Abstract
AbstractWe 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 $$\text {SO}(2,\mathbb {Q}_p)$$
SO
(
2
,
Q
p
)
is obtained by a direct application of our general formula. As for $$\text {SO}(3,\mathbb {Q}_p)$$
SO
(
3
,
Q
p
)
and $$\text {SO}(4,\mathbb {Q}_p)$$
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 $$\mathbb {Q}_p$$
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.
Funder
Universitat Autònoma de Barcelona
Publisher
Springer Science and Business Media LLC
Cited by
1 articles.
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