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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
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