The dimension of space of morhisms as the number of orbits

All groups are finite, all representations are over $\mathbb{C}$ (just in this context, of course), $G$ is a group, $K,H\subset G$ - its subgroups. By $\mathbb{C}$ we denote the trivial representation (of $K$ and $H$). My goal is prove that the dimension $\mathrm{dim}Hom_G(Ind_K^G\mathbb{C},Ind_H^G\mathbb{C})$ is the number of $K$-orbits of its action on the set of cosets $G/H$. Note that the induced representations mentioned above are the actions of $G$ on $G/K$ and $G/H$ respectively.
I do not know how to decompose $Ind_K^G\mathbb{C}$ and $Ind_H^G\mathbb{C}$ into the sum of irreducible ones, but understand that the number of orbits of the left action of $K$ on $X=\{x_1,x_2,...x_l\}$ is $\frac{1}{|K|}\sum_{s\in K} |X^s|$, where $X^s=\{x\in X| sx=x\}$. It should be useful, isn't it? Could you help to prove that fact somehow like that?

We wish to compute $\dim_{\,\Bbb C}\hom_G(\Bbb C[G/K],\Bbb C[G/H])$. Let's see what these maps look like.

A $G$-equivariant map $\Bbb C[G/K]\to\Bbb C[G/H]$ is determined by where it sends $K$, say

$$K\mapsto \sum_{aH\in G/H}c_{aH}aH. \tag{\circ}$$

We can extend this map $\Bbb C[G]$-linearly to create a map, but there are conditions: any action that preserves $K$ much preserve the $\Sigma$ on the RHS of $(\circ)$.$^\dagger$ The stabilizer of $K\in G/K$ is $K$.

The relation $k\Sigma=\Sigma$ for all $k\in K$ can be seen to be equivalent to $c_{aH}$ being constant on $K$-orbits of cosets of $H$ in $G/H$ (where $\Sigma$ is as in the RHS of $(\circ)$). Thus, our space of homomorphisms is isomorphic (as a $\Bbb C$-vector space) to the space of functions on $K$-orbits of $G/H$. The only thing left to say is that the characteristic functions of $K$-orbits of $G/H$ form a basis.

$^\dagger$This is sufficient: if $R$ is any ring and $M,N$ are $R$-modules with $M=Rm$ cyclic, $m\mapsto n$ can be extended to an $R$-module homomorphism $M\to N$ iff $rm=m\Rightarrow rn=n$ for all $r\in R$. Can you prove this? Hint: consider the cyclic modules $Rm$ and $Rn$ in light of the first isomorphism theorem, and rewrite the condition $(r-1)m=0\Rightarrow (r-1)n=0$ for all $r\in R$ using annihilators.

Another route is Frobenius reciprocity $-$ $\hom_G({\rm Ind}_K^GV,W)\cong\hom_K(V,{\rm Res}^G_KW)$ $-$ yielding

$$\dim_{\Bbb C}\hom_G({\rm Ind}_K^G\Bbb C,{\rm Ind}_H^G\Bbb C)=\dim_{\Bbb C}\hom_K(\Bbb C,\Bbb {\rm Res}^G_K\Bbb C[G/H])=\dim_{\Bbb C}\Bbb C[G/H]^K$$

where $\Bbb C[G/H]^K$ is the space of $K$-invariant elements of $\Bbb C[G/H]$. The same reasoning as above may be used to determine this is $\Bbb C$-space isomorphic to the space of functions on $K$-orbits of $H$.

• Brilliant! Thanks a lot!
– user74574
Dec 24 '13 at 19:47