ETD

• Bruhat G-order
• Bruhat order
• Hermitian symmetric varieties
• KLV polynomials
• linear algebraic group
• local system
• orbits

Let G be a connected reductive linear algebraic group over an algebraically closed field. Fix a torus T and a Borel subgroup B and suppose there is a parabolic subgroup P such that G/L is a Hermitian symmetric variety.
The Borel subgroup B acts on G/L with a finite number of orbits. The set of these orbits can be ordered
through the Bruhat order.
We give parametrizations for the B-orbits in cases where the base field has characteristic 2.
We also give a characterization for the Bruhat order among these orbits and a formula to compute the dimension.

Consider now the same situation over the complex field and for every B-orbit the set of B-equivariant rank 1 local systems up to isomorphisms.
Following Lusztig and Vogan we can order the pairs orbit-local system through the Bruhat G-order.
We will show a combinatorial characterization of the Bruhat G-order. This can be used to improve the computation of the Kazhdan-Lusztig-Vogan polynomial.