ar X iv : q - a lg / 9 50 80 17 v 2 1 6 N ov 1 99 5 ON INNER PRODUCT IN MODULAR TENSOR CATEGORIES

In this paper we study some properties of tensor categories that arise in 2-dimensional conformal and 3-dimensional topological quantum field theory – so called modular tensor categories. By definition, these categories are braided tensor categories with duality which are semisimple, have finite number of simple objects and satisfy some non-degeneracy condition. Our main example of such a category is the reduced category of representations of a quantum group U q g in the case when q is a root of unity (see [AP, GK]). The main property of such categories is that we can introduce a natural projective action of mapping class group of any 2-dimensional surface with marked points on appropriate spaces of morphisms in this category (see [Tu]). This property explains the name " modular tensor category " and is crucial for establishing relation with 3-dimensional quantum field theory and in particular, for construction of invariants of 3-manifolds (Reshetikhin-Turaev invariants). In particular, for the torus with one puncture we get a projective action of the modular group SL 2 (Z) on any space of morphisms Hom(H, U), where U is any simple object and H is a special object which is an analogue of regular representation (see [Lyu]). In the case U = C this action is well known: it is the action of modular group on the characters of corresponding affine Lie algebra. We study this action for arbitrary representation U ; in particular, we show that this action is unitary with respect to a natural inner product on the space of intertwining operators. In the special case g = sl n and U being a symmetric power of fundamental representation this is closely related with Macdonald's theory. It was shown in the paper [EK3] (though we didn't use the word " S-matrix " there) that in this case the matrix coefficients of the matrix S are some special values of Macdonald's polynomials of type A n−1. Thus, the properties of S-matrix immediately yield a number of identities for values of Macdonald's polynomials at roots of 1. In this case, the action of modular group is closely related with the difference Fourier transform defined in a recent paper of Cherednik ([Ch]). In particular, this shows that for g = sl 2 all matrix elements of S-matrix can be written in terms of q-ultraspherical polynomials. Unfortunately, we had to spend a large part of this paper recalling known …