  We describe on any finitely generated group G the space of maps G->C which
satisfy the parallelogram identity, f(xy)+f(xy^{-1})=2f(x)+2f(y).
  It is known (but not well-known) that these functions correspond to
Zariski-tangent vectors at the trivial character of the character variety of G
in SL_2(C). We study the obstructions for deforming the trivial character in
the direction given by f. Along the way, we show that the trivial character is
a smooth point of the character variety if dim H_1(G,C)<2 and not a smooth
point if dim H_1(G,C)>2.
