Twisted character of a small representation of GL(4)
| dc.creator | Flicker, Yuval Z. | |
| dc.creator | Zinoviev, Dmitrii | |
| dc.date | 2006-06-12 | |
| dc.date.accessioned | 2026-07-07T07:17:11Z | |
| dc.date.available | 2026-07-07T07:17:11Z | |
| dc.description | We compute by a purely local method the (elliptic) twisted by transpose-inverse character χ_{π_Y} of the representation π_Y=I_{(3,1)}(1_3xχ_Y) of G=GL(4,F), where F is a p-adic field, p not 2, and Y is an unramified quadratic extension of F; χ_Y is the nontrivial character of F^\x/N_{Y/F}Y^x. The representation π_Y is normalizedly induced from \pmatrix m_3&\ast 0&m_1\endpmatrix \mapstoχ_Y(m_1), m_i in GL(i,F), on the maximal parabolic subgroup of type (3,1). We show that the twisted character χ_{π_Y} of π_Y is an unstable function: its value at a twisted regular elliptic conjugacy class with norm in C_Y=``GL(2,Y)/F^x'' is minus its value at the other class within the twisted stable conjugacy class. It is zero at the classes without norm in C_Y. Moreover π_Y is the endoscopic lift of the trivial representation of C_Y. We deal only with unramified Y/F, as globally this case occurs almost everywhere. Naturally this computation plays a role in the theory of lifting of C_Y and GSp(2) to GL(4) using the trace formula. Our work extends -- to the context of nontrivial central characters -- the work of math.NT/0606262, where representations of PGL(4,F) are studied. In math.NT/0606262 a 4-dimensional analogue of the model of the small representation of PGL(3,F) introduced with Kazhdan in a 3-dimensional case is developed, and the local method of computation introduced by us in the 3-dimensional case is extended. As in math.NT/0606262 we use here the classification of twisted (stable) regular conjugacy classes in GL(4,F). | |
| dc.description | Accepted for publication by the International Journal of Number Theory | |
| dc.identifier | https://arxiv.org/abs/math/0606263 | |
| dc.identifier | http://arxiv.org/abs/math/0606263 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/113853 | |
| dc.subject | Number Theory | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Representation Theory | |
| dc.title | Twisted character of a small representation of GL(4) | |
| dc.type | text |