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Representations of quantum groups at a $p$-th root of unity and of semisimple groups in charactersitic $p$ : independence of $p$

Representations of quantum groups at a $p$-th root of unity and of semisimple groups in charactersitic $p$ : independence of $p$

H.H. ANDERSEN, J.C. JANTZEN, W. SOERGEL
Representations of quantum groups at a $p$-th root of unity and of semisimple groups in charactersitic $p$ : independence of $p$
  • Année : 1994
  • Tome : 220
  • Format : Papier, Électronique
  • Langue de l'ouvrage :
    Anglais
  • Class. Math. : 20G05, 17B35, 17B37
  • Nb. de pages : 338
  • ISSN : 0303-1179
  • DOI : 10.24033/ast.251

Consider an indecomposable finite root system $R$. associate to $R$ two families of objects : 1.) For any odd integer $p>1$ (prime to $3$ if $R$ is of type $G_2$) let $U_p$ be the quantized enveloping algebra at a $p$-th root of unity. We take here Lusztig's version. 2.) For any prime $p$ let $G_p$ be the semisimple connected and simply connected algebraic group over an algebraically closed field of characteristic $p$. Restrict to the case where $p$ is greater than the Coxeter number $h$ of $R$. Consider the block of the trivial one dimensional module for $U_p$ resp. for $G_p$. The simple modules in this block are indexed by certain elements in the affine Weyl group $W_a$ of $R$. Suppose that $L_w$ is the simple module indexed by $w$ and that $V_w$ is the Weyl module with head $L_w$. We show that there are integers $d_{w,x}$ independent of $p$ such that in the $U_p$ case (resp. in the $G_p$ case) $d_{w,x}$ is equal to the multiplicity of $L_x$ as a composition factor of $V_w$ for all $p>h$ (resp. for all $p>>0$). This implies : If the Lusztig conjecture holds in the quantum case, then it holds for $p>>0$ in the prime characteristic case

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