ON STABILITY AND ENDOSCOPIC TRANSFER, SECTION 1
for a generic element of g(2), then the fundamental relative invariants are:
f2 — x2p + 2rxy + y26
fz = zv- ut
A calculation shows that
P = h • h •
g(2) contains 16 open orbits indexed by the square classes of f\ and f2-
1.2. Some classes of examples
1. Consider the stable orbits associated with the partitions 2
n - r
Lemma 1.2.1. The Ranga Rao data associated to the partition 2rl2(n~r) is given as follows.
0(2) = j x = r B] } r :BESym(r)j^Sym(r )
: A € G L ( r ) , [ | | ] € Sp(2(n - r))
Thus the prehomogeneous vector space (M,g(2)) may be identified with (GL(r),Sym(r)) where
g-X:= gX'g , g € GL(r) , X € Sym(r)
(This P. V.S. appears as number 2 in the Sato-Kimura List (cf. ).)
• The Ranga Rao function p is given by
tp(X) = det
(B) , X and B as above.
• The open M.(F)-orbits are in one to one correspondence with the set of equivalence classes of quadratic forms
of rank r.
All of this is easy to check. •
Let's recall the classification of quadratic forms (recall that we are assuming that q = 1 mod 4).
Proposition 1.2.2. ()
Let d, T] and r denote the discriminant, Hasse-invariant and rank, respectively, of a quadratic form.
(i) lfr — \, then there are 4 equivalence classes corresponding to the square classes of the discriminant
(ii) Ifr = 2, then there are 7 equivalence classes given by the conditions: d = 1 mod (Fx)2 , or d = e, 7r, CK mod (F 1 ) 2
(iii) Ifr2, then there are 8 equivalence classes corresponding to the four possible choices of the square class of d
and the two choices ofrj.