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When $Sp(2n,\mathbb{C})$ acts on the flag variety of $SL(2n,\mathbb{C})$, theorbits are in bijection with fixed point free involutions in the symmetricgroup $S_{2n}$. In this case, the associated Kazhdan-Lusztig-Vogan polynomials$P_{v,u}$ can be indexed by pairs of fixed point free involutions $v\geq u$,where $\geq$ denotes the Bruhat order on $S_{2n}$. We prove that thesepolynomials are combinatorial invariants in the sense that if $f: [u, w_0 ]\rightarrow [u , w_0]$ is a poset isomorphism of upper intervals in the Bruhatorder on fixed point free involutions, then $P_{v,u} = P_{f(v),u}$ for all $v\geq u$.

Combinatorial Invariance of Kazhdan-Lusztig-Vogan Polyomials for Fixed Point Free Involutions