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Search: All articles in the CJM digital archive with keyword intertwining

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1. CJM Online first

Henniart, Guy; Sécherre, Vincent
 Types et contragrÃ©dientes Soit $\mathrm{G}$ un groupe rÃ©ductif $p$-adique, et soit $\mathrm{R}$ un corps algÃ©briquement clos. Soit $\pi$ une reprÃ©sentation lisse de $\mathrm{G}$ dans un espace vectoriel $\mathrm{V}$ sur $\mathrm{R}$. Fixons un sous-groupe ouvert et compact $\mathrm{K}$ de $\mathrm{G}$ et une reprÃ©sentation lisse irrÃ©ductible $\tau$ de $\mathrm{K}$ dans un espace vectoriel $\mathrm{W}$ de dimension finie sur $\mathrm{R}$. Sur l'espace $\mathrm{Hom}_{\mathrm{K}(\mathrm{W},\mathrm{V})}$ agit l'algÃ¨bre d'entrelacement $\mathscr{H}(\mathrm{G},\mathrm{K},\mathrm{W})$. Nous examinons la compatibilitÃ© de ces constructions avec le passage aux reprÃ©sentations contragrÃ©dientes $\mathrm{V}^Äe$ et $\mathrm{W}^Äe$, et donnons en particulier des conditions sur $\mathrm{W}$ ou sur la caractÃ©ristique de $\mathrm{R}$ pour que le comportement soit semblable au cas des reprÃ©sentations complexes. Nous prenons un point de vue abstrait, n'utilisant que des propriÃ©tÃ©s gÃ©nÃ©rales de $\mathrm{G}$. Nous terminons par une application Ã  la thÃ©orie des types pour le groupe $\mathrm{GL}_n$ et ses formes intÃ©rieures sur un corps local non archimÃ©dien. Keywords:modular representations of p-adic reductive groups, types, contragredient, intertwiningCategory:22E50

2. CJM 2011 (vol 63 pp. 1238)

Bump, Daniel; Nakasuji, Maki
 Casselman's Basis of Iwahori Vectors and the Bruhat Order 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 integralsCategories:20C08, 20F55, 22E50

3. CJM 1998 (vol 50 pp. 193)

Xu, Yuan
 Intertwining operator and $h$-harmonics associated with reflection groups We study the intertwining operator and $h$-harmonics in Dunkl's theory on $h$-harmonics associated with reflection groups. Based on a biorthogonality between the ordinary harmonics and the action of the intertwining operator $V$ on the harmonics, the main result provides a method to compute the action of the intertwining operator $V$ on polynomials and to construct an orthonormal basis for the space of $h$-harmonics. Keywords:$h$-harmonics, intertwining operator, reflection groupCategories:33C50, 33C45

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