A twistor correspondence and Penrose transform for odd-dimensional hyperbolic space

Abstract

For odd-dimensional hyperbolic space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper H"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we construct transforms between the cohomology of certain line bundles on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper T"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">T</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {T}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (a twistor space for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper H"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ) and eigenspaces of the Laplacian <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta"> <mml:semantics> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:annotation encoding="application/x-tex">\Delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and of the Dirac operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper H"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The transforms are isomorphisms. As a corollary we obtain that every eigenfunction of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta"> <mml:semantics> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:annotation encoding="application/x-tex">\Delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper H"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> extends as a holomorphic eigenfunction of the corresponding holomorphic operator on a certain region of the complexification of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper H"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We also obtain vanishing theorems for the cohomology of a class of line bundles on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper T"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">T</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {T}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .