## On certain Iwahori invariants in the unramified principal series.(English)Zbl 0804.22010

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 [S. Kato, J. Fac. Sci., Univ. Tokyo, Sect. I A 28, 929-943 (1981; Zbl 0499.22018)].

### MSC:

 2.2e+51 Representations of Lie and linear algebraic groups over local fields

Zbl 0499.22018
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