The invariant holonomic system on a semisimple Lie algebra

Hamiltonian reduction and nearby cycles for mirabolic D-modules

“…The mirabolic Harish-Chandra D-module. In the seminal paper [HK1], Hotta and Kashiwara have defined, for any complex semisimple group G, a holonomic D(G)-module that they called the Harish-Chandra D-module. This D-module is important, for instance, because of its close relation to the system of partial differential equations on the group G introduced by Harish-Chandra around 1960 in his study of irreducible characters of infinite dimensional representations of the group G.…”

La transformation de Fourier pour les ${\cal D}$-modules

“…Pour les LA TRANSFORMATION DE FOURIER POUR LES P-MODULES 1893 preuves de ces résultats, ainsi que pour plus de détails, commentaires, etc., nous renvoyons le lecteur à la très riche littérature classique sur ce sujet (voir [2], [3], [5], [12], [13]- [15], [19], [35], [37], [40], etc. ).…”

“…-Nous précisons ici que la notion de régularité que nous utilisons dans le cas algébrique est celle de Mebkhout (voir [35], p. 185) : si X est une variété algébrique lisse, X ^-> X une compactification d'Hironaka-Nagata de X et M. un P^-module cohérent, alors M est régulier si (j^M)^ l'est comme P^an-module. Cela revient à la notion de régularité complète de [12], p. 331 (voir aussi dans le même article l'exemple 3.4, p. 337), et c'est équivalent à la définition de [3], p. 302 (cf. [35], p. 183 et 163).…”

Untitled

“…The above result implies that for a semisimple Lie algebra over C, we have constructed an injective map from the set of nilpotent orbits of the adjoint group into the set of equivalence classes of irreducible representations of its Weyl group. At the end of this section we will use the results of [BV2, BV3,HK] to show that this correspondence is the Springer correspondence [S].…”

“…To analyze these D(~) -modules we develop a theory analogous to Howe's formalism of dual pairs, proving an equivalence of categories between an appropriate category of D(~)w-modules and the category of all W-modules over C. We show that the D(g)G-module of distributions on go supported in the nilpotent cone of go is (as a D(~)w module) in our category. Thus, to each distribution supported on the nilpotent cone we can associate a (finite dimensional) representation of W. If the distribution is the orbital integral corresponding to a fixed nilpotent element of go then we prove that the representation of W is irreducible and derive a formula for the Fourier transform of the orbital integral in terms of W -harmonic polynomials corresponding to this representation of W. In [BV2, BV3,HK] results of this nature were proved in the case when 9 0 is a Lie algebra over C looked upon as a Lie algebra over R. They prove that the indicated Fourier transform is given in terms of a harmonic polynomial transforming according to the Springer representation associated to the corresponding nilpotent G-orbit in 9 [S]. We use this theorem to prove that our general correspondence between nilpotent Go -orbits of 9 0 is given by the Springer correspondence for the corresponding G-orbits in 9.…”