Representation theoretic embedding of twisted Dirac operators

Abstract

<p>Let <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> be a non-compact connected semisimple real Lie group with finite center. Suppose <inline-formula content-type= math/mathml >\n<mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML alttext= upper L >\n <mml:semantics>\n <mml:mi>L</mml:mi>\n <mml:annotation encoding= application/x-tex >L</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is a non-compact connected closed subgroup of <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> acting transitively on a symmetric space <inline-formula content-type= math/mathml >\n<mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML alttext= upper G slash upper H >\n <mml:semantics>\n <mml:mrow>\n <mml:mi>G</mml:mi>\n <mml:mrow class= MJX-TeXAtom-ORD >\n <mml:mo>/</mml:mo>\n </mml:mrow>\n <mml:mi>H</mml:mi>\n </mml:mrow>\n <mml:annotation encoding= application/x-tex >G/H</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> such that <inline-formula content-type= math/mathml >\n<mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML alttext= upper L intersection upper H >\n <mml:semantics>\n <mml:mrow>\n <mml:mi>L</mml:mi>\n <mml:mo>∩<!-- ∩ --></mml:mo>\n <mml:mi>H</mml:mi>\n </mml:mrow>\n <mml:annotation encoding= application/x-tex >L\\cap H</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is compact. We study the action on <inline-formula content-type= math/mathml >\n<mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML alttext= upper L slash upper L intersection upper H >\n <mml:semantics>\n <mml:mrow>\n <mml:mi>L</mml:mi>\n <mml:mrow class= MJX-TeXAtom-ORD >\n <mml:mo>/</mml:mo>\n </mml:mrow>\n <mml:mi>L</mml:mi>\n <mml:mo>∩<!-- ∩ --></mml:mo>\n <mml:mi>H</mml:mi>\n </mml:mrow>\n <mml:annotation encoding= application/x-tex >L/L\\cap H</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> of a Dirac operator <inline-formula content-type= math/mathml >\n<mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML alttext= upper D Subscript upper G slash upper H Baseline left-parenthesis upper E right-parenthesis >\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>D</mml:mi>\n <mml:mrow class= MJX-TeXAtom-ORD >\n <mml:mi>G</mml:mi>\n <mml:mrow class= MJX-TeXAtom-ORD >\n <mml:mo>/</mml:mo>\n </mml:mrow>\n <mml:mi>H</mml:mi>\n </mml:mrow>\n </mml:msub>\n <mml:mo stretchy= false >(</mml:mo>\n <mml:mi>E</mml:mi>\n <mml:mo stretchy= false >)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding= application/x-tex >D_{G/H}(E)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> acting on sections of an <inline-formula content-type= math/mathml >\n<mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML alttext= upper E >\n <mml:semantics>\n <mml:mi>E</mml:mi>\n <mml:annotation encoding= application/x-tex >E</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-twist of the spin bundle over <inline-formula content-type= math/mathml >\n<mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML alttext= upper G slash upper H >\n <mml:semantics>\n <mml:mrow>\n <mml:mi>G</mml:mi>\n <mml:mrow class= MJX-TeXAtom-ORD >\n <mml:mo>/</mml:mo>\n </mml:mrow>\n <mml:mi>H</mml:mi>\n </mml:mrow>\n <mml:annotation encoding= application/x-tex >G/H</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. As a byproduct, in the case of <inline-formula content-type= math/mathml >\n<mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML alttext= left-parenthesis upper G comma upper H comma upper L right-parenthesis equals left-parenthesis upper S upper L left-parenthesis 2 comma double-struck upper R right-parenthesis times upper S upper L left-parenthesis 2 comma double-struck upper R right-parenthesis comma normal upper Delta left-parenthesis upper S upper L left-parenthesis 2 comma double-struck upper R right-parenthesis times upper S upper L left-parenthesis 2 comma double-struck upper R right-parenthesis right-parenthesis comma upper S upper L left-parenthesis 2 comma double-struck upper R right-parenthesis times upper S upper O left-parenthesis 2 right-parenthesis right-parenthesis >\n <mml:semantics>\n <mml:mrow>\n <mml:mo stretchy= false >(</mml:mo>\n <mml:mi>G</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>H</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>L</mml:mi>\n <mml:mo stretchy= false >)</mml:mo>\n <mml:mo>=</mml:mo>\n <mml:mo stretchy= false >(</mml:mo>\n <mml:mi>S</mml:mi>\n <mml:mi>L</mml:mi>\n <mml:mo stretchy= false >(</mml:mo>\n <mml:mn>2</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mrow class= MJX-TeXAtom-ORD >\n <mml:mrow class= MJX-TeXAtom-ORD >\n <mml:mi mathvariant= double-struck >R</mml:mi>\n </mml:mrow>\n </mml:mrow>\n <mml:mo stretchy= false >)</mml:mo>\n <mml:mo>×<!-- × --></mml:mo>\n <mml:mi>S</mml:mi>\n <mml:mi>L</mml:mi>\n <mml:mo stretchy= false >(</mml:mo>\n <mml:mn>2</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mrow class= MJX-TeXAtom-ORD >\n <mml:mrow class= MJX-TeXAtom-ORD >\n <mml:mi mathvariant= double-struck >R</mml:mi>\n </mml:mrow>\n </mml:mrow>\n <mml:mo stretchy= false >)</mml:mo>\n <mml:mo>,</mml:mo>\n <mml:mi mathvariant= normal >Δ<!-- Δ --></mml:mi>\n <mml:mo stretchy= false >(</mml:mo>\n <mml:mi>S</mml:mi>\n <mml:mi>L</mml:mi>\n <mml:mo stretchy= false >(</mml:mo>\n <mml:mn>2</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mrow class= MJX-TeXAtom-ORD >\n <mml:mrow class= MJX-TeXAtom-ORD >\n <mml:mi mathvariant= double-struck >R</mml:mi>\n </mml:mrow>\n </mml:mrow>\n <mml:mo stretchy= false >)</mml:mo>\n <mml:mo>×<!-- × --></mml:mo>\n <mml:mi>S</mml:mi>\n <mml:mi>L</mml:mi>\n <mml:mo stretchy= false >(</mml:mo>\n <mml:mn>2</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mrow class= MJX-TeXAtom-ORD >\n <mml:mrow class= MJX-TeXAtom-ORD >\n <mml:mi mathvariant= double-struck >R</mml:mi>\n </mml:mrow>\n </mml:mrow>\n <mml:mo stretchy= false >)</mml:mo>\n <mml:mo stretchy= false >)</mml:mo>\n <mml:mo>,</mml:mo>\n <mml:mi>S</mml:mi>\n <mml:mi>L</mml:mi>\n <mml:mo stretchy= false >(</mml:mo>\n <mml:mn>2</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mrow class= MJX-TeXAtom-ORD >\n <mml:mrow class= MJX-TeXAtom-ORD >\n <mml:mi mathvariant= double-struck >R</mml:mi>\n </mml:mrow>\n </mml:mrow>\n <mml:mo stretchy= false >)</mml:mo>\n <mml:mo>×<!-- × --></mml:mo>\n <mml:mi>S</mml:mi>\n <mml:mi>O</mml:mi>\n <mml:mo stretchy= false >(</mml:mo>\n <mml:mn>2</mml:mn>\n <mml:mo stretchy= false >)</mml:mo>\n <mml:mo stretchy= false >)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding= application/x-tex >(G,H,L)=(SL(2,{\\mathbb R})\\times SL(2,{\\mathbb R}),\\Delta (SL(2,{\\mathbb R})\\times SL(2,{\\mathbb R})),SL(2,{\\mathbb R})\\times SO(2))</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, we identify certain representations of <inline-formula content-type= math/mathml >\n<mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML alttext= upper L >\n <mml:semantics>\n <mml:mi>L</mml:mi>\n <mml:annotation encoding= application/x-tex >L</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> which lie in the kernel of <inline-formula content-type= math/mathml >\n<mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML alttext= upper D Subscript upper G slash upper H Baseline left-parenthesis upper E right-parenthesis >\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>D</mml:mi>\n <mml:mrow class= MJX-TeXAtom-ORD >\n <mml:mi>G</mml:mi>\n <mml:mrow class= MJX-TeXAtom-ORD >\n <mml:mo>/</mml:mo>\n </mml:mrow>\n <mml:mi>H</mml:mi>\n </mml:mrow>\n </mml:msub>\n <mml:mo stretchy= false >(</mml:mo>\n <mml:mi>E</mml:mi>\n <mml:mo stretchy= false >)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding= application/x-tex >D_{G/H}(E)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>.</p>