The Left Cells In The Affine Weyl Group (?)₇ With A=5, 6

Left cells were introduced by Kazhdan and Lusztig in 1979,which play a very important role in the representation of Hecke algebra and algebraic group.In this paper,we use Shi's algorithm to describe all the left cells in the two-sided cells Ω of affine Weyl group E₇ with a(Ω) = 5,6, in terms of left cells graph.Then we work out all the distinguished involutions in the two-sided cells W₅.