For a simple Lie algebra, the existence of an invariant convex cone forces the Lie algebra to have a Hermitian structure, i.e. the maximal compact subgroup has center isomorphic to the circle group.
Invariant convex cones arise in the analysis of holomorphic semigroups in the complexification of the Lie group, first studied by Grigori Olshanskii.
The semigroup is made up of those elements in the complexification which, when acting on the Hermitian symmetric space of compact type, leave invariant the bounded domain corresponding to the noncompact dual.
The unitary part of their polar decomposition is the operator corresponding to an element in the original real Lie group, while the positive part is the exponential of an imaginary multiple of the infinitesimal operator corresponding to an element in the maximal cone.
The oscillator semigroup of Roger Howe concerns the special case of this theory for the real symplectic group.
The Lie algebra of the symplectic group on R2n has a unique invariant convex cone.
[1] The cone and its properties can be derived directly using the description of the symplectic Lie algebra provided by the Weyl calculus in quantum mechanics.
The symplectic group acts naturally on this subalgebra by reparametrization and this yields the adjoint representation.
If C is any other invariant convex cone then it is invariant under the closed subgroup U of the symplectic group consisting of orthogonal transformations commuting with J. Identifying R2n with the complex inner product space Cn using the complex structure J, U can be identified with U(n).
There is a copy of SL(2,R) in the symplectic group acting only on the variables xi and yi.
Applying diagonal scaling operators in the second and subsequent copies of SL(2,R), the cone C must contain the quadratic form (x1)2.
On the other hand, if K is another compact subgroup of the symplectic group, averaging over Haar measure shows that it leaves invariant a positive element of P. Thus K is contained in a conjugate of U.
Then x+ = σ(x)^{-1} is an antiautomorphism of H which induces the inverse on the real symplectic group G. If g is in the open Olshanski semigroup H, let h = g+g.
Since G acts transitively on the interior of X, post-multiplying by an element of G if necessary, it can be assumed that h fixes 0.
Since h1 is diagonal, the theory for SU(1,1) and SL(2,C) acting on the unit disk in C shows that h1 lies in exp C. On the other hand, k = g (h1)−1 satisfies k+k = 1 so that σ(k) = k. Thus k lies in G and therefore, using the invariance of C, H admits the decomposition In fact there is a similar decomposition for the closed Olshanski symplectic semigroup: Moreover, the map (g,x) ↦ g exp x is a homeomorphism.
By the holomorphic functional calculus the exponential map on the space of operators with real spectrum defines a homeomorphism onto the space of operators with strictly positive spectrum, with an analytic inverse given by the logarithm.
[4] In fact a Möbius transformation f taking {z: ||z|| < 1, zt = z} into a compact subset has a unique fixed point z0 with fn(z) → z0 for any z. Uniqueness follows because, if f has a fixed point, after conjugating by an element of the real symplectic group, it can be assumed to be 0.
Then f has the form f(z) = az(1 + cz)−1at, where ct = c, with iterates fm(z) = amz(1 + cmz)−1(am)t with cm = c + atca + ⋅⋅⋅ + (am − 1)tcam − 1.
Thus for ||z|| ≤ r < 1, fm(z) tends to 0 uniformly, so that in particular 0 is the unique fixed point and it is obtained by applying iterates of f. Existence of a fixed point for f follows by noting that is an increasing sequence nk such that fnk and fn2k + 1 − n2k are both uniformly convergent on compacta, to h and g respectively.
By construction g ∘ h = h. So points in the image of h are fixed by g. Now g and h are either constant or have the form az(1 + cz)−1at followed by a real symplectic transformation.
is proved by first showing that any strictly larger semigroup S contains an element g sending |t| < 1 onto |t| > 1.
So if their inverses lie in the symplectic semigroup, it contains a neighbourhood of the identity and hence the whole of SL(2,C).
Precomposing with a scaling transformation, it can be assumed that g carries the closed unit disk onto a small neighbourhood of r. Pre-composing with an element of SU(1,1), the inverse image of the real axis can be taken to be the diameter joining –1 and 1.
Indeed, there is an embedding due to Harish-Chandra of the space of complex symmetric n by n matrices as a dense open subset of the compact Grassmannian of Langrangian subspaces of C2n.
[9] In fact, with the standard complex inner product on C2n, the Grassmannian of n-dimensional subspaces has a continuous transitive action of SL(2n,C) and its maximal compact subgroup SU(2n).
Taking coordinates (z1,...,zn,w1,...,wn) on C2n, the symplectic form is given by An n-dimensional subspace U is called Lagrangian if B vanishes on U.
The set of Langrangian subspaces U for which the restriction of the orthogonal projection onto U0 is an isomorphism forms an open dense subset Ω of the Lagrangian Grassmannian.
Under this correspondence elements of the complex symplectic group, viewed as block matrices
The unit ball for the operator norm and its closure are left invariant under the corresponding real form of the symplectic group.
If g(W) already lies in Ω, it will also have operator norm greater than 1 and W can be then be taken to be 0 by precomposing with a suitable element of G. Pre-composing g with a scaling transformation and post-composing g with a unitary transformation, it can be assumed that g(0) is a diagonal matrix with entries λi ≥ 0 with r = λ1 > 0 and that the image of the unit ball is contained in a small ball around this point.