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The Hecke algebra for the hyperoctahedral group contains the Hecke algebra for the symmetric group as a subalgebra. Inducing the index representation of the subalgebra gives a Hecke algebra module, which splits multiplicity free. The corresponding zonal spherical functions are calculated in terms of q-Krawtchouk polynomials using the quantised enveloping algebra for sl(2,C). The result covers a number of previously established interpretations of (q-)Krawtchouk polynomials on the hyperoctahedral group, finite groups of Lie type, hypergroups and the quantum SU(2) group.

𝑞-Krawtchouk polynomials as spherical functions on the Hecke algebra of type 𝐵