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Let $G/K$ be a symmetric space. Let $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ be a Cartan decomposition, with the odd part $\mathfrak{p}$. It is well known that the algebra of invariant differential operators in this case is commutative, and "polynomial conjecture" states that it is isomorphic to $S(\mathfrak{p})^{\mathfrak{k}}$. It was formulated by C.Torossian in a 1993 paper, but actually it is a special case of an older conjecture by Duflo (though the reference I know are proceedings of a 1986 conference, and I haven't seen them.)

Is this conjecture still open? If it is, it makes me a little curious, because there aren't many symmetric spaces. What are the known and the open cases then?

EDIT: The conjecture (in this form) was formulated in Torossian, C., Operateurs differentiels invariants sur les espaces symetriques I. Methodes des orbites. J. Funct. Anal. 117 (1993), no. 1, 118–173. Torossian made a reference to Duflo, M., in Open problems in representation theory of Lie groups, Conference on Analysis on homogeneous spaces, (T. Oshima editor), August 25-30, Kataka, Japan, 1986. (As I understand it, Duflo's conjecture is much more general; admittedly, I did not read this 1986 text). A more recent account is in "Quantification pour les paires symétriques et diagrammes de Kontsevich" A. Cattaneo, C. Torossian, Annales Sci. de l'Ecole Norm. Sup. (5) 2008, 787--852, available here http://www.math.jussieu.fr/~torossian/

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  • $\begingroup$ It would help to give one or two explicit references here. (And I corrected a typo.) $\endgroup$ – Jim Humphreys Jul 16 '14 at 13:22
  • $\begingroup$ I guess the main reference is: Journal of Functional Analysis, Volume 117, Issue 1, October 1993, Pages 174–214, Invariant Differential Operators in Symmetrical Spaces. II. Generalized Harish-Chandra Homomorphism, by C. Torossian. $\endgroup$ – Claudio Gorodski Jul 17 '14 at 1:34
  • $\begingroup$ Isn't it a bit old? $\endgroup$ – Alex Gavrilov Jul 17 '14 at 2:22
  • $\begingroup$ I mean, I know this reference, but this does not help me to answer the question. $\endgroup$ – Alex Gavrilov Jul 17 '14 at 2:23
  • $\begingroup$ It is still open. $\endgroup$ – DamienC Apr 9 '16 at 7:02
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As far as I know, Duflo's conjecture is still open.

Let me make several remarks:

  1. Duflo's conjecture actually says that the algebra of invariant differential operators on a symetric space is isomorphic to the $\mathfrak k$-invariant part of $S(\mathfrak g)/(h-\chi(h),h\in\mathfrak k)$, where $\chi$ is the character given by half the trace of the adjoint action of $\mathfrak k$ on $\mathfrak p$. this shift by a character did not appear in Cattaneo-Torossian paper and this was very surprising... there was indeed a mistake in that paper, which is corrected in Cattane-Rossi-Torossian: http://arxiv.org/pdf/1105.5973.pdf
  2. Duflo's conjecture is indeed more general. It holds for general reductive homogeneous spaces: it claims that the center of the algebra of invariant differential operators is isomorphic to the Poisson center of $\big(S(\mathfrak g)/(h-\chi(h),h\in\mathfrak k)\big)^{\mathfrak k}$.
  3. Rybnikov's result mentionned in Alexander Chervov's comment prove a localized version of it for Riemaniann reductive homogeneous spaces (i.e. it holds on the level of fraction fields).
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If I understand correctly this is basically proved in L. Rybnikov: "On the Commutativity of Weakly Commutative Riemannian Homogeneous Spaces"

Abstract

A Riemannian homogeneous space X=G/H is said to be commutative if the algebra of G-invariant differential operators on X is commutative and weakly commutative if the associated Poisson algebra is commutative. Clearly, the commutativity of X implies its weak commutativity. The converse implication is proved in this paper.

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    $\begingroup$ But symmetric spaces are well known to be commutative in this sense. So, I do not see where this helps. $\endgroup$ – Alex Gavrilov Jul 16 '14 at 10:31
  • $\begingroup$ @AlexGavrilov Oh, it seems I misunderstand the situation, it seemed to me if the algebras are known to be commutative, then they are isomorphic... $\endgroup$ – Alexander Chervov Jul 17 '14 at 6:30
  • $\begingroup$ There is another paper by Rybnikov related to this: link.springer.com/article/10.1007/s00031-004-9004-9 Structure of the Center of the Algebra of Invariant Differential Operators on Certain Riemannian Homogeneous Spaces , Abstrcat: We study Duflo's conjecture on the isomorphism between the center of the algebra of invariant differential operators on a homogeneous space and the center of the associated Poisson algebra. For a rather wide class of Riemannian homogeneous spaces, which includes the class of (weakly) commutative spaces, we prove the "weakened version" of this conjecture..... $\endgroup$ – Alexander Chervov Jul 17 '14 at 6:31
  • $\begingroup$ @AlexGavrilov These algebras are commutative - are they known/conjectured to be isomorphic to polynomials in some number of variables ? Or there can be some non-trivial relations ? $\endgroup$ – Alexander Chervov Jul 17 '14 at 6:35
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    $\begingroup$ They are subalgebras of polynomial algebras. In cases I remember they are polynomial, but I doubt it is always so. The interesting problem here is not exactly to prove that they are isomorphic, but to find a natural isomorphism. Also, both algebras have natural filtrations, and I think a decent isomorphism is supposed to preserve it. $\endgroup$ – Alex Gavrilov Jul 17 '14 at 10:39

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