Twisted character of a small representation of PGL(4)
  We compute by a purely local method the elliptic, twisted by transpose-inverse, character \chi_\pi of the representation \pi=I_{(3,1)}(1_3) of PGL(4,F) normalizedly induced from the trivial representation of the maximal parabolic subgroup of type (3,1), where F is a p-adic field. Put C=(GL(2,F)xGL(2,F))'/F^x (F^x embeds diagonally, prime means equal determinants). It is a twisted elliptic endoscopic group of PGL(4). We deduce from the computation that \chi_\pi is an unstable function: its value at one twisted regular elliptic conjugacy class with norm in C is minus its value at the other class within the twisted stable conjugacy class, and zero at the classes without norm in C. Moreover \pi is the unstable endoscopic lift of the trivial representation of C. Naturally, this computation plays a role in the theory of lifting from C (=``SO(4,F)'') and PGp(2,F) to PGL(4,F) using the trace formula. Our work develops a 4-dimensional analogue of the model of the small representation of PGL(3,F) introduced by the first author with Kazhdan in a 3-dimensional case, and it uses the classification of twisted stable and unstable regular conjugacy classes in PGL(4,F). It extends the local method of computation introduced by us in the 3-dimensional case.   An extension math.NT/0606263 of our work here to apply to similar representations of GL(4,F) whose central character is nontrivial will appear in Int. J. Number Theory. 
Yuval Z. Flicker, Dmitrii Zinoviev