A construction of the supercuspidal representations of \(GL_ n(F)\), \(F\) \(p\)-adic. (English) Zbl 0789.22032
Let \(F\) be a \(p\)-adic field. The result of this paper is the following Theorem: The irreducible supercuspidal representations of \(GL_ n(F)\) are all induced from representations of open compact mod center subgroups.
The two steps of the proof are, first, to construct a set of induced representations and then using the Matching Theorem of Deligne-Kazhdan- Vigneras (which relies on global methods) to show that the set is complete. This is by counting the representations constructed so far and to compare the numbers with corresponding results for representations of division algebras. A purely local approach to the same question which avoids the Matching Theorem but systematically uses the theory of \(K\)- types can be found in [C. Bushnell and Ph. Kutzko: “The Admissible Dual of \(GL(N)\) via Compact Open Subgroups” (Ann. Math. Studies 129) (1993; Zbl 0787.22016)].
The two steps of the proof are, first, to construct a set of induced representations and then using the Matching Theorem of Deligne-Kazhdan- Vigneras (which relies on global methods) to show that the set is complete. This is by counting the representations constructed so far and to compare the numbers with corresponding results for representations of division algebras. A purely local approach to the same question which avoids the Matching Theorem but systematically uses the theory of \(K\)- types can be found in [C. Bushnell and Ph. Kutzko: “The Admissible Dual of \(GL(N)\) via Compact Open Subgroups” (Ann. Math. Studies 129) (1993; Zbl 0787.22016)].
Reviewer: E.-W.Zink (Berlin)
MSC:
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
