Homogeneous polynomials and SL(2,R)

From: Jose Carlos Santos (jcsantos_at_fc.up.pt)
Date: 06/07/04 

Date: 7 Jun 2004 00:50:08 -0700

Hi all,

Let n be a natural number and define H_n as the real vector space of all
homogeneous polynomial functions of degree n from R^2 into R. The group
SL(2,R) of all 2x2 matrices with determinant 1 acts on H_n: if P belongs
to H_n and g belongs to SL(2,R) then put

   (g.P)(x,y) = P(g^{-1}.(x,y)).

There's a non-degenerate bilinear form B:H_n x H_n --> R which is invariant;
this means that, for every g in SL(2,R) and every P and Q in H_n, we have

   B(g.P,g.Q) = B(P,Q).

Besides, such a form is unique up to a scalar multiple. I know how to prove
all this, but I would like to know these bilinear forms explicitely. I was
able to determine that for H_1 we have

   B(ax + by,a'x + b'y) = ab' - ba'

and that for H_2 we have

   B(ax^2 + bxy + cy^2,a'x^2 + b'xy + c'y^2) = 2ac' - bb' + 2ca'.

Does anyone see how to define B for each n?

Best regards,

Jose Carlos Santos