- On cuspidal unipotent representations
- Award date
- 8 September 2015
- Number of pages
- Document type
- PhD thesis
- Faculty of Science (FNWI)
- Korteweg-de Vries Institute for Mathematics (KdVI)
The theory of unipotent representations is gradually included in the philosophy of Langlands. Lusztig's work shows that unipotent representations of a connected reductive group G correspond to unipotent classes in the Langlands L-group LG. The supercuspidal unipotent representations are those unipotent representations with compactly supported (modulo the centre of G) matrix coefficients. An important invariant called formal degree attaches to every supercuspidal representation.
The endomorphism algebra H := End(π) of a unipotent representation π of G is an affine Hecke algebra, which can be completed into a type I Hilbert algebra. Studying irreducible unipotent representations of G is equivalent to (in a fashion of Morita equivalence) studying simple H-modules. If G is a p-adic reductive connected group, the equivalence just mention respects all the notions of harmonic analysis, which enables us to use techniques from harmonic analysis to study H.
A rational function (called μ function) on the algebraic torus associated with H encodes the spectral properties of H. The residue value of μ at certain point (called residual point) equals the formal degree of π up to a non-zero rational factor.
Using a technique called spectral transfer map introduced by Opdam, we show that for every connected absolutely simple p-adic group, there is only one residual point such that the residue value of μ at that point is equal to the formal degree of π up to a rational constant. Consequently, the corresponding Langlands parameters can be determined.
- Research conducted at: Universiteit van Amsterdam