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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 avoided 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...).

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 avoided 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...).

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 avoided 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...).

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 avoided 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...).

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...).

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 avoided 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...).