Invariant measures on p-adic Lie groups: the p-adic quaternion algebra and the Haar integral on the p-adic rotation groups
arXiv (Cornell University)(2023)
摘要
We provide a general expression of the Haar measure - that is, the
essentially unique translation-invariant measure - on a p-adic Lie group.
We then argue that this measure can be regarded as the measure naturally
induced by the invariant volume form on the group, as it happens for a standard
Lie group over the reals. As an important application, we next consider the
problem of determining the Haar measure on the p-adic special orthogonal
groups in dimension two, three and four (for every prime number p). In
particular, the Haar measure on SO(2,ℚ_p) is obtained by a
direct application of our general formula. As for SO(3,ℚ_p)
and SO(4,ℚ_p), instead, we show that Haar integrals on
these two groups can conveniently be lifted to Haar integrals on certain
p-adic Lie groups from which the special orthogonal groups are obtained as
quotients. This construction involves a suitable quaternion algebra over the
field ℚ_p and is reminiscent of the quaternionic realization of the
real rotation groups. Our results should pave the way to the development of
harmonic analysis on the p-adic special orthogonal groups, with potential
applications in p-adic quantum mechanics and in the recently proposed
p-adic quantum information theory.
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关键词
invariant measures,p-adic
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