Explicit group decomposition

Question

Let $E/F$ be a quadratic extension of non-archimedean fields. Let $p$ its maximal ideal and $O$ its ring of integers. I am interested in the subgroup $A$ of invertible matrices of the form $$ \left( \begin{array}{ccc} O & O & O \\ p & O & O \\ p&p&O \end{array} \right) $$

I would like to understand its index in the subgroup of all invertible matrices with integer coefficients. However, I am stuck with a problem that seems elementary. Indeed, I could think of Bruhat decomposition but this would give double classes more than left/right-classes, and also I don't know how to deal with it in the case of non-split groups.

Is there any other way to compute this index (or the volume with respect to a well-normalised Haar measure?)

Answer 1

Let $k=O/p$ the residue class field. It is finite, say $|k|=q$.

Consider the reduction mod $p$ homomorphism $$ \phi:GL_3(O)\to GL_3(k). $$ Then

• $\phi$ is surjective.
• The order of the image is $(q^3-1)(q^3-q)(q^3-q^2)$.
• $A$ is the preimage of the subgroup $U\le GL_3(k)$ of upper triangular matrices.
• That group has order $|U|=(q-1)^3q^3$.

By the correspondence principle $$ [GL_3(O):A]=[GL_3(k):U]=\frac{|GL_3(k)|}{|U|}=(q^2+q+1)(q+1). $$