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**When is a Riesz distribution a complex measure?**
*(English.
French summary)*
Zbl 1263.43003

M. Riesz [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 in a Lorentzian space with signature \((1,n-1)\). S. G. Gindikin [Funct. Anal. Appl. 9, 50–52 (1975); translation from Funkts. Anal. Prilozh. 9, No. 1, 56–58 (1975; Zbl 0332.32022); Usp. Mat. Nauk 19, No. 4(118), 3–92 (1964; Zbl 0144.08101)] extended the results to apply to a homogeneous cone in \(\mathbb{R}^{n}\) and a simply transitive group \(T\) of linear automorphisms of the cone given by generalized Riemann-Liouville operators for certain ranges of a parameter. He determined all positive \(T\)-invariant distributions on the cone and also dealt with positivity leading to unitary representations of the group.

The Riemann-Liouville integral (also called the Riesz potential of \(f\)) on \(\mathbb{R}\) is \[ I^{\alpha}f(x) = {\Gamma(\alpha)^{-1}} \int_{0}^{\alpha}{f(t)(x-t)^{1-\alpha}\,dt} \] when \(\alpha > 0\); the \(I^{\alpha}\) form a semigroup. Riesz showed, for instance, that \(I^{2}\) is the inverse of the Laplacian \(-\Delta\). The integral defines an analytic function of \(\alpha\), a tempered distribution called a Riesz distribution (cf. [J. A. C. Kolk and V. S. Varadarajan, Math. Scand. 68, No. 2, 273–291 (1991; Zbl 0773.46016)]). For \(f\) with sufficient derivatives one has \(I^{-m}f(x) = f^{m}(x)\) since \(\Gamma\) has poles at all negative integers. If so, if \(f\) has \(n\) derivatives, the integral can be extended by analytic continuation for all \(\alpha > -n\).

J. Faraut and A. Korányi [Analysis on symmetric cones. Oxford: Clarendon Press (1994; Zbl 0841.43002)] developed a more general case where \(\Omega\) is taken to be the open cone, the interior of the set generated by sums of squares of the elements (with nonzero determinants) of a simple Euclidean Jordan algebra \(V\) (Euclidean in that the trace leads to an inner product for \(V\)). The Jordan product can be represented as the Jordan anticommutator product of matrices. The multivariate gamma-function for \(\Omega\) has a set of \(r\) complex parameters, \(r\) denoting the rank of \(V\). The author’s article essentially reproduces their proof. Faraut and Korányi deal also with the complexification of the Jordan algebra extending the group to consist of holomorphic automorphisms of complex tubular neighbourhoods in \(V + i\Omega\). The author refers to Faraut and Korányi’s use of the non-negative polynomials of I. N. Bernstein [Funct. Anal. Appl. 6, 273–285 (1973); translation from Funkts. Anal. Prilozh. 6, No. 4, 26–40 (1972; Zbl 0282.46038)] but not giving any details at all.

The author is particularly concerned with the part of Gindikin’s theorem saying that the Riesz distribution is a positive measure. Then the parameters \(\alpha\) are necessarily in the set \(\{0, d/2, \dotsc,(r-1)d/2\}\) or such that \(\alpha >(r-1)d/2\). The discrete set is called a Wallach-set [N. R. Wallach, Lect. Notes Math. 466, 226–231 (1975; Zbl 0311.22008)] (dealing with representations of semisimple Lie groups). The rank is the number of independent primitive idempotents summing up to the unit idempotent and \(d\) is the Peirce constant defined by \(n = r + {d \over 2} r(r-1)\) (C. S. Peirce decomposed representations of Jordan product algebras into eigenvalues and eigenvectors). Considering a ‘frame’ of independent irreducible idempotents summing up to the unit idempotent, \( x \in V\) has ‘spectral decomposition’ \(\sum_{1,\dotsc ,r}\lambda_{i}e_{i}\). The Jordan determinant \(\Delta(x)\) can be defined as the product of the (positive) eigenvalues of \(x\). The Riesz distribution is positive definite (c.a.d. de type positif) if and only if its Laplace transform of \(\Delta(x)\) is positive. The Riesz distribution is indeed a complex measure; when it is a positive measure it is called a Riesz measure.

T. J. Hilgert and K.-H. Neeb [J. Geom. Anal. 11, No. 1, 43–75 (2001; Zbl 0989.22021)] deal with operator-valued Riesz distributions on an \(n\)-dimensional state space. In Chapter III of their paper, they describe the scalar case bringing together, with more detail than the original, results scattered throughout [Faraut and Korányi, loc. cit.]. They assumed in their proof that the Riesz distribution is a Radon measure. The author instead assumes that the measures are Radon with support consisting of the locally integrable functions on \(\Omega\). The idea is that a distribution defined on an open subset of \(\mathbb{R}^{n}\) by a locally integrable function can be extended to the whole of \(\mathbb{R}^{n}\) as a locally finite complex measure if and only if the function \(f\) is locally integrable also at the boundary of \(\Omega\). This allows him to extend Faraut and Korányi’s results to the closed cone. The author’s proof of the extended theorem uses the same kind of argument as that of Faraut and Korányi’s but involves positivity of local Laplace transforms instead of the classical Laplace transforms. It is known from [Faraut and Korányi, loc. cit.] that the group acts on the determinant as a multiplier in that \(\Delta(gx) = \det(g)^{r \over n}\Delta(x)\). By the author’s Lemma 3.4, \(\Delta^{\alpha - {n \over r}} \in L_{loc}^{1} (\overline{\Omega}\)) if and only if Re \(\alpha > {n \over r} -1\).

A. Korányi and J. A. Wolf [Ann. Math. (2) 81, 265–288 (1965; Zbl 0137.27402)], leaning on [I. I. Pyatetski-Shapiro, Automorphic functions and the geometry of classical domains. New York-London-Paris: Gordon and Breach Science Publishers (1969; Zbl 0196.09901)] analysed the orbit structure of \(\partial D\), where \(D\) is a (non-homogeneous) Siegel domain of genus three [C. L. Siegel, Math. Ann., Berlin, 116, 617–657 (1939; Zbl 0021.20302; JFM 65.0357.01)]. The relevant (non-compact) group is \(G^{0}\), the identity component of its automorphism group. The analysis of the orbit structure of the boundary of the cone is an elementary part of the theory for a (non-homogeneous) Siegel domain of genus three [Siegel, loc. cit.]; the symmetric cone itself is of genus one. The Harish-Chandra realisation allows the theory to be applied to the symmetric cone now sitting inside the unit ball of the (half-plane) complexification \(V + i \Omega\). The boundary of \(D\) is made up of disjoint orbits, i.e., domains of lower rank lying flat on the boundary. The analysis is broken down using the \(G^{0} \backslash \Gamma\), where \(\Gamma\) is the Siegel modular group. Each of these orbits itself is a base for a fibration under the modular group. The boundary of \(D\) may be thought of as a non-commutative tubular domain [Wallach, loc. cit.]. I. Satake [Algebraic structures of symmetric domains. Tokyo: Iwanami Shoten, Publishers. Princeton, New Jersey: Princeton University Press (1980; Zbl 0483.32017)] constructs in Chapter III §8 of his book a fibration with the orbits as fibres such that there are two consecutive fibrations and the base contains double indices.

For the symmetric cone \(\Omega\), the group has \(r+1\) orbits \(O_{k}\) \((0 \leq k \leq r-1)\), each being sums of squares or of elements of rank \(k\). These orbits correspond to the Wallach parameters \((k-1)d/2\). The orbits are mutually disjoint cones, being sectors filling the boundary. The orbit \(O_{0}\) is the vertex itself and \(O_{r} = \Omega\) (Satake reverses the order of the orbits). From Proposition IV.3.2 in [Faraut and Korányi, loc. cit.], it can be shown that the boundary is split into sectors embating from the vertex T. The closure of \(O_{k}\) is the union of orbits of rank \(\leq k\). The primitive idempotents lie in a compact submanifold of \(V\) and the orbit \(O_{k}\) can be linked to a primitive idempotent of rank \(k\) and passes through all the idempotents of rank \(k\).

For a proof of the uniqueness of the measure \(\mu\) supported on \(\partial \Omega\), the author departs from [Faraut and Korányi, loc. cit.] by dealing separately with the measures \(\mu_{k}\) supported on each orbit \(O_k\). This method is due to M. Lassalle [Invent. Math. 89, 375–393 (1987; Zbl 0622.22008)]. The author proves that the decomposition of the measure \(\mu\) on the boundary as \(\sum \mu_{k}\) is unique. The proof uses relative invariance of the \(\mu_{k}\), involving multipliers \((\det (g)^{k})\) for the regular representations [R. A. Wijsman, Invariant measures on groups and their use in statistics. Hayward, CA: Institute of Mathematical Statistics (1990; Zbl 0803.62001)].

The article includes a section on the paper [G. Letac and H. Massam, J. Multivariate Anal. 99, No. 7, 1393–1417 (2008; Zbl 1140.62043)]. The Wishart distributions are relevant in that they are constructed on cones of positive semidefinite matrices.

The Riemann-Liouville integral (also called the Riesz potential of \(f\)) on \(\mathbb{R}\) is \[ I^{\alpha}f(x) = {\Gamma(\alpha)^{-1}} \int_{0}^{\alpha}{f(t)(x-t)^{1-\alpha}\,dt} \] when \(\alpha > 0\); the \(I^{\alpha}\) form a semigroup. Riesz showed, for instance, that \(I^{2}\) is the inverse of the Laplacian \(-\Delta\). The integral defines an analytic function of \(\alpha\), a tempered distribution called a Riesz distribution (cf. [J. A. C. Kolk and V. S. Varadarajan, Math. Scand. 68, No. 2, 273–291 (1991; Zbl 0773.46016)]). For \(f\) with sufficient derivatives one has \(I^{-m}f(x) = f^{m}(x)\) since \(\Gamma\) has poles at all negative integers. If so, if \(f\) has \(n\) derivatives, the integral can be extended by analytic continuation for all \(\alpha > -n\).

J. Faraut and A. Korányi [Analysis on symmetric cones. Oxford: Clarendon Press (1994; Zbl 0841.43002)] developed a more general case where \(\Omega\) is taken to be the open cone, the interior of the set generated by sums of squares of the elements (with nonzero determinants) of a simple Euclidean Jordan algebra \(V\) (Euclidean in that the trace leads to an inner product for \(V\)). The Jordan product can be represented as the Jordan anticommutator product of matrices. The multivariate gamma-function for \(\Omega\) has a set of \(r\) complex parameters, \(r\) denoting the rank of \(V\). The author’s article essentially reproduces their proof. Faraut and Korányi deal also with the complexification of the Jordan algebra extending the group to consist of holomorphic automorphisms of complex tubular neighbourhoods in \(V + i\Omega\). The author refers to Faraut and Korányi’s use of the non-negative polynomials of I. N. Bernstein [Funct. Anal. Appl. 6, 273–285 (1973); translation from Funkts. Anal. Prilozh. 6, No. 4, 26–40 (1972; Zbl 0282.46038)] but not giving any details at all.

The author is particularly concerned with the part of Gindikin’s theorem saying that the Riesz distribution is a positive measure. Then the parameters \(\alpha\) are necessarily in the set \(\{0, d/2, \dotsc,(r-1)d/2\}\) or such that \(\alpha >(r-1)d/2\). The discrete set is called a Wallach-set [N. R. Wallach, Lect. Notes Math. 466, 226–231 (1975; Zbl 0311.22008)] (dealing with representations of semisimple Lie groups). The rank is the number of independent primitive idempotents summing up to the unit idempotent and \(d\) is the Peirce constant defined by \(n = r + {d \over 2} r(r-1)\) (C. S. Peirce decomposed representations of Jordan product algebras into eigenvalues and eigenvectors). Considering a ‘frame’ of independent irreducible idempotents summing up to the unit idempotent, \( x \in V\) has ‘spectral decomposition’ \(\sum_{1,\dotsc ,r}\lambda_{i}e_{i}\). The Jordan determinant \(\Delta(x)\) can be defined as the product of the (positive) eigenvalues of \(x\). The Riesz distribution is positive definite (c.a.d. de type positif) if and only if its Laplace transform of \(\Delta(x)\) is positive. The Riesz distribution is indeed a complex measure; when it is a positive measure it is called a Riesz measure.

T. J. Hilgert and K.-H. Neeb [J. Geom. Anal. 11, No. 1, 43–75 (2001; Zbl 0989.22021)] deal with operator-valued Riesz distributions on an \(n\)-dimensional state space. In Chapter III of their paper, they describe the scalar case bringing together, with more detail than the original, results scattered throughout [Faraut and Korányi, loc. cit.]. They assumed in their proof that the Riesz distribution is a Radon measure. The author instead assumes that the measures are Radon with support consisting of the locally integrable functions on \(\Omega\). The idea is that a distribution defined on an open subset of \(\mathbb{R}^{n}\) by a locally integrable function can be extended to the whole of \(\mathbb{R}^{n}\) as a locally finite complex measure if and only if the function \(f\) is locally integrable also at the boundary of \(\Omega\). This allows him to extend Faraut and Korányi’s results to the closed cone. The author’s proof of the extended theorem uses the same kind of argument as that of Faraut and Korányi’s but involves positivity of local Laplace transforms instead of the classical Laplace transforms. It is known from [Faraut and Korányi, loc. cit.] that the group acts on the determinant as a multiplier in that \(\Delta(gx) = \det(g)^{r \over n}\Delta(x)\). By the author’s Lemma 3.4, \(\Delta^{\alpha - {n \over r}} \in L_{loc}^{1} (\overline{\Omega}\)) if and only if Re \(\alpha > {n \over r} -1\).

A. Korányi and J. A. Wolf [Ann. Math. (2) 81, 265–288 (1965; Zbl 0137.27402)], leaning on [I. I. Pyatetski-Shapiro, Automorphic functions and the geometry of classical domains. New York-London-Paris: Gordon and Breach Science Publishers (1969; Zbl 0196.09901)] analysed the orbit structure of \(\partial D\), where \(D\) is a (non-homogeneous) Siegel domain of genus three [C. L. Siegel, Math. Ann., Berlin, 116, 617–657 (1939; Zbl 0021.20302; JFM 65.0357.01)]. The relevant (non-compact) group is \(G^{0}\), the identity component of its automorphism group. The analysis of the orbit structure of the boundary of the cone is an elementary part of the theory for a (non-homogeneous) Siegel domain of genus three [Siegel, loc. cit.]; the symmetric cone itself is of genus one. The Harish-Chandra realisation allows the theory to be applied to the symmetric cone now sitting inside the unit ball of the (half-plane) complexification \(V + i \Omega\). The boundary of \(D\) is made up of disjoint orbits, i.e., domains of lower rank lying flat on the boundary. The analysis is broken down using the \(G^{0} \backslash \Gamma\), where \(\Gamma\) is the Siegel modular group. Each of these orbits itself is a base for a fibration under the modular group. The boundary of \(D\) may be thought of as a non-commutative tubular domain [Wallach, loc. cit.]. I. Satake [Algebraic structures of symmetric domains. Tokyo: Iwanami Shoten, Publishers. Princeton, New Jersey: Princeton University Press (1980; Zbl 0483.32017)] constructs in Chapter III §8 of his book a fibration with the orbits as fibres such that there are two consecutive fibrations and the base contains double indices.

For the symmetric cone \(\Omega\), the group has \(r+1\) orbits \(O_{k}\) \((0 \leq k \leq r-1)\), each being sums of squares or of elements of rank \(k\). These orbits correspond to the Wallach parameters \((k-1)d/2\). The orbits are mutually disjoint cones, being sectors filling the boundary. The orbit \(O_{0}\) is the vertex itself and \(O_{r} = \Omega\) (Satake reverses the order of the orbits). From Proposition IV.3.2 in [Faraut and Korányi, loc. cit.], it can be shown that the boundary is split into sectors embating from the vertex T. The closure of \(O_{k}\) is the union of orbits of rank \(\leq k\). The primitive idempotents lie in a compact submanifold of \(V\) and the orbit \(O_{k}\) can be linked to a primitive idempotent of rank \(k\) and passes through all the idempotents of rank \(k\).

For a proof of the uniqueness of the measure \(\mu\) supported on \(\partial \Omega\), the author departs from [Faraut and Korányi, loc. cit.] by dealing separately with the measures \(\mu_{k}\) supported on each orbit \(O_k\). This method is due to M. Lassalle [Invent. Math. 89, 375–393 (1987; Zbl 0622.22008)]. The author proves that the decomposition of the measure \(\mu\) on the boundary as \(\sum \mu_{k}\) is unique. The proof uses relative invariance of the \(\mu_{k}\), involving multipliers \((\det (g)^{k})\) for the regular representations [R. A. Wijsman, Invariant measures on groups and their use in statistics. Hayward, CA: Institute of Mathematical Statistics (1990; Zbl 0803.62001)].

The article includes a section on the paper [G. Letac and H. Massam, J. Multivariate Anal. 99, No. 7, 1393–1417 (2008; Zbl 1140.62043)]. The Wishart distributions are relevant in that they are constructed on cones of positive semidefinite matrices.

Reviewer: Aubrey Wulfsohn (Coventry)

### MSC:

43A85 | Harmonic analysis on homogeneous spaces |

32M15 | Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) |

44A10 | Laplace transform |

46F10 | Operations with distributions and generalized functions |

47G10 | Integral operators |

17A15 | Noncommutative Jordan algebras |