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Haar measure on linear groups over local skew fields. (English) Zbl 0864.22003

Every totally disconnected locally compact group has a neighborhood basis at the identity consisting of compact open subgroups. In principle, it is thus a purely algebraical task to determine Haar measure (not the Haar integral) for such a group: every compact open set is the disjoint union of finitely many cosets of such subgroups. If the group is \(\sigma\)-compact every open set is covered by a sequence of disjoint compact open sets. Therefore, it remains to determine the indices of compact open subgroups in a fixed compact open subgroup. In the article under review, this is done for the groups \(\text{GL}(n,K)\), \(\text{SL}(n,K)\), \(\text{PGL}(n,K)\), and \(\text{PSL}(n,K)\), where \(K\) is a local skew field (that is, a totally disconnected locally compact skew field). As consequences, the author obtains an easy proof of the fact that the groups \(\text{GL}(n,{\mathbf Q}_p)\) and \(\text{GL}(m,{\mathbf Q}_q)\) are isomorphic as topological groups if, and only if, \((n,p)=(m,q)\). The results of the paper under review were used by the author to compute scale functions as introduced by G. Willis [Math. Ann. 300, 341-363 (1994; Zbl 0811.22004)]. It may be worth noting that the author also includes a proof that the Dieudonné determinant is continuous, and that its kernel \(\text{SL}(n,K)\) is closed, for each local field \(K\).

MSC:

22E35 Analysis on \(p\)-adic Lie groups
22E50 Representations of Lie and linear algebraic groups over local fields
28C10 Set functions and measures on topological groups or semigroups, Haar measures, invariant measures

Citations:

Zbl 0811.22004
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