Complexity of the usual torus action on Kazhdan-Lusztig varieties

We investigate the class of Kazhdan-Lusztig varieties, and its subclass of matrix Schubert varieties, endowed with a naturally defined torus action. Writing a matrix Schubert variety (X (w)) over bar as (X (w)) over bar = Y-w x C-d (where d is maximal possible), we show that Y-w can be of complexity-k exactly when k not equal 1. Also, we give a combinatorial description of the extremal rays of the weight cone of a Kazhdan-Lusztig variety, which in particular turns out to be the edge cone of an acyclic directed graph. As a consequence we show that given permutations v and w , the complexity of Kazhdan-Lusztig variety indexed by (v, w) is the same as the complexity of the Richardson variety indexed by (v, w). Finally, we use this description to compute the complexity of certain Kazhdan-Lusztig varieties.