Let G be an F-algebra group(G=1+J , where J is the jacobson radical of a finite dimensional F-algebra ,where F is a field of prime characteristic) In a paper of Isaacs ("Characters of groups associated with finite algebras" from 1995) there is a claim of Gutkin with a wrong proof.It says: Let x be an irreducible character of G ,then $ x=a^G $ ,where a is a linear character of some subgroup H<=G of the form : H=1+U where U is multiplicativly closed F-subspace of J. This result would be a generalisation of the result of Isaacs paper,but its unclear to me if it has been proven until today.Does someone know if the result is true and can recommand me a paper/link for more information?
I think the result you are referring to is over a finite field $F$ (Gutkin's claim was also stated over local fields, however).
The result over finite fields was proved by Z. Halasi in On the characters and commutators of finite algebra groups, J. Algebra 275 (2004), 481-487. Halasi's proof uses some results from the paper by Isaacs in a counting argument, and therefore only proves Gutkin's claim for algebra groups over finite fields.
Recently, M. Boyarchenko has freed Halasi's proof from its dependence on Isaacs's results, and has proved Gutkin's claim also for algebra groups over local fields (see Representations of unipotent groups over local fields and Gutkin's conjecture, arXiv:1003.2742v1).