TY - JOUR

T1 - Computation of a twisted character of a small representation of GL(3, E)

AU - Flicker, Yuval Z.

AU - Zinoviev, Dmitrii

N1 - Funding Information:
The first author was partially supported by the Humboldt Stiftung, MPIM-Bonn, CRC 701 at Universität Bielefeld and the Fulbright Foundation. The second author was partially supported by the Russian Foundation for Basic Research, project no. 12-01-00905. We are very grateful to the referee for very careful reading of this work.

PY - 2012/8

Y1 - 2012/8

N2 - 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 tḡ -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.

AB - 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 tḡ -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.

KW - Admissible representations of a p-adic group

KW - endoscopy

KW - stable conjugacy

KW - twisted characters

UR - http://www.scopus.com/inward/record.url?scp=84863767627&partnerID=8YFLogxK

U2 - 10.1142/S1793042112500704

DO - 10.1142/S1793042112500704

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AN - SCOPUS:84863767627

SN - 1793-0421

VL - 8

SP - 1153

EP - 1230

JO - International Journal of Number Theory

JF - International Journal of Number Theory

IS - 5

ER -