Let $ H < G$ be a subgroup of a finite group $ G$ . Let $ X:=G/H$ and $ \mathcal{F} \in Sh_G(X)$ be an equivariant sheaf on $ X$ (w.r.t. left multiplication) associated to a finite dimensional representation $ \pi$ of $ H$ with character $ \chi$ . Let $ Ind_H^G(\chi)$ be the character of $ Ind_H^G(\pi)$ and for any $ g \in G$ define $ X_g=\{x\in X :gx=x \}$ .

Recently I found out that the basic formula for the induced character $ Ind_H^G(\chi)$ can be interpreted very naturally from the point of view equivariant sheaves giving the following elegant equation:

$ $ Ind_H^G(\chi)(g)=\Sigma_{x\in X_g} Tr(g^* ,\mathcal{F_x})$ $

Where $ Tr(g^*,\mathcal{F}_x)$ is the trace of the induced action of $ g$ on stalk of $ \mathcal{F}$ at $ x \in X_g$ .

The formula above is surprisingly elegant compared to the one I derived it from (which involved either choosing representatives for cosets or dividing by the order of $ H$ while here we avoid both).

Question 1: Is there a reasonably geometric argument for why this formula is true?

Question 2: In what kind of generality does this formula hold? (continuous representations, locally compact groups, distributional characters etc…).