Cylindric versions of specialised Macdonald functions and a deformed Verlinde algebra

Understand

We define cylindric generalisations of skew Macdonald functions when one of their parameters is set to zero.

• We define these functions as weighted sums over cylindric skew tableaux: fixing two integers n>2 and k>0 we shift an ordinary skew diagram of two partitions, viewed as a subset of the two-dimensional integer lattice, by the period vector (n,-k).
• Imposing a periodicity condition one defines cylindric skew tableaux as a map from the periodically continued skew diagram into the integers.
• The resulting cylindric Macdonald functions appear in the coproduct of a commutative Frobenius algebra, which is a particular quotient of the spherical Hecke algebra.