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

Abstract

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 q -Krawtchouk polynomials using the quantised enveloping algebra for s l ( 2 , C ) {\mathfrak {sl}}(2,\mathbb {C}) . The result covers a number of previously established interpretations of ( q q -)Krawtchouk polynomials on the hyperoctahedral group, finite groups of Lie type, hypergroups and the quantum S U ( 2 ) SU(2) group.