I am considering an element $x \in GL(2, \mathbb{Q}_p)$ in the double coset $GL(2, \mathbb{Z}_p) T(p) GL(2, \mathbb{Z}_p)$, where $T(p)$ is the diagonal matrix $(p, 1)$. I wonder about what can be said of his determinant and Weyl discriminant, and something is troubling me.

Assume $x$ is semi-simple, so that it is conjugate (over an algebraic closure) to a diagonal matrix $(a,b)$. Then $$\det(x) = ab \qquad \text{and} \qquad D(x) = \left( 1 - \frac{a}{b} \right)\left( 1 - \frac{b}{a} \right)$$

Assume $x$ is non-central, otherwise $a=b$. The characteristic polynomial has integer coefficients, hence the eigenvalues are integral and hence in $\mathbb{Z}_p$, since it is integrally closed.

Since $\det(x)$ is $p$ (by the assumption of belonging to the double class), $a$ has to be associated to $p$ and $b$ a unit (or the converse) by looking at the valuations. Based on it, can I bound $D(x)$ in function of $p$. Or the $p$-valuation of $D(x)$?


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