## 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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