On certain Iwahori invariants in the unramified principal series (Q1207874)

Let \(G\) denote a reductive \(p\)-adic group, \(\tau\) an unramified character of a minimal parabolic subgroup \(P\) and \(I(\tau) = \text{Ind}^ G_ P \tau\) the induced principal series representation. The space \(I(\tau)^ B\) of fixed vectors under the Iwahori group \(B\) is a finite dimensional module under the Hecke algebra \({\mathcal H}\) of the pair \((G,B)\). For a good maximal compact subgroup \(K\) which contains \(B\) the Hecke algebra \(\theta\) of \((G,K)\) is a subalgebra of \({\mathcal H}\) and is, by the Satake map isomorphic to the Weyl-invariants in the coordinate ring of a maximal torus \(T\) of the Langlands dual group. The contents of the paper is an explicit description of (most of) the eigenfunctions of \(\theta\) in \(I(\tau)^ B\) which sheds some light on the structure of the module \(I(\tau)^ B\). For example one gets a new proof of the irreducibility criterion of Kato and Müller [\textit{S. Kato}, J. Fac. Sci., Univ. Tokyo, Sect. I A 28, 929-943 (1981; Zbl 0499.22018)].