Let π be an irreducible supercuspidal representation of GL n (F), where F is a p-adic field. By a result of Bushnell and Kutzko, the group of unramified self-twists of π has cardinality n/e, where e is the o F-period of the principal o F-order in M n (F) attached to π. This is the degree of the local Rankin-Selberg L-function L(s, π × π ∨). In this paper, we compute the degree of the Asai, symmetric square, and exterior square L-functions associated to π. As an application, assuming p is odd, we compute the conductor of the Asai lift of a supercuspidal representation, where we also make use of the conductor formula for pairs of supercuspidal representations due to Bushnell, Henniart, and Kutzko (1998). L(s, ρ, r) = 1 det 1 − (r • ρ)(Frob)| (Ker N) I q −s where Frob is the geometric Frobenius and I is the inertia subgroup of the Weil group of F. Thus, L(s, ρ, r) = P(q −s) −1 for some polynomial P(X) with P(0) = 1, and by the degree of L(s, ρ, r) we mean the degree of P(X). If π = π(ρ) denotes the L-packet of irreducible admissible representations of G(F) corresponding to ρ under the conjectural Langlands correspondence, then its Langlands L-function, denoted by L(s, π, r), is expected to coincide with L(s, ρ, r). In many cases, candidates for L(s, π, r) can also be obtained either via the Rankin-Selberg method
Let G be a split classical group over a non-Archimedean local field F with the cardinality of the residue field q F > 5. Let M be the group of Fpoints of a Levi factor of a proper F-parabolic subgroup of G. Let [M, σ M ] M be an inertial class such that σ M contains a depth-zero Moy-Prasad type of the form (K M , τ M), where K M is a hyperspecial maximal compact subgroup of M. Let K be a hyperspecial maximal compact subgroup of G(F) such that K contains K M. In this article, we classify s-typical representations of K. In particular, we show that the s-typical representations of K are precisely the irreducible subrepresentations of ind K J λ, where (J, λ) is a level-zero G-cover of (K ∩ M, τ M).
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