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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