Examples of leading term cycles of Harish-Chandra modules (Q2827854)

Let \(G_0\) be a real reductive Lie group and \(K_0\) a maximal compact subgroup. We denote their complexifications by \(G\) and \(K\) respectively, and put \(\mathfrak{g} = \mathrm{Lie}\,G\), the complexified Lie algebra of \(G_0\).NEWLINENEWLINELet \(X\) be an irreducible Harish-Chandra \((\mathfrak{g},K)\) module (HC-module for short). Then, there are several geometric invariants associated with \(X\). Perhaps a simpler one is the \textit{associated variety} \(\mathrm{AV}(X)\), which is a union of the closure of nilpotent \(K\)-orbits in \((\mathfrak{g}/\mathfrak{k})^{\ast}\). One can even refine it to include multiplicities and get a notion of \textit{associated cycles}.NEWLINENEWLINEIn the following, for simplicity, we assume the infinitesimal character of \(X\) is the same as the trivial representation. Let \(B\) be a Borel subgroup and put \(\mathfrak{B} = G/B\). Then, one can localize \(X\) on \(\mathfrak{B}\) to get a \(D\)-module \(\mathcal{X}\). Such an irreducible \(D\)-module can be classified by \(K\)-orbits in \(\mathfrak{B}\) and \(K\)-equivariant local systems on them. The attached \(K\)-orbit \(Q\) is called the \textit{support} of \(X\). As a \(D\)-module, \(\mathcal{X}\) has a \textit{characteristic cycle} \(\mathbb{C}{X}\), which is a union of conormal bundles of various \(K\)-orbits in \(\mathfrak{B}\) with multiplicities. Note that the relevant \(K\)-orbits here are contained in the closure of the support \(\overline{Q}\).NEWLINENEWLINEA most subtle geometric invariant of \(X\) is the leading term cycle \(\mathrm{LTC}(X)\). This is a union of conormal bundles of \(K\)-orbits in \(\mathfrak{B}\) appearing in \(\mathbb{C}(X)\) (with positive multiplicity) which are projected to an irreducible component of \(\mathrm{AV}(X)\) via the moment map. In fact, the moment map image of \(\mathbb{C}(X)\) is just \(\mathrm{AV}(X)\), and the multiplicities in the associated cycle of \( X \) are given by using the multiplicities in \(\mathrm{LTC}(X)\) and the Euler characteristic of the Springer fiber, although it is difficult to calculate them in practice.NEWLINENEWLINEThere are similar notions in the category of highest weight modules (in fact, the notions for HC-modules above originally stemmed from them). In the case of highest weight modules, already \textit{M. Kashiwara} and \textit{T. Tanisaki} [Invent. Math. 77, 185--198 (1984; Zbl 0611.22008); \textit{T. Tanisaki}, Adv. Stud. Pure Math. 14, 1--30 (1988; Zbl 0703.17002)] proved that \(\mathbb{C}(X)\) can be reducible for type B and C, and a reducible example for type A was given by \textit{M. Kashiwara} and \textit{Y. Saito} [Duke Math. J. 89, No. 1, 9--36 (1997; Zbl 0901.17006)]. For leading term cycles, though Kashiwara and Tanisaki's results give (possible) reducibility, it was largely believed that \(\mathrm{LTC}(X)\) is irreducible for type A until \textit{G. Williamson} [Prog. Math. 312, 517--532 (2015; Zbl 1344.22006)] gave an example of reducible \(\mathrm{LTC}(X)\).NEWLINENEWLINEIn the paper under review, the reducibility of \(\mathrm{LTC}(X)\) (and hence \(\mathbb{C}(X)\)) is studied in detail for \(G_0 = \mathrm{Sp}_{2n}(\mathbb R)\) and \(\mathrm{Sp}(p,q)\). Here, the authors are specially interested in the highest weight modules of \(\mathfrak{g}\) which are at the same time Harish-Chandra modules.NEWLINENEWLINEThe main results state that, for \(G_0 = \mathrm{Sp}(2n, \mathbb R)\), there is a systematic way to produce many examples of HC-modules (or highest weight modules) with reducible leading term cycles and hence with reducible characteristic cycles. The authors also get lower estimates of the number of such HC-modules. It is proved that there are many reducible leading term cycles! This result implies that there exist such HC-modules for \(G_0 = \mathrm{Sp}(n,n)\) too by using \textit{P. E. Trapa}'s argument [Compos. Math. 143, No. 2, 515--540 (2007; Zbl 1165.22013)].NEWLINENEWLINEThis paper includes many interesting ideas and strategies to attack the problem of determining leading term cycles.NEWLINENEWLINEThe most interesting one among them is that, since the HC-modules treated here are highest weight modules, their supports are the closure of a \(K\)-orbit as well as a \(B\)-orbit in the full flag variety \(G/B\), where \(B\) is a suitably chosen Borel subgroup. Their associated varieties (nilpotent \(K\)-orbits) and orbital varieties (nilpotent \(B\)-orbits) also coincide, and so on.NEWLINENEWLINECohomological induction is one of the authors' main tools, and a detailed knowledge of Harish-Chandra cells [\textit{W. M. McGovern}, Am. J. Math. 120, No. 1, 211--228 (1998; Zbl 0965.22016)], and the theory of Springer representations associated to nilpotent orbits are used in an essential way together with counting arguments.NEWLINENEWLINEA more detailed study in this direction is given by the same authors in [``Characteristic cycles of highest weight Harish-Chandra modules for \(\mathrm{Sp}(2n,\mathbb R)\)'', Preprint, \url{arXiv:1509.00141}]. However, the paper under review reveals most important ideas and techniques in a short and concise form.