2016/05/19 by Joseph A. Wolf, Wolf, Joseph A.
Mathematics · #Advanced Algebra and Geometry #Finite Group Theory Research #Advanced Operator Algebra Research
paper · pdf · doi:10.48550/arxiv.1605.06191
In a series of recent papers we extended the notion of square integrability,\nfor representations of nilpotent Lie groups, to that of stepwise square\nintegrability. There we discussed a number of applications based on the fact\nthat nilradicals of minimal parabolic subgroups of real reductive Lie groups\nare stepwise square integrable. Here, in Part I, we prove stepwise square\nintegrability for nilradicals of arbitrary parabolic subgroups of real\nreductive Lie groups. This is technically more delicate than the case of\nminimal parabolics. We further discuss applications to Plancherel formulae and\nFourier inversion formulae for maximal exponential solvable subgroups of\nparabolics and maximal amenable subgroups of real reductive Lie groups.\nFinally, in Part II, we extend a number of those results to (infinite\ndimensional) direct limit parabolics. These extensions involve an infinite\ndimensional version of the Peter-Weyl Theorem, construction of a direct limit\nSchwartz space, and realization of that Schwartz space as a dense subspace of\nthe corresponding L2 space.\n