While working on my thesis, I wanted to bound the dimension of a Jacquet module of a uniform character (i.e. a combination of Deligne-Lusztig characters) using an idea of Gelfand, appearing in ‘Representations of the full linear group over finite fields’ (1970). In modern terms, Gelfand shows that the cuspidal characters of are generic/regular/have a Whittaker model.
For a cuspidal representation , and , a non-trivial additive character, Gelfand shows that (1): is a rational function of of degree , and (2): it equals for all large enough . Hence it is always .
The first property is proved by a rather explicit computation. Gelfand shows that the number of upper triangular unipotent matrices in a given conjugacy class is a polynomial in , and computes its degree. This is enough, since the character values of a cuspidal character at unipotent elements was already known, due to Green (I think). The second property essentially follows from the first and a multiplicity one result.
I wanted to do the same with a representation of : show that the dimension of the Jacquet module with respect to some unipotent subgroup was a polynomial and that I can compute its degree. Specifically, I needed to compute the number of matrices such that
for any positive integers , and . (The symplectic form is defined in the linked file below.)
So I did. I knew that I didn’t have a formula for the values of my character at unipotent elements, but I knew that the values are polynomials, and I thought I knew their degrees. This would be enough to prove what I needed. But I was wrong. I did not know the degrees of the polynomials. I couldn’t find a way to compute them without doing some heavy lifting, like in Lusztig’s ‘On the Green Polynomials of Classical Groups’ (1976). Essentially, I was missing the following result: degrees of values at unipotent elements of “small” representations are smaller than degress of values from cuspidal representations. Small here means a semisimple representation that is not also regular. I still don’t know if this is true or not.
But then I found a different way to bound the dimension of my Jacquet module: compute the character’s wave front set and use Lusztig’s results from ‘A unipotent support for irreducible representations’ (1992).
Back to counting matrices. I use the same ideas from Gelfand’s paper, only extending the amount of variables I need to keep track of (not just rank). If you’re interested, here it is: