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\(q\)-Krawtchouk polynomials as spherical functions on the Hecke algebra of type \(B\). (English) Zbl 0957.33014
The author unifies several interpretations of Krawtchouk and \(q\)-Krawtchouk polynomials by deriving them as zonal spherical functions for the Hecke algebra for the hyperoctahedral group. Let \({\mathcal H}_n\) be the Hecke algebra for the hyperoctahedral group and \({\mathcal F}_n\) the subalgebra corresponding to the Hecke algebra for the symmetric group. Consider the representation of \({\mathcal H}_n\) induced from the index representation of \({\mathcal F}_n\), and let \(V_n\) be the induced module. \(V_n\) is a commutative algebra carrying a non-degenerate bilinear form. The \({\mathcal F}_n\)-invariant elements of \(V_n\) can be identified with an irreducible module of \(U_{q^{1/2}}(\mathfrak{sl}(2,\mathbb{C}))\), the quantized universal enveloping algebra for \(\mathfrak{sl}(2,\mathbb{C})\) acting on \(V_n\). The author calculates the characters of \(V_n\) and gives an explicit orthogonal basis for the \({\mathcal F}_n\)-invariant elements of \(V_n\) as eigenvectors of a particular selfadjoint operator. The author finds the second order \(q\)-difference equation satisfied by the corresponding zonal spherical functions and uses it to connect these to previously studied \(q\)-Krawtchouk polynomials.

33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics
20C08 Hecke algebras and their representations
43A90 Harmonic analysis and spherical functions
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