Orbit decomposition of the restriction of an equivariant sheaf?

Question

All sets and groups in the question are finite.

In order to understand equivariant sheaves better I'm trying to prove some basic facts from Mackey theory using equivariant sheaves. The main obstacle i've been faced with is the difficulty of keeping track of all the different equivalences.

Let $G$ be a group and $K \lt G$ a subgroup. Perhaps one of the most basic canonical equivalences is between $Sh_G(G/K)$ and $Sh_K(pt)$. This can be implemented as follows:

Let $p: G \to pt$ and $q : G \to G/K$ be the projections. Then one has the following compositions:

$$Sh_K(pt) \overset{k \mapsto k^{-1}}{\longrightarrow} Sh_{K^{op}}(pt) \overset{p^*}{\to} Sh_{G \times K^{op}}(G) \overset{q^{K^{op}}_*}{\to} Sh_G(G/K)$$

Explanations:

1. First arrow is the inversion isomorphism from $K$ to $K^{op}$. It is obviously an equivalence.
2. The second arrow is the pullback which remembers the left $G$-equivariant structure coming from the fact that the projection $G/K \to pt$ is left $G$-equivariant. This is an equivalence since the $G$-action is free.
3. The third arrow is $K^{op}$-invariant sections of the pushforward along $q: G \to G/K$. This is an equivalence since the right $K^{op}$ action is free.

The functor in the other direction is the composition of the pseudo inverses.

This is all very neat until one tries to use this equivalence to prove stuff. I'll try to explain with an example. Suppose we wanted Mackey's induction formula. In other words we'd like to describe the functor $Res^K_G \circ Ind^G_H$ in terms of other functors.

Let $p: G/H \to pt$ and $q: G/K \to pt$. In this notation the composition above admits the following description:

$$Sh_G(G/H) \overset{p_*}{\to} Sh_G(pt) \overset{Res^K_G}{\to} Sh_K(pt)$$

By commutativity of the corresponding diagram this composition is isomorphic to the composition:

$$Sh_G(G/H) \overset{Res^K_G}{\to} Sh_K(G/H) \overset{p_*}{\to} Sh_K(pt)$$

Once one realizes that the category $Sh_K(G/H)$ splits as a direct sum $\bigoplus_{\mathcal{O}} Sh_K(\mathcal{O})$ over all $K$-orbits in $G/H$, then the second functor above $p_*$ becomes easy to describe as it is just a direct sum of induction functors.

However, consider now the following functor:

$$F: Sh_K(pt) \to Sh_G(G/H) \overset{Res^K_G}{\to} Sh_K(G/H) \to \bigoplus_{\mathcal{O}} Sh_K(\mathcal{O}) $$

This functor obtained by composing the first functor with equivalences seems rather mysterious.

It looks like it should be a sum over certain restriction functors (a fact we already know if we know about the classical mackey formula). Beyond that however this functor is completely mysterious to me. In order to compute it I found no other choice other than composing all the functors involved and which resulted in a huge expression I didn't manage to decipher.

How can one describe $F$ using the language of equivariant sheaves?

How does one compute similar functors in practice?

Of course I already know what $F$ should be from the classical Mackey formula but i'm looking for a way to "grab hold" of this functor using the language equivariant sheaves.

Answer 1

In my opinion the Mackey formula is most easily seen if you think in terms of groupoids/stacks. This idea makes sense in many situations, but for simplicity let's restrict to representations of discrete groups.

The basic idea is:

1) There is a Cartesian diagram of groupoids: $$ \require{AMScd} \begin{CD} [K\backslash G/H] @>{\widetilde{f}}>> [pt/K];\\ @V{\widetilde{g}}VV @VV{g}V \\ [pt/H] @>>{f}> [pt/G]; \end{CD} $$ 2) There is a base-change formula: $$g^\ast f_! \simeq \widetilde{f}_!\widetilde{g}^\ast : Rep(H) = Sh([pt/H]) \to Sh([pt/K]) = Rep(K)$$

3)There is a decomposition of groupoids: $$[K\backslash G/H] \simeq \bigsqcup [pt/({}^gH \cap K)]$$

Then the Mackey formula

$$Res^G_K Ind^G_H \simeq Ind^K_{{}^gH \cap K} Res^{{}^gH}_{{}^gH \cap K} {}^g(-): Rep(H) \to Rep(K)$$

follows immediately from these three facts (once you unravel how $[pt/({}^gH \cap K)]$ maps to $[pt/H]$ in the above decomposition).

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To add a bit more detail/fix notation: Suppose $G$ is a (discrete) group acting on a set $X$. We write $[X/G]$ for the quotient/action groupoid. Note that we can write the quotient stack as a disjoint union of over orbits $$[X/G] \simeq \bigsqcup_{Gx} [pt/C_G(x)]$$ Given a groupoid $\mathcal G$ we define $$Sh(\mathcal G) = Fun(\mathcal G,Vect)$$ Note that $$Sh([X/G]) \simeq Sh_G(X) \simeq \bigoplus_{Gx} Rep(C_G(x))$$ (the expression for $[K\backslash G/H]$ above is a special case of this). Given a map (functor) of groupoids $f:\mathcal G_1 \to \mathcal G_2$, we have adjoint functors $$f_! : Sh(\mathcal G_1) \leftrightarrows Sh(\mathcal G_2): f^\ast$$ where $f^\ast$ is the obvious functor and $f_!$ is given by left Kan extension (which is a fancy way of saying sum over the fibers).

Consider the map $f:[pt/H] \to [pt/G]$ corresponding to the inclusion of a subgroup in a group. In this language the induction/restriction functors are given by $$f_! = Ind_H^G : Rep(H) = Sh([pt/H]) \leftrightarrows Sh([pt/G]) = Rep(G): Res^G_H = f^\ast$$ Thus composites of induction and restriction functors as appear in the Mackey formula can be thought of as pushing and pulling sheaves on groupoids.