[Paper Review] Changements de base explicites des représentations supercuspidales de U(1,1)(F)

Laure Blasco|ArXiv.org|Sep 18, 2009

Advanced Algebra and Geometry11 references3 citations

TL;DR

This paper explicitly computes the unstable (labile) base change for supercuspidal representations of the unitary group $U(1,1)(F_0)$, where $F_0$ is a non-archimedean local field of characteristic zero and odd residual characteristic. Using the theory of types and explicit character computations, it determines how these representations transfer to $GL(2,F)$, completing a key step toward understanding endoscopic base change for $U(2,1)(F_0)$, with results derived via character identities and Galois invariance.

ABSTRACT

Let F be a nonarchimedean local field of characterisitic 0 and odd residual characteristic. We describe explicitly the two base change lifts of supercuspidal representations of U(1,1)(F). This represents a step towards the goal of describing base change of endoscopic supercuspidal L-packets of U(2,1)(F).

Motivation & Objective

• To complete the description of base change for endoscopic supercuspidal packets of $U(2,1)(F_0)$ by computing the unstable base change for $U(1,1)(F_0)$.
• To determine the image of supercuspidal representations of $U(1,1)(F_0)$ under base change to $GL(2,F)$, where $F$ is a quadratic extension of $F_0$.
• To resolve the ambiguity in distinguishing between stable and unstable base change images for supercuspidal representations, especially in singleton endoscopic packets.
• To provide an explicit, computable description of base change using character identities and the theory of types, under minimal assumptions (characteristic zero, odd residual characteristic).
• To extend prior results on $U(2,1)$ base change to the $U(1,1)$ case, forming a foundational step for the full endoscopic packet transfer.

Proposed method

• Uses the classification of supercuspidal representations of $U(1,1)(F_0)$ via the theory of types developed by Bushnell and Kutzko.
• Applies explicit character computations on compact open subgroups to verify base change identities, particularly using the trace formula on strata and stabilizers.
• Employs the method of Y. Flicker for computing stable base change, combined with results from J. Rogawski on character identities and endoscopic transfer.
• Relies on the compatibility of base change with twisting by characters, allowing reduction to minimal level supercuspidal representations.
• Uses the structure of endoscopic packets—singleton or cardinality 2—to distinguish cases, especially in the ramified vs. unramified extension setting.
• Applies technical lemmas on character traces over subgroups and Galois-fixed subgroups to compare representations across $U(1,1)(F_0)$ and $GL(2,F)$.

Experimental results

Research questions

• RQ1 How does the unstable base change map for supercuspidal representations of $U(1,1)(F_0)$ to $GL(2,F)$ behave, and how can it be explicitly described?
• RQ2 What distinguishes the images of stable and unstable base change in the case of singleton endoscopic packets?
• RQ3 How can character identities be used to verify the base change correspondence between $U(1,1)(F_0)$ and $GL(2,F)$ representations?
• RQ4 What role does the Galois action play in distinguishing base change images, particularly for level-zero supercuspidals?
• RQ5 How does the structure of the endoscopic packet (singleton vs. cardinality 2) affect the base change computation?

Key findings

• The unstable base change for supercuspidal representations of $U(1,1)(F_0)$ to $GL(2,F)$ is explicitly determined using character trace identities and type theory.
• For level-zero supercuspidals, the base change image is fully determined: all but two are in singleton endoscopic packets when $F$ is ramified over $F_0$, and all are in such packets when $F$ is unramified.
• The image of the base change map consists of admissible, $G_{F/F_0}$-invariant, central character trivial representations of $GL(2,F)$.
• The method successfully distinguishes between stable and unstable base change images, particularly in singleton endoscopic packets, by verifying character identities between $U(1,1)(F_0)$ and $GL(2,F)$.
• The computation relies on a technical lemma on character traces over subgroups, which allows comparison of representations across $U(1,1)$ and $GL(2)$ via stabilizer structures and group actions.
• The results are valid under minimal assumptions: $F_0$ of characteristic zero and odd residual characteristic, with no further ramification restrictions.