Quantization of branching coefficients for classical Lie groups

We study natural quantizations K of branching coefficients corresponding to the restrictions of the classical Lie groups to the Levi subgroups of their standard parabolic subgroups. The polynomials obtained can be regarded as generalizations of Lusztig q-analogues of weight multiplicities. For GLn they coincide with Poincaré polynomials previously studied by Shimozono and Weyman. They also appear in the Hilbert series of the Euler characteristic of certain graded virtual G-modules and, by a result of Broer, admit nonnegative coefficients providing that restrictive conditions are verified. When the Levi subgroup considered is isomorphic to a direct product of linear groups, we prove that these polynomials admit a stable limit which decomposes as nonnegative integer combination of Poincaré polynomials. For a general Levi subgroup, it is conjectured that the polynomials K have nonnegative coefficients when they are parametrized by two partitions. When G=GLn, the polynomials K can be interpreted as quantizations of the Littlewood–Richardson coefficients. We show that there also exists a duality between tensor product coefficients for types B, C, D (defined as the analogues of the Littlewood–Richardson coefficients) and branching coefficients corresponding to the restriction of SO2n to subgroups defined from orthogonal decompositions of the root system Dn (which are not Levi subgroups). These coefficients can also be quantified but the q-analogues obtained may have negative coefficients. Given a tensor product Π of irreducible GLN-modules, we then introduce for each classical group G=SON or SpN some q-analogues of the multiplicities obtained by decomposing Π into its G-irreducible components. We establish a duality between the polynomials and . According to a conjecture by Shimozono, the stable one-dimensional sums for nonexceptional affine crystals are expected to coincide with the polynomials associated to a sequence of rectangular partitions of decreasing heights.