Hua measures on the space of \(p\)-adic matrices and inverse limits of Grassmannians (Q2866962)

The author constructs, for any \(s>-1\), a measure \(\mu_s^n\) on the space \(\mathrm{Mat}(n,\mathbb Q_p)\) of all \(n\times n\) matrices with \(p\)-adic entries, which is the unique Borel probability measure \(\nu\) satisfying the equation NEWLINE\[CARRIAGE_RETURNNEWLINE d\nu ((a+zc)^{-1}(b+zd))=|\det (a+zc)|^sd\nu (z),\quad z\in \text{Mat}(n,\mathbb Q_p), CARRIAGE_RETURNNEWLINE\]NEWLINE for any block matrix \(\left( \begin{smallmatrix} a&b\\ c&d\end{smallmatrix}\right) \in GL(2n,\mathbb Z_p)\). For \(s=0\), this measure can be interpreted as a unique \(\mathrm{GL}(2n,\mathbb Q_p)\)-invariant measure on the Grassmannian \(\mathrm{Gr}_{2n}^n\) of \(n\)-dimensional subspaces in \(\mathbb Q_p^{2n}\).NEWLINENEWLINEConsidering the chain NEWLINE\[CARRIAGE_RETURNNEWLINE \ldots \leftarrow (\mathrm{Mat}(n,\mathbb Q_p),\mu_s^n)\leftarrow (\mathrm{Mat}(n+1,\mathbb Q_p),\mu_s^{n+1})\leftarrow \ldots CARRIAGE_RETURNNEWLINE\]NEWLINE with natural mappings, we obtain the inverse limit \((\mathrm{Mat}(\infty ,\mathbb Q_p),\mu_s^\infty )\) of measure spaces. This measure is proved to be quasi-invariant with respect to the action of the inductive limit of the groups \(\mathrm{GL}(2n,\mathbb Z_p)\). The author studies the action of this symmetry group in detail. Similar problems are discussed for the symplectic Lagrangian Grassmannian and the isotropic orthogonal Grassmannian.