  This is the third paper in the series. Here we define a few combinatorial
orders on Young tableaux. The first order is obtained from induced Duflo order
by the extension with the help of Vogan T_{\alpha, \beta} procedure. We call it
Duflo-Vogan order. The second order is obtained from the generalization of
Spaltenstein's construction by consideration of an orbital variety as a double
chain of nilpotent orbits. We call it the chain order. Again, we use Vogan's
T_{\alpha, \beta} procedure, however, this time to restrict the chain order. We
call it Vogan-chain order. The order on Young tableaux defined by the inclusion
of orbital variety closures is called a geometric order and the order on Young
tableaux defined by inverse inclusion of primitive ideals is called an
algebraic order.
  We get the following relations between the orders: Duflo-Vogan order is an
extension of the induced Duflo order; the algebraic order is an extension of
Duflo-Vogan order; the geometric order is an extension of the algebraic order;
Vogan-chain order is an extension of the geometric order; and, finally, the
chain order is an extension of Vogan-chain order. The computationsshow that
Duflo-Vogan and Vogan-chain orders coincide on sl_n for n<10 and in n=10 there
is one case (up to T_{\alpha,\beta} procedure and transposition) where
Vogan-chain order is a proper extension of Duflo-Vogan order. In this only case
the algebraic order coincides with Vogan-chain order. These computations permit
us to conjecture that in sl_n the algebraic order coincides with the geometric
order. As well we conjecture that the combinatorics of both the inclusions on
primitive ideals and on orbital variety closures is defined by Vogan-chain
order on Young tableaux.
