View source for Coadjoint representation

The representation of a Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c0227401.png" /> contragredient to the adjoint representation Ad of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c0227402.png" /> (cf. [[Adjoint representation of a Lie group|Adjoint representation of a Lie group]]). The coadjoint representation acts on the dual <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c0227403.png" /> of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c0227404.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c0227405.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c0227406.png" /> is a real matrix group, i.e. a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c0227407.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c0227408.png" /> is a subspace of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c0227409.png" /> of real matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274010.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274011.png" /> be the orthogonal complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274012.png" /> relative to the bilinear form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274013.png" /></td> </tr></table>

let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274014.png" /> be some subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274015.png" /> complementary to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274016.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274017.png" /> be the projection onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274018.png" /> parallel to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274019.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274020.png" /> is identified with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274021.png" /> and the coadjoint representation is given by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274022.png" /></td> </tr></table>

The corresponding representation of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274023.png" /> is also called the coadjoint representation. In the case above it is defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274024.png" /></td> </tr></table>

The coadjoint representation plays a fundamental role in the [[Orbit method|orbit method]] (see [[#References|[2]]]). Each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274025.png" />-orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274026.png" /> in the coadjoint representation carries a canonical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274027.png" />-invariant [[Symplectic structure|symplectic structure]]. In other words, on each orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274028.png" /> there is a uniquely defined non-degenerate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274029.png" />-invariant closed differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274030.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274031.png" /> (whence it follows that all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274032.png" />-orbits in the coadjoint representation are even-dimensional). To obtain an explicit expression for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274033.png" /> one proceeds as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274034.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274035.png" /> be the orbit through the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274036.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274037.png" /> be tangent vectors to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274038.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274039.png" />. There exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274041.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274042.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274043.png" /></td> </tr></table>

Then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274044.png" /></td> </tr></table>

For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274045.png" />, the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274046.png" /> is Hamiltonian with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274047.png" />; as its generator (generating function) one can take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274048.png" /> itself, considered as a linear function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274049.png" />.

The stabilizer of a point with orbit of maximal dimension in the coadjoint representation is commutative [[#References|[1]]]. The Poisson bracket arising on each orbit generates a single Berezin bracket, which defines the structure of a local Lie algebra (cf. [[Lie algebra, local|Lie algebra, local]]), in the space of smooth functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274050.png" /> (see [[#References|[3]]]). The coordinate expression for the Berezin bracket is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274051.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274052.png" /> are the [[structure constant]]s of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022740/c02274053.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Bernal,   et al.,   "Représentations des groupes de Lie résolubles" , Dunod  (1972)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Kirillov,   "Elements of the theory of representations" , Springer  (1976)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.A. Kirillov,   "Local Lie algebras"  ''Russian Math. Surveys'' , '''31''' :  4  (1976)  pp. 55–75  ''Uspekhi Mat. Nauk'' , '''31''' :  4  (1976)  pp. 57–76</TD></TR></table>