There are two types of parabolic Kazhdan Lusztig polynomials, namely, of type -1: $$P_{x,w}^{I,-1}$$ and of type $$q$$: $$P_{x,w}^{I,q}$$. See Kazhdan–Lusztig and R-Polynomials, Young’s Lattice, and Dyck Partitions

My question: What is the meaning of $$-1$$ and $$q$$?

• It woulod be helpful to refer to Deodhar's 1987 paper and Spergel's 1997 paper for the broader context. – Jim Humphreys Jun 3 '19 at 1:45

These polynomials are connected to the canonical basis of the induction from a parabolic subalgebra $$H_I$$ up to the whole Hecke algebra $$H$$. The difference is which module is being induced: The $$q$$-variant induces the trivial module (on which the generators $$T_s$$ of $$H_I$$ act as multiplication by $$q$$) while the $$(-1)$$ variant induces the sign module (on which the generators act as multiplication by $$-1$$).
• 1.I would like to how to induce from $H_I$ to $H$? 2. Is $H_I$ the Hecke algebra of Coxeter system $(W_I,I)$. 3. I would like to know is $H_I$-module, or the parabolic subalgebra $H_I$, or both being induced? – James Cheung May 4 '19 at 7:07
these are polynomials in $$q$$ of two types, which satisfy either of the two recursions: $$P_{v,w}^{I,q}=-P_{v,ws}^{I,q}\;\;\text{or}\;\;P_{v,w}^{I,-1}=qP_{v,ws}^{I,-1},$$ see for example these lecture notes.