Abstract
Abstract 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.
Highlights
The local Langlands correspondence for a general linear group over a non-Archimedean local field F is, roughly speaking, a parametrization of its irreducible admissible representations in terms of representations of the Weil–Deligne group of F
In the strongly ramified case, Q ı NE=F is conjugate-orthogonal, and so the two self-dual candidates have the same parity. This explains why the previous method no longer works, and we have to rely on the complete structure of the modules over the Hecke algebra
We will see that this condition forces our unitary group to be ramified, i.e., F=F is ramified, and our inducing types take a simple form
Summary
The local Langlands correspondence for a general linear group over a non-Archimedean local field F is, roughly speaking, a parametrization of its irreducible admissible representations in terms of representations of the Weil–Deligne group of F. We compute a special case of base change for ramified unitary groups, which complements the previous result in [53] for describing the local Langlands correspondence for packets of supercuspidal representations of unramified quasi-split unitary groups. In the non-strongly ramified case, the two self-dual candidates have opposite parities because they differ by an unramified character Q such that Q ı NE=F is conjugate-symplectic In this case we can determine the correct base change between the two by computing their parities using Asai L-functions for example [25]. In the strongly ramified case, Q ı NE=F is conjugate-orthogonal, and so the two self-dual candidates have the same parity (see Section 3.4 for a detailed discussion) This explains why the previous method no longer works, and we have to rely on the complete structure of the modules over the Hecke algebra. If ƒ W Z ! S is a sequence into a set S , we extend ƒ from Z to R by putting ƒ.r/ D ƒ.dre/ and ƒ.rC/ D ƒ.drCe/, where dre and drCe are the smallest integers r and > r, respectively
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