Representations of quantum groups at a \(p\)-th root of unity and of semisimple groups in characteristic \(p\): independence of \(p\).

*(English)*Zbl 0802.17009
Astérisque. 220. Paris: Société Mathématique de France, 321 p. (1994).

Let \(R\) be an indecomposable finite root system, and let \(p\) be an odd prime (\(>3\) if \(R\) is of type \(G_ 2\)). Let \(U_ p\) be the quantized universal enveloping algebra associated to \(R\) at a \(p\)th root of unity [defined using divided powers, as in G. Lusztig, Introduction to quantum groups, Birkhäuser, Boston (1993; Zbl 0788.17010)]. Also, let \(G_ p\) be the semisimple, connected, simply-connected algebraic group associated to \(R\) and defined over an algebraically closed field of characteristic \(p\). Assume that \(p>h\), the Coxeter number of \(R\).

It is known that the simple modules in the block of the trivial one- dimensional \(U_ p\)-module (resp. \(G_ p\)-module) are indexed by certain elements in the affine Weyl group \(W_ a\) of \(R\). Let \(L_ w\) be the simple module indexed by \(w\in W_ a\), and let \(V_ w\) be the Weyl module with head \(L_ w\). The authors’ main result is that there exist integers \(d_{w,n}\) \((n\in W_ a)\), independent of \(p\), such that \(d_{w,n}\) is equal to the multiplicity of \(L_ n\) as a composition factor of \(V_ w\) for all \(p >h\) in the \(U_ p\) case (resp. for all \(p\gg 0\) in the \(G_ p\) case).

In 1980, Lusztig made a conjecture on the characters of the representations of \(G_ p\). Recent work by D. Kazhdan and G. Lusztig [see the papers reviewed above] has shown that a similar conjecture about \(U_ p\) is equivalent to a conjecture about the affine Lie algebra associated to \(R\), a proof of which has recently been announced by M. Kashiwara and T. Tanisaki. Combining these results with the main theorem of this paper, one has a proof of Lusztig’s conjecture for \(p\gg 0\) (it is expected that the conjecture should hold for \(p>h\)).

It is known that the simple modules in the block of the trivial one- dimensional \(U_ p\)-module (resp. \(G_ p\)-module) are indexed by certain elements in the affine Weyl group \(W_ a\) of \(R\). Let \(L_ w\) be the simple module indexed by \(w\in W_ a\), and let \(V_ w\) be the Weyl module with head \(L_ w\). The authors’ main result is that there exist integers \(d_{w,n}\) \((n\in W_ a)\), independent of \(p\), such that \(d_{w,n}\) is equal to the multiplicity of \(L_ n\) as a composition factor of \(V_ w\) for all \(p >h\) in the \(U_ p\) case (resp. for all \(p\gg 0\) in the \(G_ p\) case).

In 1980, Lusztig made a conjecture on the characters of the representations of \(G_ p\). Recent work by D. Kazhdan and G. Lusztig [see the papers reviewed above] has shown that a similar conjecture about \(U_ p\) is equivalent to a conjecture about the affine Lie algebra associated to \(R\), a proof of which has recently been announced by M. Kashiwara and T. Tanisaki. Combining these results with the main theorem of this paper, one has a proof of Lusztig’s conjecture for \(p\gg 0\) (it is expected that the conjecture should hold for \(p>h\)).

Reviewer: A.N.Pressley (London)

##### MSC:

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |

17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |

20G40 | Linear algebraic groups over finite fields |

17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |