In the definition of [pre-quantization](http://en.wikipedia.org/wiki/Geometric_quantization) on a [symplectic manifold](https://en.wikipedia.org/wiki/Symplectic_manifold) $(M,\omega)$, we represent a function $f\in C^{\infty}(M)$(with Lie algebra structure) to $\hat{f}$ in the Hilbert space $L^2(M,L,\mu)$ associated to $M$. But we always assume that the Hilbert space should not be too big. But why do we assume this?. Also we consider only [irreducible representations](http://en.wikipedia.org/wiki/Group_representation) for definition of prequantization ? 


PS: see page 1 , line 9 of this [reference](http://www.maths.ed.ac.uk/~jthomas7/GeomQuant/Lecture2.pdf)



Prequantization and Hilbert space
OK, so here are just a few thought on this large topic of quantization. First of all, the question of irreducibility can equally well be asked for deformation quantization (as mentioned by other answers) and thus does not refer to any particular feature/problem of geometric quantization. The physical idea behind this requirement, that the obsevables should act irreducibly on a (pre-) Hilbert space, is, I guess, due to the fact that there are no nontrivial observables which can be measured in a compatible way with all others. A reducible representation would have some non-trivial commutant (that is the most resonable definition of "reducible" among many, partially inequivalent ones). And the elements in the commutant can be measured simultaneously to all other observables (as they commute with all of them). 


It is simply a fact from nature that such quantities are not known.


Or, perhaps one should be more specific here: suppose that you have some non-trivial commutant. Then start measuring these particular observables, which leads in you physically realized state to certain values of them. As time evolution is believed to originate from an observable (the Hamiltonian), no physically realizable time-evolution can change the values of these special observables in the commutant. Thus for all things we can do starting from one physically realizable state, we can never reach a state in which the elements of the commutant have different values. Thus, they behave as constants and we can restrict our considerations to the subspace generated by the other observables out of the starting one. On this subspace, the observables then act in an irreducible way (by construction). 


In other words: the elements in the commutant play to role of super-selection rules. Once physics is found to be in one super-selection sector (the above subsapce), we now that it will always stay there.


As a last remark: one of the big deficits of geometric quantization is that it tries to construct *one* irreducible representation from the beginning. This way, one will never see the super-selection structure of the theory and might miss some interesting and perhaps even the physically relevant super-selection sector. In deformation quantization this can not happen, as here you construct first the algebra of obsevables independently of any Hilbert space representation. Only in a second step, you look for representations. But now you are able to look for *all* of them, thus finding the super-selection structure. Surprisingly enough, it turns out that some established physical effects can be recovered this way: the Aharonov-Bohm effect is my personal favorite, it arises as different representations of the same observable algebra, each representation is irreducible and two of them are equivalent, iff the two vector potentials differ by a vector potential satisfying a certain integrality condition. 


