Abstract: The famous Kazhdan-Lusztig positivity states that the transition matrix between the canonical basis and the standard basis of the Hecke algebra (of a Weyl group) is a matrix with polynomial entries with non-negative integer coefficients. In this talk I will describe a factorization of this positivity via certain intermediate bases called the hybrid bases. I will also present a reformulation of the statements in terms of a natural restriction map on Hecke algebras. This framework also subsumes the positivity of parabolic Kazhdan-Lusztig polynomials corresponding to the sign representations. This is joint work with Ashish Mishra and Shraddha Shrivastava.