# James Parkinson (University of Sydney)

## On Lusztig's conjectures P1-P15

The theory of Kazhdan-Lusztig cells plays a fundamental role in the representation theory of Coxeter groups and Hecke algebras. The theory is relatively well understood for finite and affine types in the "equal parameter" case, primarily due to the geometric interpretation of Kazhdan-Lusztig theory in this setting and the strong positivity properties that follow from this interpretation. In contrast much less is known in the unequal parameter case, where these positivity properties no longer hold.
In 2003 Lusztig made 15 conjectures, now known as P1-P15, to capture essential properties of cells for arbitrary parameters. In this talk we discuss our recent proof of P1-P15 for affine types $$G_2$$ and $$C_2$$ with arbitrary parameters, providing the first infinite Coxeter groups other than the infinite dihedral group for which conjectures P1-P15 are known to hold for arbitrary unequal parameters. Our approach is based on two concepts: (a) a balanced system of cell representations for the Hecke algebra, and (b) the asymptotic Plancherel formula. In this talk we will focus on (a), and show that the existence of such a system of representations is sufficient to compute Lusztig's a-function.
This is joint work with Jeremie Guilhot, Universite de Tours.