We introduce a novel ultrametric on the set of equivalence classes of cuspidal irreducible representations of a general linear group GL N over a non-archimedean local field, based on distinguishability by twisted gamma factors. In the case that N is prime and the residual characteristic is greater than or equal to N 2 , we prove that, for any natural number i ≤ N 2 , there are pairs of cuspidal irreducible representations whose logarithmic distance in this ultrametric is precisely −i. This implies that, under the same conditions on N , the bound N 2 in the Local Converse Theorem for GL N is sharp.

Let G be an unramified quasi-split unitary group over a p-adic field of odd residual characteristic. The goal of this paper is to describe the supercuspidal representations within certain L-packets of G, which are classified by Arthur and Mok using the theory of endoscopy. The description is given in terms of the cuspidal types constructed by Bushnell-Kutzko and Stevens. As a starting example, we require the parameters of our packets to satisfy certain regularity conditions, such that these packets consist of very cuspidal representations in the sense of Adler and Reeder. To achieve our goal, we first interpret the question as to study the reducibilities of some parabolically induced representations, using a theory of Moeglin and Shahidi; we then apply a relation, given by Blondel, between these reducibilities and the structures of some Hecke algebras, where the latter can be computed using a Theorem of Lusztig. We can interpret our final result as explicitly describing the local Langlands correspondence for G.

Based on the ideas of Bushnell-Henniart and Paskunas-Stevens, we construct explicit Whittaker data for an essentially tame supercuspidal representation of GLn(F ), where F is a non-Archimedean local field.

We compute a special case of base change of certain supercuspidal representations from a ramified unitary group to a general linear group, both defined over a p-adic field of odd residual characteristic. In this special case, we require the given supercuspidal representation to contain a skew maximal simple stratum, and the field datum of this stratum to be of maximal degree, tamely ramified over the base field, and quadratic ramified over its subfield fixed by the Galois involution that defines the unitary group. The base change of this supercuspidal representation is described by a canonical lifting of its underlying simple character, together with the base change of the level-zero component of its inducing cuspidal type, modified by a sign attached to a quadratic Gauss sum defined by the internal structure of the simple character. To obtain this result, we study the reducibility points of a parabolic induction and the corresponding module over the affine Hecke algebra, defined by the covering type over the product of types of the given supercuspidal representation and of a candidate of its base change.

We describe a special case of base change of certain supercuspidal representations from a ramified unitary group to a general linear group, both defined over a p-adic field of odd residual characteristic. Roughly speaking, we require the underlying stratum of a given supercuspidal representation to be skew maximal simple, and the field datum of this stratum to be of maximal degree, tamely ramified over the base field, and quadratic ramified over its subfield fixed by the Galois involution that defines the unitary group. The base change of this supercuspidal representation is described by a canonical lifting of its underlying simple character, together with the base change of the level-zero component of its inducing cuspidal type, modified by a sign attached to a quadratic Gauss sum defined by the internal structure of the simple character. To obtain this result, we study the reducibility points of a parabolic induction and the corresponding module over the affine Hecke algebra, defined by the covering type over the product of types of the given supercuspidal representation and of a candidate of its base change.