# Hermitian sublattices of a given type

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.