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On a uniqueness property of supercuspidal unipotent representations
- Source :
- Advances in Mathematics, 375:107406. Academic Press Inc.
- Publication Year :
- 2020
-
Abstract
- The formal degree of a unipotent discrete series character of a simple linear algebraic group over a non-archimedean local field (in the sense of Lusztig [17] ), is a rational function of q evaluated at q = q , the cardinality of the residue field. The irreducible factors of this rational function are q and cyclotomic polynomials. We prove that the formal degree of a supercuspidal unipotent representation determines its Lusztig-Langlands parameter, up to twisting by weakly unramified characters. For split exceptional groups this result follows from the work of M. Reeder [28] , and for the remaining exceptional cases this is verified in [7] . In the present paper we treat the classical families. The main result of this article characterizes unramified Lusztig-Langlands parameters which support a cuspidal local system in terms of formal degrees. The result implies the uniqueness of so-called cuspidal spectral transfer morphisms (as introduced in [22] ) between unipotent affine Hecke algebras (up to twisting by unramified characters). In [23] the essential uniqueness of arbitrary unipotent spectral transfer morphisms was reduced to the cuspidal case.
- Subjects :
- Linear algebraic group
Pure mathematics
General Mathematics
Mathematics::Number Theory
010102 general mathematics
Rational function
Unipotent
16. Peace & justice
01 natural sciences
Transfer (group theory)
Character (mathematics)
Residue field
0103 physical sciences
010307 mathematical physics
Uniqueness
0101 mathematics
Mathematics::Representation Theory
Cyclotomic polynomial
Mathematics
Subjects
Details
- Language :
- English
- ISSN :
- 00018708
- Database :
- OpenAIRE
- Journal :
- Advances in Mathematics, 375:107406. Academic Press Inc.
- Accession number :
- edsair.doi.dedup.....3d2450b55f7986bbfbe2ff058d9e8ad4