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**The invariant holonomic system on a semisimple Lie algebra.**
*(English)*
Zbl 0538.22013

Let \(G\) be a complex semisimple Lie group. Harish-Chandra’s study of the distribution characters of infinite dimensional representations of \(G\) led him to consider certain systems of differential equations on the Lie algebra \(\mathfrak g\) of \(G\): those for “invariant eigendistributions”. Here “invariant” means annihilated by the infinitesimal conjugation action of \(\mathfrak g\); and “eigendistribution” means that each \(G\)-invariant constant coefficient differential operator acts by a scalar. There is one such system for each possible family of eigenvalues for the invariants. Harish-Chandra showed that the space of distribution solutions to such a system has dimension the order of the Weyl group \(W\). An important step in his analysis is the observation that the tempered distribution solutions have their Fourier transforms supported in the cone \(\mathcal N\) of nilpotent elements in \(\mathfrak g^*\). More recent work, beginning with J. C. Jantzen [Math. Z. 140, 127–149 (1974; Zbl 0279.20036)] has exhibited and studied an action of \(W\) on the space of solutions of one of these systems, and related it to the nature of the nilpotent \(G\) orbits appearing in the support on the Fourier transform side.

The present paper synthesizes and substantially extends all of these ideas. Each of the Harish-Chandra systems (and, in an appropriate sense, all of them at once) is shown to be a regular holonomic system. Such systems have a deep and powerful geometric structure (the Riemann-Hilbert correspondence). The effect in this setting is to relate these differential equations to the variety \(\tilde{\mathfrak g}\) of pairs \((X,\mathfrak b)\). (Here \(X\) is an element of \(\mathfrak g\) and \(\mathfrak b\) is a Borel subalgebra of \(\mathfrak g\) containing \(X\).) This variety, which was first studied by A. Grothendieck and T. Springer, is in some respects quite well understood. Perhaps the main consequence of the analysis is a relationship between the decomposition of the Harish-Chandra systems under \(W\), and the decomposition of \(\mathcal N\) into nilpotent orbits. This amounts to an analytic realization of the Springer correspondence between Weyl group representations and nilpotent orbits, and it has a variety of delightful applications in representation theory and harmonic analysis.

The paper is addressed to those with some serious understanding of regular holonomic systems, making it rather dense for most group representers. There are introductory discussions of the relationship between algebraic and analytic holonomic systems (Serre’s GAGA for differential equations), and of the Fourier transform of a holonomic system. These can help ease the uninitiated into the main parts of the paper. In any case, the reader is amply warned on the first page by the ominous sentence, “We are going into more details”.

The present paper synthesizes and substantially extends all of these ideas. Each of the Harish-Chandra systems (and, in an appropriate sense, all of them at once) is shown to be a regular holonomic system. Such systems have a deep and powerful geometric structure (the Riemann-Hilbert correspondence). The effect in this setting is to relate these differential equations to the variety \(\tilde{\mathfrak g}\) of pairs \((X,\mathfrak b)\). (Here \(X\) is an element of \(\mathfrak g\) and \(\mathfrak b\) is a Borel subalgebra of \(\mathfrak g\) containing \(X\).) This variety, which was first studied by A. Grothendieck and T. Springer, is in some respects quite well understood. Perhaps the main consequence of the analysis is a relationship between the decomposition of the Harish-Chandra systems under \(W\), and the decomposition of \(\mathcal N\) into nilpotent orbits. This amounts to an analytic realization of the Springer correspondence between Weyl group representations and nilpotent orbits, and it has a variety of delightful applications in representation theory and harmonic analysis.

The paper is addressed to those with some serious understanding of regular holonomic systems, making it rather dense for most group representers. There are introductory discussions of the relationship between algebraic and analytic holonomic systems (Serre’s GAGA for differential equations), and of the Fourier transform of a holonomic system. These can help ease the uninitiated into the main parts of the paper. In any case, the reader is amply warned on the first page by the ominous sentence, “We are going into more details”.

Reviewer: David Vogan

### MSC:

22E46 | Semisimple Lie groups and their representations |

14F10 | Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials |

32C38 | Sheaves of differential operators and their modules, \(D\)-modules |

35Q15 | Riemann-Hilbert problems in context of PDEs |

### Keywords:

invariant holonomic system; geometric Weyl group representations; Springer representations; invariant eigendistributions; intersection cohomology complex; regular holonomic system; Riemann-Hilbert correspondence
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\textit{R. Hotta} and \textit{M. Kashiwara}, Invent. Math. 75, 327--358 (1984; Zbl 0538.22013)

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