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Mod p Reducibility of Unramified Representations of Finite Groups of Lie Type
Author(s) -
Tiep Pham Huu,
Zalesskii A. E.
Publication year - 2002
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/84.2.439
Subject(s) - mathematics , algebraically closed field , rank (graph theory) , type (biology) , infinitesimal , prime (order theory) , block (permutation group theory) , pure mathematics , element (criminal law) , field (mathematics) , simple (philosophy) , modulo , combinatorics , mathematical analysis , ecology , philosophy , epistemology , political science , law , biology
Dedicated to the memory of Professor A. I. Kostrikin The main problem under discussion is to determine, for quasi‐simple groups of Lie type G , irreducible representations ϕ of G that remain irreducible under reduction modulo the natural prime p . The method is new. It works only for p >3 and for representations ϕ that can be realized over an unramified extension of Q p , the field of p ‐adic numbers. Under these assumptions, the main result says that the trivial and the Steinberg representations of G are the only representations in question provided G is not of type A1. This is not true for G =SL(2, p ). The paper contains a result of independent interest on infinitesimally irrreducible representations ρ of G over an algebraically closed field of characteristic p . Assuming that G is not of rank 1 and G ≠ G 2 (5), it is proved that either the Jordan normal form of a root element contains a block of size d with 1< d < p ‐1 or the highest weight of ρ is equal to p ‐1 times the sum of the fundamental weights. 2000 Mathematical Subject Classification : 20C33, 20G15.