There is a well-known orthogonality property of $U(N)$ group characters
$$ \int d U \chi_{\mu}(U)\chi_\lambda(U^\dagger V)=\delta_{\mu\lambda}\frac{\chi_\mu(V)}{\dim_\mu} $$
where the integral is over unitary group, $\chi_\lambda$ is a character, labeled by the partition $\lambda$ and $\dim_\mu$ is the dimension of the correspondent representation, namely $\dim_\lambda=\chi_\lambda(\bf{1})$, the value of the character on the trivial group element.
In mathematical physics, in particular in topological strings (for example topological vertex) there appears the q-deformation of the dimension, namely $\dim^q_\lambda=\chi_\lambda(\rho)$ where $\rho$ is the diagonal matrix with the entries $1,q,q^2,\ldots$. The question:
is there any natural deformation of the unitary integral, which gives q-deformed dimensions in the r.g.s.?
$$ \left[\int d U \right]^q \chi_{\mu}(U)\chi_\lambda(U^\dagger V)=\delta_{\mu\lambda}\frac{\chi_\mu(V)}{\dim^q_\mu} $$
