Representation Theory of the American Mathematical Society | 2021
Invariant measures on nilpotent orbits associated with holomorphic discrete series
Abstract
<p>Let <inline-formula content-type= math/mathml >\n<mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML alttext= upper G Subscript double-struck upper R >\n <mml:semantics>\n <mml:msub>\n <mml:mi>G</mml:mi>\n <mml:mrow class= MJX-TeXAtom-ORD >\n <mml:mi mathvariant= double-struck >R</mml:mi>\n </mml:mrow>\n </mml:msub>\n <mml:annotation encoding= application/x-tex >G_\\mathbb R</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> be a real form of a complex, semisimple Lie group <inline-formula content-type= math/mathml >\n<mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML alttext= upper G >\n <mml:semantics>\n <mml:mi>G</mml:mi>\n <mml:annotation encoding= application/x-tex >G</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. Assume <inline-formula content-type= math/mathml >\n<mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML alttext= upper G Subscript double-struck upper R >\n <mml:semantics>\n <mml:msub>\n <mml:mi>G</mml:mi>\n <mml:mrow class= MJX-TeXAtom-ORD >\n <mml:mi mathvariant= double-struck >R</mml:mi>\n </mml:mrow>\n </mml:msub>\n <mml:annotation encoding= application/x-tex >G_\\mathbb R</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> has holomorphic discrete series. Let <inline-formula content-type= math/mathml >\n<mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML alttext= script upper W >\n <mml:semantics>\n <mml:mrow class= MJX-TeXAtom-ORD >\n <mml:mi class= MJX-tex-caligraphic mathvariant= script >W</mml:mi>\n </mml:mrow>\n <mml:annotation encoding= application/x-tex >\\mathcal W</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> be a nilpotent coadjoint <inline-formula content-type= math/mathml >\n<mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML alttext= upper G Subscript double-struck upper R >\n <mml:semantics>\n <mml:msub>\n <mml:mi>G</mml:mi>\n <mml:mrow class= MJX-TeXAtom-ORD >\n <mml:mi mathvariant= double-struck >R</mml:mi>\n </mml:mrow>\n </mml:msub>\n <mml:annotation encoding= application/x-tex >G_\\mathbb R</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-orbit contained in the wave front set of a holomorphic discrete series. We prove a limit formula, expressing the canonical measure on <inline-formula content-type= math/mathml >\n<mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML alttext= script upper W >\n <mml:semantics>\n <mml:mrow class= MJX-TeXAtom-ORD >\n <mml:mi class= MJX-tex-caligraphic mathvariant= script >W</mml:mi>\n </mml:mrow>\n <mml:annotation encoding= application/x-tex >\\mathcal W</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> as a limit of canonical measures on semisimple coadjoint orbits, where the parameter of orbits varies over the positive chamber defined by the Borel subalgebra associated with holomorphic discrete series.</p>