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Orbit structure of the closure of a homogeneous cone. (English) Zbl 1341.43002

The article under review is not easy reading so a considerable part of the review is introductory background.
M. Riesz [in: C. R. Congr. internat. Math., Oslo 1936, 2, 44–45 (1937; JFM 63.0477.01)] developed an \(n\)-dimensional fractional integration for the forward light-cone arising from physics (originally in 4-dimensional space) in a Minkowski space with signature \((1,n-1)\). The Lorentz group acts as isometries preserving the Minkowski metric. It can also be taken to be the group of isometries of \((n-1)\)-dimensional hyperbolic geometry, time-preserving in the sense that the unit time vector \((1,0,0,0)\) is sent to another vector with positive first coordinate. L. Gårding [Ann. Math. (2) 48, 785–826 (1947; Zbl 0029.21601)] represented the points of \(n\)-dimensional Lorentz space as symmetric matrices; subsequent authors often use this approach. For infinite-dimensional (or infinitesimal) representations of the Lorentz group instigated by E. P. Wigner [Ann. Math. (2) 40, 149–204 (1939; Zbl 0020.29601; JFM 65.1129.01)], see also Harish-Chandra [Proc. R. Soc. Lond., Ser. A 189 (1947)], V. Bargmann [Ann. Math. (2) 48, 568–640 (1947; Zbl 0045.38801)].
Let \(\Omega\) denote an open convex cone, usually a ‘forward’ cone (no straight line), and that the \(\mathrm{GL}(\Omega) \subset \mathrm{GL}_{n}(\mathbb{R}^n)\) acts on \(\Omega\) simply (i.e., isotropy free) and transitively. The closure \(\overline{\Omega}\) is not homogeneous (for example, the vertex is on the boundary of \(\Omega\)). The Riemann-Liouville integral on \(\mathbb{R}\), also called the Riesz potential, has integrand \(f(t)(x-t)^{1-\alpha}\) where \(\alpha \in\mathbb{C}\) and \(f\) is rapidly decreasing at infinity. The integral defines an analytic function of \(\alpha\), multiplied by a \(\Gamma\)-factor, so defining a tempered distribution called the Riesz distribution. Associated with a homogeneous convex cone is a semigroup of generalised Riemann-Liouville differential operators (these preserve the homogeneity). It is known (cf. [C. B. Chua, SIAM J. Optim. 14, No. 2, 500–506 (2003; Zbl 1046.90058)]) that every homogeneous cone is isomorphic to a ‘slice’ of a cone of positive definite matrices. Much of the method originated in articles by Harish-Chandra in the American Journal of Mathematics during the 50’s.
A bounded domain of a complex affine space is called symmetric (E. Cartan [La géométrie des éspaces de Riemann. Paris: Gauthier-Villars (1925; JFM 51.0566.01)]) if for every point there is an involutive biholomorphism for which the point is an isolated fixed point. All homogeneous bounded domains in \(\mathbb{C}^m\), \(0 < m \leq 3\), are symmetric. È. B. Vinberg and I. I. Piatetski-Shapiro [Trans. Mosc. Math. Soc. 12, 404–437 (1963; Zbl 0137.05603); translation from Tr. Mosk. Mat. Obšč. 12, 359–388 (1963)] proved that every bounded homogeneous domain in \(\mathbb{C}^n\) is holomorphically homeomorphic with a Siegel domain (a generalisation of the upper half-plane in \(\mathbb{C}\), see C. L. Siegel [Am. J. Math. 65, 1–86 (1943; Zbl 0138.31401)]). The Siegel domains of genus II were introduced by Piatetski-Shapiro in his study of open bounded homogeneous domains, showing that an injective symplectic representation induces a homogeneous embedding of a homogeneous bounded domain into the Siegel domains of genus II (including genus I).
A. Kirillov’s method of orbits [Russ. Math. Surv. 17, No. 4, 53–104 (1962); translation from Usp. Mat. Nauk 17, No. 4(106), 57–110 (1962; Zbl 0106.25001)] observes that the orbit of any irreducible representation should correspond to a symplectic manifold under the action of a nilpotent group and the irreducible representations could be indexed by its co-adjoint orbits in the dual Lie algebra. The assumption of symplecticity is restrictive entailing the dimension for the cone in \(\mathbb{R}^n\) must be an even integer. However the dimension of an orbit on the boundary of a cone is not necessarily even.
J. L. Koszul [Can. J. Math. 7, 562–576 (1955; Zbl 0066.16104)] proved that a homogeneous bounded domain in \(\mathbb{C}^m\) which admits a simply transitive real solvable group \(H\) of analytic automorphisms is symmetric. This can be shown to result in the corresponding Lie algebra being a \(j\)-algebra (defined below). See also his [Bull. Soc. Math. Fr. 89, 515–533 (1961; Zbl 0144.34002)].
A \(j\)-algebra (I. I. Piatetski-Shapiro [Sov. Math., Dokl. 2, 1460–1463 (1961); translation from Dokl. Akad. Nauk SSSR 141, 316–319 (1961; Zbl 0121.06801)]) is defined as a solvable Lie algebra \(\mathfrak{h}\) and a transformation \(j\) such that
\[ j^{2} =- I,\quad [jx:y] = [x:y] \text{ and }[x:y] + [jx:y] + [x:jy] - [jx:jy] = 0. \]
There is also a skew-symmetric bilinear (symplectic) 2-form \(\omega\) such that \(\omega ([jx:jy]) = \omega([x:y])\) and \(\omega([jx:x]) > 0\) for all \(x \neq 0\).
È. B. Vinberg [Sov. Math., Dokl. 2, 1470–1473 (1961); translation from Dokl. Akad. Nauk SSSR 141, 521–524 (1961; Zbl 0114.38402)] showed that \(j\)-algebras can be expressed as triangular matrices with positive diagonal elements. I. I. Piatetski-Shapiro [Izv. Akad. Nauk SSSR, Ser. Mat. 26, 453–484 (1962; Zbl 0109.05601)] showed that the embedding of a \(j\)-algebra to Siegel domains is many to one. È. B. Vinberg [Trans. Mosc. Math. Soc. 12, 340–403 (1963); translation from Tr. Mosk. Mat. Obšč 12, 303–358 (1963; Zbl 0138.43301)] gave an inductive description of \(\Omega\) and proved the existence and uniqueness of connected normal solvable split subgroups \(H \subset \mathrm{GL}(\Omega)\). The closed cone does not correspond to a normal \(j\)-algebra. There is an inductive process (to determine orbits) since a \(j\)-algebra can be written as a direct sum of the \(j\)-algebra corresponding to the sphere in \(\mathbb{C}^k, k \leq r\), and a lower dimensional \(j\)-algebra, used by Gindikin in 1964.
The Wallach set (see N. R. Wallach, broaching the concept in [Trans. Am. Math. Soc. 158, 107–113 (1971; Zbl 0227.22016)] and more explicitly in [J. Lepowsky and N. R. Wallach, Trans. Am. Math. Soc. 184, 223–246 (1974; Zbl 0279.17001)]) is a set of increasing positive rational parameter, at equal intervals along a line starting at 0 and ending with a half-line from the largest discrete point onwards. It consists of those parameters for which the powers \(P^{\alpha}\) for Bergman type kernels are self-reproducing kernels for the Hilbert space of square-integrable holomorphic functions on a Hermitian symmetric domain. They are powers of the Bergman kernel ending within the Cauchy type Szegő kernel; the half-line corresponds to the trivial representation which is itself connected to the Hilbert space of holomorphic functions by the Wiener-Paley theorem. The Wallach set is indeed used for the structure of symmetric cones; cf. [M. Vergne and H. Rossi, Acta Math. 136, 1–59 (1976; Zbl 0356.32020)]. The reviewer views the Wallach set, and its generalisation by Gindikin, as a spectrum of \(\mathrm{GL}(\Omega)\) as it is the set of irreducible representations; these may be characterised as the orbits of the nilpotent subgroup \(N\) of the KAN-decomposition of the (connected semi-simple Lie) group \(\mathrm{GL}(\Omega)\); cf. K. Iwasawa [Ann. Math. (2) 50, 507–558 (1949; Zbl 0034.01803)].
S. G. Gindikin [Usp. Mat. Nauk 19, No. 4(118), 3–92 (1964; Zbl 0144.08101)] constructed, by induction, non-symmetric homogeneous cones \(\Omega\) in \(\mathbb{R}^n\). The rank \(r\) of \(\Omega\) can be taken as the number of eigenvectors for the diagonal subgroup of \(\mathrm{GL}(\Omega)\). With matrices in mind he decomposed \(\mathbb{R}^n\) as a direct sum of subsets \(R_{i,j}\) of dimensions \(n_{i,j}\) where \(1 \leq i \leq j \leq r\) so as to build a non-associative algebra structure for the group. He then uses the inner product determined by the trace of the relative matrices. One can fix the vertex so that it projects orthogonally onto the Dirac \(\delta_{i,j}\) in each \(R_{i,j}\). He then provides a matrix multiplication and a transpose of the group actions. All this leads to a system of rational-valued coordinate functions \(\chi_{s}\) on \(\Omega\) so that the cone is characterised as those elements whose coordinates are positive. The dual cone consists of elements of \(\mathbb{R}^n\) for which the inner product with its elements in \(\Omega\) are non-negative; the cone is called self-dual or self-adjoint if it is the same as its dual cone. Non self-dual cones have to be of rank 3 or higher. Indeed Vinberg [loc. cit. (1963)] constructed a pair of non-selfdual cones of rank 3. È. B. Vinberg [Sov. Math., Dokl. 1, 787–790 (1960); translation from Dokl. Akad. Nauk SSSR 133, 9–12 (1960; Zbl 0143.05203); Tr. Mosk. Mat. Obšč. 13, 56–83 (1965; Zbl 0224.17010)] was also involved in this construction at that time using Lie algebras and weights. He used suitable rectangular matrix (and vector) valued entries in order to cope with the decompositions of the integer \(n\). \(R_{s}\) is again defined by the analytic continuation, now with respect to the parameter \(s = (s_1,\ldots, s_r) \in \mathbb{C}^r\). Gindikin also defined generalised Riemann-Liouville operators and power functions \(\Delta_{s}\) (related to determinants) on the cone. The case where all the \(s_i\) are equal indicates self-duality; there are then \(r-1\) equally spaced intervals and a half-line in each \(s_i\)-portion of the spectrum. The cone can alternatively be obtained as the number of orthogonal primitive idempotents of \(\mathrm{GL}(\Omega)\), cf. the Peirce [1870] decomposition of an algebra. The idempotents of \(\mathrm{GL}(\Omega)\) determine the eigenvectors for the group action; this enables one to decompose a non-symmetric cone into symmetric sub-cones.
In [Funct. Anal. Appl. 9, 50–52 (1975); translation from Funkts. Anal. Prilozh. 9, No. 1, 56–58 (1975; Zbl 0332.32022)], S. G. Gindikin was looking for unitary representations in order to determine an extended Wallach set, denoted by \(\Xi\) or \(\Xi (\Omega)\). His setting is in \(\mathbb{C}^r\) but to determine \(\Xi\) he needs only real-valued matrices. The inductive process with \(r = 1\). When \(r = n = 1 \) the Riesz distribution is positive on the positive half-line. He constructs a cone \(\tilde{\Omega}\) as the orthogonal projection of the component matrices of \(\Omega\) onto the subspace of those having their last column all zeros. The corresponding (triangular matrix) Lie algebra denoted \(\tilde{\mathfrak{h}}\) is produced in the same way. The parameters of \(\tilde{s}\) all have \(r-1\) components. There is also a parameter \(\hat{s}\) such that each \(\hat{s}_{i} = \tilde{s}_{i} - {n_{i,r} \over {2}}\). The induction process uses his Lemma that either \(\tilde{s}_{r} = 0\) and \(\tilde{s} \in \Xi(\tilde{\Omega})\) or \(\tilde{s}_{r} > 0\) and \(\hat{s} \in \Xi(\tilde{\Omega})\). Gindikin also deals with the Riesz distributions using power functions \(\Delta_{s}: s \in \Xi(\Omega)\) on \(\Omega\) and measures which are relatively invariant under \(H\). The Riesz distribution is a positive measure where and only where it has support in \(\Xi\), already known for \(n=1\).
In [Proc. Japan Acad., Ser. A 74, No. 8, 132–134 (1998; Zbl 0921.43004)], the author related the structure of \(\Xi\) to the orbits \(\mathcal{O}\) in the closure of the cone, the orbits themselves are open cones in \(\Omega\), and described the use of the \(\varepsilon = \sum_i^r \varepsilon_{i}\), the \(\varepsilon_{i}\) take values either 0 or 1. Since \(\mathrm{GL}(\Omega)\) is an algebraic Lie group (in the sense that it can be expressed locally by polynomials) the boundary of \({\mathrm{GL}(\Omega)}\) decomposes into a sum of mutually disjoint orbits. Indeed \(\overline{\Omega}\) is a disjoint union of \(2^{r}\) \(H\)-orbits \(\mathcal{O}_{\varepsilon}\) where the labelling \(\varepsilon_{i,j}\), where \(1\leq i,j\leq r\), takes values either 0 or 1. In [H. Ishi, J. Math. Soc. Japan 52, No. 1, 161–186 (2000; Zbl 0954.43003)], he had already set \(\Omega\) in a real \(n\)-dimensional vector space \(V\), not necessarily \(\mathbb{R}^n\). He defined \(\overline{\Omega} \subset V\) to be the non-negativity set in \(V\) of a \(j\)-algebra inner product of \(V\) and elements of the dual cone. The author’s notation is a bit confusing as he uses \(\underline{\varepsilon}\) to denote \(r\)-vectors having domains in \(\{0,1 \}\) and having domains in \(\mathbb{Z}\). The reviewer distinguishes between these using \(\varepsilon\) for \(\{0,1\}\) and \(\underline{\varepsilon}\) for \(\mathbb{Z}\); the latter are used to arrange the orbits in order. The author first demonstrates his theorem using matrices and then gives a more abstract proof. Other results in this article are improvements on his theorems on Riesz distributions and an attempt to know more about the structure of \(\Xi\). The orbits themselves are open cones in \(\overline{\Omega}\), each made up of subcones of the same rank. His main theorem gives a criterion for the ordering \(\overline{\mathcal{O}_1} \subset \overline{\mathcal{O}_2} \subset \overline{\Omega}\). Other results in this article are improvements on his theorems on Riesz distributions and an attempt to know more about the structure of \(\Xi\).
The crux of the proof of the main theorem is from partitions of the integer \(n\) as \(n_{1} +n_{2}+\dots+n_{k}, k=1,\ldots,r\); he partitions \(n\) as sums in all possible combinations to determine the different subcones of \(\overline{\Omega}\), cf. [I. M. Gel’fand and M. I. Graev, Tr. Moskov. Mat. Obšč. 8, 321–390 (1959; Zbl 0105.35102); correction ibid. 9, 562 (1960)]. The author constructs systems of vector spaces \(\Upsilon_{i,j}\) of \(n_i \times n_j\) matrices which satisfy conditions to enable matrix products of the \(\Upsilon\) and transposes of component block matrices. The \(\Upsilon_{i,i}\) are to be scalar multiples of identity matrices. The conditions needed were described in [H. Ishi, Differ. Geom. Appl. 24, No. 6, 588–612 (2006; Zbl 1156.17300), p. 589–590]. The question is as to when an \(\overline{\mathcal{O}_{\varepsilon}}\) is contained in another \(\overline{\mathcal{O}_{\varepsilon}'}\). To do this the author defines \(\sigma_{k}(\underline{\varepsilon})\) to be \(\sum_i^k n_{k,i} \varepsilon_{i}, k= 1,\ldots,r\). He fixes \(H\)-orbits in space by constructing \(r \times r\) diagonal matrices \(E _{\varepsilon_i, i = 1,\ldots,r}\) with entries \(\varepsilon_{i}{n_{i}}\) in the vector space of symmetric \(r \times r\) symmetric matrices for the matrix realisation of \(\Omega\).
The main theorem states that for \(\varepsilon^1, \varepsilon^2\), one has \(\overline{\mathcal{O}_{\varepsilon^1}} \subset \overline{\mathcal{O}_{\varepsilon^2}}\) if and only if \(\sigma_{k} (\varepsilon^1) \leq \sigma_{k} (\varepsilon^2)\) for all \(k = 1, \ldots, r\), i.e., where the \(\sigma_{k}\) are in lexicographic order. The author also constructs vector spaces \(\tau_{k}(\underline{\varepsilon})\) with norms \(\sigma \underline{\varepsilon_{i}}\) respectively. The two orbits indicated by vectors will be hyperboloids. To do this he uses the matrix dimension of bottom block-row of lower diagonal matrices, \(T_{r,k} \in \Upsilon_ {r,k}\), the \(T\) being rectangular matrices in \(\mathfrak{h}\).
As an illustration consider the Vinberg cone of rank 3 constructed as pairs of positive definite \(2 \times 2\) real matrices composed to be expressed as symmetric \(4 \times 4\) matrices. Partitioning \(n=4\) as \(2+1+1\) the indices \(\varepsilon_{i}\) in lexicographic order are 000 (the vertex) 001 010 011 and 100 (the cone). Here \(\Upsilon_{2,1}\) is a row vector \((v.0)\) and \(\Upsilon_{3,1}\) is a column vector \((0,v), v \in \mathbb{R}\).
The author gives an interesting example in his Section 2 of Lorentz cones \(\Omega \subset \mathbb{R}^n\) realised as positive-definite symmetric \(n-1 \times n-1\) matrices and Lorentz groups \(G\) acting on \(\Omega\). When \(n=3\) this is the cone composed of \(2 \times 2\) matrices. Since he does not have an underlying symplectic form he starts by using decompositions of \(n -1\) to construct a triangular matrix group \(H\), with positive diagonal elements. Since \(H\) can be expressed as \(T^{t}T: T \backslash H\), where \(^{t}T\) denotes the transpose. \(H\) acts simply transitively on the cone and one can construct irreducible representations of \(G\) induced from \(H\). He constructs also a (commutative) diagonal group \(A \subset G\) generated by \(2 \times 2\) velocity boosts; \(A\) will have positive exponential components. He constructs also a nilpotent subgroup \(N\) which is essentially \(H\) with each of the diagonal elements replaced by 1 (so \(N\) is a commutative nilpotent subgroup). He lands up with an Iwasawa type AN subgroup, a solvable nilpotent subgroup which induces a simply transitive action of the Lorentz group on the cone and induces the spectrum. The corresponding Iwasawa group \(K\) will be the isotropy subgroup of \(G\) at the vertex. The orbits will be hyperboloids in the boundary of the cone.

MSC:

43A35 Positive definite functions on groups, semigroups, etc.
22E27 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
22E46 Semisimple Lie groups and their representations
32M10 Homogeneous complex manifolds
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
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