The Distinguished Involutions Of The Weyl Group Of Type E₇

Let W be the Weyl group of type E7. In this paper, we study the left cells in W, and describe all the distinguished involutions of W. We get about half of the distinguished involutions by finding the elements of the form wj, applying thoerem 1.7.3.5 and computing the right and left {s, t｝- star operation in matlab. In order to search the other distinguished involutions, we compute the Kazhdan-Lusztig polynomials of the possible elements. Besides that, we conclude the result about the relation between d and wod which are in the two-sided cells of the same orbit.