Polynomiality of parahoric fixed-space dimensions for mod-ℓ representations in full generality

Determine whether, for every connected reductive group G over a non-archimedean local field F with residue field cardinality q, every parahoric subgroup P ≤ G with nth congruence subgroup P_n, and every irreducible admissible representation π of G over a perfect field C of characteristic ℓ ≠ p, there exists a polynomial h_P(X) ∈ Q[X] such that dim_C(π^{P_{2n}}) = h_P(q^{2n}) for all sufficiently large integers n, without imposing the additional hypotheses used to prove this result in restricted cases.

Background

The paper proves that, under certain hypotheses (either those ensuring a Harish–Chandra–Howe local character expansion in characteristic 0 or those of Waldspurger–DeBacker), the dimension of the P_{2n}-fixed subspace of an irreducible admissible mod-ℓ representation π eventually equals a polynomial h_P evaluated at q^{2n}. This is a local character expansion consequence that controls the asymptotics of parahoric-invariant dimensions.

The authors note that this polynomiality is well known in characteristic 0 and that Henniart–Vignéras established it for G = GL_n(D) and for G = SL_2(F) (without restriction on p) in the mod-ℓ setting. They indicate that the assertion in its general form (i.e., without the additional structural hypotheses they use to prove their corollary) has been communicated as a conjecture by Marie-France Vignéras, highlighting that a complete, hypothesis-free result remains to be settled.