The Cells Of Affine Weyl Group (?)₄in Quasi-split Case

The affine Weyl group (Cn, S) can be realized as the fixed point set of the affine Weyl group (A2n-1,S) under a certain group automorphism, the weight function L on Cn is the restriction of the length function I of A2n-1to Cn, we say Coxeter group (Cn, L) is in the quasi-split case. We will consider all cells of weighted Coxeter group (C4, L) in quasi-split case in this paper. Our main result is as follows:for any partition λ of8, we describe all left cells and two-sided cells in Eλ and research their properties in structure, e.g. left-connectedness, infinite-ness, generalized τ-invariants and so on.