  Quantization of the Teichm\"uller space of a non-compact Riemann surface has
emerged in 1980's as an approach to three dimensional quantum gravity. For any
choice of an ideal triangulation of the surface, Thurston's shear coordinate
functions on the edges form a coordinate system for the Teichm\"uller space,
and they should be replaced by suitable self-adjoint operators on a Hilbert
space. Upon a change of triangulations, one must construct a unitary operator
between the Hilbert spaces intertwining the quantum coordinate operators and
satisfying the composition identities up to multiplicative phase constants. In
the well-known construction by Chekhov, Fock and Goncharov, the quantum
coordinate operators form a family of reducible representations, and the phase
constants are all trivial. In the present paper, we employ the
harmonic-analytic theory of the Shale-Weil intertwiners for the Schr\"odinger
representations, as well as Faddeev-Kashaev's quantum dilogarithm function, to
construct a family of irreducible representations of the quantum shear
coordinate functions and the corresponding intertwiners for the changes of
triangulations. The phase constants are explicitly computed and described by
the Maslov indices of the Lagrangian subspaces of a symplectic vector space,
and by the pentagon relation of the flips of triangulations. The present work
may generalize to the cluster $\mathscr{X}$-varieties.
