The large scale space N\G and the support of Plancherel measure
The Plancherel decomposition of functions invariant up to a unitary character of the unipotent subgroup of G.
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In his second book Real Reductive Groups II, Nolan Wallach obtained the Plancherel formula for the space of functions invariant up to a unitary character of the maximal unipotent subgroup and essentially square integrable on the large scale real homogeneous space NG. A similar result for p-adic groups should hold true as well but had not (at the point of writing) been published in the literature. This work by the present author is a modest attempt in addressing this gap. The issues arising in proving the Plancherel formula in the p-adic case are resolved by algebraic arguments and are relative...
In his second book Real Reductive Groups II, Nolan Wallach obtained the Plancherel formula for the space of functions invariant up to a unitary character of the maximal unipotent subgroup and essentially square integrable on the large scale real homogeneous space NG. A similar result for p-adic groups should hold true as well but had not (at the point of writing) been published in the literature. This work by the present author is a modest attempt in addressing this gap. The issues arising in proving the Plancherel formula in the p-adic case are resolved by algebraic arguments and are relatively elementary in nature. We hope that these may yield insights into solving issues that still arise in the real case.
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