On the index of the quotient of a Borel subalgebra by an ad-nilpotent ideal

In this paper, we give upper bounds for the index of the quotient of the Borel subalgebra of a simple Lie algebra or its nilpotent radical by an ad-nilpotent ideal. For the nilpotent radical quotient, our bound is a generalization of the formula for the index given by Panov in the type A case. In general, this bound is not exact. Using results from Panov, we show that the upper bound for the Borel quotient is exact in the type $A$ case, and we conjecture that it is exact in general.