Quantized Howe-type dualities via Koornwinder polynomials and the X=K phenomenon

Abstract

We derive the equality between one-dimensional sums associated with tensor products of Kirillov-Reshetikhin column crystals of classical affine types and Lusztig q-analogues of weight multiplicities. The matching of the corresponding root systems is suggested by Howe duality. Our main tool is the dual Cauchy formula for Koornwinder polynomials due to Mimachi, which we combine with specializations in these polynomials. The mentioned dualities are proved for one-dimensional sums of all (twisted and untwisted) classical affine types except types B_n^(1) and D_n^(1). On another hand, all the Lusztig q-analogues of classical type are covered by our dualities, but they may have different parameters depending on the length of the roots in the underlying root system.