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# Casselman's Basis of Iwahori Vectors and the Bruhat Order

Published:2011-06-26
Printed: Dec 2011
• Daniel Bump,
Department of Mathematics Stanford University California 94305-2125 USA
• Maki Nakasuji,
Department of Mathematics Stanford University California 94305-2125 USA
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## Abstract

W. Casselman defined a basis $f_u$ of Iwahori fixed vectors of a spherical representation $(\pi, V)$ 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 $W$. On the other hand, there is a natural basis $\psi_u$, and one seeks to find the transition matrices between the two bases. Thus, let $f_u = \sum_v \tilde{m} (u, v) \psi_v$ and $\psi_u = \sum_v m (u, v) f_v$. Using the Iwahori-Hecke algebra we prove that if a combinatorial condition is satisfied, then $m (u, v) = \prod_{\alpha} \frac{1 - q^{- 1} \mathbf{z}^{\alpha}}{1 -\mathbf{z}^{\alpha}}$, where $\mathbf z$ are the Langlands parameters for the representation and $\alpha$ runs through the set $S (u, v)$ of positive coroots $\alpha \in \hat{\Phi}$ (the dual root system of $G$) such that $u \leqslant v r_{\alpha} < v$ with $r_{\alpha}$ the reflection corresponding to $\alpha$. The condition is conjecturally always satisfied if $G$ is simply-laced and the Kazhdan-Lusztig polynomial $P_{w_0 v, w_0 u} = 1$ with $w_0$ the long Weyl group element. There is a similar formula for $\tilde{m}$ conjecturally satisfied if $P_{u, v} = 1$. This leads to various combinatorial conjectures.
 Keywords: Iwahori fixed vector, Iwahori Hecke algebra, Bruhat order, intertwining integrals
 MSC Classifications: 20C08 - Hecke algebras and their representations 20F55 - Reflection and Coxeter groups [See also 22E40, 51F15] 22E50 - Representations of Lie and linear algebraic groups over local fields [See also 20G05]

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