Fourier inversion of invariant integrals on semisimple real Lie groups

Abstract

Let <italic>G</italic> be a connected, semisimple real Lie group with finite center. Associated with every regular semisimple element <italic>g</italic> of <italic>G</italic> is a tempered invariant distribution <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda Subscript g"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Λ<!-- Λ --></mml:mi> <mml:mi>g</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{ \Lambda _g}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> given by an orbital integral. This paper gives an inductive formula for computing the Fourier transform of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda Subscript g"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Λ<!-- Λ --></mml:mi> <mml:mi>g</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{ \Lambda _g}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in terms of the space of tempered invariant eigendistributions of <italic>G</italic>.