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Hua measures on the space of \(p\)-adic matrices and inverse limits of Grassmannians. (English. Russian original) Zbl 1284.22013
Izv. Math. 77, No. 5, 941-953 (2013); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 77, No. 5, 95-108(2013).
The author constructs, for any \(s>-1\), a measure \(\mu_s^n\) on the space \(\text{Mat}(n,\mathbb Q_p)\) of all \(n\times n\) matrices with \(p\)-adic entries, which is the unique Borel probability measure \(\nu\) satisfying the equation \[ d\nu ((a+zc)^{-1}(b+zd))=|\det (a+zc)|^sd\nu (z),\quad z\in \text{Mat}(n,\mathbb Q_p), \] 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 \(GL(2n,\mathbb Q_p)\)-invariant measure on the Grassmannian \(\text{Gr}_{2n}^n\) of \(n\)-dimensional subspaces in \(\mathbb Q_p^{2n}\).
Considering the chain \[ \ldots \leftarrow (\text{Mat}(n,\mathbb Q_p),\mu_s^n)\leftarrow (\text{Mat}(n+1,\mathbb Q_p),\mu_s^{n+1})\leftarrow \ldots \] with natural mappings, we obtain the inverse limit \((\text{Mat}(\infty ,\mathbb Q_p),\mu_s^\infty )\) of measure spaces. This measure is proved to be quasi-invariant with respect to the action of the inductive limit of the groups \(GL(2n,\mathbb Z_p)\). 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.

MSC:
22E66 Analysis on and representations of infinite-dimensional Lie groups
43A80 Analysis on other specific Lie groups
46G12 Measures and integration on abstract linear spaces
Full Text: DOI arXiv
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