element which separately acts on { 1 , . . . , p } and { p + 2 , . . . , n } as required, we have that w $^{′}$u = w . As an example, suppose that p = q = 3, and let u be the signed permutation 316254. To unscramble the 312, we must multiply on the left by 1 ↦→ 2, 2 ↦→ 3, 3 ↦→ 1, and to unscramble the 65 we must multiply on the left by 5 ↦→ 6 , 6 ↦→ 5. Thus we multiply u on the left by w ′ = 231465 to get w $^{′}$u = w = 125364. Note that a permutation w having the properties above is completely determined by the positions (in the one-line notation) of 1 , . . . , p among the first n − 1 spots, which can be chosen freely. Thus there are ( n − 1 p ) such w , and hence ( n − 1 p ) closed ˜ K -orbits, as claimed. Definition 5.3.2. Let Q ∈ ˜ K \ X be a closed orbit. Call the flag wB ∈ Q , where w has the properties listed in the proof of Proposition 5.3.1, the standard representative of Q . For w ∈ W such that wB is the standard representative of some closed orbit Q , define I$_{w}$ := { i ∈ { 1 , . . . , n − 1 } | w ( i ) > p + 1 } . For each i ∈ I$_{w}$ , define Finally, define C ( i ) := # { j | i < j ≤ n − 1 , w ( j ) ≤ p } . f ( w ) := ∑ i ∈ I$_{w}$ C ( i ) . Then we have the following formula for the S -equivariant class of the closed orbit Q : Proposition 5.3.3. Let Q = ˜ K · wB be any closed orbit, with wB the standard representative. 124