Computation of a twisted character of a small representation of GL(3, E)

Yuval Z. Flicker, Dmitrii Zinoviev

Research output: Contribution to journalArticlepeer-review


Let E/F be a quadratic extension of p-adic fields, p ≠ 2. Let x → x̄ be the involution of E over F. The representation π of GL(3, E) normalizedly induced from the trivial representation of the maximal parabolic subgroup is invariant under the involution σ(g) = J t-1J. We compute - by purely local means - the σ-twisted character χ pi σ of π. We show that it is σ-unstable, namely its value at one σ-regular-elliptic conjugacy class within a stable such class is equal to negative its value at the other such conjugacy class within the stable class, or zero when the σ-regular-elliptic stable conjugacy class consists of a single such conjugacy class. Further, we relate this twisted character to the twisted endoscopic lifting from the trivial representation of the "unstable" twisted endoscopic group U(2, E/F) of GL(3, E). In particular π is σ-elliptic, that is, χ pi σ is not identically zero on the σ-elliptic set.

Original languageEnglish
Pages (from-to)1153-1230
Number of pages78
JournalInternational Journal of Number Theory
Issue number5
StatePublished - Aug 2012
Externally publishedYes


  • Admissible representations of a p-adic group
  • endoscopy
  • stable conjugacy
  • twisted characters


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