On a uniqueness property of supercuspidal unipotent representations

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 [Lus3]), 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 [R3], and for the remaining exceptional cases this is verified in [Fe2]. 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 [Opd2]) between unipotent affine Hecke algebras (up to twisting by unramified characters). In [Opd3] the essential uniqueness of arbitrary unipotent spectral transfer morphisms was reduced to the cuspidal case.

The Wavefront Sets of Unipotent Supercuspidal Representations

“…A Langlands correspondence for unipotent supercuspidal representations has been obtained by [Mor96] when G is simple and adjoint, see also [Lus95]. For arbitrary reductive K-split groups, this correspondence is available by [FOS19;FO20]. Let IrrpG ω pkqq cusp,unip denote the set of equivalence classes of irreducible unipotent supercuspidal G ω pkq-representations.…”

On Formal Degrees of Unipotent Representations

“…For supercuspidal representations, both φ HII and [49] boil down to the same source, namely [16,17]. There it is shown that on the cuspidal level for a Levi subgroup M of G, in the bijection…”

“…Another way of expressing this is that the linearly extended parameter function m ∨ R on the affine Kac roots D(ĝ,θ) is constant and equal to 1 (see [16,Proposition 4.2.1]).…”

“…Unipotent representations of simple adjoint groups over K were classified by Lusztig [30,31]. The classification has also been worked out in several papers when G splits over an unramified extension of K. We have exhibited a local Langlands correspondence for supercuspidal unipotent representations of reductive groups over K in [16,17]. Then the second author generalised this to a Langlands parametrisation of all tempered unipotent representations in [39].…”

“…In the remainder of the introduction, we assume that G splits over an unramified field extension. The conjectures by Hiraga, Ichino and Ikeda ('HII-conjectures') were proven for supercuspidal unipotent representations in [43,16,15,17], for unipotent representations of simple adjoint groups in [38] and for tempered unipotent representations in [39]. However, in the last case the method only sufficed to establish the desired formulas up to a constant.…”