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 $latex G=GL_n(\mathbb{F}_q)$ are generic/regular/have a Whittaker model.
For a cuspidal representation $latex \tau$, and $latex \psi\in \hat{\mathbb{F}_q}$, a non-trivial additive character, Gelfand shows that (1): $latex (\textup{res}_U^G\ \tau,\psi)_U$ is a rational function of $latex q$ of degree $latex 0$, and (2): it equals $latex 1$ for all large enough $latex q$. Hence it is always $latex 1$.
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 $latex q$, 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 $latex Sp_{2n}(\mathbb{F}_q)$: 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
$latex u=\begin{pmatrix}
u’&M&Z \\
&I_{2\ell}&M’ \\
& &u’^*
\end{pmatrix}\in Sp_{2m}(\mathbb{F}_q),$
$latex \textup{rank}(u-1)=k,$
$latex u’_{1,2}+u’_{2,3}+…+u’_{m-\ell,m-\ell+1}=c$
for any positive integers $latex m, k, \ell$, and $latex c\in\mathbb{F}_q$. (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:
http://www.math.tau.ac.il/~drorspei/Unipotent Symplectic Matrices Satisfying Something.pdf