| It is still unknown if we can apply the philosophy of the orbit method to the algebraic group {dollar}Gsb{lcub}n{rcub}(q){dollar} of the unipotent upper triangular matrices of dimension n with entries in a finite field with q elements {dollar}{lcub}rm I!F{rcub}sb{lcub}q{rcub}{dollar} and 1's on the main diagonal. In this context, A. A. Kirillov formulated two conjectures. The first gives an explicit formula for the irreducible characters of {dollar}Gsb{lcub}n{rcub}(q){dollar} and the second, called main conjecture, states the identity of several sequences of polynomials defined in terms of combinatorial objects or related to the representation theory of {dollar}Gsb{lcub}n{rcub}(q).{dollar} A counterexample to the main conjecture is known only in even characteristic.; We investigate the general case and study in detail a more manageable case, where the group {dollar}Gsb{lcub}n{rcub}(q){dollar} is replaced by a 2-step nilpotent group, obtained as a quotient of {dollar}Gsb{lcub}n{rcub}(q).{dollar} For this new group it is possible to give a complete description of the co adjoint orbits. It is proven that there exists a one-to-one correspondence between coadjoint orbits and irreducible representations, satisfying Kirillov character formula, if the characteristic of {dollar}{lcub}rm I!F{rcub}sb{lcub}q{rcub}{dollar} is odd. For arbitrary characteristic, and contrary to the general case, we find the identity of two of the corresponding sequences of polynomials, those related to the distribution of the coadjoint orbits and the number of involutions of the quotient group. A fourth new sequence of polynomials, describing the distribution of the half-rank of general two-diagonal antisymmetric matrices, is proven to be identical to the sequence describing the distribution of the coadjoint orbits. |
