ON STABILITY AND ENDOSCOPIC TRANSFER, SECTION 1
for a generic element of g(2), then the fundamental relative invariants are:
fi=Sp-r2
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
r
l
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))
M =
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. [25]).)
The Ranga Rao function p is given by
tp(X) = det
n
"
r
(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.
Proof
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. ([29])
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
andne {±1}.
(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.
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