Periodic Kazhdan-Lusztig polynomials (for an affine Weyl group) are polynomials that control Jordan-Holder multiplicities for certain representations ("baby Verma modules") of an algebraic group in positive characteristic, roughly analogously to how KL polynomials control Jordan-Holder multiplicities of Verma modules in category O. I believe they were introduced in Lusztig's paper "Hecke algebras and Jantzen's generic decomposition patterns"; a reference is Soergel, "Kazhdan-Lusztig polynomials and a combinatoric for tilting modules".

I need to be able to compute periodic Kazhdan-Lusztig polynomials. Are there programs that do this or papers doing computations in specific cases (e.g. do we have a combinatorial interpretation in type A?)

I am aware of many such resources for ordinary KL polynomials, but my impression is that turning such computations into computations of periodic KL polynomials is quite hard to do by hand. If this is wrong, please tell me.

inverseK-L polynomials for an affine Weyl group. He refers to papers by Andersen and Kato which study these polynomials theoretically: in the affine case, unlike the finite case, they differ a lot from the usual K-L polynomials. But Lusztig's "periodic" constructions look really periodic and apply to modular Lie algebra representations, not algebraic group representations. All are hard to compute explicitly. $\endgroup$ – Jim Humphreys Jan 2 '15 at 18:41Representation Theory. $\endgroup$ – Jim Humphreys Jan 2 '15 at 18:43