Hello friends of mathematics :)
I have a question about the induced representation. Suppose $G$ is a group and $H$ a subgroup of $G$. Suppose $\rho$ is a representation of $H$ on the vectorspace $V$, then we can define the induced representation $\psi$ of $G$ on $W$ by:
- $W=Ind_H^G(V)=\{f:G\rightarrow V: f(gh)=\rho(h^{-1})f(g)\ \ \ \forall g\in G,h\in H\}$
- $\psi(g)(f(x))=f(g^{-1}x)$
Now suppose we have a familiy of vectorspaces $(E,X,\pi)$, thus we have the total space $E$, basisspace $X$ and projectionmap $\pi$. My first question:
- Per definition is the familiy of vectorspaces $(E,X,\pi)$ a vectorbundle if E is locally trivial. We say a familiy is trivial if it is isomorphic to a product space. But what means locally trivial precisely??
Also suppose we are looking to the space of all sections $\Gamma(E)$. If a group G acts transitive on $X$, then we call the vectorbundle homogeneous (or is there also another definition). Then we can define a representation $r$ of $G$ on $\Gamma(E)$ by:
$$r(g)s(x)=g\cdot s(g^{-1}x)$$
Now have a look to $E=G$ and $X=G/H$ with $H$ subgroup of $G$. How can we make clear that the induced representation is the same as the action given above?
I hope for explanations (because I have to use it for my script).
Thank you.