TITLE: Why do we want *unitary* representations of locally compact groups into $B(H)$?
QUESTION [7 upvotes]: This is related to a previous question of mine but I have a more philosophical issue with the material. Everywhere I have looked for representations of locally compact groups into $B(H)$, everyone says that we want unitary representations but do not say what motivates this. Unitarity is definitely a nice property, but why should we hamper ourselves so much? Why not just map into $GL(H)$ (the invertible operators on $H$)? This is most closely aligned with the representation theory of finite groups and seems completely reasonable. I'm not looking for an "ends justify the means" kind of argument since when developing mathematics we don't even know what the "end" is necessarily but a more intuitive or lower level argument.

REPLY [3 votes]: Note that people don't only study unitary representations.  If $G$ is a real or $p$-adic Lie group, then the study of representations of $G$ is a big part of the theory of automorphic forms and representations (it is the ``local'' part of the theory), and one studies representations that are not necessarily unitary (or, perhaps better, not necessarily unitarizable).  
In this context there is another adjective, admissible, discovered by Harish-Chandra, which is of crucial importance, and a key theorem of Harish-Chandra shows that topologically irreducible unitary representations are admissible.  

In practice, the emphasis on unitary representations, to the extent it exists, probably arose because one of the first goals of rep. theory of locally compact groups was to generalize the Plancherel theorem from Fourier theory on locally compact abelian groups, and the Peter--Weyl theorem from the theory of compact groups,
by determining the structure of $L^2(G)$ as a $G$-representation.  
More generally, the theory of automorphic forms naturally leads to the question
of determining the structure of $L^2(\Gamma\ G)$ as a $G$-representation, where $\Gamma$ is a discrete subgroup of  $G$.
Since $L^2$ spaces are unitary, this leads to a focus on unitary reps.  

You might find this post useful for obtaining more background on some of the motivations of the theory.