Definition of induced representation

Question

Definition. Suppose that $H$ is a subgroup of the finite group $G$ and $\sigma \colon H \to \operatorname{GL}(W)$ is a representation of $H$. Then the induced representation from $H$ up to $G$ denoted $\pi = \operatorname{Ind}_H^G \sigma$ is a group homomorphism $\pi \colon G \to \operatorname{GL}(V)$, where $$ V = \{ f \colon G \to W \mid \text{$f(hg) = \sigma(h)f(g)$, for all $h \in H$, $g \in G$} \}. $$ The representation $\pi(g)$ is then defined on $f \in V$ by $$ \text{$[\pi(g) f](x) = f(xg)$, for all $x,g \in G$} $$

Can someone explain this definition to me? I don't understand what $\pi$ is given by? Its a function that maps $g$ to what in $\text{Aut}(V)$? Can someone explain the line $[\pi(g)f](x)=f(xg)$? I don't know what I am missing, but what is the representation given by?

Answer 1

Given any vector space $W$, there is a natural representation of the group $G$ on the space $W^G$ of all set maps $f:G\to W$. This representation is given by $(g\cdot f)(\tilde g):=f(\tilde gg)$ for all $\tilde g\in G$. So $g$ acts on $f$ by precomposition with the right translation by $g$. (It is easy to see that this defines a representation without any assumptions on $W$.)

Now if you have given a subgroup $H\subset G$ and a representation of $H$ on $W$, then this gives rise to a $G$-invariant subspace $V\subset W^G$. This is the subspace of functions which are $H$-equivariant in the sense that $f(h\tilde g)=h\cdot(f(\tilde g))$ for all $h\in H$ and $\tilde g\in G$. So you can restrict the above action of $G$ to this invariant subspace, and this is exactly the induced representation in your question.

The construction of an induced representation is easier to understand in the setting of Lie groups. The equivaraint functions can be considered as functions defined on the space of cosets $H\backslash G$. If $G$ is a Lie group and $H\subset G$ is a closed subgroup, then this coset space is a smooth manifold and the representation $W$ of $H$ gives rise a vector bundle over the coset space endowed with a natural $G$-action. This gives rise to a representation of $G$ on the space of smooth sections of the vector bundle, which can be identified with smooth functions $G\to W$ which are equivariant in a similar sense as above.

Answer 2

We start with a known representation of the subgroup $H$, and construct a representation of the group $G$. In this sense, the latter representation will be "induced" by the former; hence the term.

The known representation of $H$ is acting in the space of linear operators ${\rm{GL}}(W)$, where $W$ is a linear space (say, that of all square-integrable functions). However, the induced representation of $G$ will be acting on a subspace $V\subset W$ of the {\it{Mackey functions}}, those defined by the subsidiary condition $$ V = \{ f \in W \mid \text{$f(hg) = [\sigma(h)f](g)$, for all $h \in H$, $g \in G$} \}. $$

The induced representation $G \to {\rm{GL}}(V)$ is implemented by the operators $$ \pi_g f(x) = f(xg)\quad\mbox{for all}\quad g,\,x\in G $$ Had the function $f$ belonged to the space $W$, we would have said that this is a map $G\to {\rm{GL}}(W)$, and would have stopped there. However, we agreed that $f$ obeys the subsidiary condition; so $f\in V$, wherefrom $$\pi:\quad G\to{\rm{GL}}(V).$$

This enables us to make one more step. Consider the quotient space $H\backslash G$ of cosets $Y = Hg$ and choose a ``representative'' $r(Y)$ for each coset $Y$. Then a coset $Y$ can be written as $Y = H~r(Y)$. As soon as we apply the definition of the induced representation to a representative, $$ \pi_g f(r(Y)) = f(r(Y)~g), $$ we can employ the subsidiary condition to find the representation for the entire coset. To do so, we start out with $$ \pi_g f(r(Y)) = f(r(Y)~g) = f( r(Y)~g~r^{-1}(Yg)~r(Yg) ) $$ and note that the coset $Yg$ has a representative of its own, $r(Yg)$, so it can be presented as $$ Yg = h(Y,g)~r(Yg)\quad\mbox{where}\quad h(Y,g)\in H~. $$ Combining the last two equations, we see that $h(Y,g) = r(Y)~g~r^{-1}(Yg)$, and we finally write down: $$ \pi_g f(r(Y)) = f(r(Y)~g) = f(~h(Y,g)~r(Yg)~) = \sigma(h(Y,g)) f(r(Yg))~. $$ It is therefore sufficient to define the induced representation on the cosets (technically, on the cosets' representatives), whereafter the subsidiary condition automatically extends the definition to the entire group $G$.

With $r$ denoting a section, i.e., a map from cosets to representatives, the above equation assumes the form $$ \pi_g f\circ r (Y) = \sigma(h(Y,g))~f\circ r(Yg)~. $$ Obviously, $$ \omega = f\circ r $$ is a function acting on the coset space. Such functions are named Wigner functions, and in their terms our equation can be written down as $$ \tilde{\pi}_g\omega (Y) = \tilde{\sigma}(h(Y,g))~\omega(Yg)~, $$ where $\tilde{\pi}_g\omega (Y) \equiv \pi_g f(r(Y))$ and $\tilde\sigma(h)\omega(Y)\equiv\sigma(h)f(r(Y))~$.

We now see that this is a transformation law of the Wigner functions. The multiplier $\tilde{\sigma}(h(Y,g))$, called the Wigner rotation, is responsible for the Thomas precession in quantum physics.