Comparing the projective modules of higher Frobenius kernels and finite Chevalley groups
by
Zongzhu Lin
Department of Mathematics, Kansas State University, Manhattan, KS 66506

In 1986, Parshall raised the question of relating the support varieties of modules for a semi-simple algebraic group G as a module for its Lie algebra with the support varieties for the finite groups G(\F_p^r). Here p is the defining characteristic of the group G.

He raised an easier question of whether each G-module which is projective as a module for its Lie algebra is also projective for the finite group G(\F_p). Question is answered by Lin and Nakano using a May spectral sequence and Quillen's isomorphism for p-groups for all p which is larger than the edge multiplicities of its Dynkin diagram. It is expected that the restrictions on p . Carlson, Lin, and Nakano have found a way to relate the support varieties for r=1.

In this talk we will discuss the natural generalization of Parshall's question: For any positive integer r and any finite dimensional G-module, if M is projective as a module for the rth Frobenius kernel G_r, is M also a projective module for the finite group G(\F_p^r)? There is a different approach to this question. Instead of using the Quillen's isomorphism of associated graded algebra of the group algebra a Sylow subgroup to the restricted enveloping algebra of the Lie algebra of the unipotent group, the question is reduced to the Borel subgroup B. We conjecture that if M is a B module that projective as a B_r module, then M is also projective as a B(\F_p^r)-module. We expect this conjecture is true (and has been verified for lower ranks groups). This conjecture will give an affirmative answer the generalized Parshall question.

Date received: July 28, 2004

Copyright © 2004 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caoh-33.