Lower boundary hyperplanes of the canonical left cells in the affine Weyl group \(W_a(\widetilde A_{n-1})\). (Q997114)

Let \(\Gamma\) be any canonical left cell of the affine Weyl group \(W_a\) of type \(\widetilde A_{n-1}\) for \(n>1\). In this paper, the author describes the lower boundary hyperplanes for \(\Gamma\). His results answer two questions of J. E. Humphreys: (1) Can one find the set \(B(L)\) of all lower boundary hyperplanes for any canonical left cell \(L\) of \(W_a\)? (2) How does the partition \(\varphi(L)\) determine the set \(B(L)\), and in which case does the set \(B(L)\) determine the partition \(\varphi(L)\) also?