Editing Maximal compact subgroup (section)

==Existence and uniqueness==
The '''Cartan-Iwasawa-Malcev theorem''' asserts that every connected Lie group (and indeed every connected [[locally compact group]]) admits maximal compact subgroups and that they are all conjugate to one another. For a [[semisimple Lie group]] uniqueness is a consequence of the '''[[Cartan fixed point theorem]]''', which asserts that if a compact group acts by isometries on a complete simply connected [[negative curvature|nonpositively curved]] [[Riemannian manifold]] then it has a fixed point.

Maximal compact subgroups of connected Lie groups are usually ''not'' unique, but they are unique up to conjugation, meaning that given two maximal compact subgroups ''K'' and ''L'', there is an element ''g'' ∈ ''G'' such that<ref>Note that this element ''g'' is not unique – any element in the same coset ''gK'' would do as well.</ref> ''gKg''<sup>−1</sup> = ''L''. Hence a maximal compact subgroup is [[essentially unique]], and people often speak of "the" maximal compact subgroup.

For the example of the general linear group GL(''n'', '''R'''), this corresponds to the fact that ''any'' [[inner product]] on '''R'''<sup>''n''</sup> defines a (compact) orthogonal group (its isometry group) – and that it admits an orthonormal basis: the change of basis defines the conjugating element conjugating the isometry group to the classical orthogonal group O(''n'', '''R''').

===Proofs===
For a real semisimple Lie group, Cartan's proof of the existence and uniqueness of a maximal compact subgroup can be found in {{harvtxt|Borel|1950}} and {{harvtxt|Helgason|1978}}. {{harvtxt|Cartier|1955}} and {{harvtxt|Hochschild|1965}} discuss the extension to connected Lie groups and connected locally compact groups.

For semisimple groups, existence is a consequence of the existence of a compact [[Complexification (Lie group)|real form]] of the noncompact semisimple Lie group and the corresponding [[Cartan decomposition]]. The proof of uniqueness relies on the fact that the corresponding [[Riemannian symmetric space]] ''G''/''K''  has [[negative curvature]] and 
Cartan's fixed point theorem. {{harvtxt|Mostow|1955}} showed that the derivative of the exponential map at any point of ''G''/''K'' satisfies |d exp ''X''| ≥ |X|. This implies that ''G''/''K'' is a [[Hadamard space]], i.e. a [[complete metric space]] satisfying a weakened form of the parallelogram rule in a Euclidean space. Uniqueness can then be deduced from the [[Bruhat-Tits fixed point theorem]]. Indeed, any bounded closed set in a Hadamard space is contained in a unique smallest closed ball, the center of which is called its [[circumcenter]]. In particular a compact group acting by isometries must fix the circumcenter of each of its orbits.

===Proof of uniqueness for semisimple groups===
{{harvtxt|Mostow|1955}} also related the general problem for semisimple groups to the case of GL(''n'', '''R'''). The corresponding symmetric space is the space of positive symmetric matrices. A direct proof of uniqueness relying on elementary properties of this space is given in {{harvtxt|Hilgert|Neeb|2012}}.

Let <math>\mathfrak{g}</math> be a real semisimple Lie algebra with [[Cartan involution]] σ. Thus the [[fixed point subgroup]] of σ is the maximal compact subgroup ''K'' and there is an eigenspace decomposition

:<math>\displaystyle{\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p},}</math>

where <math>\mathfrak{k}</math>, the Lie algebra of ''K'', is the +1 eigenspace. The Cartan decomposition gives

:<math>\displaystyle{G=K\cdot \exp \mathfrak{p} = K\cdot P = P\cdot K.}</math>

If ''B'' is the [[Killing form]] on <math>\mathfrak{g}</math> given by ''B''(''X'',''Y'') = Tr (ad X)(ad Y), then

:<math>\displaystyle{(X,Y)_\sigma=-B(X,\sigma(Y))}</math>

is a real inner product on <math>\mathfrak{g}</math>. Under the adjoint representation, ''K'' is the subgroup of ''G'' that preserves this inner product.

If ''H'' is another compact subgroup of ''G'', then averaging the inner product over ''H'' with respect to the Haar measure gives an inner product invariant under ''H''. The operators Ad ''p'' with ''p'' in ''P'' are positive symmetric operators. This new inner produst can be written as

:<math>(S\cdot X,Y)_\sigma,</math>

where ''S'' is a positive symmetric operator on <math>\mathfrak{g}</math> such that 
Ad(''h'')<sup>''t''</sup>''S'' Ad ''h'' = ''S'' for ''h'' in ''H'' (with the transposes computed with respect to the inner product). Moreover, for ''x'' in ''G'',

:<math>\displaystyle{\mathrm{Ad}\, \sigma(x)=(\mathrm{Ad}\,(x)^{-1})^t.}</math>

So for ''h'' in ''H'',

:<math>\displaystyle{S\circ \mathrm{Ad}(\sigma(h))= \mathrm{Ad}(h)\circ S.}</math>

For ''X'' in <math>\mathfrak{p}</math> define

:<math>\displaystyle{f(e^X)=\mathrm{Tr}\, \mathrm{Ad}(e^X) S.}</math>

If ''e''<sub>''i''</sub> is an orthonormal basis of eigenvectors for ''S'' with ''Se''<sub>''i''</sub> = λ<sub>''i''</sub> ''e''<sub>''i''</sub>, then

:<math>\displaystyle{f(e^X)=\sum \lambda_i (\mathrm{Ad}(e^X)e_i,e_i)_\sigma \ge (\min \lambda_i)\cdot \mathrm{Tr}\,e^{\mathrm{ad}\,X},}</math>

so that ''f'' is strictly positive and tends to ∞ as |''X''| tends to ∞. In fact this norm is equivalent to the operator norm on the symmetric operators ad ''X'' and each non-zero eigenvalue occurs with its negative, since i ad ''X'' is a ''skew-adjoint operator'' on the compact real form <math>\mathfrak{k}\oplus i\mathfrak{p}</math>.

So ''f'' has a global minimum at ''Y'' say. This minimum is unique, because if ''Z'' were another then

:<math>\displaystyle{e^Z=e^{Y/2} e^X e^{Y/2},}</math>

where ''X'' in <math>\mathfrak{p}</math> is defined by the Cartan decomposition

:<math>\displaystyle{e^{Z/2}e^{-Y/2}=k\cdot e^{X/2}.}</math>

If ''f''<sub>''i''</sub> is an orthonormal basis of eigenvectors of ad ''X'' with corresponding real eigenvalues μ<sub>''i''</sub>, then

:<math>\displaystyle{g(t)= f(e^{Y/2} e^{tX} e^{Y/2})= \sum e^{\mu_i t} \|Ad(e^{Y/2})f_i\|^2_\sigma.}</math>

Since the right hand side is a positive combination of exponentials, the real-valued function ''g'' is [[Strictly convex function|strictly convex]] if ''X'' ≠ 0, so has a unique minimum. On the other hand, it has local minima at ''t'' = 0 and ''t'' = 1, hence ''X'' = 0 and ''p'' = exp ''Y'' is the unique global minimum. By construction
''f''(''x'') = ''f''(σ(''h'')''xh''<sup>−1</sup>) for ''h'' in ''H'', so that ''p'' = σ(''h'')''ph''<sup>−1</sup> for ''h'' in ''H''. Hence σ(''h'')= ''php''<sup>−1</sup>. Consequently, if ''g'' = exp ''Y''/2, ''gHg''<sup>−1</sup> is fixed by σ and therefore lies in ''K''.