Types for tame $p$-adic groups

Eisenstein series for $G_2$ and the symmetric cube Bloch--Kato conjecture

“…This study is essential to showing that, on the Galois side, the family of Galois representations we obtain is no more ramified at places |N than ρ F itself. This is where the assumptions that 4 N and 9 N come in to play; by results of Fintzen [Fin21], we have a satisfactory theory of types for G 2 (Q ), but only at primes not dividing the order of the Weyl group of G 2 , which is 12. However, if N is divisible exactly once by a prime , then π F is an unramified twist of Steinberg at , and we have a way to circumvent having to use the general theory of types in this case.…”

“…Proof. As in the previous proposition, we invoke the theory of types as developed in [Fin21] to conclude that there is a neighborhood U of y 0 in U such that, this time, for every y ∈ Σ with w(y) in U , and Π y = {σ} a singleton, we have that σ v is a constituent of a principal series representation of the form…”

Comparison of Bushnell-Kutzko and Yu's constructions of supercuspidal representations

“…Kim [24] has proven that Yu's construction is exhaustive when the residual characteristic of F is sufficiently big. Fintzen has a better exhaustion result [19] using another method. In the following, we describe Yu's construction and introduce his notations, closely following Yu's paper [39].…”

“…This implies that one cannot expect a factorable construction « ρ d = ⊗κ i », as in Yu's work. In others words, if one tries to obtain by combining [15] and Fintzen's work [19] on Yu's construction, then one has to expect to work with a non-increasing (for ⊂) sequence of twisted Levi G 0 , . .…”

“…, G d (F )), Yu associated a group K d compact modulo the center of G(F ) and an irreducible supercuspidal representation ρ d such that the compactly induced representation to G(F ) is irreducible and supercuspidal. Fintzen showed that Yu's construction is exhaustive if p does not divide the order of the Weyl group of G [19].…”