  We compute the topological partition function (twisted index) of
$\mathcal{N}=2$ $U(N)$ Chern-Simons theory with an adjoint chiral multiplet on
$\Sigma_g \times S^1$. The localization technique shows that the underlying
Frobenius algebra is the equivariant Verlinde algebra which is obtained from
the canonical quantization of the complex Chern-Simons theory regularized by
$U(1)$ equivariant parameter $t$. Our computation relies on a Bethe/Gauge
correspondence which allows us to represent the equivariant Verlinde algebra in
terms of the Hall-Littlewood polynomials $P_\lambda(x_B, t)$ with a
specialization by Bethe roots $x_B$ of the $q$-boson model. We confirm a
proposed duality to the Coulomb branch limit of the lens space superconformal
index of four dimensional $\mathcal{N}=2$ theories for $SU(2)$ and $SU(3)$ with
lower levels. In $SU(2)$ case we also present more direct computation based on
Jeffrey-Kirwan residue operation.
