Harmonic analysis on real reductive symmetric spaces.

*(English)*Zbl 1018.43009
Li, Ta Tsien (ed.) et al., Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20-28, 2002. Vol. II: Invited lectures. Beijing: Higher Education Press. 545-554 (2002).

Let \(G\) be a Lie group, \(H\) a closed subgroup. Assume that \(G/H\) has a \(G\)-invariant measure \(dx\). Then the representation \(U\) of \(G\) by translations on \(L^2(G/H,dx)\) (the quasiregular representation) is unitary. The main problem of harmonic analysis on \(G/H\) is to decompose \(U\) (as explicitly as possible) into irreducible ones.

Very often there are differential operators on \(G/H\) which commute with the action of \(G\) and which are essentially self-adjoint on \(L^2(G/H,dx)\). In this case \(G\) preserves their spectral decomposition, and the decomposition of \(L^2(G/H,dx)\) in terms of eigenfunctions of these operators is often an important step in the decomposition of \(U\).

It is the latter problem which is solved in the paper under review for reductive symmetric spaces \(G/H\). Here \(G\) is a reductive Lie group of Harish-Chandra class (for example, a semisimple Lie group with finite center), and \(H\) is an open subgroup of the subgroup \(G^{\sigma}\) of fixed points of an involution \(\sigma\). The decomposition is obtained for the Schwartz space \({\mathcal C}(G/H,\tau)\) of \(\tau\)-spherical functions \(f(x)\) of \(K\)-type \(\tau\) (i.e. \(f(kx)=\tau(k)f(x)\), \(k\in K\), \(x\in G/H\)) rapidly decreasing as well as their derivatives by elements of the universal enveloping algebra of \(\operatorname {Lie}G\). Here \(\tau\) is an irreducible unitary representation of \(K\), a maximal compact subgroup of \(G\). The above-mentioned problem of harmonic analysis on semisimple (reductive) symmetric spaces has a long history. For compact (hence Riemannian) symmetric spaces harmonic analysis was developed by E. Cartan in his classical work of 1929. For arbitrary Riemannian symmetric spaces (\(H\) compact) of non-compact type, a decomposition of the quasiregular representation was given by Harish-Chandra in 1958. The explicit expression of the Plancherel measure was found by S. Gindikin and F. Karpelevich in 1962. The non-Riemannian case (\(H\) non-compact) is much more difficult. Harish-Chandra’s grandiose twenty-years lasting project was crowned with remarkable success in 1975-76, when he obtained the Plancherel formula (the expansion of the delta function concentrated at the neutral element of \(G\) into characters of irreducible unitary representations) for semisimple Lie groups which are viewed as symmetric spaces. Besides, in the 1980s the Plancherel formula was obtained for some classes of semisimple symmetric spaces by different authors: S. Sano, N. Bopp, P. Harinck for \(G/H\) where \(G\) is a complex semisimple Lie group, \(H\) a real form, V. F. Molchanov for all rank one spaces.

Recently (in the last decade) essential achievements were obtained for arbitrary \(G/H\) by E. van den Ban and H. Schlichtkrull on the one hand, and by P. Delorme on the other hand: they obtained some versions of a Plancherel formula. Notice that the problem is still not completely solved: there is no formula giving the decomposition of the delta function concentrated at the origin of \(G/H\) into spherical (\(H\)-invariant) distributions on \(G/H\) corresponding to irreducible unitary representations.

The paper under review is a short exposition of an invited lecture given by the author in ICM-2002, Beijing, China. A similar lecture was given by H. Schlichtkrull in ECM-2000, Barcelona, Spain.

For the entire collection see [Zbl 0993.00022].

Very often there are differential operators on \(G/H\) which commute with the action of \(G\) and which are essentially self-adjoint on \(L^2(G/H,dx)\). In this case \(G\) preserves their spectral decomposition, and the decomposition of \(L^2(G/H,dx)\) in terms of eigenfunctions of these operators is often an important step in the decomposition of \(U\).

It is the latter problem which is solved in the paper under review for reductive symmetric spaces \(G/H\). Here \(G\) is a reductive Lie group of Harish-Chandra class (for example, a semisimple Lie group with finite center), and \(H\) is an open subgroup of the subgroup \(G^{\sigma}\) of fixed points of an involution \(\sigma\). The decomposition is obtained for the Schwartz space \({\mathcal C}(G/H,\tau)\) of \(\tau\)-spherical functions \(f(x)\) of \(K\)-type \(\tau\) (i.e. \(f(kx)=\tau(k)f(x)\), \(k\in K\), \(x\in G/H\)) rapidly decreasing as well as their derivatives by elements of the universal enveloping algebra of \(\operatorname {Lie}G\). Here \(\tau\) is an irreducible unitary representation of \(K\), a maximal compact subgroup of \(G\). The above-mentioned problem of harmonic analysis on semisimple (reductive) symmetric spaces has a long history. For compact (hence Riemannian) symmetric spaces harmonic analysis was developed by E. Cartan in his classical work of 1929. For arbitrary Riemannian symmetric spaces (\(H\) compact) of non-compact type, a decomposition of the quasiregular representation was given by Harish-Chandra in 1958. The explicit expression of the Plancherel measure was found by S. Gindikin and F. Karpelevich in 1962. The non-Riemannian case (\(H\) non-compact) is much more difficult. Harish-Chandra’s grandiose twenty-years lasting project was crowned with remarkable success in 1975-76, when he obtained the Plancherel formula (the expansion of the delta function concentrated at the neutral element of \(G\) into characters of irreducible unitary representations) for semisimple Lie groups which are viewed as symmetric spaces. Besides, in the 1980s the Plancherel formula was obtained for some classes of semisimple symmetric spaces by different authors: S. Sano, N. Bopp, P. Harinck for \(G/H\) where \(G\) is a complex semisimple Lie group, \(H\) a real form, V. F. Molchanov for all rank one spaces.

Recently (in the last decade) essential achievements were obtained for arbitrary \(G/H\) by E. van den Ban and H. Schlichtkrull on the one hand, and by P. Delorme on the other hand: they obtained some versions of a Plancherel formula. Notice that the problem is still not completely solved: there is no formula giving the decomposition of the delta function concentrated at the origin of \(G/H\) into spherical (\(H\)-invariant) distributions on \(G/H\) corresponding to irreducible unitary representations.

The paper under review is a short exposition of an invited lecture given by the author in ICM-2002, Beijing, China. A similar lecture was given by H. Schlichtkrull in ECM-2000, Barcelona, Spain.

For the entire collection see [Zbl 0993.00022].

Reviewer: V.F.Molchanov (Tambov)

##### MSC:

43A85 | Harmonic analysis on homogeneous spaces |

22E46 | Semisimple Lie groups and their representations |

22F30 | Homogeneous spaces |

22E30 | Analysis on real and complex Lie groups |

22E50 | Representations of Lie and linear algebraic groups over local fields |

33C67 | Hypergeometric functions associated with root systems |

##### Keywords:

reductive symmetric spaces; Plancherel formula; Eisenstein integrals; temperedness; Maass-Selberg relations
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\textit{P. Delorme}, in: Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20--28, 2002. Vol. II: Invited lectures. Beijing: Higher Education Press; Singapore: World Scientific/distributor. 545--554 (2002; Zbl 1018.43009)

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