Whittaker functions for Steinberg representations of GL(n) over a p-adic field

2023/11/05 by Ehud Moshe Baruch, Baruch, Ehud Moshe, Markos Karameris +1
Mathematics · #20C08 #22E50 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2311.02706

openalex publication_date 2023/11/05 · openalex created_date 2023/11/10 · openalex updated_date 2026/07/28

Abstract

Let G=GL_n(F) and let (π_St,V) be a (generalized) Steinberg representation of G. It is well known that the space of Iwahori fixed vectors in V is one dimensional. The Iwahori Hecke algebra acts on this space via a character. We determine the value of this character on a particular Hecke algebra element and use this action to determine in full the Whittaker function associated with an Iwahori fixed vector generalizing a result of Baruch and Purkait for GL₂(F). We show that the Iwahori fixed vector is "new" in the sense that it is not fixed by any larger parahoric. We also show that the restriction of the (generalized) Steinberg representation to SL_n(F) remains irreducible hence we get the Whittaker function attached to a Steinberg representation of SL_n(F).

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