# Papers/Programs for computing periodic KL polynomials?

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.

• It would be helpful to indicate some background material on the "periodic" stuff, for instance this freely available paper by Lusztig: ams.org/journals/ert/1997-001-11/S1088-4165-97-00033-2 (also maybe add a tag 'kazhdan-lusztig'). Most important, clarify which Coxeter groups are involved here. – Jim Humphreys Jan 2 '15 at 14:34
• Thanks! I have made the changes you suggest. I'm working here with affine Weyl groups, though having an answer for affine type A would already be really good. – dhy Jan 2 '15 at 15:59
• @JimHumphreys: I am confused; Soergel, in his remark 4.4, calls the polynomials p_{B,A} "periodic polynomials" (and refers to the generic decompositions pattern paper). Has the notation changed since Soergel wrote that paper? I am not too familiar with the field. – dhy Jan 2 '15 at 16:53
• It's true that Soergel for some reason used the "periodic" language, but he is actually just working with slightly modified inverse K-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. – Jim Humphreys Jan 2 '15 at 18:41
• [edited] To clarify your added references, the Lusztig paper (sciencedirect.com/science/article/pii/0001870880900316) deals with inverse K-L polynomials (not periodic versions) and the Soergel paper (ams.org/mathscinet-getitem?mr=1444322) also involves the usual K-L polynomials, especially for an affine Weyl group. Lusztig's periodic polynomials are defined in his papers in Representation Theory. – Jim Humphreys Jan 2 '15 at 18:43