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Some results on the unramified principal series of \(p\)-adic groups. (English) Zbl 0804.22007

Let \(G\) denote an unramified quasisplit reductive \(p\)-adic group, \(\chi\) an unramified character of a minimal parabolic subgroup \(P\) and \(I(\chi) = \text{Ind}^ G_ P \chi\) the induced principal series representation. Let \(\pi(\chi)\) be the unique irreducible unramified subquotient of \(I(\chi)\). The first result of the paper says that \(\pi(\chi)\) has a Whittaker model iff \(\chi\) is not annihilated by any nondivisible root. Under a certain condition to the group this gives that \(\pi(\chi)\) has a Whittaker model iff \(I(\chi)\) is irreducible, i.e.: \(I(\chi) = \pi(\chi)\). Further some explicit formulas for the spherical functions and the Whittaker functions are given.

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
43A90 Harmonic analysis and spherical functions
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References:

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