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href="/nlab/show/homology">homology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/chain">chain</a>, <a class="existingWikiWord" href="/nlab/show/cycle">cycle</a>, <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+characteristic+class">universal characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/secondary+characteristic+class">secondary characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+characteristic+class">differential characteristic class</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>/<a class="existingWikiWord" href="/nlab/show/long+exact+sequence+in+cohomology">long exact sequence in cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/twisted+%E2%88%9E-bundle">twisted ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a></p> </li> </ul> <h3 id="special_and_general_types">Special and general types</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>, <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+group+cohomology">nonabelian group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+group+cohomology">Lie group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+cohomology">Galois cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+cohomology">groupoid cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+groupoid+cohomology">nonabelian groupoid cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a>, <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/taf">taf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+cohomology">Dolbeault cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/etale+cohomology">etale cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a>, <a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a>, <a class="existingWikiWord" href="/nlab/show/Brauer+group">Brauer group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crystalline+cohomology">crystalline cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/syntomic+cohomology">syntomic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+cohomology">motivic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+of+operads">cohomology of operads</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+cohomology">cyclic cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+topology">string topology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>/<a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">covering ∞-bundle</a>/<a class="existingWikiWord" href="/nlab/show/local+system">local system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-vector+bundle">(∞,n)-vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/Spin+structure">Spin structure</a>, <a class="existingWikiWord" href="/nlab/show/Spin%5Ec+structure">Spin^c structure</a>, <a class="existingWikiWord" href="/nlab/show/String+structure">String structure</a>, <a class="existingWikiWord" href="/nlab/show/Fivebrane+structure">Fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+with+constant+coefficients">cohomology with constant coefficients</a> / <a class="existingWikiWord" href="/nlab/show/cohomology+with+a+local+system+of+coefficients">with a local system of coefficients</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra+extensions">Lie algebra extensions</a>, <a class="existingWikiWord" href="/nlab/show/Gelfand-Fuks+cohomology">Gelfand-Fuks cohomology</a>,</li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gerstenhaber-Schack+cohomology">bialgebra cohomology</a></p> </li> </ul> <h3 id="special_notions">Special notions</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%3Fech+cohomology">?ech cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin+structure">twisted spin structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin%5Ec+structure">twisted spin^c structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structures">twisted differential c-structures</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structure">twisted differential string structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+differential+fivebrane+structure">twisted differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p>differential cohomology</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+elliptic+cohomology">differential elliptic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative cohomology</a></p> </li> </ul> <h3 id="extra_structure">Extra structure</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">in generalized cohomology</a></p> </li> </ul> <h3 id="operations">Operations</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+operations">cohomology operations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a>, <a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a>, <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+localization">cohomology localization</a></p> </li> </ul> <h3 id="theorems">Theorems</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a>, <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a>, <a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+theory">Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+theorem">Hodge theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Hodge+theory">nonabelian Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+Hodge+theory">noncommutative Hodge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">hypercovering theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eckmann-Hilton+duality">Eckmann-Hilton-Fuks duality</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/cohomology+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="representation_theory">Representation theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/geometric+representation+theory">geometric representation theory</a></strong></p> <h2 id="ingredients">Ingredients</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/linear+algebra">linear algebra</a>, <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>, <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></li> </ul> <h2 id="sidebar_definitions">Definitions</h2> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/2-representation">2-representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+algebra">group algebra</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, <a class="existingWikiWord" href="/nlab/show/n-vector+space">n-vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/affine+space">affine space</a>, <a class="existingWikiWord" href="/nlab/show/symplectic+vector+space">symplectic vector space</a></p> </li> <li> <p><a class="existingWikiWord" 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href="/nlab/new/Be%3Flinson-Bernstein+localization">?</a></span></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kazhdan-Lusztig+theory">Kazhdan-Lusztig theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BBDG+decomposition+theorem">BBDG decomposition theorem</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/representation+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#RelationToTheOrbitMethod'>Relation to the orbit method</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a suitable (<a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact</a>) <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/equivariant+K-theory">equivariant K-theory</a> of the point is the <a class="existingWikiWord" href="/nlab/show/representation+ring">representation ring</a> of the group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>Rep</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> K_G(\ast) \simeq Rep(G) \,. </annotation></semantics></math></div> <p>Accordingly the construction of an <a class="existingWikiWord" href="/nlab/show/index">index</a> (<a class="existingWikiWord" href="/nlab/show/push-forward+in+generalized+cohomology">push-forward</a> to the point) in equivariant K-theory is a way of producing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/representations">representations</a> from <a class="existingWikiWord" href="/nlab/show/equivariant+vector+bundles">equivariant vector bundles</a>. Specifically with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>↪</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">K \hookrightarrow G</annotation></semantics></math> a suitable subgroup for the push-forward from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>-equivariant to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-equivariant K-theory/representations, this method is also called <em>Dirac induction</em> since it is analogous to the construction of <em><a class="existingWikiWord" href="/nlab/show/induced+representations">induced representations</a></em>.</p> <p>Applied to equivariant <a class="existingWikiWord" href="/nlab/show/complex+line+bundles">complex line bundles</a> on <a class="existingWikiWord" href="/nlab/show/coadjoint+orbits">coadjoint orbits</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>, Dirac induction is a <a class="existingWikiWord" href="/nlab/show/K-theory">K-theoretic</a> formulation of the <a class="existingWikiWord" href="/nlab/show/orbit+method">orbit method</a>.</p> <h2 id="properties">Properties</h2> <h3 id="RelationToTheOrbitMethod">Relation to the orbit method</h3> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact Lie group</a> with <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\mathfrak{g}^\ast</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/push-forward+in+generalized+cohomology">push-forward</a> in compactly supported <a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/equivariant+K-theory">equivariant K-theory</a> to the point (the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-equivariant <a class="existingWikiWord" href="/nlab/show/index">index</a>) produces the <a class="existingWikiWord" href="/nlab/show/Thom+isomorphism">Thom isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ind</mi> <mrow><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msubsup><mi>K</mi> <mi>G</mi> <mrow><mi>σ</mi><mo>+</mo><mi>dim</mi><mi>G</mi></mrow></msubsup><mo stretchy="false">(</mo><msup><mi>𝔤</mi> <mo>*</mo></msup><msub><mo stretchy="false">)</mo> <mi>cpt</mi></msub><mover><mo>→</mo><mo>≃</mo></mover><msubsup><mi>K</mi> <mi>G</mi> <mn>0</mn></msubsup><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>Rep</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> ind_{\mathfrak{g}^\ast} \;\colon\; K_G^{\sigma + dim G}(\mathfrak{g}^\ast)_{cpt} \stackrel{\simeq}{\to} K_G^0(\ast) \simeq Rep(G) \,. </annotation></semantics></math></div> <p>Moreover, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>𝒪</mi><mo>↪</mo><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">i \colon \mathcal{O} \hookrightarrow \mathfrak{g}^\ast</annotation></semantics></math> a regular <a class="existingWikiWord" href="/nlab/show/coadjoint+orbit">coadjoint orbit</a>, <a class="existingWikiWord" href="/nlab/show/push-forward+in+generalized+cohomology">push-forward</a> involves a <a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twist</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Rep</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>≃</mo><msubsup><mi>K</mi> <mi>G</mi> <mn>0</mn></msubsup><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mover><mo>←</mo><mrow><msub><mi>ind</mi> <mi>𝒪</mi></msub></mrow></mover><msubsup><mi>K</mi> <mi>G</mi> <mrow><mi>σ</mi><mo stretchy="false">(</mo><mi>𝒪</mi><mo stretchy="false">)</mo><mo>+</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>𝒪</mi><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>𝒪</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>i</mi> <mo>!</mo></msub></mrow></mover><msubsup><mi>K</mi> <mi>G</mi> <mrow><mi>σ</mi><mo>+</mo><mi>dim</mi><mi>G</mi></mrow></msubsup><mo stretchy="false">(</mo><msup><mi>𝔤</mi> <mo>*</mo></msup><msub><mo stretchy="false">)</mo> <mi>cpt</mi></msub></mrow><annotation encoding="application/x-tex"> Rep(G) \simeq K_G^0(\ast) \stackrel{ind_{\mathcal{O}}}{\leftarrow} K_G^{\sigma(\mathcal{O}) + dim(\mathcal{O})}(\mathcal{O}) \stackrel{i_!}{\to} K_G^{\sigma + dim G}(\mathfrak{g}^\ast)_{cpt} </annotation></semantics></math></div> <p>and</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mo>!</mo></msub></mrow><annotation encoding="application/x-tex">i_!</annotation></semantics></math> is surjective</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ind</mi> <mi>𝒪</mi></msub><mo>=</mo><msub><mi>ind</mi> <mrow><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow></msub><mo>∘</mo><msub><mi>i</mi> <mo>!</mo></msub></mrow><annotation encoding="application/x-tex">ind_{\mathcal{O}} = ind_{\mathfrak{g}^\ast} \circ i_!</annotation></semantics></math>.</p> </li> </ol> </div> <p>This is (<a href="#FHT">FHT II, (1.27), theorem 1.28</a>). Related results are in (<a href="#Hochs12">Hochs 12, section 2.2</a>). For more background see at <em><a class="existingWikiWord" href="/nlab/show/orbit+method">orbit method</a></em>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/Schubert+calculus">Schubert calculus</a></li> </ul> <h2 id="references">References</h2> <p>The idea of Dirac induction goes back to <a class="existingWikiWord" href="/nlab/show/Raoul+Bott">Raoul Bott</a>‘s formulation in the 1960s of <a class="existingWikiWord" href="/nlab/show/index+theory">index theory</a> in the <a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant</a> context.</p> <ul id="Bott65"> <li><a class="existingWikiWord" href="/nlab/show/Raoul+Bott">Raoul Bott</a>, <em>The index theorem for homogeneous differential operators</em>, In: <em>Differential and combinatorial topology</em> (A symposium in honor of Marston Morse), 1965, Princeton Univ. Press, Princeton, NJ, 167–186</li> </ul> <ul id="AtiyahBott68"> <li><a class="existingWikiWord" href="/nlab/show/Michael+Atiyah">Michael Atiyah</a>, <a class="existingWikiWord" href="/nlab/show/Raoul+Bott">Raoul Bott</a>, <em>A Lefschetz fixed point formula for</em> <p>elliptic complexes. II_. Applications. Ann. of Math. (2), 88:451– 491, 1968.</p> </li> </ul> <p>A generalization to <a class="existingWikiWord" href="/nlab/show/super+Lie+groups">super Lie groups</a> is discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Gregory+Landweber">Gregory Landweber</a>, <em>Twisted representation rings and Dirac induction</em>, J. Pure Appl. Algebra 206 (2006), no. 1-2, 21-54 (<a href="http://arxiv.org/abs/math/0403524">arXiv:math/0403524</a>)</li> </ul> <p>An inverse to Dirac induction, hence a construction of good equivariant vector bundles that push to a given representation, is discussed in</p> <ul id="FHT"> <li><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Constantin+Teleman">Constantin Teleman</a>, part II, section 1 of <em><a class="existingWikiWord" href="/nlab/show/Loop+Groups+and+Twisted+K-Theory">Loop Groups and Twisted K-Theory</a></em></li> </ul> <ul id="Hochs12"> <li><a class="existingWikiWord" href="/nlab/show/Peter+Hochs">Peter Hochs</a>, section 2.2 of <em>Quantisation of presymplectic manifolds, K-theory and group representations</em> (<a href="http://arxiv.org/abs/1211.0107">arXiv:1211.0107</a>)</li> </ul> <p>The analog of Dirac induction for K-theory replaced by <a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a> is discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Nora+Ganter">Nora Ganter</a>, <em>The elliptic Weyl character formula</em> (<a href="http://arxiv.org/abs/1206.0528">arXiv:1206.0528</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 29, 2013 at 10:47:49. 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