The \(c\)-function expansion of a basic hypergeometric function associated to root systems. (English) Zbl 1290.33015

An explicit expansion in symmetric Macdonald polynomials of a basic hypergeometric function associated to a root system was obtained by Cherednik in the case the associated twisted affine root system is reduced. An extension of Cherednik’s construction in the form of a meromorphic Weyl group invariant solution of the spectral problem of the Macdonald \(q\)-difference operators was obtained by the author of the present paper. Now, the author derives an explicit \(c\)-function expansion of a basic hypergeometric function \(\mathcal{E}_+\) associated to root systems. The obtained explicit \(c\)-function expansion is an expansion in terms of the basis of the space of meromorphic solutions of the spectral problem with coefficients expressed in terms of a \(q\)-analog of the Harish-Chandra \(c\)-function. The basic Harish-Chandra series \(\hat \Phi _\eta (\cdot, \gamma )\) with base point given by a torus element \(\eta \) is a meromorphic common eigenfunction of the Macdonald \(q\)-difference operators. The obtained \(c\)-function expansion of a basic hypergeometric function \(\mathcal{E}_+\) has the form \(\mathcal{E}_+(t,\gamma )=\hat c_\eta (\gamma _0)^{-1}\sum_{w\in W_0}\hat c_\eta (w\gamma )\hat \Phi _\eta (t,w\gamma )\), where \(W_0\) is the Weyl group of the underlying finite root system.


33C67 Hypergeometric functions associated with root systems
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