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In mathematics, an invariant convex cone is a closed convex cone in a Lie algebra of a connected Lie group that is invariant under inner automorphisms. The study of such cones was initiated by Ernest Vinberg and Bertram Kostant. 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. The invariant convex cone generated by a generator of the Lie algebra of the center is closed and is the minimal invariant convex cone (up to a sign). The dual cone with respect to the Killing form is the maximal invariant convex cone. Any intermediate cone is uniquely determined by its intersection with the Lie algebra of a maximal torus in a maximal compact subgroup. The intersection is invariant under the Weyl group of the maximal torus and the orbit of every point in the interior of the cone intersects the interior of the Weyl group invariant cone. For the real symplectic group, the maximal and minimal cone coincide, so there is only one invariant convex cone. When one is properly contained in the other, there is a continuum of intermediate invariant convex cones. Invariant convex cones arise in the analysis of holomorphic semigroups in the complexification of the Lie group, first studied by Grigori Olshanskii. They are naturally associated with Hermitian symmetric spaces and their associated holomorphic discrete series. 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 semigroup acts by contraction operators on the holomorphic discrete series; its interior acts by Hilbert–Schmidt operators. 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. A similar decomposition already occurs in the semigroup. The oscillator semigroup of Roger Howe concerns the special case of this theory for the real symplectic group. Historically this has been one of the most important applications and has been generalized to infinite dimensions. This article treats in detail the example of the invariant convex cone for the symplectic group and its use in the study of the symplectic Olshanskii semigroup. ==Invariant convex cone in symplectic Lie algebra== The Lie algebra of the symplectic group on R2''n'' has a unique invariant convex cone. It is self-dual.〔See: * * 〕 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.〔See: * *〕 Let the variables in R2''n'' be ''x''1, ..., ''x''''n'', ''y''1, ..., ''y''''n''. Taking the standard inner product on R2''n'', the symplectic form corresponds to the matrix : The real polynomials on R2''n'' form an infinite-dimensional Lie algebra under the Poisson bracket : The polynomials of degree ≤ 2 form a finite-dimensional Lie algebra with center the constant polynomials. The homogeneous polynomials of degree 2 form a Lie subalgebra isomorphic to the symplectic Lie algebra. The symplectic group acts naturally on this subalgebra by reparametrization and this yields the adjoint representation. Homogeneous polynomials of degree 2 on the other hand are just symmetric bilinear forms on R2''n''. They therefore correspond to symmetric 2''n'' × 2''n'' matrices. The Killing form on the Lie algebra is proportional to the trace form Tr ''AB''. The positive definite symmetric bilinear forms give an open invariant convex cone with closure the set ''P'' of positive semi-definite symmetric bilinear forms. Because the Killing form is the trace form, the cone ''P'' is self-dual. Any positive symmetric bilinear form defines a new inner product on R2''n''. The symplectic from defines an invertible skew-adjoint operator ''T'' with respect to this inner product with –''T''2 a positive operator. An orthonormal basis can be chose so that ''T'' has 2 × 2 skew-symmetric matrices down the diagonal. Scaling the orthonormal basis, it follows that there is a symplectic basis for R2''n'' diagonalizing the original positive symmetric bilinear form. Thus every positive symmetric bilinear form lies in the orbit of a diagonal form under the symplectic group. 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 R2''n'' with the complex inner product space C''n'' using the complex structure ''J'', ''U'' can be identified with ''U''(''n''). Taking any non-zero point in ''C''. the average over ''U'' with respect to Haar measure lies in ''C'' and is non-zero. The corresponding quadratic form is a multiple of the standard inner product. Replacing ''C'' by –''C'' this multiple can be taken to be positive. There is a copy of SL(2,R) in the symplectic group acting only on the variables ''x''''i'' and ''y''''i''. These operators can be used to transform into with 0 < ''t'' < 2. It follows that ''C'' contains the point . Applying diagonal scaling operators in the second and subsequent copies of SL(2,R), the cone ''C'' must contain the quadratic form . By invariance ''C'' must also contain the quadratic forms and . By convexity it contains all diagonal positive symmetric bilinear forms. Since any positive symmetric bilinear form is in the orbit of a diagonal form, ''C'' contains the cone of non-negative symmetric bilinear forms. By duality the dual cone ''C'' * is contained in ''P''. If ''C'' is a proper cone, the previous argument shows that ''C'' * = ''P'' and hence that ''C'' = ''P''. This argument shows that every positive definite symmetric form is in the orbit of a form with corresponding quadratic form : with ''a''''i'' > 0. This corresponds to a cone in the Lie algebra of the (diagonal) maximal torus of ''U''. Since every element of ''P'' is diagonalizable, the stabilizer of a positive element in the symplectic group is contained in a conjugate of ''U''. 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''. It follows that ''U'' is a maximal compact subgroup of the symplectic group and that any other such subgroup must be a conjugate of ''U''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Invariant convex cone」の詳細全文を読む スポンサード リンク
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