Let $F$ be a $p$-adic field, $E / F$ a quadratic extension, and $n \ge 1$. Let $V = E^n$ with the obvious diagonal Hermitian form, $$ \langle (u_1, \dots, u_n), (v_1, \dots, v_n) \rangle = \sum_{i = 1}^n u_i \overline{v_i}, $$ where the bar is the nontrivial Galois automorphism of $E / F$.

Let $\mathcal{L}$ be the $\mathcal{O}_E$-span of the basis vectors. This is a Hermitian integral lattice; that is, $\langle x, x \rangle \in \mathcal{O}_F$ for all $x \in \mathcal{L}$. If $E / F$ is unramified, it is clearly maximal with respect to this property; but if $E / F$ is ramified this is not always the case (I think it is never the case for $n \ge 3$).

If $\mathcal{L}'$ is a maximal integral lattice containing $\mathcal{L}$, are the stabilisers of $\mathcal{L}$ and $\mathcal{L}'$ in the unitary group $U(V)$ attached to $V$ equal? If not, do they have the same volume (i.e. Haar measure)?

(According to Gan, Hanke and Yu, "On an exact mass formula of Shimura", the group $U(V)$ acts transitively on the maximal lattices $\mathcal{L}'$, so their stabilisers are all conjugate. Hence the answer to my second question doesn't depend on the choice of $\mathcal{L}'$, although the first might.)

  • $\begingroup$ Might this be in Shimura's book Euler products and Eisenstein series (he has a section on Hermitian spaces over local fields), or maybe one of his papers on the arithmetic of unitary groups? $\endgroup$ – Rob Harron Aug 24 '11 at 20:50
  • $\begingroup$ I can't see it there, although that book does have a lot of useful related stuff (e.g. the result that all maximal lattices are conjugate is Shimura's Lemma I.5.9) $\endgroup$ – David Loeffler Aug 24 '11 at 21:24

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