  Noncommutative multivariable versions of weighted shifts arise naturally as
`weighted' left creation operators acting on Fock space. We investigate the
unital weak operator topology closed algebras they generate. The unweighted
case yields noncommutative analytic Toeplitz algebras. The commutant can be
described in terms of weighted right creation operators when the weights
satisfy a condition specific to the noncommutative setting. We prove these
algebras are reflexive when the eigenvalues for the adjoint algebra include an
open set in complex $n$-space, and provide a new elementary proof of
reflexivity for the unweighted case. We compute eigenvalues for the adjoint
algebras in general, finding geometry not present in the single variable
setting. Motivated by this work, we obtain general information on the spectral
theory for noncommuting $n$-tuples of operators.
