Supercuspidal representations of GL\(_n\): Explicit Whittaker functions. (English) Zbl 0913.22013
Let us consider \((\pi, V)\) as an irreducible supercuspidal representation of \(G= GL_n (F)\) \((n \geq 1)\) over a non-Archimedean local field \(F\). Consider further a \(G\)-homomorphism \(V \rightarrow W\), where the space \(W = W (G, \chi)\) consists of functions \(f: G \rightarrow \mathbb C\) which are right \(G\)-smooth and satisfy \(f(ug)=\chi(u) f(g), u \in U\), and a maximal unipotent subgroup \(V\) of \(G\), and where \(\chi\) is a smooth nondegenerate character of \(U\). Its image is a space of Whittaker functions of \(\pi\) and the pair \((U, \chi)\). The main result of the present work is to describe this pair \((U, \chi)\) and to further construct an explicit Whittaker function on \(\pi\).
Reviewer: Eugene Kryachko (Kiev)
MSC:
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
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