A mirror symmetric construction of qH*_T(G/P)_(q)

Let G be a simple simply connected complex algebraic group. We give a Lie theoretic construction of a conjectural mirror family associated to a general flag variety G/P, and show that it recovers the Peterson variety presentation for the T-equivariant quantum cohomology rings qH*T(G/P)(q) with quantum parameters inverted. For SL_n/B we relate our construction to the mirror family defined by Givental and its T-equivariant analogue due to Joe and Kim.