The Weyl group W plays a fundamental role in Lie theory. For example, for the complex special linear group G=SL_n the Weyl group is the symmetric group on n letters. In general W is defined to be a subquotient of the Lie group G. When can the Weyl group be lifted to a subgroup of the group G? Technically this is asking if a certain short exact sequence splits The answer for Sl_n is yes if n is odd but no if n is even.
This question was raised by J. Tits in 1966. I give a complete answer in the case of simply connected complex groups . The result is the same over a finite field of odd order.