Let
$\mathcal{X}=\left[X\u2215G\right]$
be a smooth Deligne–Mumford quotient stack. In a previous paper
we constructed a class of exotic products called
inertial products on
$K\left(I\mathcal{X}\right)$,
the Grothendieck group of vector bundles on the inertia stack
$I\mathcal{X}$. In this paper
we develop a theory of Chern classes and compatible power operations for inertial products.
When
$G$
is diagonalizable these give rise to an augmented
$\lambda $ring
structure on inertial Ktheory.
One wellknown inertial product is the
virtual product. Our
results show that for toric Deligne–Mumford stacks there is a
$\lambda $ring
structure on inertial Ktheory. As an example, we compute the
$\lambda $ring
structure on the virtual Ktheory of the weighted projective lines
$\mathbb{P}\left(1,2\right)$ and
$\mathbb{P}\left(1,3\right)$. We prove that, after tensoring
with
$\u2102$, the augmentation
completion of this
$\lambda $ring
is isomorphic as a
$\lambda $ring
to the classical Ktheory of the crepant resolutions of singularities of the coarse moduli spaces of the
cotangent bundles
${\mathbb{T}}^{\ast}\mathbb{P}\left(1,2\right)$
and
${\mathbb{T}}^{\ast}\mathbb{P}\left(1,3\right)$,
respectively. We interpret this as a manifestation of mirror symmetry in the spirit of
the hyperKähler resolution conjecture.
