Let $G$ be a connected, reductive group over a local field $F$ of characteristic zero, and $H$ a closed subgroup of $G$ which is defined over $F$. Let $\mu_H, \mu_G$ be right Haar measures on $H(F), G(F)$ with modular functions $\delta_H, \delta_G$.

In papers, notes, and textbooks on automorphic forms and representation theory, I am often a little frustrated when I get to material with induced representations. Not only does the definition change depending on the context, I encounter different authors using different definitions for the same context. I have to think about the things they are writing afterward in order to figure out what definition they must be using.

So my question is, what are the different kinds of representations of (rational points of) algebraic groups commonly encountered in automorphic forms? For such representations, what are the usual definitions of induced representation? And how can one recognize which definition to use?

For example, if $F$ is nonarchimedean, then a common situation is to study smooth, irreducible, admissible representations of $H(F)$. Here we have a complex vector space $V$ with the discrete topology, appearing as the representation space of a representation $(\pi,V)$ of $H(F)$. One assumes that $V$ has no nontrivial proper $H(F)$-invariant subspaces, that the mapping $H(F) \times V \rightarrow V$ is continuous (this is equivalent to being smooth), and that for any compact open subgroup $K$ of $H(F)$, the space of $K$-fixed vectors is finite dimensional.

Then I have seen $\textrm{Ind}_{H(F)}^{G(F)} \pi$ to be defined as the space of functions $f: G \rightarrow V$ such that $f(hg) = \pi(h) f(g)$ for all $h \in H(F), g \in G(F)$, and $f$ is right $K$-invariant for some open compact subgroup $K$ of $G(F)$. (as defined in Fiona Murnaghan's notes http://www.math.toronto.edu/murnaghan/courses/mat1197/notes.pdf)

On the other hand, I have also seen the functions in $\textrm{Ind}_{H(F)}^{G(F)} \pi$ defined differently, in that one instead requires $f(hg) = \delta_{G/H}(h)^{\frac{1}{2}}\pi(h)f(g)$, where $\delta_{G/H}(h) = \frac{\delta_H(h)}{\delta_G(h)}$. Since $G(F)$ is unimodular, $\delta_{G/H}(h) = \delta_H(h)$.

When $F = \mathbb{R}$, one usually assumes the representation space is a Banach space, or even a Hilbert space, and irreducible here means that there are no nontrivial proper *closed* invariant subspaces. Here, the definition of induced representation may depend on a choice of quasi-invariant measure on $H \setminus G$, although for Hilbert space representations the idea is to make all such induced representations unitarily equivalent for different quasi-invariant measures.