# q-deformation of the unitary group integral

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.?

$$\left[\int d U \right]^q \chi_{\mu}(U)\chi_\lambda(U^\dagger V)=\delta_{\mu\lambda}\frac{\chi_\mu(V)}{\dim^q_\mu}$$

-
One natural candidate to do everything with the Hopf algebra which gives the usual $q$-variant of $\mathrm{SU}_2(n)$. –  Mariano Suárez-Alvarez Oct 13 '10 at 17:12