3
$\begingroup$

Consider an unramified quadratic extension $E/F$ of non-archimedean local fields, and suppose that $\langle\cdot,\cdot\rangle$ is a fixed Hermitian form on $E^d$ such that $\mathcal{O}_E^d$ is self-dual, where $\mathcal{O}_E\subset E$ is the ring of integers. For simplicity, we can assume that $\langle\cdot,\cdot\rangle$ may be represented by the identity matrix.

For any sublattice $\Lambda\subset \mathcal{O}_E^d$, recall that the form $\langle\cdot,\cdot\rangle|_\Lambda$ may be diagonalized so that it may be represented by the matrix $$\varpi^\lambda=\left(\begin{array}{ccc}\varpi^{\lambda_1}&&\\&\ddots&\\&&\varpi^{\lambda_n}\end{array}\right),$$ where $\varpi$ is a uniformizer of $F$ and $\lambda=(\lambda_1,\ldots, \lambda_d)$ is a partition: $\lambda_i\geq\lambda_{i+1}\geq0$. Let's say that $\Lambda$ is of (Hermitian) type $\lambda$. Here is my question:

Fixing the lattice $\mathcal{O}_E^d$ with its self-dual form, how many sublattices $\Lambda\subset \mathcal{O}_E^d$ of (Hermitian) type $\lambda$ are there?

Let’s call these numbers $a_\lambda$. My hope is that there is a polynomial in $q=|\mathcal{O}_F/(\varpi)|$ counting such lattices, maybe in terms of Hall(-Littlewood) polynomials. This question arises while computing certain orbital integrals on unitary groups I wish to understand. I can explicitly work these values out for low rank, but I wonder if these quantities are known.

EDIT: I've added a reference-request tag. Even if this precise question is not addressed, I'd appreciate any reference where similar questions are considered.

Some simple observations:

1) The assumptions force the weight $|\lambda|=\sum\lambda_i=2n$ to be even;

2) If a sublattice $\Lambda\subset\ \mathcal{O}_E^d$ is of type $\lambda$, then $$ [\mathcal{O}_E^d:\Lambda]=q^{\frac{1}{2}|\lambda|}.$$

Using 1) and 2), one can compute the sum $\sum_{|\lambda|=2n}a_\lambda$ as a sum of Hall-Littlewood polynomials since it counts the total number of sublattices of a fixed index. I don't see how to isolate the terms.

Jacobowitz, Ronald, Hermitian forms over local fields, Am. J. Math. 84, 441-465 (1962). ZBL0118.01901.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.