# Reference Request: Definition of Induced Representation for reductive groups over a local field

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.

Of course, you are asking several related-but-different questions... I would grant you that it is hard to "see the general pattern", but/and by this point I think that the reason it is hard to see the general pattern is that there isn't one, except at a fairly philosophical, or else extremely utilitarian, level. That is, there is usually not a good explanation "from first principles" of why a choice is good, or why we do what we do.

For real Lie groups, there are non-trivial issues about what kind of space a repn should "be on". One viewpoint (for non-compact, semi-simple or reductive groups) is that, instead of $G$-repns, we should consider $\mathfrak g,K$-modules. And for optimal coherence, from a $G$-repn we can only keep the smooth vectors, and, indeed, only $K$-finite ones. "Why?" Well, because this game allows us to prove useful theorems. Not obvious. (The issue about whether or not to include the modular function as a renormalization is just a detail: it does not affect the mathematical reality, only the notation.) But/and, no, spaces of smooth vectors are not Hilbert spaces, etc. The Casselman and/or Wallach "globalization" functors (left/right adjoints to the forgetful functor from $G$-modules to $\mathfrak g,K$-modules) in general produce different $G$-repns, and not necessarily Hilbert spaces (unless somehow we have side information that the $\mathfrak g,K$-module came from a Hilbert space repn of $G$...)

For p-adic groups, or, more generally, totally disconnected ones, often the most tractable repn space is a colimit of finite-dimensional vector spaces. Although often said, it is simply not correct that this has a "discrete" topology. This is a misrepresentation of the fact that, indeed, every linear map from a (locally convex, topological vector space) colimit of finite-dimensional vector spaces to another (locally convex) topological vector space is continuous...

In all cases, one wants a (true) version of Frobenius Reciprocity to hold, well, actually, as the defining feature of an induced repn, namely, as left/right adjoint to the forgetful functor of restriction of a repn to a subgroup. Except for finite groups, it turns out that there is not generally a good left adjoint to restriction (hence, a false Frobenius Reciprocity), but there is a legitimate right adjoint (hence, a true Frobenius Reciprocity). Nevertheless, in P. Cartier's Corvallis-Conference article on repns of p-adic groups, he shows that the candidate for left adjoint, compact-induction, while not an adjoint to the forgetful functor, is an adjoint to some functor, so is right-exact. It took me many years to understand why he wrote that passage! :)

For principal series repns, for example, for either Lie groups or p-adic groups, it is simply not possible to make any definition of general principal series as Hilbert spaces (or Banach...) because for various reasons there can be no such unitary repn, nor anything close. Thus, for Lie groups, the safe defn of an induced repn is the Frechet space of smooth functions with the requisite left-equivariance. For some values of the inducing character, the space has an invariant inner product, so can be extended to a unitary repn on a Hilbert space.

Similarly, the generally-legitimate form of an induced repn for p-adic (or totally-disconnected) groups is a colimit of finite-dimensional vector spaces, so is a special kind of LF-space, but certainly not Hilbert or Banach. But/and for special characters, there is an invariant pre-Hilbert structure, and there-you-go.

So, to recap, one must (apparently) think both in terms of what the goals are, and what is a legitimate, coherent construction.

For $F$ a non-archimedean local field:

One can define the induction functor in several ways. First, there is parabolic induction--this is an example of induction in general. Depending on the context, one can either choose to normalize the induction (by adding the twist by $\delta^{1/2}$ as you mentioned). This would send unitary representations to unitary representations. In Murnaghan's notes, she is using non-normalized induction. Different authors may use normalized induction or non-normalized induction, depending on the context. The normalization will simplify certain statements like the contragredient of a representation, but it will complicate certain statements like Frobenius reciprocity.

By the way, any irreducible, smooth complex representation of any reductive $p$-adic group is admissible. I think this is due to Jacquet but some sources attribute a proof to Bernstein.

Here he focuses on $GL_n(F)$ but he defines induction in general.