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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Weylsche Integralformel</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>In der <a href="Mathematik" title="Mathematik">Mathematik</a> ist die <b>Weylsche Integralformel</b> oder <b>Integralformel von Weyl</b> eine Formel zur Berechnung des Integrals von Funktionen auf kompakten Lie-Gruppen, mit der insbesondere die Berechnung des Integrals von Klassenfunktionen auf eine Integration über den maximalen Torus reduziert werden kann. Sie ist nach <a href="Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a> benannt.
</p>

<div class="mw-heading mw-heading2"><h2 id="Aussage">Aussage</h2></div>
<p>Sei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>G</mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span> eine <a href="Kompakter_Raum" title="Kompakter Raum">kompakte</a>, <a href="Zusammenh%C3%A4ngender_Raum" title="Zusammenhängender Raum">zusammenhängende</a> <a href="Lie-Gruppe" title="Lie-Gruppe">Lie-Gruppe</a>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\subset G}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>T</mi>
<mo>⊂<!-- ⊂ --></mo>
<mi>G</mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle T\subset G}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd1b0d224a714650873a9aca9bae7c054783ca83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.561ex; height:2.176ex;" alt="{\displaystyle T\subset G}" loading="lazy"></span> ein <a href="Maximaler_Torus" title="Maximaler Torus">maximaler Torus</a> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon G\to \mathbb {C} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:<!-- : --></mo>
<mi>G</mi>
<mo stretchy="false">→<!-- → --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
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</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f\colon G\to \mathbb {C} }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/267426f12da9959539878b9c1436bbf07aa3db9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.431ex; height:2.509ex;" alt="{\displaystyle f\colon G\to \mathbb {C} }" loading="lazy"></span> eine <a href="Stetige_Funktion" title="Stetige Funktion">stetige Funktion</a>. Dann ist
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{G}f(g)\,dg={\frac {1}{\#W}}\int _{T}\det(\operatorname {Id} -\operatorname {Ad} _{G/T}(t^{-1}))\int _{G/T}f(gtg^{-1})\,dg\,dt}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
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<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>g</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mi mathvariant="normal">#<!-- # --></mi>
<mi>W</mi>
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<msub>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
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<mo movablelimits="true" form="prefix">det</mo>
<mo stretchy="false">(</mo>
<mi>Id</mi>
<mo>−<!-- − --></mo>
<msub>
<mi>Ad</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>G</mi>
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<mo>/</mo>
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<mi>T</mi>
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<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<msup>
<mi>t</mi>
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<mo stretchy="false">)</mo>
<msub>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>G</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
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<mi>T</mi>
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</msub>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mi>t</mi>
<msup>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
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</msup>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>g</mi>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>t</mi>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle \int _{G}f(g)\,dg={\frac {1}{\#W}}\int _{T}\det(\operatorname {Id} -\operatorname {Ad} _{G/T}(t^{-1}))\int _{G/T}f(gtg^{-1})\,dg\,dt}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c28b3ecd33ce880a2636f212da11b5e227e41e17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:62.387ex; height:6.009ex;" alt="{\displaystyle \int _{G}f(g)\,dg={\frac {1}{\#W}}\int _{T}\det(\operatorname {Id} -\operatorname {Ad} _{G/T}(t^{-1}))\int _{G/T}f(gtg^{-1})\,dg\,dt}" loading="lazy"></span>,</dd></dl>
<p>wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>W</mi>
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<annotation encoding="application/x-tex">{\displaystyle W}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"></span> die <a href="Weyl-Gruppe" title="Weyl-Gruppe">Weyl-Gruppe</a> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>G</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle G}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Ad} _{G/T}\colon T\to \operatorname {Aut} (T_{e}G/T)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Ad</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>G</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
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<mi>T</mi>
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<mo>:<!-- : --></mo>
<mi>T</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>Aut</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>T</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>e</mi>
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<mi>G</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
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<mi>T</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Ad} _{G/T}\colon T\to \operatorname {Aut} (T_{e}G/T)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef43ccb91d7455ac5d31edaa3b5063fb105f7de4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:25.554ex; height:3.176ex;" alt="{\displaystyle \operatorname {Ad} _{G/T}\colon T\to \operatorname {Aut} (T_{e}G/T)}" loading="lazy"></span> die Einschränkung der <a href="Adjungierte_Darstellung" title="Adjungierte Darstellung">adjungierten Darstellung</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Ad} \mid _{T}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Ad</mi>
<msub>
<mo stretchy="false">∣<!-- ∣ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Ad} \mid _{T}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7642ece82464fa55a46ec4c2be1e56b1a1a004e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.072ex; height:2.843ex;" alt="{\displaystyle \operatorname {Ad} \mid _{T}}" loading="lazy"></span> auf den ersten Summanden der <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Ad} \mid _{T}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Ad</mi>
<msub>
<mo stretchy="false">∣<!-- ∣ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Ad} \mid _{T}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7642ece82464fa55a46ec4c2be1e56b1a1a004e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.072ex; height:2.843ex;" alt="{\displaystyle \operatorname {Ad} \mid _{T}}" loading="lazy"></span>-invarianten Zerlegung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}=T_{e}(G/T)\oplus {\mathfrak {t}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="fraktur">g</mi>
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<mo>=</mo>
<msub>
<mi>T</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>e</mi>
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</msub>
<mo stretchy="false">(</mo>
<mi>G</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
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<mi>T</mi>
<mo stretchy="false">)</mo>
<mo>⊕<!-- ⊕ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="fraktur">t</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}=T_{e}(G/T)\oplus {\mathfrak {t}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5500404c3f0ae7909d8a1e8940378bf73a1b2fa0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.711ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {g}}=T_{e}(G/T)\oplus {\mathfrak {t}}}" loading="lazy"></span> bedeutet.
</p>
<div class="mw-heading mw-heading3"><h3 id="Spezialfall">Spezialfall</h3></div>
<p>Insbesondere erhält man für eine stetige <a href="Charakter_(Mathematik)#Klassenfunktionen" title="Charakter (Mathematik)">Klassenfunktion</a>
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{G}f(g)\,dg={\frac {1}{\#W}}\int _{T}\det(\operatorname {Id} -\operatorname {Ad} _{G/T}(t^{-1}))f(t)\,dt}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>G</mi>
</mrow>
</msub>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>g</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mi mathvariant="normal">#<!-- # --></mi>
<mi>W</mi>
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</mfrac>
</mrow>
<msub>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
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</msub>
<mo movablelimits="true" form="prefix">det</mo>
<mo stretchy="false">(</mo>
<mi>Id</mi>
<mo>−<!-- − --></mo>
<msub>
<mi>Ad</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>G</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
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<mi>T</mi>
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<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
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</msup>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \int _{G}f(g)\,dg={\frac {1}{\#W}}\int _{T}\det(\operatorname {Id} -\operatorname {Ad} _{G/T}(t^{-1}))f(t)\,dt}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c37d3093c6f3b376f7052155334f0d852a567eef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:49.532ex; height:5.676ex;" alt="{\displaystyle \int _{G}f(g)\,dg={\frac {1}{\#W}}\int _{T}\det(\operatorname {Id} -\operatorname {Ad} _{G/T}(t^{-1}))f(t)\,dt}" loading="lazy"></span>,</dd></dl>
<p>man braucht also nur über den maximalen Torus zu integrieren.
</p>
<div class="mw-heading mw-heading3"><h3 id="Erläuterungen"><span id="Erl.C3.A4uterungen"></span>Erläuterungen</h3></div>
<p>Es gilt
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(\operatorname {Id} -\operatorname {Ad} _{G/T}(t^{-1}))=\prod _{\alpha >0}\left(e^{\alpha (t)/2}-e^{-\alpha (t)/2}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo movablelimits="true" form="prefix">det</mo>
<mo stretchy="false">(</mo>
<mi>Id</mi>
<mo>−<!-- − --></mo>
<msub>
<mi>Ad</mi>
<mrow class="MJX-TeXAtom-ORD">
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<mrow class="MJX-TeXAtom-ORD">
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<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
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</msup>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munder>
<mo>∏<!-- ∏ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
<mo>&gt;</mo>
<mn>0</mn>
</mrow>
</munder>
<mrow>
<mo>(</mo>
<mrow>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>α<!-- α --></mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \det(\operatorname {Id} -\operatorname {Ad} _{G/T}(t^{-1}))=\prod _{\alpha &gt;0}\left(e^{\alpha (t)/2}-e^{-\alpha (t)/2}\right)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12e93fb1c86f5d5970a3e360a9e4abbcec387255.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:46.575ex; height:6.009ex;" alt="{\displaystyle \det(\operatorname {Id} -\operatorname {Ad} _{G/T}(t^{-1}))=\prod _{\alpha >0}\left(e^{\alpha (t)/2}-e^{-\alpha (t)/2}\right)}" loading="lazy"></span>,</dd></dl>
<p>wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha (t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>α<!-- α --></mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha (t)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a6cfa6e844c065b7c29c0a39cc4dcf40b237ea80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle \alpha (t)}" loading="lazy"></span> vom Eigenwertproblem abhängt.
</p>
<div class="mw-heading mw-heading2"><h2 id="Beispiel">Beispiel</h2></div>
<p>Für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=\mathbb {U} (n)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>G</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">U</mi>
</mrow>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle G=\mathbb {U} (n)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04026e8fb736ee58906b0624658eebc78cc82229.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.807ex; height:2.843ex;" alt="{\displaystyle G=\mathbb {U} (n)}" loading="lazy"></span> ergibt sich
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{G}f(g)\mathrm {d} g={\frac {1}{n!}}\int _{T}f(\operatorname {diag} (x_{1},\ldots ,x_{n}))|\Delta |^{2}\prod _{i=1}^{n}{\frac {\mathrm {d} x_{i}}{x_{i}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>G</mi>
</mrow>
</msub>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mi>g</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mi>n</mi>
<mo>!</mo>
</mrow>
</mfrac>
</mrow>
<msub>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msub>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>diag</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<munderover>
<mo>∏<!-- ∏ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \int _{G}f(g)\mathrm {d} g={\frac {1}{n!}}\int _{T}f(\operatorname {diag} (x_{1},\ldots ,x_{n}))|\Delta |^{2}\prod _{i=1}^{n}{\frac {\mathrm {d} x_{i}}{x_{i}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/71adcb69e618c0cf60c5c3ce0e4f59d5e299f594.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:50.805ex; height:6.843ex;" alt="{\displaystyle \int _{G}f(g)\mathrm {d} g={\frac {1}{n!}}\int _{T}f(\operatorname {diag} (x_{1},\ldots ,x_{n}))|\Delta |^{2}\prod _{i=1}^{n}{\frac {\mathrm {d} x_{i}}{x_{i}}}}" loading="lazy"></span>,</dd></dl>
<p>wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta ^{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/78b80e8f2a6394e3258ba49317756d06bb402188.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.99ex; height:2.676ex;" alt="{\displaystyle \Delta ^{2}}" loading="lazy"></span> die <a href="Vandermonde-Determinante" class="mw-redirect" title="Vandermonde-Determinante">Vandermonde-Determinante</a> ist, außerdem ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \#W=n!}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">#<!-- # --></mi>
<mi>W</mi>
<mo>=</mo>
<mi>n</mi>
<mo>!</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \#W=n!}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03df2a49840b6d4951fea1e93cdacb13d918bbb5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.511ex; height:2.509ex;" alt="{\displaystyle \#W=n!}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Beweis">Beweis</h2></div>
<p>Der Beweis folgt aus den Eigenschaften der durch
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(g,t)=gtg^{-1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>q</mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mo>,</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>g</mi>
<mi>t</mi>
<msup>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle q(g,t)=gtg^{-1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6cb5e3b906e37e672077ee5b1610b72dbcc8b7dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.373ex; height:3.176ex;" alt="{\displaystyle q(g,t)=gtg^{-1}}" loading="lazy"></span></dd></dl>
<p>definierten Abbildung
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\colon G/T\times T\to G}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>q</mi>
<mo>:<!-- : --></mo>
<mi>G</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>T</mi>
<mo>×<!-- × --></mo>
<mi>T</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>G</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle q\colon G/T\times T\to G}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c79d6dd4e7588a44dacbd833af3796601eea405c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.646ex; height:2.843ex;" alt="{\displaystyle q\colon G/T\times T\to G}" loading="lazy"></span>,</dd></dl>
<p>nämlich
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \deg(q)=\#W}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>deg</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>q</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi mathvariant="normal">#<!-- # --></mi>
<mi>W</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \deg(q)=\#W}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f7bb8b77592e6ca27d44fce0c33c69b1608d5bc9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.836ex; height:2.843ex;" alt="{\displaystyle \deg(q)=\#W}" loading="lazy"></span></dd></dl>
<p>für den <a href="Abbildungsgrad" title="Abbildungsgrad">Abbildungsgrad</a> und
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(\mathrm {d} q(gT,t))=\det(\operatorname {Ad} _{G/T}(t^{-1})-\operatorname {Id} )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo movablelimits="true" form="prefix">det</mo>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mi>q</mi>
<mo stretchy="false">(</mo>
<mi>g</mi>
<mi>T</mi>
<mo>,</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo movablelimits="true" form="prefix">det</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>Ad</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>G</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>T</mi>
</mrow>
</msub>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>Id</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \det(\mathrm {d} q(gT,t))=\det(\operatorname {Ad} _{G/T}(t^{-1})-\operatorname {Id} )}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3efcb6c7383535fe560c26cee59ea126aa6cd0d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:38.466ex; height:3.509ex;" alt="{\displaystyle \det(\mathrm {d} q(gT,t))=\det(\operatorname {Ad} _{G/T}(t^{-1})-\operatorname {Id} )}" loading="lazy"></span></dd></dl>
<p>für die <a href="Determinante" title="Determinante">Determinante</a> des Differentials von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>q</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle q}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li>T. Bröcker, T. tom Dieck: <i>Representations of compact Lie groups.</i> Springer Verlag New York 1985.</li>
<li>M. Sepanski: <i>Compact Lie groups.</i> Springer Verlag New York 2007.</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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