Question

For a finite group $G$ and a finite-dimensional real representation $V$ of $G$, denote by $S^V$ the one-point compactification of $V$, with basepoint at infinity.

What is the reduced chain complex $C_*(S^V,\infty)$ as an object of the derived category of $G$-representations?

Admittedly this is a somewhat open-ended question. One could regard $C_*(S^V,\infty)$ itself already as an "explicit" chain complex of $G$-representations. One can also write down a "more explicit" presentation of it in terms of the lattice of subgroups of $G$ and the subrepresentations of $V$ that they fix. Is there a nice succinct answer here, perhaps identifying $C_*(S^V,\infty)$ with another known object in the derived category $G$-representations which would be natural from other (e.g. purely representation theoretic) points of view?

Answer 1

Exercise 10 of section 1 of Chapter II of tom Dieck's book Transformation Groups gives you one answer to your question. It reads:

Let $S(V)$ be the representation sphere of a finite group $G$. Show directly that $S(V)$ admits a $G$-equivariant triangulation by looking at the convex hull of $\{\pm ge_i \ | \ g \in G; e_1, \dots, e_m \in S(V) \text{ basis for } V \}$.

So he is describing an easy-to-define $G$-CW structure on $S(V)$. Then $S^V$ will be two cones on $S(V)$ attached together, so one can easily read off its $G$-CW structure as well.

In degree $i$, the associated cellular chain complex $C^{CW}_*(S^V)$ will have one copy of the induced representation $1_H^G$ for each $i$ cell of the form $G/H \times D^i$. Exploring a detailed answer to his exercise will reveal a more detailed answer to your question, as the stablizers of families of the elements $e_i$ will relate to the representation $V$.