Quantization of branching coefficients for classical Lie groups

Abstract

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 GL n 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 K ˜ 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 = GL n , 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 SO 2 n to subgroups defined from orthogonal decompositions of the root system D n (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 GL N -modules, we then introduce for each classical group G = SO N or Sp N some q-analogues D of the multiplicities obtained by decomposing Π into its G-irreducible components. We establish a duality between the polynomials D and K ˜ . According to a conjecture by Shimozono, the stable one-dimensional sums for nonexceptional affine crystals are expected to coincide with the polynomials D associated to a sequence of rectangular partitions of decreasing heights.