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.