Casselman's basis of iwahori vectors and the bruhat order

Abstract

W. Casselman defined a basis fu of Iwahori fixed vectors of a spherical representation (π, V) of a split semisimple p-adic group G over a nonarchimedean local field F by the condition that it be dual to the intertwining operators, indexed by elements u of the Weyl group IV. On the other hand, there is a natural basis i/iu, and one seeks to find the transition matrices between the two bases. Thus, let fu =matamtical equation repersenting and matamtical equation repersenting Using the Iwahori-Hecke algebra we prove that if a combinatorial condition is satisfied, then m(u,v) = matamtical equation repersenting where z are the Langlands parameters for the representation and a runs through the set S(u, v) of positive coroots a ε (the dual root system of G) such that u ≤ vrα < v with rα the reflection corresponding to a. The condition is conjecturally always satisfied if G is simply-laced and the Kazhdan-Lusztig polynomial matamtical equation repersenting = 1 with wo the long Weyl group element. There is a similar formula for m conjecturally satisfied if P u,v = 1. This leads to various combinatorial conjectures. © Canadian Mathematical Society 2011.