## A proof of Lusztig's conjectures for affine type $$G_2$$ with arbitrary parameters

### J. Guilhot and J. Parkinson

#### Abstract

We prove Lusztig's conjectures $${\bf P1}$$–$${\bf P15}$$ for the affine Weyl group of type $$\tilde{G}_2$$ for all choices of parameters. Our approach to compute Lusztig's $$\mathbf{a}$$-function is based on the notion of a "balanced system of cell representations" for the Hecke algebra. We show that for arbitrary Coxeter type the existence of balanced system of cell representations is sufficient to compute the $$\mathbf{a}$$-function and we explicitly construct such a system in type $$\tilde{G}_2$$ for arbitrary parameters. We then investigate the connection between Kazhdan-Lusztig cells and the Plancherel Theorem in type $$\tilde{G}_2$$, allowing us to prove $${\bf P1}$$ and determine the set of Duflo involutions. From there, the proof of the remaining conjectures follows very naturally, essentially from the combinatorics of Weyl characters of types $$G_2$$ and $$A_1$$, along with some explicit computations for the finite cells.

Keywords: Lusztig conjectures, Hecke algebra, Kazhdan-Lustig polynomial.

: Primary 20C08; secondary 05E10.

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 Tuesday, December 5, 2017