¹

1 Introduction↩︎

As a sequel of our previous work ([1]), we prove in this paper a large and essential part of the wavefront set conjecture ([1]) for real classical groups. This brings substantial progress for a large family of irreducible admissible representations of real classical groups in the investigation of long-standing open problems in the representation theory of real reductive Lie groups on relations among the wavefront set of an irreducible admissible representation and other relevant invariants. The approach we took is based on the theory of local descents for irreducible admissible representations, which has been well developed for representations with generic local Langlands parameters, and puts for the first time the study of the long-standing open problems on wavefront sets and related problems within the framework of the local Langlands program.

Around 1970, L. Hörmander introduced an influential notion of wavefront sets for distributions over smooth manifolds in the theory of Fourier integral operators ([2], [3]). For any unitary representation \(\pi\) of a real Lie group \(G\), R. Howe introduced around 1980 the notion of the wavefront set of \(\pi\) ([4]). He first proved that the wavefront set \(\mathrm{WF}(\pi)\) of the Harish-Chandra distribution character \(\Theta_\pi\) ([5], [6]) is \(G\)-invariant under the adjoint action and is determined by its image under the projection \(G\times{\mathfrak {g}}^*\to{\mathfrak {g}}^*\), where \({\mathfrak {g}}\) is the Lie algebra of \(G\) and \({\mathfrak {g}}^*\) is the dual space of \({\mathfrak {g}}\). Hence the wavefront set of \(\pi\) as defined by R. Howe is a subset of the nilpotent cone of \({\mathfrak {g}}\) if \(G\) is reductive and \({\mathfrak {g}}^*\) is identified with \({\mathfrak {g}}\) via the Killing form.

As expected in [4], [7], for a real reductive group \(G\), the microlocal structure of the wavefront set of an irreducible admissible representation \(\pi\) of \(G\) should be detected by the germ expression near the identity of the distribution character \(\Theta_\pi\). Meanwhile, D. Vogan introduced the notion of associated variety \({\mathrm{AV}}(\pi)\) of a Harish-Chandra module of a reductive Lie group and conjectured that \({\mathrm{AV}}(\pi)=\mathrm{WF}(\pi)\), which provides an algebraic approach to understand the microlocal structure of the distribution character \(\Theta_\pi\). It remains a difficult problem in theory to explicitly determine the structure of either the associated varieties \({\mathrm{AV}}(\pi)\) or the wavefront sets \(\mathrm{WF}(\pi)\).

The expectation that an irreducible representation should have a well-defined maximal nilpotent support goes back to the Kostant-Kirillov orbit method philosophy and to the early work on primitive ideals. Duflo’s work on primitive ideals [8] provided one of the algebraic foundations for this viewpoint. For complex semisimple Lie groups, the relationship between primitive ideals, orbital integrals, local character expansions, and nilpotent orbits was developed in the work of Barbasch–Vogan and Joseph; see, for example, [7], [9]–[12]. In this setting the maximal nilpotent support is governed by the closure of a single nilpotent orbit. We shall refer to this principle as Vogan’s maximal-orbit principle. This viewpoint is also consistent with the Kirillov–Kostant orbit method for the unitary dual of real reductive groups, as explained for instance in [13]–[16].

In general, let \(F\) be an arbitrary local field of characteristic zero and \(G\) be the \(F\)-rational points of a reductive algebraic group \(\mathbb{G}\) over \(F\). Denote by \(\Pi_F(G)\) the set of equivalence classes of irreducible admissible representations of \(G\), which in the archimedean case are of Casselman-Wallach type. When \(F\) is non-archimedean, for any \(\pi \in \Pi_F(G)\), an asymptotic expansion of the distribution character \(\Theta_\pi\) near the identity was established by R. Howe ([17]; see also [18], [19]) and Harish-Chandra ([20]): \[\begin{align} \label{thetapi} \Theta_\pi=\sum_{{\mathcal{O}}\in{\mathcal{N}}_F({\mathfrak {g}})_\circ}c_{\mathcal{O}}\widehat{\mu}_{\mathcal{O}},\quad c_{\mathcal{O}}\in{\mathbb{C}}, \end{align}\tag{1}\] where \({\mathcal{N}}_F({\mathfrak {g}})_\circ\) denotes the set of \(F\)-rational nilpotent orbits in the Lie algebra \({\mathfrak {g}}\) of the group \(G\) under the adjoint action of \(G\) and \(\widehat{\mu}_{\mathcal{O}}\) denotes the Fourier transform of the (normalized) measure \(\mu_{\mathcal{O}}\) on the nilpotent orbit \({\mathcal{O}}\). From 1 , one defines the analytic wavefront set of \(\pi\in\Pi_F(G)\) to be \[\begin{align} \label{wf-tr} \mathrm{WF}_{\mathrm{tr}}(\pi):=\set{{\mathcal{O}}\in{\mathcal{N}}_F({\mathfrak {g}})_\circ | c_{\mathcal{O}}\neq 0}, \end{align}\tag{2}\] where \(c_{\mathcal{O}}\)’s are as in 1 . When \(F\) is an archimedean local field, following the work of D. Barbasch and D. Vogan ([7]), one may define \(\mathrm{WF}_{\mathrm{tr}}(\pi)\) in a similar way.

Over a non-archimedean local field \(F\), C. Mœglin and J.-L. Waldspurger define in [21] (see also [22]) the algebraic wavefront set \(\mathrm{WF}_\mathrm{wm}(\pi)\) of \(\pi\in\Pi_F(G)\) by means of the generalized Whittaker models of \(\pi\) associated with any nilpotent element \(f\) in the set \({\mathcal{N}}_F({\mathfrak {g}})\) of all nilpotent elements in \({\mathfrak {g}}\) (see Section 2.2 for some details). One of the main results in [21], [23] shows that under the \(F\)-rational topological order (Definition 9), the set of the maximal members, which is denoted by \(\mathrm{WF}_{\mathrm{tr}}(\pi)^{r-\mathrm{max}}\), of the analytic wavefront set \(\mathrm{WF}_{\mathrm{tr}}(\pi)\) coincides with the set of maximal members, which is denoted by \(\mathrm{WF}_\mathrm{wm}(\pi)^{r-\mathrm{max}}\), of the algebraic wavefront set \(\mathrm{WF}_\mathrm{wm}(\pi)\), that is, \[\begin{align} \label{alg-ana-wfs} \mathrm{WF}_{\mathrm{tr}}(\pi)^{r-\mathrm{max}}=\mathrm{WF}_\mathrm{wm}(\pi)^{r-\mathrm{max}} \end{align}\tag{3}\] holds for any \(\pi\in \Pi_F(G)\). It is important to mention that over any non-archimedean local field \(F\), a wavefront set may not be the closure of a single nilpotent orbit, due to the examples constructed by C.-C. Tsai ([24], [25]), although it remains the closure of a single nilpotent orbit in many cases.

It is natural to extend the notion of the generalized Whittaker models to any irreducible Casselman-Wallach representations of real reductive groups and to conjecture that the assertion as in 3 holds for the archimedean case. However, only a few special situations in the archimedean case have been studied (see [26]–[32] for instance). It should be mentioned that when \(F\) is archimedean, there are more invariants (associated variety, characteristic cycle and wavefront cycle, etc.) of \(\pi\) closely related to the wavefront set of \(\pi\) ([33], [34]). These invariants form a rich theory. We refer to the papers of W. Rossmann ([35], [36]), of D. Vogan ([34]), of J. Adams and D. Vogan ([16]), and of W. Schmid and K. Vilonen ([33]) for excellent surveys and detailed discussions of the theory, in particular, for the relation between the explicit computation of the wavefront sets and the construction of the unitary dual of the real reductive groups via the Kirillov-Kostant orbit method ([15], [37]–[41]).

It is important to point out that in [16], Adams and Vogan found, when \(F\) is archimedean, a sophisticated and comprehensive algorithm, based on certain profound geometric structures behind the wavefront sets, to compute the \(\overline{F}\)-structure of the wavefront set \(\mathrm{WF}_{\mathrm{tr}}(\pi)\) for \(\pi\in\Pi_F(G)\) via the complex-geometrical parametrization of \(\Pi_F(G)\). It remains a difficult problem to determine the \(F\)-rational structure of the wavefront sets in general.

For families of special unipotent representations of \(G\), some substantial progress on determination of the \(F\)-rational structure of the wavefront sets was made by J.-L. Waldspurger ([42]–[44]) and by D. Ciubotaru, L. Mason-Brown, E. Okada and others ([45]–[48]) for the non-archimedean case, and by D. Barbasch, J.-J. Ma, B. Sun and C.-B. Zhu ([49], [50]) for the archimedean case. Those results are essential to the explicit construction of the unipotent part of the unitary dual \(\widehat{G}\) of \(G\), which is regarded as the building block of the whole unitary dual. For other families of representations, we refer to relevant papers ([51]–[57]), for instance.

It is worthwhile to mention that even weaker information on the wavefront sets than the complete \(F\)-rational structure may find substantial applications. For instance, in the twisted automorphic descent theory of D. Jiang and L. Zhang ([58]), the information of wavefront sets played a substantial role in their construction of explicit automorphic modules for irreducible cuspidal automorphic representations of classical groups and in their proof of the global Gan-Gross-Prasad conjecture. It is also used in the work of D. Jiang and B. Liu ([59], [60]) to determine which global Arthur packets have no cuspidal members. The local upper bound conjecture on the wavefront sets by D. Jiang ([61]) and its variants have been extensively studied by B. Liu, C.-H. Lo and F. Shahidi ([62]–[64]) and by H. Atobe, D. Ciubotaru and J.-L. Kim ([65]–[67]). Those local results are substantial towards the understanding of the local Arthur packets and the unitary dual problem for \(G\).

1.1 Wavefront set and Langlands reciprocity↩︎

In our previous paper ([1]), we assume that \(F\) is a local field of characteristic zero. In order to discover a practical way to determine the \(F\)-rational structure of the wavefront sets, based on our previous work ([68]), we developed the theory of local descents for enhanced local \(L\)-parameters and proposed in [1] to utilize the local Langlands correspondence and the local Gan-Gross-Prasad conjecture as a bridge to transfer the problem of explicit computation of wavefront sets for a representation \(\pi\) to that on the Langlands dual side and consider the structure of the arithmetic wavefront set associated to the enhanced local \(L\)-parameter of \(\pi\), which seems quite productive and fruitful.

In [1], for any classical group \({\mathbb{G}}\) defined over \(F\) and \(G={\mathbb{G}}(F)\), we introduced the notion of arithmetic wavefront set \(\mathrm{WF}_{\mathrm{ari}}(\pi)\) ([1]) for any \(\pi\in\Pi_F(G)\) with generic local \(L\)-parameter (see Section 2.3) and made the following Wavefront Set Conjecture.

To illustrate the spirit of the local Langlands reciprocity in the theory of local descents for both representations \(\pi\) of \(G(F)\) and their associated enhanced local \(L\)-parameters we established (joint with C. Chen) in [69] the coincidence of the first occurrence indices of the local descents for the representations and their enhanced local \(L\)-parameters when the local \(L\)-parameters are generic. In order to understand Conjecture 1, we introduce in this paper a new notion of the wavefront set of \(\pi\) when a \(\pi\in\Pi_F(G)\) has a generic local \(L\)-parameter by the local descents. It is called the descent wavefront set of \(\pi\) and is denoted by \(\mathrm{WF}_{\mathrm{des}}(\pi)\). We refer to Section 2.3 for details. The spirit of the local Langlands reciprocity for the wavefront sets is illustrated in the following theorem.

This theorem will be restated as Theorem 12 and proved in Section 2.3. By the composition law of descents as established in Theorem 16, we obtain in Corollary 18 that \[\label{des-wm} \mathrm{WF}_{\mathrm{des}}(\pi)\subset\mathrm{WF}_\mathrm{wm}(\pi),\quad {\rm for\;any}\;\pi\in\Pi_F(G),\tag{4}\] with a generic \(L\)-parameter and over all local fields \(F\) of characteristic zero. Conjecture 1 naturally leads to the following assertion.

It is worthwhile to point out that Conjecture 3 is of representation-theoretic nature and is expected to be proved through an approach based on representation theory and combinatorics. It is even more mysterious if one considers the relations between \(\mathrm{WF}_{\mathrm{des}}(\pi)\) and \(\mathrm{WF}_\mathrm{wm}(\pi)\) for the boundary nilpotent orbits of the maximal ones.

In Section 1.2, we explain the indispensable roles played by this reciprocity of wavefront sets (Theorem 2) in our approach to the Wavefront Set Conjecture (Conjecture 1) in the archimedean case.

1.2 Main results over \({\mathbb{R}}\)↩︎

As indicated in our previous work ([1]), the arithmetic wavefront set \(\mathrm{WF}_{\mathrm{ari}}(\pi)\) for \(\pi\in\Pi_F(G)\) with a generic local \(L\)-parameter is defined by means of the consecutive descents of the enhanced local Langlands datum \((\varphi,\chi)\) associated with \(\pi\), based on the \(F\)-rational structure of the local Langlands correspondence and the local Gan-Gross-Prasad conjecture. The \(F\)-rational structure of \(\mathrm{WF}_{\mathrm{ari}}(\pi)\) is expected to be completely determined and explicitly computed by the arithmetic information of \((\varphi,\chi)\). This expectation was completely realized in [1] for all classical groups over the archimedean local fields. For later use, we record the following consequence of [1] and [1].

It is important to mention that over a non-archimedean local field \(F\), the \(F\)-rational structures of the arithmetic wavefront set \(\mathrm{WF}_{\mathrm{ari}}(\pi)\) can be explicitly computed for various lower rank cases, however, it is more complicated as indicated in [1], [24], [25] for instance. It is a subject matter of our ongoing project. We refer to [1] for some relevant discussions.

The objective of this paper is to prove a large part of Conjecture 1 (the Wavefront Set Conjecture) for all classical groups over archimedean local fields. When \(F={\mathbb{C}}\), the field of complex numbers, Conjecture 1 holds trivially, and hence for the archimedean case considered in this paper we assume that \(F={\mathbb{R}}\), the field of real numbers. Here is the main result of this paper (Theorem 28).

With Theorem 5, the analytic wavefront set \(\mathrm{WF}_{\mathrm{tr}}(\pi)^{r-\mathrm{max}}\) can be computed explicitly and effectively, and produces members of \(\mathrm{WF}_{\mathrm{tr}}(\pi)^{r-\mathrm{max}}\) through the arithmetic wavefront set \(\mathrm{WF}_{\mathrm{ari}}(\pi)^{r-\mathrm{max}}\) by means of the enhanced local \(L\)-parameter \((\varphi,\chi)\) of \(\pi\), for a large family of \(\pi\in\Pi_F(G)\), i.e. for \(\pi\) having a generic local \(L\)-parameter. Hence the microlocal structure of the Harish-Chandra distribution character \(\Theta_\pi\) can be completely determined by the arithmetic information associated with \(\pi\) via the local Langlands reciprocity. We strongly believe that with a full development of the compatibility theory of the local descents and the local Langlands conjecture, this arithmetic approach to determine the wavefront sets \(\mathrm{WF}_\mathrm{wm}(\pi)^{r-\mathrm{max}}\) and \(\mathrm{WF}_{\mathrm{tr}}(\pi)^{r-\mathrm{max}}\) could be extended to the general situation.

By the reciprocity of wavefront sets (Theorem 2), we obtain the Wavefront Set Conjecture (Conjecture 1) if we assume that Conjecture 3 holds.

Combining Theorems 4 and 5, we obtain an arithmetic refinement of Vogan’s maximal-orbit principle. In the generic real classical case considered here, the maximal analytic wavefront support is determined by the arithmetic wavefront set attached to the enhanced local \(L\)-parameter. In particular, in the nontrivial component-group case, the maximal analytic wavefront orbit is a single real rational nilpotent orbit, and this orbit is computed explicitly from the enhanced local \(L\)-parameter.

Let us also point out that this maximal-orbit principle is a feature of the real-group setting and should not be interpreted as a literal statement over all local fields. In the \(p\)-adic case, Tsai has shown that the geometric wavefront set need not be a singleton [24]. This does not contradict the orbit method philosophy; rather, it indicates that the \(p\)-adic form of the philosophy is more genuinely arithmetic. The leading asymptotic behavior of the Harish-Chandra distribution character \(\Theta_\pi\) for some \(\pi\in\Pi_F(G)\) is still governed by nilpotent orbital data, as in the local character expansion and in the relation with generalized Whittaker models, but the passage from rational nilpotent orbits to geometric nilpotent orbits need not collapse to a single maximal orbit. Thus the correct \(p\)-adic analogue of the orbit method should allow a finite set of maximal orbits, reflecting the \(p\)-adic nature of the rational structures associated with representations \(\pi\in\Pi_F(G)\). From this viewpoint, Tsai’s examples refine rather than refute the orbit method picture: they show that over non-archimedean fields the leading nilpotent support is orbital but not necessarily geometrically irreducible.

The idea of the proof of Theorem 5 can be briefly outlined as follows. As mentioned earlier, the general comparison between \(\mathrm{WF}_\mathrm{wm}(\pi)\) and \(\mathrm{WF}_{\mathrm{tr}}(\pi)\) in the archimedean case is a long-standing open problem. The work of W. Schmid and K. Vilonen ([33]) compares wavefront cycles with associated cycles, and thereby relates \(\mathrm{WF}_{\mathrm{tr}}(\pi)\) to the associated variety \({\rm AV}(\pi)\) of \(\pi\). The arithmetic wavefront set \(\mathrm{WF}_{\mathrm{ari}}(\pi)\) introduced in [1] builds up an important bridge connecting \(\mathrm{WF}_\mathrm{wm}(\pi)\) and \(\mathrm{WF}_{\mathrm{tr}}(\pi)\), which at the same time is also tied to \({\rm AV}(\pi)\). As we explained above, the arithmetic wavefront set \(\mathrm{WF}_{\mathrm{ari}}(\pi)\) introduced in [1] is defined by the consecutive \(L\)-parameter descents from the enhanced local \(L\)-parameter \((\varphi, \chi)\) of \(\pi\) and hence is an arithmetic object. In Definition 11, we introduce the descent wavefront set \(\mathrm{WF}_{\mathrm{des}}(\pi)\) of \(\pi\) on the representation side and prove in Theorem 12 the reciprocity of wavefront sets: \(\mathrm{WF}_{\mathrm{des}}(\pi)=\mathrm{WF}_{\mathrm{ari}}(\pi)\) holds for any \(\pi\in\Pi_F(G)\) with a generic \(L\)-parameter. Then we prove a composition law (Theorem 16) for generalized Whittaker models using a generalized root exchange lemma (Lemma 14), which implies that (Corollary 18) \[\label{subset} \mathrm{WF}_{\mathrm{ari}}(\pi)\subset \mathrm{WF}_\mathrm{wm}(\pi).\tag{5}\] We emphasize that the results in Section 2 and Section 3 hold for arbitrary local fields of characteristic zero, and also have applications for the wavefront sets in the non-archimedean case.

By comparing the algorithm for \(\mathrm{WF}_{\mathrm{ari}}(\pi)^{r-\mathrm{max}}\) in terms of enhanced \(L\)-parameters given in [1] with the algorithm for associated varieties of \(A_{\mathfrak {q}}(\lambda)\) modules in [70], we show that (Theorem 26) \[\label{AV} {\rm AV}(\pi) = \bigcup_{{\mathcal{O}}\in \mathrm{WF}_{\mathrm{ari}}(\pi)^{r-\mathrm{max}}}\overline{{\mathcal{O}}_{\mathfrak p}},\tag{6}\] where \({\mathcal{O}}_{\mathfrak p}\) denotes the image of \({\mathcal{O}}\) under the Kostant-Sekiguchi correspondence. Combined with the fundamental result of Schmid and Vilonen ([33]), it follows that (Corollary 27) \[\mathrm{WF}_{\mathrm{tr}}(\pi)^{r-\mathrm{max}}=\mathrm{WF}_{\mathrm{ari}}(\pi)^{r-\mathrm{max}}.\]

Let \({\mathcal{V}}({\rm Ann}\,\pi)\subset \frak g_{\mathbb{C}}\) be the associated variety of the annihilator ideal in \(U(\frak g_{\mathbb{C}})\) of \(\pi\), where \(\frak g_{\mathbb{C}}\cong {\mathfrak {g}}_{\mathbb{C}}^*\) is the complexification of \({\mathfrak {g}}\) with Cartan decomposition \(\frak g_{\mathbb{C}}=\frak k \oplus \frak p\). A result of H. Matumoto ([27]) shows that the members of \(\mathrm{WF}_\mathrm{wm}(\pi)\) all lie in \({\mathcal{V}}({\rm Ann}\,\pi)\). By [71], \({\mathcal{V}}({\rm Ann}\,\pi)\) is the Zariski closure of a single stable nilpotent orbit \({\mathcal{O}}^{\rm st}(\pi)\) and \[{\mathrm{AV}}(\pi) = \overline{{\mathcal{O}}^{\rm st}(\pi)\cap {\mathrm{AV}}(\pi)} \subset {\mathcal{V}}({\rm Ann}\,\pi) \cap \frak p.\] From above discussions and 5 , 6 , it follows that \[\mathrm{WF}_{\mathrm{ari}}(\pi)^{r-\mathrm{max}} \subset \mathrm{WF}_\mathrm{wm}(\pi)^{s-\mathrm{max}},\] which completes the proof of Theorem 5.

In summary, the strategy of the proof can be roughly depicted as follows: \[\xymatrix{ & \mathrm{WF}_{\mathrm{des}}(\pi)=\mathrm{WF}_{\mathrm{ari}}(\pi) \ar@{-->}[dl]_{\text{Composition Law \;}} \ar@{<->}[dd]^{\text{Algorithms}} \ar@{<-->}[dr] & \\ \mathrm{WF}_\mathrm{wm}(\pi) \ar[dr]_{\text{Matumoto, Vogan \;}} & & \mathrm{WF}_{\mathrm{tr}}(\pi) \\ & \mathrm{AV}(\pi) \ar@{<->}[ur]_{\text{ \;Schmid-Vilonen}} & }\]

This paper is organized as follows. In Section 2 we formulate the classification of nilpotent orbits and various versions of wavefront sets for classical groups. In Section 3 we present a generalized root exchange lemma and use it to prove a composition law for generalized Whittaker models of classical groups over any local field of characteristic zero. We explicate the arithmetic wavefront sets for real classical groups based on our previous work [1] in Section 4, and explain the local Langlands correspondence and associated varieties for discrete series in Section 5. The comparison between arithmetic wavefront sets and associated varieties is established in Section 6. Combining these results, we prove the main theorem in Section 7.

2 Wavefront Sets of Classical Groups↩︎

In this section, we formulate the classification of nilpotent orbits and various versions of wavefront sets for classical groups, in order to make the paper self-contained. Let \(F\) be a local field of characteristic zero as before. Let \(F(\boldsymbol{\delta})\) be a quadratic extension of \(F\), where \(\boldsymbol{\delta}=\sqrt{d}\) for a non-square element \(d\in F^\times\). Assume that \(E=F\) or \(F(\boldsymbol{\delta})\), and that \(V\) is an \(\epsilon\)-Hermitian vector space of dimension \({\mathfrak {n}}\) over \(E\), where \(\epsilon\in \{\pm 1\}\). Let \[{\mathbb{G}}(V):={\mathrm{Isom}}(V)^\circ, \quad (\text{\circ indicates the identity component)}\] and denote by \(G(V)\) the \(F\)-rational points of \({\mathbb{G}}(V)\).

In this paper, we assume that \(G\) equals either \(G(V)\) or its metaplectic double cover \({\mathrm{Mp}}(V)\) when \(E=F\) and \(\epsilon=-1\). The Lie algebra of \(G\) is denoted by \({\mathfrak {g}}={\mathfrak {g}}(V)\), and similar notation may be used without further explanation. Let \({\mathcal{N}}_F({\mathfrak {g}})\) be the set of all nilpotent elements in the Lie algebra \({\mathfrak {g}}\) and \({\mathcal{N}}_F({\mathfrak {g}})_\circ\) be the set of \(F\)-rational nilpotent orbits under the adjoint action of \(G\). For a nilpotent element \(f\in {\mathcal{N}}_F({\mathfrak {g}})\), write \({\mathcal{O}}_f\in {\mathcal{N}}_F({\mathfrak {g}})_\circ\) for the \(F\)-rational nilpotent orbit containing \(f\).

2.1 Rational nilpotent orbits↩︎

It is a classical theorem that there are only finitely many nilpotent orbits of \({\mathfrak {g}}\) over the algebraic closure \(\overline{F}\) of \(F\) by certain partitions determined by \({\mathfrak {g}}\) ([72]). For the \(F\)-rational nilpotent orbits in \({\mathfrak {g}}\), we follow [73] to parametrize the set \({\mathcal{N}}_F({\mathfrak {g}})_\circ\) by sesquilinear Young tableaux. For \(k\geq 1\), denote by \(S_k\) the \(k\)-dimensional irreducible \(\frak{sl}_2\)-module over \(E\), equipped with a fixed invariant \((-1)^{k-1}\)-Hermitian form. When \(k\) is odd, we require that the discriminant of this \((-1)^{k-1}\)-Hermitian form equals \(1\). Let \[\underline{p} = [p_1^{m_1}, \ldots, p_r^{m_r}]\] be a partition of \({\mathfrak {n}}=\dim_E V\), where \(p_1>\cdots>p_r>0\) and \(m_1,\ldots, m_r\geq 0\), subject to the parity conditions as in [72]. Let \(V_{p_i}\) be an \(m_i\)-dimensional \((-1)^{p_i-1}\epsilon\)-Hermitian space over \(E\), where \(i=1,\ldots, r\). The datum \(\left(\underline{p}, \{V_{p_i}\}\right)\) is called a sesquilinear Young tableau of sign \(\epsilon\). In particular if \(r=1\), the sesquilinear Young tableau is simply written as \[\label{onepart} ([p_1^{m_1}], \{V_{p_1}\}) = (p_1^{m_1}, V_{p_1}).\tag{7}\]

If the \(\epsilon\)-Hermitian vector space \(V\) of dimension \({\mathfrak {n}}\) over \(E\) has an isometry \[\label{isom} V\cong \bigoplus^r_{i=1} S_{p_i}\otimes V_{p_i}, \quad (\text{the orthogonal direct sum})\tag{8}\] then we say the sesquilinear Young tableau \((\underline{p}, \{V_{p_i}\})\) is admissible for \(V\). By [73], there is a one-to-one correspondence between the \(F\)-rational nilpotent orbits in \({\mathfrak {g}}\) and the equivalence classes of admissible sesquilinear Young tableaux for \(V\), except that when \(G\) is an even special orthogonal group, two \(F\)-rational nilpotent orbits may correspond to an admissible sesquilinear Young tableau with only even parts. See [1] for more details.²

If \(\underline{p}' = [p_1^{m_1'}, \ldots, p_r^{m_r'}]\) and \(\left(\underline{p}', \{V'_{p_i}\}\right)\) is an admissible sesquilinear Young tableau for another \(\epsilon\)-Hermitian space \(V'\) over \(E\), we define the direct sum \[\label{directsum} \left(\underline{p}, \{V_{p_i}\}\right) \oplus \left(\underline{p}', \{V'_{p_i}\}\right):= \left(\underline{p}\cup \underline{p}', \{V_{p_i}\oplus V'_{p_i}\}\right),\tag{9}\] where \(\underline{p}\cup \underline{p}' := [p_1^{m_1+m_1'},\ldots, p_r^{m_r+m_r'}]\). Clearly 9 is an admissible sesquilinear Young tableau for the \(\epsilon\)-Hermitian space \(V\oplus V'\) (the orthogonal direct sum). Thus the sesquilinear Young tableaux of the same sign form a monoid, and the following result is clear by construction.

We recall the topological orders on \({\mathcal{N}}_F({\mathfrak {g}})_\circ\) following [1].

2.2 Generalized Whittaker models and algebraic wavefront set↩︎

Fix a non-trivial additive character \(\psi_F\) of \(F\), and denote by \(\kappa\) the Killing form of \({\mathfrak {g}}\). Let \({\mathcal{O}}\) be an \(F\)-rational nilpotent orbit in \({\mathfrak {g}}\). Fix \(f\in {\mathcal{O}}\) and extend it to an \(\frak{sl}_2\)-triple \(\{h, e, f\}\) in \({\mathfrak {g}}\), where \(h=[e,f]\) is semisimple with \([h,e]=2e\) and \([h,f]=-2f\). When \({\mathcal{O}}=\{0\}\) it is understood that \(h=0\). For \(i\in \mathbb{Z}\), put \[\label{grade} {\mathfrak {g}}_i:=\set{x\in {\mathfrak {g}}| [h,x]=ix},\quad {\mathfrak {g}}_{\geq i}:=\bigoplus_{j\geq i} {\mathfrak {g}}_j\quad\text{and}\quad {\mathfrak {g}}_{\leq i}:=\bigoplus_{j\leq i}{\mathfrak {g}}_j.\tag{10}\] Then we have a parabolic subgroup \(P=MU\) of \(G\) with Levi subgroup \(M\) and unipotent radical \(U\), whose Lie algebras are \(\frak m={\mathfrak {g}}_0\) and \({\mathfrak {u}}= {\mathfrak {g}}_{\geq 1}\) respectively. Moreover, if we set \[V_{[i]}:=\set{v\in V | hv=iv},\] then we obtain that \[\label{levi} {\mathfrak {m}}= {\mathfrak {g}}(V_{[0]})\oplus \bigoplus_{i>0} \frak{gl}(V_{[i]}).\tag{11}\]

Define a character \(\psi_f\) of \(U_2:=\exp({\mathfrak {g}}_{\geq 2})\) by \[\psi_f(\exp(z)) := \psi_F(\kappa(f, z)),\quad z\in {\mathfrak {g}}_{\geq 2}.\] where \(\kappa(\cdot,\cdot)\) is the Killing form on \({\mathfrak {g}}\). Following [73], \[\kappa_f(x,y) := \kappa(f, [x,y]),\quad x, y\in {\mathfrak {g}}_1\] defines a symplectic form on \({\mathfrak {g}}_1\). Let \(\omega_{\psi_f}\) be the irreducible unitarizable smooth representation of \(U\) such that \(U_2\) acts by the character \(\psi_f\), which is unique up to isomorphism by the Stone-von Neumann theorem. Note that if \({\mathfrak {g}}_1=\{0\}\), then \(\omega_{\psi_f} = \psi_f\) is a character of \(U=U_2\).

When \({\mathfrak {g}}_1\neq\{0\}\), fix a Lagrangian subspace \({\mathfrak {l}}\) of \({\mathfrak {g}}_1\) with respect to the symplectic form \(\kappa_f\), and denote \(U_{1.5}:=\exp({\mathfrak {g}}_{\geq 1.5})\), where \({\mathfrak {g}}_{\geq 1.5}:={\mathfrak {l}}\oplus {\mathfrak {g}}_{\geq 2}\). Then \(\psi_f\) extends to a character of \(U_{1.5}\) by making it trivial on the \({\mathfrak {l}}\)-part. By [31] we have that \[\omega_{\psi_f}\cong {\mathrm{ind}}^U_{U_{1.5}}\psi_f.\] Here and thereafter, \({\mathrm{ind}}\) denotes the smooth compactly supported induction ([74]) when \(F\) is non-archimedean, and the Schwartz induction ([75]) when \(F\) is archimedean.

As before, denote by \(\Pi_F(G)\) the set of equivalence classes of irreducible admissible representations of \(G\), which are of the Casselman-Wallach type if \(F\) is archimedean and are genuine if \(G\) is metaplectic. For any \(\pi\in\Pi_F(G)\), define the generalized Whittaker model of \(\pi\) associated to \(f\in{\mathcal{O}}\) by \[\mathrm{Wh}_{\mathcal{O}}(\pi) :={\mathrm{Hom}}_{U}(\omega_{\psi_f}^\vee, \pi^*),\] where \(\pi^*\) is the strong dual of \(\pi\). By [31], \[\mathrm{Wh}_{\mathcal{O}}(\pi) \cong {\mathrm{Hom}}_{U_{1.5}}(\pi, \psi_f).\] Define the twisted Jacquet module \[\label{Jac} {\mathcal{J}}_{\mathcal{O}}(\pi):= \left(\pi\,\widehat\otimes\, \omega_{\psi_f}^\vee\right)_U \cong \left(\pi\otimes \psi_f^{-1}\right)_{U_{1.5}},\tag{12}\] where the subscripts indicate the maximal coinvariant Hausdorff quotients and the last isomorphism is given by [76]. Note that in the definition of \(\mathrm{Wh}_{\mathcal{O}}(\pi)\) and \({\mathcal{J}}_{\mathcal{O}}(\pi)\), we may not require that \(\pi\) is irreducible.

2.3 Descent and arithmetic wavefront sets↩︎

In [1] we introduced the arithmetic wavefront set \(\mathrm{WF}_{\mathrm{ari}}(\pi)\) for \(\pi\in \Pi_F(G)\) with a generic local \(L\)-parameter. It is defined by means of consecutive descents of enhanced \(L\)-parameter of \(\pi\), based on the local Langlands correspondence (established through [77]–[80]) and the local Gan-Gross-Prasad conjecture ([81]) (which is now a theorem thanks to the works of many authors). We refer to [1] for a list of relevant references.

In this paper, we introduce a new wavefront set \(\mathrm{WF}_{\mathrm{des}}(\pi)\) for each \(\pi\in \Pi_F(G)\) with a generic local \(L\)-parameter by means of the local descents of \(\pi\) and prove that \(\mathrm{WF}_{\mathrm{des}}(\pi)=\mathrm{WF}_{\mathrm{ari}}(\pi)\) for each \(\pi\in \Pi_F(G)\) with a generic local \(L\)-parameter (Theorem 12). The point is that \(\mathrm{WF}_{\mathrm{des}}(\pi)\) is more suitable for our proof of Theorem 5. To this end, we put \[{\mathcal{Z}}:=F^\times/ \set{ x\cdot c(x) | x\in E^\times},\] where \(c\) is the generator of \({\rm Gal}(E/F)\). Let \(0< p_1\leq {\mathfrak {n}}\) such that \(p_1\) is odd if \(G\) is special orthogonal and is even if \(G\) is symplectic or metaplectic, and finally there is no parity constraint on \(p_1\) if \(G\) is unitary. Suppose that \({\mathcal{O}}\) is an \(F\)-rational nilpotent orbit in \({\mathfrak {g}}\) corresponding to the partition \([p_1, 1^{{\mathfrak {n}}-p_1}]\). Then we have two cases.

• If \(p_1>1\) and \({\mathcal{O}}\) corresponds to the (unique) admissible sesquilinear Young tableau \[([p_1, 1^{{\mathfrak {n}}-p_1}], \{V_{p_1}, V_1\})\] such that the one-dimensional \((-1)^{p_1-1}\epsilon\)-Hermitian space \(V_{p_1}\) has discriminant \(a_1\in {\mathcal{Z}}\), we write \({\mathcal{O}}={\mathcal{O}}_{p_1, a_1}\). As in 8 , we may assume that \[V=V'\oplus V'':= S_{p_1}\otimes V_{p_1}\oplus S_1\otimes V_1\] under the action of the \(\frak{sl}_2\)-triple \(\{h, e, f\}\), where \(f\in {\mathcal{O}}_{p_1, a_1}\). Then \({\mathcal{J}}_{{\mathcal{O}}_{p_1, a_1}}(\pi)\) in 12 is a smooth (and nuclear Fréchet of moderate growth if \(F\) is archimedean) representation of \[H:=\begin{cases} G(V''), & \text{if G is not symplectic}, \\ {\mathrm{Mp}}(V''), & \text{if G is symplectic}, \end{cases}\] where in the case that \(G\) is unitary and \(p_1\) is even, we take a splitting \[G(V'') \hookrightarrow {\mathrm{Mp}}({\mathrm{Res}}_{E/F}(\boldsymbol{\delta}^{(\epsilon+1)/2}\cdot V''))\] as in [81]. Note that the Lie algebra of \(H\) is \({\mathfrak {h}}={\mathfrak {m}}^f\), where \({\mathfrak {m}}\) as in 11 is the Lie algebra of the corresponding Levi subgroup \(M\).

• If \(p_1=1\), then \({\mathcal{O}}=\{0\}\). In this case, if \(V'\) is an anisotropic line in \(V\) with discriminant \(a_1\in {\mathcal{Z}}\) and \(V=V'\oplus V''\) is the orthogonal decomposition, we write \[{\mathcal{J}}_{{\mathcal{O}}_{1,a_1}}(\pi):=\pi|_H,\quad \text{where}\quad H:= G(V'').\] In this case we define the sesquilinear Young tableau \((p_1, V_{p_1}):=(1, V')\).

It is clear that \((G,H)\) form a relevant pair in the setting of the local Gan-Gross-Prasad conjecture ([81]).

Note that if \(p_1=1\), the \(L\)-descent orbit is the zero nilpotent orbit. In view of the non-tempered Gan-Gross-Prasad conjecture ([82]), the above approach may be also applied to study the wavefront sets of Arthur type representations by relaxing the conditions [item:generic] and [item:decreasing] in Definition 11.

3 Composition Law for Generalized Whittaker Models↩︎

The results in this section hold for all local fields \(F\) of characteristic zero. We establish a root exchange lemma (Lemma 14) for general affine algebraic groups over \(F\), and as an application we prove a composition law (Theorem 16) for generalized Whittaker models of classical groups.

3.1 Root exchange lemma↩︎

In this section, we formulate a version of the so-called root exchange lemma ([83], [84]), which works for all local fields of characteristic zero following the arguments in [31], and then prove a generalization of it that will be a key technical ingredient in our proof of Theorem 5 and will be useful in future applications.

For a group \(G\), if \(x,y\in G\), write \([x,y]:=xyx^{-1}y^{-1}\) for their commutator.

Applying Lemma 13 repeatedly, we have the following useful generalization, a global analog of which can be found in [85].

3.2 Composition law↩︎

In this section, by using the root exchange lemma (Lemma 14), we prove a composition law for generalized Whittaker models of classical groups over any local field of characteristic zero. We mention that a special case of its global analog can be found in [86]. Our proof is Lie algebra theoretical, which avoids matrix computations and should be more conceptual.

Assume that \({\mathcal{O}}'\in {\mathcal{N}}_F({\mathfrak {g}})_\circ\) corresponds to a sesquilinear Young tableau of the form \[([p_1^{m_1},\ldots, p_r^{m_r}, 1^{{\mathfrak {n}}''}], \{V'_{p_i}, V'_1\}),\] where \(p_1> \cdots > p_r>1\), such that \[V = \bigoplus^r_{i=1} S_{p_i}\otimes V'_{p_i} \oplus S_1\otimes V'_1\] under the action of an \(\frak{sl}_2\)-triple \(\{h',e',f'\}\subset {\mathfrak {g}}\) with \(f'\in {\mathcal{O}}'\). Write \(\underline{p}:=[p_1^{m_1},\ldots, p_r^{m_r}]\). Then \({\mathfrak {n}}={\mathfrak {n}}'+{\mathfrak {n}}''\), where \({\mathfrak {n}}' =|\underline{p}|:= \sum^r_{i=1}m_i p_i\). Put \[V':= \bigoplus^r_{i=1} S_{p_i}\otimes V'_{p_i} \quad \text{and}\quad V'':=S_1\otimes V'_1.\] There is an involution \(\theta\) of \({\mathfrak {g}}\) such that \[\label{inv} {\mathfrak {g}}^\theta = {\mathfrak {g}}'\oplus {\mathfrak {g}}'':={\mathfrak {g}}(V')\oplus {\mathfrak {g}}(V''),\tag{14}\] and \[\label{inv-} {\mathfrak {g}}^{-\theta}\cong V'\otimes V''\text{ as (\frak g'\oplus \frak g'')-modules}.\tag{15}\]

Further assume that \({\mathcal{O}}'' \in {\mathcal{N}}_F({\mathfrak {g}}'')_\circ\) corresponds to a sesquilinear Young tableau \[(\underline{q}, \{V''_{q_j}\}),\quad \text{where}\quad \underline{q}=[q_1^{n_1}, \ldots, q_s^{n_s}], \;q_1>\cdots>q_s >0\] such that \[V''= \bigoplus^s_{j=1} S_{q_j}\otimes V''_{q_j}\] under the action of an \(\frak{sl}_2\)-triple \(\{h'', e'', f''\}\subset {\mathfrak {g}}''\) with \(f''\in{\mathcal{O}}''\).

The \(\frak{sl}_2\)-triples \(\{h,e,f\}, \{h',e',f'\}\) and \(\{h'',e'',f''\}\) are related as in the following lemma.

Now we impose the assumption that \[\label{pq} p_r \geq q_1>1.\tag{16}\] The main result of this section is the following composition law for generalized Whittaker models.

In the rest of this section, we prove Theorem 16 and give Corollary 18. Keep the notation in 10 and 14 and write \[{\mathfrak {g}}'_i:=\set{ x\in {\mathfrak {g}}| [h',x]=ix}\quad{\rm and}\quad {\mathfrak {g}}''_i:=\set{x\in {\mathfrak {g}}'' | [h'',x]=ix}, \quad {\rm for}\;i\in {\mathbb{Z}}.\] Similar to 10 , define \({\mathfrak {g}}'_{\geq i}, {\mathfrak {g}}'_{\leq i}, {\mathfrak {g}}''_{\geq i}\) and \({\mathfrak {g}}''_{\leq i}\) in the obvious way. We may fix an \(h''\)-stable Lagrangian subspace \({\mathfrak {l}}' \subset {\mathfrak {g}}'_1\) such that \({\mathfrak {g}}'_1\cap {\mathfrak {g}}_{\geq 2}\subset {\mathfrak {l}}'\), and also fix a Lagrangian subspace \({\mathfrak {l}}'' \subset {\mathfrak {g}}_1''\). Put \[{\mathfrak {g}}'_{\geq 1.5}:={\mathfrak {l}}' \oplus {\mathfrak {g}}'_{\geq 2},\qquad {\mathfrak {g}}''_{\geq 1.5}:={\mathfrak {l}}''\oplus {\mathfrak {g}}''_{\geq 2}.\]

We introduce \[\begin{align} {\mathfrak {c}}:&= {\mathfrak {g}}''_{\geq 1.5}\oplus ({\mathfrak {g}}_{\geq 2}\cap {\mathfrak {g}}'_{\geq 2})\oplus {\mathfrak {l}}' \\ & =\left ( ({\mathfrak {g}}'_{\geq 1.5}\oplus {\mathfrak {g}}''_{\geq 1.5})\cap {\mathfrak {g}}^\theta \right) \oplus \left({\mathfrak {g}}_{\geq 2}\cap {\mathfrak {g}}'_{\geq 2} \cap {\mathfrak {g}}^{-\theta}\right)\oplus {\mathfrak {l}}', \end{align}\] where \({\mathfrak {g}}^\theta\) and \({\mathfrak {g}}^{-\theta}\) are as in 14 and 15 respectively. For \(1\leq i\leq p_1-2\), define \[\begin{align} {\mathfrak {x}}_i:& ={\mathfrak {g}}_{2-i}\cap {\mathfrak {g}}'_{\geq 2}\cap {\mathfrak {g}}^{-\theta}, \\ {\mathfrak {y}}_i:&={\mathfrak {g}}_{i}\cap{\mathfrak {g}}'_{\leq 0}\cap{\mathfrak {g}}^{-\theta}, \\ {\mathfrak {c}}_i:& =\bigoplus_{k=1}^{i-1}{\mathfrak {x}}_k\oplus {\mathfrak {c}}\oplus \bigoplus_{k=i+1}^{p_1-2}{\mathfrak {y}}_k. \end{align}\] Put \(C= \exp({\mathfrak {c}})\), \(X_i =\exp({\mathfrak {x}}_i)\), \(Y_i=\exp({\mathfrak {y}}_i)\), and \(\psi_C:=\psi_f|_C\).

We shall verify that \(C, X_i\), \(Y_i\) and \(\psi_C\) satisfy the conditions in Lemma 14. It is straightforward to verify condition [RE1]. By the above definitions, for \(1\leq i\leq p_1-2\) we have \[\begin{align} & [{\mathfrak {x}}_i,{\mathfrak {y}}_j]\subset {\mathfrak {g}}_{j-i+2}\cap {\mathfrak {g}}^\theta \subset {\mathfrak {c}}\qquad \text{ if } 1\leq i\leq j\leq p_1-2,\\ & [{\mathfrak {x}}_i, {\mathfrak {c}}]=[{\mathfrak {x}}_i, ({\mathfrak {g}}'_{\geq 1.5}\oplus {\mathfrak {g}}''_{\geq 1.5})\cap {\mathfrak {g}}^\theta]\subset {\mathfrak {g}}_{\geq 3-i}\cap {\mathfrak {g}}'_{\geq 2}\cap {\mathfrak {g}}^{-\theta} \subset {\mathfrak {c}}\oplus \bigoplus_{k<i}{\mathfrak {x}}_k, \\ &[{\mathfrak {y}}_i, {\mathfrak {c}}]\subset \left({\mathfrak {g}}_{\geq i+1}\cap {\mathfrak {g}}^{-\theta}\right)\oplus \left({\mathfrak {g}}_{\geq i+2}\cap {\mathfrak {g}}^\theta\right) \subset {\mathfrak {c}}\oplus \bigoplus_{k>i}{\mathfrak {y}}_k. \end{align}\] This validates conditions [RE2], [RE3] and [RE4].

For a subspace \({\mathfrak {s}}\subset {\mathfrak {g}}\), denote by \({}^c {\mathfrak {s}}\) the image of \({\mathfrak {s}}\) under the Chevalley involution on \({\mathfrak {g}}\). Then condition [RE5] is immediate from the following lemma.

It is easy to see that \[{\mathfrak {c}}\oplus \bigoplus^{p_1-2}_{i=1} {\mathfrak {x}}_i = {\mathfrak {g}}'_{\geq 1.5}\oplus {\mathfrak {g}}''_{\geq 1.5} \quad \text{and} \quad {\mathfrak {c}}\oplus \bigoplus^{p_1-2}_{i=1}{\mathfrak {y}}_i = {\mathfrak {l}}\oplus {\mathfrak {g}}_{\geq 2}=:{\mathfrak {g}}_{\geq 1.5}\] for a Lagrangian subspace \({\mathfrak {l}}\subset {\mathfrak {g}}_1\). Applying the root exchange Lemma 14, we obtain that \[\mathrm{Wh}_{{\mathcal{O}}''}({\mathcal{J}}_{{\mathcal{O}}'}(\pi))\cong {\mathrm{Hom}}_{U'_{1.5}U''_{1.5}}(\pi, \psi_{f'}\psi_{f''})\cong {\mathrm{Hom}}_{U_{1.5}}(\pi, \psi_f)\cong \mathrm{Wh}_{\mathcal{O}}(\pi),\] where \(U'_{1.5} :=\exp({\mathfrak {g}}'_{\geq 1.5})\) and \(U''_{1.5}:=\exp({\mathfrak {g}}''_{\geq 1.5})\). This proves Theorem 16.

By induction, we have the following consequence of Theorem 16.

4 Arithmetic Wavefront Set: the Archimedean Case↩︎

We consider real classical groups in the rest of this paper. In this section, we recall the algorithm of [1] that computes the maximal elements \({\rm WF}_{\rm ari}(\pi)^{\max}\) in terms of enhanced \(L\)-parameters. We refer to loc. cit. for more details.

4.1 Enhanced \(L\)-parameters↩︎

Let \(E ={\mathbb{R}}\) or \({\mathbb{C}}\), and write \({\rm Gal}(E/\mathbb{R})=\langle c \rangle\). Let \(V\) be an \({\mathfrak {n}}\)-dimensional \(\epsilon\)-hermitian vector space over \(E\) with \(\epsilon = \pm 1\), i.e. \(V\) is equipped with a non-degenerate pairing \[\langle\, ,\,\rangle: V\times V\to E\] that is linear in the first variable, \(c\)-linear in the second variable, and satisfies \[\langle y, x\rangle = \epsilon\cdot c ( \langle x, y\rangle), \quad x, y\in V.\]

Let \(G\) be the group of real points of the connected algebraic group \({\rm Isom}(V)^\circ\), or the metaplectic double cover of \({\mathrm{Sp}}(V)\) in the case that \(E=\mathbb{R}\) and \(\epsilon=-1\). Then \[\label{cg} G={\mathrm {U}}(p,q),\quad {\mathrm{SO}}(p,q),\quad {\rm Sp}_{2n}(\mathbb{R}) \quad \textrm{or}\quad {\rm Mp}_{2n}(\mathbb{R}).\tag{19}\] Fix a quasi-split pure inner form \(G^*\) of \(G\). Note that \(G^*\) is unique unless \(G ={\mathrm {U}}(p,q)\) with \(p+q=2n+1\), or \(G={\mathrm{SO}}(p,q)\) with \(p+q=2n\) and \(p\equiv n+1 ({\rm mod}\;2)\). Set \(\frak n = \dim_E V (=p+q \textrm{ or }2n)\). In the first two cases of 19 , we also write \(G={\mathrm {U}}_{\frak n}\) or \({\mathrm{SO}}_{\frak n}\) for convenience and for the consistency of notation with that in [1]. Define \[\label{disc} {\rm disc}(V):=(-1)^{\frac{(p-q)(p-q-1)}{2}} \in \{\pm 1\}.\tag{20}\]

The \(L\)-group of \(G\) is \({}^LG=G^\vee\rtimes W_{\mathbb{R}}\), where \(G^\vee\) is the complex dual group of \(G\) and \(W_\mathbb{R}\) is the real Weil group. An \(L\)-parameter for \(G\) is an admissible homomorphism \(W_{\mathbb{R}}\to {}^LG\). Following [1], [81], we realize \(L\)-parameters as \(G\)-relevant conjugate self-dual semisimple representations of \(W_E\).

When \(E=\mathbb{C}\), for \(\alpha\in {\mathbb{Z}}\) define the character \[\varsigma_\alpha: W_{\mathbb{C}}= {\mathbb{C}}^\times \to {\mathbb{C}}^\times, \quad z\mapsto \left(|z|/z \right)^\alpha.\] When \(E=\mathbb{R}\), for \(\alpha \in {\mathbb{Z}}\) define the two-dimensional representation \(\sigma_\alpha:={\rm Ind}^{W_\mathbb{R}}_{W_\mathbb{C}}\,\varsigma_\alpha\) of \(W_\mathbb{R}\). Then \(\sigma_\alpha\cong \sigma_{-\alpha}\) is irreducible when \(\alpha\neq 0\), and \(\sigma_0 \cong \mathbb{1}\oplus {\mathrm{sgn}},\) where \(\mathbb{1}\) and \({\rm sgn}\) denote the trivial and sign characters of \(\mathbb{R}^\times\) respectively, viewed as characters of \(W_\mathbb{R}\) via local class field theory. Set \[\rho_\alpha:=\begin{cases} \varsigma_\alpha, & \textrm{if }E=\mathbb{C}, \\ \sigma_\alpha, & \textrm{if }E=\mathbb{R}. \end{cases}\]

A generic \(L\)-parameter of \(G^*\) can be viewed as a conjugate self-dual semisimple representation of \(W_E\) of the form \[\label{varphi} \varphi:=\tau \oplus \varphi_{\rm gp} \oplus {}^c\tau^\vee\tag{21}\] where \({}^c\tau^\vee\) is the conjugate dual of \(\tau\), and \(\varphi_{\rm gp}\) is the direct sum of all irreducible conjugate self-dual summands of \(\varphi\) having the same parity as \(\varphi\).

More precisely, \[\label{gp} \varphi_{\rm gp} = m_1\rho_{\alpha_1}\oplus m_2 \rho_{\alpha_2}\oplus \cdots\oplus m_r\rho_{\alpha_r} \oplus \varphi_{\rm quad},\tag{22}\] where

• \(m_1, \ldots, m_r\) are positive integers;

• \(\alpha_1<\alpha_2<\cdots <\alpha_r\) are integers satisfying the following conditions: they are nonnegative when \(E=\mathbb{R}\); even when \(G^\vee\) is orthogonal; odd when \(G^\vee\) is symplectic, and of the same parity as \({\frak n}-1\) when \(G^* ={\mathrm {U}}_{\frak n}\);

• \(\varphi_{\rm quad} = {\mathrm{sgn}}^{m_1+\cdots+m_r}\) when \(G^* ={\rm Sp}_{2n}(\mathbb{R})\) and zero otherwise;

• \(\det\varphi(-1) = {\rm disc}(V)\) when \(G^*={\mathrm{SO}}_{2n}\);

• \(\dim \varphi\) is the dimension of the natural representation of \(G^\vee\).

For convenience, in the unitary group case, if we drop the assumption that \(\varphi\) has parity \((-1)^{{\frak n}-1}\) in the above definition, then we refer to \(\varphi\) as a modified \(L\)-parameter.

We may assume that \(\tau\) in 21 is of the form \[\label{tau} \tau = |\cdot |_E^{s_1} \tau_1 \oplus \cdots \oplus |\cdot |^{s_l}_E \tau_l,\tag{23}\] where \(\tau_1,\ldots, \tau_l\) are irreducible and bounded, and \(s_1\geq \cdots \geq s_l\geq 0\).

Associated to \(\varphi\) is the component group \({\mathcal{S}}_\varphi \cong {\mathcal{S}}_{\varphi_{\rm gp}}\cong {\mathbb{Z}}_2^r\), and an enhanced \(L\)-parameter is a pair \((\varphi,\chi)\) where \(\chi\in \widehat{{\mathcal{S}}_\varphi}\). Thus \(\chi\) is determined by a sequence of signs \[\label{eq:chi-alpha} \chi(\alpha_1), \ldots, \chi(\alpha_r).\tag{24}\] Fixing the Whittaker datum of \(G^*\) as in [87], the local Langlands correspondence established through [77]–[80] yields a bijection \[\widehat{{\mathcal{S}}_\varphi} \xrightarrow{\;\sim \;}\Pi_\varphi[G^*], \qquad \chi \longmapsto \pi(\varphi,\chi),\] where the Vogan packet \(\Pi_\varphi[G^*]\) of \(\varphi\) is a disjoint union of irreducible representations of pure inner forms of \(G^*\).

4.2 Arithmetic wavefront set↩︎

For the proof of the main result, we will reduce to the case that \(\pi\) is a discrete series representation; see Section 6. To keep notation minimal, we only explicate \({\rm WF}_{\rm ari}(\pi)^{\max}\) for the discrete \(L\)-parameters introduced in [1]. We refer to [1] for the complete definition of \({\rm WF}_{\rm ari}(\pi)\).

Assume that \(\varphi\) is a discrete \(L\)-parameter of \(G\), so that \(\varphi = \varphi_{\rm gp}\) and is multiplicity-free, that is, \[\label{eq:varphi-discrete} \varphi = \rho_{\alpha_1} \oplus \cdots \oplus \rho_{\alpha_n} \oplus \varphi_{\rm quad},\tag{25}\] where \(\alpha_1<\cdots <\alpha_n\) and \[n := \left\lfloor \dfrac{\frak n}{[{\mathbb{C}}: E]}\right\rfloor \quad (\text{\lfloor\cdot\rfloor denotes the integer part and {\frak n}=\dim_{E}V}),\] so that \(n\) is the rank of \(G^\vee\). In particular \({\mathcal{S}}_\varphi \cong {\mathbb{Z}}_2^n\).

In this case \(G\) is of equal rank (see [88]), excluding the pure inner class of \({\mathrm{SO}}(p,q)\) with \(p,q\) odd. For a discrete series \(\pi=\pi(\varphi,\chi)\) of \(G\), the set \({\rm WF}_{\rm ari}(\pi)^{\max}\) is a singleton consisting of a real nilpotent orbit \({\mathcal{O}}(\varphi,\chi)\) parametrized by a signed Young tableau. Following [1], we write \[\label{eq:awf} {\mathcal{O}}(\varphi,\chi)=((p_1,\epsilon_1),\dots,(p_k,\epsilon_k)),\tag{26}\] where \(p_1\ge \cdots \ge p_k\) is a partition of \(\frak n\) and \(\epsilon_i\in\{\pm\}\), subject to:

• \(p_i\)’s are odd if \(G= {\mathrm{SO}}(p,q)\) and even if \(G = {\rm Sp}_{2n}({\mathbb{R}})\) or \({\rm Mp}_{2n}({\mathbb{R}})\);

• each row \((p_i, \epsilon_i)\) consists of \(p_i\) alternating signs ending with \(\epsilon_i\);

• there are \(p\) pluses and \(q\) minuses in \({\mathcal{O}}(\varphi,\chi)\) if \(G={\mathrm {U}}(p, q)\) or \({\mathrm{SO}}(p,q)\).

The algorithm of [1] computes \({\mathcal{O}}(\varphi,\chi)\) inductively via consecutive descents of enhanced \(L\)-parameters. Motivated by the local Gan–Gross–Prasad conjecture (see [81]), we introduced the first descent of \((\varphi,\chi)\), denoted \({\mathcal{D}}_{p_1}(\varphi,\chi)\), which is a set of enhanced discrete (modified) \(L\)-parameters \((\phi,\chi')\) of opposite parity ([1] and [69]). The obtained set \({\mathcal{D}}_{p_1}(\varphi,\chi)\) can be explicated as follows.

We first define the sign alternating set \({\rm sgn}(\chi)\). Define \[\label{sgn} {\rm sgn}(\chi):= \begin{cases} \set{ 1\leq i <n | \chi(\alpha_{i+1}) = - \chi(\alpha_i)}, & \textrm{if }G^* \neq {\mathrm{SO}}_{2n+1}, \\ \set{ 0\leq i <n | \chi(\alpha_{i+1}) = - \chi(\alpha_i)}, & \textrm{if }G^* = {\mathrm{SO}}_{2n+1}, \end{cases}\tag{27}\] where in the case \(G^*={\mathrm{SO}}_{2n+1}\) we set \(\alpha_0:=-1\) and \(\chi(\alpha_0):=1\). Write \[s_\chi:= \#{\mathrm{sgn}}(\chi),\] and \[{\mathrm{sgn}}(\chi)= \set{ i_1< i_2< \cdots < i_{s_\chi}}.\] For convenience, we record the first row \((p_1,\epsilon_1)\) of \({\mathcal{O}}(\varphi,\chi)\) and the elements \((\phi,\chi')\in {\mathcal{D}}_{p_1}(\varphi,\chi)\) for each group type in the following form. Define \[\label{phi} \phi = \begin{cases} \rho_{\beta_1}\oplus \cdots \oplus \rho_{\beta_{s_\chi}} &\text{ if } G^*\ne {\mathrm{Mp}}_{2n}({\mathbb{R}}),\\ \rho_{\beta_1}\oplus \cdots \oplus \rho_{\beta_{s_\chi}} \oplus {\mathrm{sgn}}^{s_\chi} &\text{ if } G^*= {\mathrm{Mp}}_{2n}({\mathbb{R}}), \end{cases}\tag{28}\] with \(\alpha_{i_j}<\beta_j <\alpha_{i_j+1}\) for all \(1\leq j\leq s_\chi\). The signs in \(\chi'\) are the values \(\chi'(\beta_j)\), arranged analogously to 24 . The relevant data for the first descent are summarized as follows:

Here \(\chi_n:=\chi(\alpha_n)\) and the last column \(\chi'\) is called the \(L\)-descent of \(\chi\). In addition, to parametrize the pure inner class when \(G^*={\mathrm{SO}}_{2n+1}\), we introduce the auxiliary sign \[\label{delta0} \delta_0:= (-1)^n{\rm disc}(V) \qquad (\text{see }\eqref{disc}).\tag{29}\] The auxiliary sign for \({\mathrm{SO}}_{2s_\chi+1}\), the descent of \(G^*={\mathrm{SO}}_{2n}\), is given by \[\delta_0' =(-1)^{n-s_\chi-1}\chi_n.\]

Next, we apply the same procedure to \((\phi,\chi')\) to produce the second row \((p_2,\epsilon_2)\), which is independent of the choice of the \(\beta_j\) in 28 . Finally, by induction, we obtain the signed Young tableau \[{\mathcal{O}}(\varphi,\chi)=((p_1,\epsilon_1),(p_2,\epsilon_2),\dots,(p_k,\epsilon_k)).\] More detailed discussions on our algorithm will be given in the next section via examples.

4.3 Examples of \(L\)-parameter descent↩︎

To illustrate the algorithm in Section 4.2, we give examples for unitary and orthogonal groups. More examples can be executed on the website [89].

4.3.1 Unitary groups↩︎

We first carry out the inductive algorithm for a discrete enhanced \(L\)-parameter \((\varphi,\chi)\) for \({\mathrm {U}}(10,8)\). Following the notation in 22 and 24 , write \[\varphi=\rho_{\alpha_1} \oplus \cdots \oplus \rho_{\alpha_{18}}\] where \(\alpha_1<\alpha_2<\cdots<\alpha_{18}\) are odd integers, and the character \(\chi\) of \({\mathcal{S}}_{\varphi}\) as \[-+-+--+-+--+-++---.\] First, we compute the first descent \({\mathcal{D}}_{p_1}(\varphi,\chi)\) of \((\varphi,\chi)\) with \(p_1=6\). By 27 , we have \[{\rm sgn}(\chi)=\{1,2,3,4,6,7,8,9,11,12,13,15\}.\] By Table 1, \({\mathcal{D}}_{p_1}(\varphi,\chi)\) consists of modified \(L\)-parameters \((\phi,\chi')\) such that \(\phi = \rho_{\beta_1}\oplus \cdots \oplus \rho_{\beta_{12}}\), where \(\beta_j\) are even integers for all \(1\leq j\leq 12\) satisfying that \[\begin{align} & \alpha_1<\beta_1<\alpha_2<\beta_2<\alpha_3<\beta_3<\alpha_4<\beta_4<\alpha_5<\alpha_6<\beta_5<\alpha_7<\beta_6<\alpha_8 \\ <\, &\beta_7<\alpha_9<\beta_8<\alpha_{10}<\alpha_{11}<\beta_9<\alpha_{12}<\beta_{10}<\alpha_{13}<\beta_{11}<\alpha_{14}<\alpha_{15}<\beta_{12}<\alpha_{16}, \end{align}\] and the character \(\chi'\) of \({\mathcal{S}}_\phi\) is given by \[-+-++-+--+--.\] It is easy to note that the cardinality \(\#{\mathcal{D}}_{p_1}(\varphi,\chi)\) is \(\prod_{j\in{\mathrm{sgn}}(\chi)} \left(\frac{\alpha_{j+1}-\alpha_j}{2}\right)\).

Continuing the process, we obtain the following table of consecutive descents: \[{ \setlength{\arraycolsep}{6pt} \renewcommand{1.5}{1.2} \begin{array}{@{}lcl@{}} \chi & \epsilon_1 & p \\ \hline -+-+--+-+--+-++--- & & \\ -+-++-+--+-- & - & 6 = 18-12 \\ -+--+--+ & - & 4 = 12-8 \\ +--++ & + & 3 = 8-5 \\ -- & + & 3 = 5-2 \\ & - & 2 \end{array} }\] In this table, the third column gives the row length \(p_i\) of the signed Young diagram \({\mathcal{O}}(\varphi,\chi)\); the second column gives the sign \(\epsilon_i\), which equals the rightmost sign of the character being descended at each step; and the first column records the successive descended characters.

An alternative presentation is the following triangular array of signs: \[{ \setlength{\arraycolsep}{2pt} \renewcommand{1.5}{1.15} \scriptsize \begin{array}{@{}cccc cccc cccc cccc cccc ccccc@{}} -&&+&&-&&+&&--&&+&&-&&+&&--&&+&&-&&++&&---\\ &-&&+&&-&&+&&+&&-&&+&&-&&-&&+&&-&&-&\\ &&-&&+&&-&&&&-&&+&&-&&&&-&&+&&&&\\ &&&+&&-&&&&&&-&&+&&&&&&+&&&&&\\ &&&&-&&&&&&&&-&&&&&&&&&&&&\\ \end{array} }\] From Table 1, we read off the signed Young tableau \({\mathcal{O}}(\varphi,\chi)\) as \[\begin{ytableau} +&- & + & - & + & - \\ + & - & + & - \\ + & - & + \\ + & - & + \\ + & - \\ \end{ytableau}\]

4.3.2 Special orthogonal groups↩︎

Since the descent process for special orthogonal groups involves an auxiliary sign, we include one example to clarify this feature. Consider a discrete enhanced \(L\)-parameter \((\varphi,\chi)\) for \({\mathrm{SO}}(16,15)\) with character \[-++-+-++-+--+++.\] Since the group is \({\mathrm{SO}}(16,15)\), we have \({\mathrm{disc}}(V)=1\), hence the auxiliary sign in 29 equals \(\delta_0=-1\). If instead \(\delta_0=1\), then this corresponds to \({\mathrm{SO}}(15,16)\), which is isomorphic to \({\mathrm{SO}}(16,15)\) but yields a different signed Young tableau.

We compute \({\mathrm{sgn}}(\chi)=\{0,1,3,4,5,6,8,9,10,12\}\) with \(p_1=11\) and \(\epsilon_1=+\). Then the character \(\chi'\) of elements in the first descent is \[+--+-++-++.\] Proceeding as above, we obtain the table of consecutive descents \[{ \setlength{\arraycolsep}{6pt} \renewcommand{1.5}{1.2} \begin{array}{@{}lccc@{}} \chi & \epsilon_1 & \delta_0 & p \\ \hline -++-+-++-+--+++ & & - & \\ +--+-++-++ & + & & 11 \\ --+--+ & + & - & 7 \\ ++-- & - & & 5 \\ + & - & - & 5 \\ & + & & 3 \end{array} }\] and the signed Young diagram \({\mathcal{O}}(\varphi,\chi)\) \[\begin{ytableau} +&- & + & - & + & - & + & - & + & - & + \\ +&- & + & - & + & - & + \\ - & + & - & + & - \\ - & + & - & + & - \\ + & - & +\\ \end{ytableau}\]

5 Explicit Local Langlands Correspondence↩︎

In this section, we give the correspondence between discrete enhanced \(L\)-parameters and discrete series representations following [88] and review the algorithm for the associated varieties of these modules following [70]. From now on, we fix a choice \(\mathrm{i}=\sqrt{-1}\), the square root of \(-1\).

5.1 Pure inner form and Vogan packet↩︎

Let \(G\) be as in 19 , with Lie algebra \(\frak{g}\); write \({\mathcal{G}}\) and \(\frak g_{\mathbb{C}}\) for their complexifications respectively. Let \(Z_{\mathcal{G}}\) be the center of \({\mathcal{G}}\). Fix an antiholomorphic involution \(\varsigma\) on \({\mathcal{G}}\) such that \({\mathcal{G}}^\varsigma\) is a compact real form of \({\mathcal{G}}\). If \(\theta\) is an algebraic involution on \({\mathcal{G}}\) commuting with \(\varsigma\), then \({\mathcal{G}}^{\varsigma\circ \theta}\) is a real form of \({\mathcal{G}}\) with \(\theta\) as its Cartan involution.

Fix a maximal \(\varsigma\)-stable torus \(T\) of \({\mathcal{G}}\). The Weyl group of \({\mathcal{G}}^\varsigma\) (and of \({\mathcal{G}}\)) with respect to the Cartan subgroup \(T^\varsigma\) is \(W := N_{{\mathcal{G}}^\varsigma}(T^\varsigma)/T^{\varsigma}\). Denote by \((X^*(T),R, X_*(T),R^\vee)\) the root datum of \({\mathcal{G}}\). Take \(G^\vee\) to be the complex dual group of \(G\), defined by the dual root datum \((X_*(T),R^\vee,X^*(T),R)\) when \(G\) is linear, and the complex symplectic group when \(G\) is the metaplectic group. Let \(T^\vee\) be the dual torus of \(T\) in \(G^\vee\) so that \(X^*(T^\vee)=X_*(T)\) and \(X_*(T^\vee)=X^*(T)\).

The exponential map \[\label{eq:exp} \exp: X_*(T)\otimes_{{\mathbb{Z}}} {\mathbb{C}}\to T({\mathbb{C}})\tag{30}\] sends \(\chi \otimes_{{\mathbb{Z}}} a \mapsto \chi(\exp(a))\), where \((\chi : {\mathbb{C}}^\times \to T) \in X_*(T)\) and \(a\in {\mathbb{C}}\).

Fix a Borel subgroup \(B^\vee\) of \(G^\vee\) containing \(T^\vee\), and let \(\rho^\vee \in X^*(T^\vee)\otimes_{\mathbb{Z}}{\mathbb{R}}= X_*(T)\otimes_{\mathbb{Z}}{\mathbb{R}}\) be the half-sum of the positive roots in \(B^\vee\), and set \[x_b := \exp(\pi \mathrm{i}\rho^\vee)\in T({\mathbb{C}})\] as the “base point” for what follows. Since \(z_b := x_b^2 \in Z_{\mathcal{G}}\), the conjugation by \(x_b\) defines an involution on \({\mathcal{G}}\). In fact, the conjugation by \(x_b\) is a Cartan involution of the quasi-split real form \({\mathcal{G}}^{\varsigma \circ \mathrm{Inn}(x_b)}\) of \({\mathcal{G}}\).

Define \[\label{eq:CX} {\mathcal{X}}:= \set{x\in T | x^2 = z_b},\tag{31}\] which carries a natural \(W\)-action by conjugation. Because \(z_b^2 = 1\), the set \({\mathcal{X}}\) is contained in the maximal compact subgroup of \(T\), namely \(T^\varsigma\).

Suppose \(x\) and \(x'\) lie in the same \(W\)-conjugacy class. Choose \(\dot{w} \in N_{{\mathcal{G}}^\varsigma}(T)\) with \({}^{\dot{w}} x = x'\). Then conjugation by \(\dot{w}\) induces an isomorphism of real reductive groups \[\label{eq:iota1} \iota_{x,x'} := \mathrm{Inn}(\dot{w}) : {\mathcal{G}}^{\varsigma \circ \mathrm{Inn}(x)} \longrightarrow {\mathcal{G}}^{\varsigma \circ \mathrm{Inn}(x')}.\tag{32}\] If \(\dot{w}'\) is another element in \(N_{{\mathcal{G}}^\varsigma}(T)\) satisfying \({}^{\dot{w}'} x = x'\), then \[\label{eq:conj-in-K} \dot{w} ^{-1} \dot{w}' \in {\mathcal{G}}^\varsigma \cap {\mathcal{G}}^{\varsigma \circ \mathrm{Inn}(x')}.\tag{33}\] For linear groups we set \[G_{x} := {\mathcal{G}}^{\varsigma \circ \mathrm{Inn}(x)} \text{ for } x \in {\mathcal{X}}\quad \text{ and } \quad G^* := G_{x_b}.\]

Turning to the metaplectic case: let \(G\) be the metaplectic double cover of \({\mathcal{G}}^{\varsigma \circ \mathrm{Inn}(x_b)}\) (with projection map \(\mathrm{pr}\)). Since \(z_b = -1\), we have that \({\mathcal{X}}\) is a single \(W\)-orbit. For \(x\in {\mathcal{X}}\), fix a lift \(\dot{w}_x \in N_{{\mathcal{G}}^\varsigma}(T)\) such that \({}^{\dot{w}_x} x_b = x\), and define \(G_{x}\) by the cartesian diagram \[\label{eq:iota2} \begin{tikzcd} G_{x} \ar[r,"\iota_{x,x_b}"] \ar[d]& G_{x_b} \ar[r,"\cong"]\ar[d,"\mathrm{pr}"]& {\mathrm{Mp}}_{2n}({\mathbb{R}}) \\ {\mathcal{G}}^{\varsigma \circ \mathrm{Inn}(x)} \ar[r, "\mathrm{Inn}(\dot{w}_x)"] & {\mathcal{G}}^{\varsigma \circ \mathrm{Inn}(x_b)} \\ \end{tikzcd}\tag{34}\]

In all cases, the pure inner forms (also known as strong inner forms) of \(G\) are classified by the \(W\)-conjugacy classes of elements in \({\mathcal{X}}\) (see [90] and [88]). Let \[\label{eq:strong-inner-forms} \set{x_1,\cdots, x_k}\tag{35}\] be fixed representatives of the \(W\)-conjugacy classes of elements in \({\mathcal{X}}\). Then \[\Set{G_{x_i} | i=1,2,\cdots, k}\] is the set of all pure inner forms of \(G\) (up to isomorphism).

The Vogan packets are defined as follows. For \(x\in {\mathcal{X}}\), let \(\Pi_{\varphi}(G_x)\) be the \(L\)-packet consisting of the elements in \(\Pi_{\mathbb{R}}(G_x)\) with \(L\)-parameter \(\varphi\); we will define this precisely for discrete series parameters in Section 5.3.

Suppose \(x\) and \(x'\) are in the same \(W\)-conjugacy class. Let \(\dot{w} \in N_{{\mathcal{G}}^\varsigma}(T)\) be such that \({}^{\dot{w}} x = x'\) as before, and let \(\iota_{x,x'}\) be defined by 32 and its natural extension to the metaplectic case via 34 . We say \[(x,\pi) \sim (x',\pi') \text{ if and only if } \pi \cong \pi' \circ \iota_{x,x'}.\] Thanks to 33 , the definition of \(\sim\) is independent of the choice of \(\dot{w}\). As usual, we denote the equivalence class of \((x,\pi)\) under the relation \(\sim\) by \([x,\pi]\).

The Vogan packet \(\Pi_\varphi[G^*]\) is defined to be \[\label{Vp} \Pi_\varphi[G^*] := \bigsqcup_{i=1}^k \Pi_{\varphi}(G_{x_i})\tag{36}\] for a fixed choice of the representatives 35 . On the other hand, it is more convenient to work symmetrically, following the Atlas convention. Under the equivalence relation \(\sim\), we have the following natural identification \[\Pi_\varphi[G^*] = \left.\bigsqcup_{x\in {\mathcal{X}}} \Pi_{\varphi}(G_x)\right\slash \sim.\]

5.2 Discrete series representations↩︎

When \(G\) is linear, consider the element \[z^\vee := \exp(2\pi \mathrm{i}\rho)\] in \(Z^\vee\) (the center of \(G^\vee\)) where \(\exp\) is defined in the same spirit as 30 and \(\rho\) is the half-sum of the positive roots for a choice of positive root system of \({\mathcal{G}}\). The element \(z^\vee\) is independent of any choice. It is given by the following formula in our case: \[\label{eq:z-vee} z^\vee = \begin{cases} (-1)^{n-1} & \text{if G^\vee is a general linear group}, \\ -1 & \text{if G^\vee is a symplectic group}, \\ 1 & \text{if G^\vee is an orthogonal group}. \end{cases}\tag{37}\] For the metaplectic group, we define \(z^\vee = -1\) by convention.

Recall that an element \(\lambda\) in \(X^*(T)\otimes {\mathbb{R}}\) is called regular if \[\left\langle \lambda, \alpha^\vee\right\rangle\neq 0 \text{ for all } \alpha^\vee \in R^\vee.\] Now define the positive system \(\Delta_\lambda^+\) by \[\Delta^+_\lambda := \Set{\alpha \in R | \left\langle \lambda, \alpha^\vee\right\rangle > 0}.\] Let \(\rho_\lambda\) be the half-sum of the positive roots for the positive system \(\Delta^+_\lambda\) and \(\mathfrak{b}_\lambda\) be the Borel subalgebra of \(\mathfrak{g}_{\mathbb{C}}\) corresponding to \(\Delta^+_\lambda\). For \(x\in {\mathcal{X}}\), let \(K_x := G_{x}^{\mathrm{Inn}(x)}\) be the maximal compact subgroup of \(G_{x}\) when \(G\) is linear and the preimage of \(G^{\mathrm{Inn}(x)}_x\) in \(G_{x}\) when \(G\) is metaplectic. Let \[\rho_{x,\lambda} := \frac{1}{2} \sum_{\alpha \in \Delta^+_\lambda, \alpha(x) = 1} \alpha\] be the half-sum of the positive roots in the complexified Lie algebra of \(K_x\).

Under this notation, we have the following fact, see [88] for example \[(x,\pi_x(\lambda)) \sim ({}^w x, \pi_{{}^w x}(w\cdot \lambda)) \text{ for all } w \in W.\]

5.3 Discrete Vogan packet↩︎

Let \(B\) be the Borel subgroup of \({\mathcal{G}}\) corresponding to \(B^\vee\). Let \(\Delta^+\) be the set of roots in \({\rm Lie}\,B\); then the corresponding positive coroots \((\Delta^\vee)^+\) form the set of roots in \({\rm Lie}\,B^\vee\).

A discrete \(L\)-parameter \(\phi: W_{\mathbb{R}}\to G^\vee \rtimes \mathrm{Gal}({\mathbb{C}}/{\mathbb{R}})\), up to conjugation, may be assumed to be of the form: \[\label{HC} \phi(z) = (z/\bar z)^\lambda, \quad z\in {\mathbb{C}}^\times \subset W_{\mathbb{R}}\tag{38}\] where \(\lambda\in X^*(T)\otimes {\mathbb{R}}\) satisfies

• \(\lambda\) is regular;

• \(\lambda\) is \(B\)-dominant;

• \(\exp(2\pi \mathrm{i}\lambda)= z^\vee\), see 37 for the definition of \(z^\vee\).

In this case \(\phi(j) = \exp(-\pi \mathrm{i}\lambda)\rtimes c\), where as before \(c\in \mathrm{Gal}({\mathbb{C}}/{\mathbb{R}})\) is the non-trivial element. Moreover, the centralizer in \(G^\vee\) of the image of \(\phi\) is \[\Set{h\in T^\vee | h^2=1}.\] This set is disconnected and can be identified with the component group \({\mathcal{S}}_\phi\) of \(\phi\). Let \(s\) be the map \[s : X^*(T)/ 2X^*(T) \rightarrow \Set{h\in T^\vee | h^2=1} = {\mathcal{S}}_\phi\] defined by \(\gamma + 2X^*(T) \mapsto \exp(\pi \mathrm{i}\gamma)\) for \(\gamma \in X^*(T) = X_*(T^\vee)\).

For each \(x\in {\mathcal{X}}\), we define \(\tau_x \in \widehat{{\mathcal{S}}_\phi}\) by \[\tau_x(s(\gamma + 2X^*(T))) := \gamma(xx_b^{-1}), \quad \gamma \in X^*(T).\] Note that \((x x_b^{-1})^2 = 1\) for all \(x\in {\mathcal{X}}\). Hence \(\gamma(x x_b^{-1}) = 1\) for all \(\gamma \in 2X^*(T)\), so \(\tau_x\) is well-defined. By the duality for algebraic tori, \(\tau\) gives the following bijection \[\tau : {\mathcal{X}}\xrightarrow{\; \sim\; }\widehat{{\mathcal{S}}_\phi},\qquad x\longmapsto \tau_x.\]

Recall that an irreducible Casselman-Wallach representation \(\pi\) of \(G^*\) is called large if the annihilator variety of \(\pi\) is the whole nilpotent cone of \(\frak g_{\mathbb{C}}\) ([26]). In particular, such large representations are generic with respect to the Whittaker datum determined by their wavefront sets ([29]).

The following proposition gives a parameterization of the Vogan packet \(\Pi_\phi[G^*]\). See [88], [87] and [92].

5.4 Associated variety↩︎

Let \(\theta\) be the Cartan involution of \(G\), which induces the Cartan decomposition \(\frak{g}_{\mathbb{C}}= \frak{k}\oplus \frak{p}\). Let \(K = G^\theta\) and \({\mathcal{K}}= {\mathcal{G}}^\theta\) be the corresponding subgroups of \(G\) and \({\mathcal{G}}\) respectively. For a finite length \((\frak{g}_{\mathbb{C}}, K)\)-module \(X\), its associated variety \({\rm AV}(X)\) is a finite union of closures of nilpotent \(K\)-orbits on \(\frak{p}\), which is an important invariant of \(X\) (see e.g. [16]). More precisely, given a \(K\)-stable good filtration on \(X\), the associated graded \({\rm gr}(X)\) is an \(S(\frak g_{\mathbb{C}})\)-module, where \(S(\frak g_{\mathbb{C}})\) is the symmetric algebra of \({\mathfrak {g}}_{\mathbb{C}}\). By definition, the associated variety \({\rm AV}(X)\) of \(X\) is the support of \({\rm gr}(X)\), viewed as a subvariety of \((\frak g_{\mathbb{C}}/\frak k)^*\cong\frak p\). For an irreducible Casselman-Wallach representation \(\pi\) of \(G\), define the associated variety of \(\pi\) to be: \(\mathop{\mathrm{AV}}(\pi):=\mathop{\mathrm{AV}}(\pi_{\rm HC})\), where \(\pi_{\rm HC}\) is the Harish-Chandra module of \(\pi\).

Now we proceed to compute the associated variety of a discrete series representation. Since the discrete series representation \(\pi_x(\lambda)\) is realized as a cohomologically induced representation, we can use the following theorem to compute its associated variety. This theorem is well known to experts (see, for example, the introduction of [70]) but is difficult to find in a proper reference. The following very general version is from [93].

For classical groups, \({\mathcal{G}}\) is viewed as a subgroup of \(\mathrm{GL}(V)\), where \(V\) is a complex vector space. We always choose a basis of \(V\) such that \(T\) is a diagonal matrix and \(B\) consists of upper triangular matrices.

In the following discussion, we let \(\mathrm{Sign}(n)\) be the set of all sequences of signs of length \(n\). An element \(\delta \in \mathrm{Sign}(n)\) is written as \[\delta = (\delta_1, \cdots, \delta_n)\] with \(\delta_i \in \{\pm 1\}\). We write \(p_\delta = \#\set{i\in [n] | \delta_i = 1},\) and \(q_\delta = \#\set{i\in [n] | \delta_i = -1}\).

The group \({\mathcal{S}}_\phi\) is identified with \({\mathbb{Z}}_2^n\) with generators \(e_i\). Then \(\widehat{{\mathcal{S}}_\phi}\) is identified with \(\mathrm{Sign}(n)\) by setting \[\chi(e_i) := \chi_i \quad \text{for } i=1, \cdots, n\] where \(\chi = (\chi_1, \cdots, \chi_n) \in \mathrm{Sign}(n)\). The bijection \(\tau : {\mathcal{X}}\xrightarrow{ \; \sim \; }\widehat{{\mathcal{S}}_\phi}\) is now described by a map \(\mathrm{Sign}(n)\to \mathrm{Sign}(n)\) sending \(\delta\) to \(\chi_\delta\) so that \(\tau_{x_\delta} = \chi_\delta\).

We make the following choices to explicate the various maps defined above and compute the wavefront set of \(\pi_{x_b}(\lambda)\) using Theorem 25 below.

1. When \({\mathcal{G}}= \mathrm{GL}_n({\mathbb{C}})\), \({\mathcal{X}}= \set{x \in T | x^2 = (-1)^{n-1}}\) and it is in bijection with \(\mathrm{Sign}(n)\) via the map \[\delta \mapsto x_\delta = (-{\rm i})^{n-1}\mathrm{diag}(\delta_1, \cdots, \delta_n).\] Let \((p_x, q_x) = (p_\delta, q_\delta)\). The real form \(G_{x}\) is identified with \({\mathrm {U}}(p_x, q_x)\). In this case, \[x_b = {\rm i}^{n-1}\mathrm{diag}(1,-1,\cdots, (-1)^{n-1})\] and \(G^* = G_{x_b} ={\mathrm {U}}(\lceil n/2\rceil, \lfloor n/2\rfloor)\). The bijection \(\tau: {\mathcal{X}}\xrightarrow{ \; \sim \; }\widehat{{\mathcal{S}}_\phi}\) is given by \[(\chi_\delta)_i = (-1)^{n-i}\delta_i \text{ for } i=1, \cdots, n.\] In this case, we obtain \[\mathrm{WF}(\pi_{x_b}(\lambda)) = \begin{ytableau} \pm & \mp & \pm & \cdots &+ \end{ytableau}.\]

2. When \({\mathcal{G}}= {\mathrm{SO}}_{2n}({\mathbb{C}})\), \({\mathcal{X}}= \set{x \in T | x^2 = 1}\) and it is in bijection with \(\mathrm{Sign}(n)\) via the map \[\delta \mapsto x_\delta = \mathrm{diag}(\delta_1, \cdots, \delta_n, \delta_n, \cdots, \delta_1).\] Let \((p_x, q_x) = (2 p_\delta, 2 q_\delta)\). The real form \(G_{x}\) is identified with \({\mathrm{SO}}(p_x, q_x)\). In this case, \[x_b = \mathrm{diag}((-1)^{n-1},\cdots,-1, 1, 1, -1, \cdots,(-1)^{n-1})\] and \(G^* = G_{x_b} = {\mathrm{SO}}(n,n)\). The bijection \(\tau: {\mathcal{X}}\xrightarrow{ \; \sim \; }\widehat{{\mathcal{S}}_\phi}\) is given by \[(\chi_\delta)_i = (-1)^{n-i}\delta_i \text{ for } i=1, \cdots, n.\] In this case, we obtain \[\mathrm{WF}(\pi_{x_b}(\lambda)) = \begin{ytableau} - &+ & \cdots &+&- \\ + \end{ytableau} .\]

3. When \({\mathcal{G}}= {\mathrm{SO}}_{2n+1}({\mathbb{C}})\), \({\mathcal{X}}= \set{x \in T | x^2 = 1}\). Then \({\mathcal{X}}\) is in bijection with \(\mathrm{Sign}(n)\) via the map \[\delta \mapsto x_\delta = \mathrm{diag}( \delta_1, \cdots, \delta_n, 1, \delta_n, \cdots, \delta_1).\] Recall that \(\delta_0 = (-1)^n {\rm disc}(V)\) determines the pure inner class of \(V\). Let \[(p_x, q_x) = \begin{cases} (2 p_\delta +1 , 2 q_\delta ) & \text{if } \delta_0 = 1. \\ (2 p_\delta , 2 q_\delta +1 ) & \text{if } \delta_0 = -1. \end{cases}\] The real form \(G_{x}\) is identified with \({\mathrm{SO}}(p_x, q_x)\). In this case, \[x_b = \mathrm{diag}((-1)^n, \cdots, -1, 1, -1, \cdots,(-1)^n).\] When \(\delta_0=1\), \[G^* = \begin{cases} {\mathrm{SO}}(n+1,n) & \text{if n is even;} \\ {\mathrm{SO}}(n,n+1) & \text{if n is odd.} \\ \end{cases}\] When \(\delta_0=-1\), \[G^* = \begin{cases} {\mathrm{SO}}(n,n+1) & \text{if n is even;} \\ {\mathrm{SO}}(n+1,n) & \text{if n is odd.} \\ \end{cases}\] The bijection \(\tau: {\mathcal{X}}\xrightarrow{ \; \sim \; }\widehat{{\mathcal{S}}_\phi}\) is given by \[(\chi_\delta)_i = (-1)^{n-i} {\rm disc}(V) \delta_i = (-1)^i \delta_0\delta_i \text{ for } i=1, \cdots, n.\] In this case, we obtain \[\mathrm{WF}(\pi_{x_b}(\lambda)) = \begin{cases} \begin{ytableau} \pm &\mp & \cdots & + &\cdots &\mp & \pm \\ \end{ytableau} & \text{if \delta_0 = 1, and n is even (resp. odd);} \\ \begin{ytableau} \mp &\pm & \cdots & - &\cdots &\pm & \mp \\ \end{ytableau} & \text{if \delta_0 = -1, and n is even (resp. odd).} \\ \end{cases}\]

4. When \({\mathcal{G}}= {\mathrm{Sp}}_{2n}({\mathbb{C}})\), \[{\mathcal{X}}= \set{x \in T | x^2 = -1}.\] It is in bijection with \(\mathrm{Sign}(n)\) via the map \[\delta \mapsto x_\delta = {\rm i}^{-1} \mathrm{diag}(\delta_1, \cdots, \delta_n, \delta_n, \cdots, \delta_1).\] Let \[(p_x, q_x) = (2 p_\delta, 2 q_\delta).\] Then \(p_x = q_x =n\) and the real form \(G_{x}\) is identified with \({\mathrm{Sp}}_{2n}({\mathbb{R}})\). In this case, \[x_b = {\rm i}^{2n-1}\mathrm{diag}(1,-1,\cdots, (-1)^{n-1}, (-1)^{n-1}, \cdots, -1, 1)\] and \(G^* = {\mathrm{Sp}}_{2n}({\mathbb{R}})\). The bijection \(\tau: {\mathcal{X}}\xrightarrow{ \; \sim \; }\widehat{{\mathcal{S}}_\phi}\) is given by \[(\chi_\delta)_i = (-1)^{n - i + 1} \delta_i \text{ for } i=1, \cdots, n.\] In this case, \[\mathrm{WF}(\pi_{x_b}(\lambda)) = \begin{ytableau} + & - & \cdots & + & \cdots & + & - \\ \end{ytableau}.\]

Following [70], the Levi part of \({\frak b}_{x_\delta}\) is given by \[{\frak u}(p_1, q_1)\times {\frak u}(p_2,q_2)\times \cdots \times {\frak u}(p_n, q_n),\] where \[(p_i, q_i)= \begin{cases} (1,0) & \text{if } \delta_i = 1, \\ (0,1) & \text{if } \delta_i = -1. \end{cases}\] Write \({\mathcal{O}}(\delta)\) for the \({\mathcal{K}}\)-orbit such that \[\mathop{\mathrm{AV}}(\pi_{x_\delta}) = \overline{{\mathcal{O}}(\delta)}.\]

The signed Young tableau that parametrizes \({\mathcal{O}}(\delta)\) is given case by case in [70] for \(G^* = {\mathrm {U}}_n\), \({\mathrm{Sp}}_{2n}({\mathbb{R}})\), and \({\mathrm{SO}}_{\frak n}\) respectively. The result also works for \(G = {\mathrm{Mp}}_{2n}({\mathbb{R}})\) as the computation of \({\mathcal{K}}\cdot(\overline{\mathfrak{u}}\cap \mathfrak{p})\) has nothing to do with the covering.

Denote by \(c_{r, s}\) a column with \(r\) pluses and \(s\) minuses. For a signed Young tableau \(T\), let \(T\oplus c_{r, s}\) be the equivalence class of the signed Young tableau formed by placing the entries of \(c_{r, s}\) at the row-ends of \(T\) such that

• at most one sign is added to each row-end;

• each sign is added to as high a row as possible.

One can similarly define \(c_{r, s} \oplus T\), but the entries of \(c_{r, s}\) are added to the row-starts of \(T\). It can be shown that the operation \(\oplus\) is well-defined and associative in a suitable sense (see [70] for detailed explanations). Denote by \(c_{\delta_i}\) the singleton Young tableau with the signature \((1,0)\) if \(\delta_i = 1\) and the signature \((0,1)\) if \(\delta_i = -1\).

5.5 Examples↩︎

To illustrate the algorithm above, we give examples for unitary and orthogonal groups. More examples can be executed on the website [89].

5.5.1 Unitary groups↩︎

We first carry out the algorithm for a discrete Harish-Chandra parameter \((\lambda, \delta)\) for \({\mathrm {U}}(10,8)\). Write \[\lambda=\lambda_{\alpha_1} \oplus \cdots \oplus \lambda_{\alpha_{18}}\] where \(\alpha_1<\alpha_2<\cdots<\alpha_{18}\) are half of odd integers, and \(\delta\) is \[+++++-----++++--+-.\] Then by the associativity of the operation \(\oplus\), one has \[\begin{align} {\mathcal{O}}(\delta) &= c_{\delta_1} \oplus c_{\delta_2}\oplus \cdots \oplus c_{\delta_{18}} = c_{5, 0} \oplus c_{0, 5} \oplus c_{4, 0} \oplus c_{0, 2} \oplus c_{1, 0} \oplus c_{0, 1} \\ & = \begin{ytableau} + & - & + & - & + & - \\ + & - & + & - \\ + & - & + \\ + & - & + \\ + & - \\ \end{ytableau} \end{align}\]

5.5.2 Special orthogonal groups↩︎

Since the formula for special orthogonal groups involves an auxiliary sign, we include one example to clarify this feature. Consider a discrete Harish-Chandra parameter \((\lambda,\delta)\) for \({\mathrm{SO}}(16,15)\) with character \[--+++++----++-+.\] By the same argument as in Subsection 4.3.2, \(\delta_0 = -1\). Then \[\begin{align} {\mathcal{O}}(\delta) &= c_{\delta_{15}} \oplus \cdots \oplus c_{\delta_2} \oplus c_{\delta_1} \oplus c_{\delta_0}\oplus c_{\delta_1} \oplus c_{\delta_2}\oplus \cdots \oplus c_{\delta_{15}} \\ &= c_{1, 0} \oplus c_{0, 1} \oplus c_{2, 0} \oplus c_{0, 4} \oplus c_{5, 0} \oplus c_{0, 2} \oplus c_{0, 1} \oplus c_{0, 2} \oplus c_{5, 0} \oplus c_{0, 4} \oplus c_{2, 0} \oplus c_{0, 1} \oplus c_{1, 0} \\ &= \begin{ytableau} +&- & + & - & + & - & + & - & + & - & + \\ +&- & + & - & + & - & + \\ - & + & - & + & - \\ - & + & - & + & - \\ + & - & +\\ \end{ytableau} \end{align}\]

6 Arithmetic Wavefront Set and Associated Variety↩︎

Following Theorem 4 ([1]), when the Vogan packet \(\Pi_\varphi[G^*]\) is not a singleton, associated to each \(\pi =\pi(\varphi,\chi)\in \Pi_\varphi[G^*]\), there is a real distinguished nilpotent orbit \[{\mathcal{O}}(\pi) = {\mathcal{O}}(\varphi,\chi) \subset {\frak g},\] which is the unique maximal element of the arithmetic wavefront set \({\rm WF_{ari}}(\pi)\) defined in loc. cit. under the real topological order. The case of discrete series representations has been recalled in Section 4.2.

In terms of the Kostant-Sekiguchi correspondence (see [16] for instance) given by \[\begin{align} \label{KSC} \Set{\text{nilpotent G-orbits on \frak g}} & \stackrel{\sim}{\longrightarrow} & \Set{\text{nilpotent {\mathcal{K}}-orbits on \frak p}}, \\ {\mathcal{O}}& \longmapsto & {\mathcal{O}}_{\frak p}, \nonumber \end{align}\tag{39}\] we prove the following result.

By the relation between the associated cycle and the wavefront cycle established in [33], we have the following immediate consequence of Theorem 26. Note that if \(\Pi_\varphi[G^*]\) is a singleton, then \(G\) is quasi-split and \(\pi\) is generic with respect to every Whittaker datum of \(G\), in which case \({\rm AV}(\pi)\) is the whole nilpotent cone in \(\frak{p}\).

6.1 Reduction↩︎

We first reduce the proof of Theorem 26 to the case where \(\pi\) is a discrete series representation. Recall that \(\varphi\) is given by 21 , 22 and 23 . First assume that \(G\) is linear. Since \(\varphi\) is generic, it follows from [94] that \(\pi(\varphi,\chi)\) is an irreducible normalized parabolic induction \[\label{ind} \pi(\varphi,\chi) \cong |\cdot|_E^{s_1}\pi_{\tau_1} \times \cdots \times |\cdot|_E^{s_l}\pi_{\tau_l} \times \pi(\varphi_{\rm gp},\chi),\tag{40}\] where \(\pi_{\tau_j}\) is the (limit of) discrete series representation of \({\rm GL}_1(E)\) or \({\rm GL}_2(E)\) with \(L\)-parameter \(\tau_j\), and \(\pi(\varphi_{\rm gp}, \chi)\) is a representation of a classical group \(G'\) in the Witt tower of \(G\), with the \(L\)-parameter \(\varphi_{\rm gp}\) as in 22 . A slight modification of 40 also holds when \(G={\mathrm{Mp}}_{2n}({\mathbb{R}})\).

Consider a discrete \(L\)-parameter \(\varphi'\) of \(G'\) of the form \[\varphi' = \rho_{\alpha_1'}\oplus \cdots \oplus \rho_{\alpha_m'} \oplus \varphi_{\rm quad},\] where \(m= m_1+\cdots+m_r\) and \(\alpha'_1<\cdots < \alpha'_m\), with \(\varphi_{\rm quad}\) as in 22 . There is a natural embedding of centralizers \[C_{G'^\vee}({\rm im}\, \varphi') \hookrightarrow C_{G'^\vee}({\rm im}\, \varphi_{\rm gp}),\] which in turn induces a surjective homomorphism of component groups \[p: {\mathcal{S}}_{\varphi'} \twoheadrightarrow {\mathcal{S}}_{\varphi_{\rm gp}}.\] Let \(\chi ' = \chi\circ p \in \widehat{{\mathcal{S}}_{\varphi'}}\). Then \(\pi(\varphi_{\rm gp},\chi)\) and \(\pi(\varphi',\chi')\) lie in the same coherent family (see [95]), and by the translation principle ([96]) it holds that \[\mathop{\mathrm{AV}}(\pi(\varphi_{\rm gp},\chi))= \mathop{\mathrm{AV}}(\pi(\varphi',\chi')).\]

Assume that Theorem 26 holds for the discrete series representation \(\pi(\varphi',\chi')\), so that \(\mathop{\mathrm{AV}}(\pi(\varphi',\chi'))\) is the closure of \({\mathcal{O}}(\varphi',\chi')_{\frak p}\). By the algorithm in [1], we have \({\mathcal{O}}(\varphi_{\rm gp},\chi) = {\mathcal{O}}(\varphi',\chi')\), hence \(\mathop{\mathrm{AV}}(\pi(\varphi_{\rm gp},\chi))\) is the closure of \({\mathcal{O}}(\varphi_{\rm gp},\chi)_{\frak p}\). Write \[{\mathcal{O}}(\varphi_{\rm gp},\chi) = ((p_1,\epsilon_1), (p_2,\epsilon_2), \ldots, (p_k, \epsilon_k)).\] Put \(t =\dim \tau\). The algorithm also gives that \[{\mathcal{O}}(\varphi, \chi) = ((p_1+2t,\epsilon_1), (p_2,\epsilon_2), \ldots, (p_k, \epsilon_k)).\]

By [1], \({\mathcal{O}}(\varphi_{\rm gp},\chi)\) is distinguished. In other words, it holds that if \(p_i = p_j\) then \(\epsilon_i = \epsilon_j\). It follows that the Lusztig-Spaltenstein induction ([97]) of the regular nilpotent orbit in \(\frak{gl}_t(E)\) and the nilpotent orbit \({\mathcal{O}}(\varphi_{\rm gp},\chi)\) in \(\frak{g}'\) is the nilpotent orbit \({\mathcal{O}}(\varphi, \chi)\) in \({\frak g}\). Then by 40 and [98], \(\mathop{\mathrm{AV}}(\pi(\varphi, \chi))\) is the closure of \({\mathcal{O}}(\varphi,\chi)_{\frak p}\). This proves that Theorem 26 holds for \(\pi(\varphi, \chi)\).

6.2 Proof for discrete series↩︎

The rest of this section is devoted to the proof of Theorem 26 for discrete series representations. Let \(\pi = \pi(\varphi, \chi)\) be a discrete series representation of \(G\). We need to show that \(\mathop{\mathrm{AV}}(\pi)\) is the closure of \({\mathcal{O}}(\varphi,\chi)_{\frak p}\).

The proof is case-by-case. We first outline the common strategy.

1. Translate the algorithm in Section 5.4 into a recursive algorithm parallel to the one in Section 4.2. Concretely, this produces the first row of the signed Young tableau \({\mathcal{O}}(\delta)\) together with a new Harish-Chandra parameter \(\delta'\), which then generates the remaining rows of \({\mathcal{O}}(\delta)\). We call \(\delta'\) an HC-descent of \(\delta\).

2. Show that the first rows of \({\mathcal{O}}(\delta)\) and \({\mathcal{O}}(\varphi,\chi)\) produced by the two algorithms agree.

3. Show that the HC-descent \(\delta'\) of \(\delta\) and the \(L\)-descent \(\chi'\) of \(\chi\) match as in Section 5.4.

Then by induction, the signed Young tableaux \({\mathcal{O}}(\delta)\) and \({\mathcal{O}}(\varphi,\chi)\) generated by both algorithms are the same.

We present the details for each family of real classical groups.

The unitary groups:

(i) Recursive algorithm. We first divide the sequence \(\delta\) into consecutive subsequences, where the signs in each subsequence are the same: \[\delta = (m_1 \delta_{k_1}, m_2 \delta_{k_2}, \dots, m_{\ell} \delta_{k_\ell}).\] For example, if \(\delta = (+1, +1, -1, -1, -1, +1)\), then \((m_{1}, m_{2}, m_{3}) = (2, 3, 1)\) and \((\delta_{k_1}, \delta_{k_2}, \delta_{k_3}) = (\delta_1, \delta_3, \delta_6) = (+1, -1, +1)\). By the associativity of the operation \(\oplus\), we can rewrite \({\mathcal{O}}(\delta)\) as \[{\mathcal{O}}(\delta) = c_{m_1 \delta_{k_1}} \oplus \cdots \oplus c_{m_\ell \delta_{k_\ell}},\] where \(c_{m_i \delta_{k_i}}\) denotes the column with \(m_i\) signs \(\delta_{k_i}\). Then the first row of \({\mathcal{O}}(\delta)\) is given by picking one sign from each \(m_i \delta_{k_i}\), and the remaining rows of \({\mathcal{O}}(\delta)\) are generated by the same algorithm for the HC-descent \[\label{HCD} \delta': = ((m_1-1)\delta_{k_1}, (m_2-1)\delta_{k_2}, \ldots, (m_\ell-1)\delta_{k_\ell}).\tag{41}\] This indeed gives a recursive algorithm for the associated variety.

(ii) Matching the first rows. We now show that the first rows of \({\mathcal{O}}(\delta)\) and \({\mathcal{O}}(\varphi,\chi)\) are the same. It suffices to check that their lengths and last signs are respectively the same. Recall from Section 5.4 that \(\delta\) and \(\chi\) are related by \[\delta_i = (-1)^{n-i} \chi_i,\quad i=1,2,\dots, n.\] It follows that \[\label{sgnHC} {\mathrm{sgn}}(\chi) = \set{ 1\leq i< n | \delta_i \delta_{i+1}=1}.\tag{42}\] Hence the length of the first row of \({\mathcal{O}}(\delta)\) is \[\ell = 1 + \# {\mathrm{sgn}}(\delta) = 1 + (n - 1) - \# {\mathrm{sgn}}(\chi) = n- s_\chi = p_1.\] The last sign of the first row of \({\mathcal{O}}(\delta)\) is clearly \(\delta_n = \chi_n = \epsilon_1\). This proves that the first rows of \({\mathcal{O}}(\delta)\) and \({\mathcal{O}}(\varphi, \chi)\) are the same.

(iii) Matching HC-descent and \(L\)-descent. Finally, we show that \(\delta'\) and \(\chi'\) are related as in Section 5.4. From 41 and 42 it is clear that \[\delta' = (\delta_j : j\in {\rm sgn}(\chi)).\] Recall from Section 4.2 that \[\chi' = ((-1)^{n-j+1} \chi_n : j\in {\rm sgn}(\chi)).\] Take \(j\in {\rm sgn}(\chi)\), and suppose that it is the \(k\)-th largest element in \({\mathrm{sgn}}(\chi)\). It suffices to verify that \[\label{HCLD} \delta_j = (-1)^{k - 1} \cdot (-1)^{n-j+1}\chi_n = (-1)^{n-j-k}\chi_n.\tag{43}\] In fact, one has \[n - j - k = \# \set{ j \leq i < n | i \notin {\mathrm{sgn}}(\chi)} = \# \set{ j \leq i < n | i \in {\mathrm{sgn}}(\delta)}.\] Therefore, \(\delta_j\) and \(\delta_n = \chi_n\) differ by \((-1)^{n-j-k}\), which shows that 43 holds.

The even special orthogonal groups:

(i) Recursive algorithm. We first write \(\delta = (m_1 \delta_{k_1}, m_2 \delta_{k_2}, \dots, m_{\ell} \delta_{k_\ell})\) as in the unitary case. By the associativity of the operation \(\oplus\), we can rewrite \({\mathcal{O}}(\delta)\) as \[{\mathcal{O}}(\delta) = c_{m_\ell \delta_{k_\ell}} \oplus \cdots \oplus c_{m_2 \delta_{k_2}} \oplus c_{2m_1 \delta_{k_1}} \oplus c_{m_2 \delta_{k_2}} \oplus \cdots \oplus c_{m_\ell \delta_{k_\ell}},\] Then the first row of \({\mathcal{O}}(\delta)\) is given by picking one sign from each \(m_i \delta_{k_i}, i \geq 2\) and \(2m_1 \delta_{k_1}\), and the remaining rows of \({\mathcal{O}}(\delta)\) are generated by the same algorithm for the HC-descent \[\label{HCDSo} \delta': = ((m_1-1)\delta_{k_1}, (m_2-1)\delta_{k_2}, \ldots, (m_\ell-1)\delta_{k_\ell}), \quad \delta_0' = \delta_1.\tag{44}\] This indeed gives a recursive algorithm for the associated variety.

(ii) Matching the first rows. We now show that the first rows of \({\mathcal{O}}(\delta)\) and \({\mathcal{O}}(\varphi,\chi)\) are the same. It suffices to check that their lengths and last signs are respectively the same. Recall from Section 5.4 that \(\delta\) and \(\chi\) are related by \[\delta_i = (-1)^{n-i} \chi_i,\quad i=1,2,\dots, n.\] It follows that \[\label{sgnHCSo} {\mathrm{sgn}}(\chi) = \set{ 1\leq i< n | \delta_i \delta_{i+1}=1}.\tag{45}\] Hence the length of the first row of \({\mathcal{O}}(\delta)\) is \[2\ell - 1 = 1 + 2 \# {\mathrm{sgn}}(\delta) = 1 + 2 (n - 1 - \# {\mathrm{sgn}}(\chi)) = 2(n- s_\chi) - 1 = p_1.\] The last sign of the first row of \({\mathcal{O}}(\delta)\) is \(\delta_n = \chi_n = \epsilon_1\). This proves that the first rows of \({\mathcal{O}}(\delta)\) and \({\mathcal{O}}(\varphi, \chi)\) are the same.

(iii) Matching HC-descent and \(L\)-descent. Finally, we show that the auxiliary signs are also matched, i.e., \(\delta_0' = (-1)^{n - s_\chi - 1}\chi_n\), and \(\delta'\) and \(\chi'\) are related as in Section 5.4.

For the auxiliary sign, note that \(n - s_\chi - 1 = s_\delta\), so \(\delta_0' = (-1)^{n - s_\chi - 1}\chi_n\).

From 44 and 45 it is clear that \[\delta' = (\delta_j : j\in {\rm sgn}(\chi)).\] Recall from Section 4.2 that \[\chi' = ((-1)^{j} : j\in {\rm sgn}(\chi)).\] Take \(j\in {\rm sgn}(\chi)\), and suppose that it is the \(k\)-th smallest element in \({\mathrm{sgn}}(\chi)\). It suffices to verify that \[\label{HCLDSo} \delta_j = (-1)^{k} \cdot \delta_0' \cdot (-1)^{j} = (-1)^{j-k} \delta_{1}.\tag{46}\] In fact, one has \[j - k = \# \set{ 1 \leq i < j | i \notin {\mathrm{sgn}}(\chi)} = \# \set{ 1 \leq i < j | i \in {\mathrm{sgn}}(\delta)}.\] Therefore, \(\delta_j\) and \(\delta_1\) differ by \((-1)^{j-k}\), which shows that 46 holds.

The odd special orthogonal groups:

(i) Recursive algorithm. We first write \(\delta = (m_1 \delta_{k_1}, m_2 \delta_{k_2}, \dots, m_{\ell} \delta_{k_\ell})\) as in the unitary case. By the associativity of the operation \(\oplus\), we can rewrite \({\mathcal{O}}(\delta)\) as \[{\mathcal{O}}(\delta) = c_{m_\ell \delta_{k_\ell}} \oplus \cdots \oplus c_{m_1 \delta_{k_1}} \oplus c_{\delta_0} \oplus c_{m_1 \delta_{k_1}} \oplus \cdots \oplus c_{m_\ell \delta_{k_\ell}},\] Then:

• if \(\delta_0\delta_1 = -1\), then the first row of \({\mathcal{O}}(\delta)\) is given by picking one sign from each \(m_i \delta_{k_i}, i \geq 1\) and \(\delta_0\), and the remaining rows of \({\mathcal{O}}(\delta)\) are generated by the same algorithm for the HC-descent \[\label{HCDSoOdd1} \delta': = ((m_1-1)\delta_{k_1}, (m_2-1)\delta_{k_2}, \ldots, (m_\ell-1)\delta_{k_\ell}).\tag{47}\]

• if \(\delta_0\delta_1 = 1\), then the first row of \({\mathcal{O}}(\delta)\) is given by picking one sign from each \(m_i \delta_{k_i}, i \geq 2\) and one \(\delta_0 = \delta_1\) from \((2m_1 + 1)\delta_1\), and the remaining rows of \({\mathcal{O}}(\delta)\) are generated by the same algorithm for the HC-descent \[\label{HCDSoOdd2} \delta': = (m_1\delta_{k_1}, (m_2-1)\delta_{k_2}, \ldots, (m_\ell-1)\delta_{k_\ell}).\tag{48}\]

These indeed give a recursive algorithm for the associated variety.

(ii) Matching the first rows. We now show that the first rows of \({\mathcal{O}}(\delta)\) and \({\mathcal{O}}(\varphi,\chi)\) are the same. It suffices to check that their lengths and last signs are respectively the same. Recall from Section 5.4 that \(\delta\) and \(\chi\) are related by \[\delta_i = (-1)^{i} \delta_0 \chi_i,\quad i=1,2,\dots, n.\] It follows that \[\label{sgnHCSoOdd} {\mathrm{sgn}}(\chi) = \set{ 0\leq i< n | \delta_i \delta_{i+1}=1}.\tag{49}\] Hence the length of the first row of \({\mathcal{O}}(\delta)\) is \[2\ell + 1 = 1 + 2 \# {\mathrm{sgn}}(\delta) = 1 + 2 (n - \# {\mathrm{sgn}}(\chi)) = 2(n- s_\chi) + 1 = p_1.\] The last sign of the first row of \({\mathcal{O}}(\delta)\) is \(\delta_n = (-1)^n \delta_0 \chi_n = \epsilon_1\). This proves that the first rows of \({\mathcal{O}}(\delta)\) and \({\mathcal{O}}(\varphi, \chi)\) are the same.

(iii) Matching HC-descent and \(L\)-descent. Finally, we show that \(\delta'\) and \(\chi'\) are related as in Section 5.4.

From 47 , 48 and 49 it is clear that \[\delta' = (\delta_j : j\in {\rm sgn}(\chi)).\] Recall from Section 4.2 that \[\chi' = ((-1)^{j - 1} \delta_0 \chi_n : j\in {\rm sgn}(\chi)).\] Take \(j\in {\rm sgn}(\chi)\), and suppose that it is the \(k\)-th largest element in \({\mathrm{sgn}}(\chi)\). It suffices to verify that \[\label{HCLDSoOdd} \delta_j = (-1)^{k - 1} \cdot (-1)^{j - 1} \delta_0 \chi_n = (-1)^{n-j-k} \delta_{n}.\tag{50}\] In fact, one has \[n - j - k = \# \set{ j \leq i < n | i \notin {\mathrm{sgn}}(\chi)} = \# \set{ j \leq i < n | i \in {\mathrm{sgn}}(\delta)}.\] Therefore, \(\delta_j\) and \(\delta_n\) differ by \((-1)^{n-j-k}\), which shows that 50 holds.

The symplectic/metaplectic groups:

(i) Recursive algorithm. We first write \(\delta = (m_1 \delta_{k_1}, m_2 \delta_{k_2}, \dots, m_{\ell} \delta_{k_\ell})\) as in the unitary case. By the associativity of the operation \(\oplus\), we can rewrite \({\mathcal{O}}(\delta)\) as \[{\mathcal{O}}(\delta) = c_{m_\ell \delta_{k_\ell}} \oplus \cdots \oplus c_{m_1 \delta_{k_1}} \oplus c_{-m_1 \delta_{k_1}} \oplus \cdots \oplus c_{-m_\ell \delta_{k_\ell}},\] Then the first row of \({\mathcal{O}}(\delta)\) is given by picking one sign from each \(m_i \delta_{k_i}\) and \(-m_i \delta_{k_i}\), and the remaining rows of \({\mathcal{O}}(\delta)\) are generated by the same algorithm for the HC-descent \[\label{HCDSp} \delta': = ((m_1-1)\delta_{k_1}, (m_2-1)\delta_{k_2}, \ldots, (m_\ell-1)\delta_{k_\ell}).\tag{51}\] This indeed gives a recursive algorithm for the associated variety.

(ii) Matching the first rows. We now show that the first rows of \({\mathcal{O}}(\delta)\) and \({\mathcal{O}}(\varphi,\chi)\) are the same. It suffices to check that their lengths and last signs are respectively the same. Recall from Section 5.4 that \(\delta\) and \(\chi\) are related by \[\delta_i = (-1)^{n-i+1} \chi_i,\quad i=1,2,\dots, n.\] It follows that \[\label{sgnHCSp} {\mathrm{sgn}}(\chi) = \set{ 1\leq i< n | \delta_i \delta_{i+1}=1}.\tag{52}\] Hence the length of the first row of \({\mathcal{O}}(\delta)\) is \[2\ell = 2 + 2 \# {\mathrm{sgn}}(\delta) = 2 + 2 (n - 1 - \# {\mathrm{sgn}}(\chi)) = 2(n- s_\chi) = p_1.\] The last sign of the first row of \({\mathcal{O}}(\delta)\) is \(-\delta_n = \chi_n = \epsilon_1\). This proves that the first rows of \({\mathcal{O}}(\delta)\) and \({\mathcal{O}}(\varphi, \chi)\) are the same.

(iii) Matching HC-descent and \(L\)-descent. Finally, we show that \(\delta'\) and \(\chi'\) are related as in Section 5.4. From 51 and 52 it is clear that \[\delta' = (\delta_j : j\in {\rm sgn}(\chi)).\] Recall from Section 4.2 that \[\chi' = ((-1)^{n-j+1} \chi_n : j\in {\rm sgn}(\chi)).\] Take \(j\in {\rm sgn}(\chi)\), and suppose that it is the \(k\)-th largest element in \({\mathrm{sgn}}(\chi)\). It suffices to verify that \[\label{HCLDSp} \delta_j = (-1)^{k} \cdot (-1)^{n-j+1}\chi_n = (-1)^{n-j-k + 1}\chi_n.\tag{53}\] In fact, one has \[n - j - k = \# \set{ j \leq i < n | i \notin {\mathrm{sgn}}(\chi)} = \# \set{ j \leq i < n | i \in {\mathrm{sgn}}(\delta)}.\] Therefore, \(\delta_j\) and \(\delta_n = -\chi_n\) differ by \((-1)^{n-j-k}\), which shows that 53 holds.

6.3 Examples↩︎

To illustrate the proof in Subsection 6.2, we give examples for unitary and orthogonal groups.

6.3.1 Unitary groups↩︎

We first carry out the algorithm for a discrete series representation \(\pi = \pi(\varphi, \chi)\) for \({\mathrm {U}}(10,8)\). Let the character \(\chi\) be \[-+-+--+-+--+-++---,\] Then by Section 5.4, the corresponding character \(\delta\) of Harish-Chandra parameter is \[+++++-----++++--+-.\] We can rewrite \({\mathcal{O}}(\delta)\) as \[{\mathcal{O}}(\delta) = c_{5, 0} \oplus c_{0, 5} \oplus c_{4, 0} \oplus c_{0, 2} \oplus c_{1, 0} \oplus c_{0, 1}\] Then the first row of \({\mathcal{O}}(\delta)\) is given by picking one sign from each entry, and the remaining rows of \({\mathcal{O}}(\delta)\) are generated by the same algorithm for the HC-descent \[\delta' = ++++----+++-.\] The first rows of \({\mathcal{O}}(\delta)\) and \({\mathcal{O}}(\varphi, \chi)\) are the same, and the HC-descent \(\delta'\) and the L-descent \(\chi' = -+-++-+--+--\) (as computed in Subsection 4.3.1) are related by the same formula as in Section 5.4.

6.3.2 Special orthogonal groups↩︎

Let \(\pi = \pi(\varphi, \chi)\) be a discrete series representation for \({\mathrm{SO}}(16, 15)\) with character \[-++-+-++-+--+++,\] and the auxiliary sign \(\delta_0 = -1\). Then by Section 5.4, the corresponding character \(\delta\) of Harish-Chandra parameter is \[--+++++----++-+.\] We can rewrite \({\mathcal{O}}(\delta)\) as \[{\mathcal{O}}(\delta) = c_{1, 0} \oplus c_{0, 1} \oplus c_{2, 0} \oplus c_{0, 4} \oplus c_{5, 0} \oplus c_{0, 2} \oplus c_{0, 1} \oplus c_{0, 2} \oplus c_{5, 0} \oplus c_{0, 4} \oplus c_{2, 0} \oplus c_{0, 1} \oplus c_{1, 0}.\] Since \(\delta_0 \delta_1 = 1\), the first row of \({\mathcal{O}}(\delta)\) is given by picking one sign from each row for index \(\geq 2\), and one sign from \(c_{0, 2} \oplus c_{0, 1} \oplus c_{0, 2}\), and the remaining rows of \({\mathcal{O}}(\delta)\) are generated by the algorithm for even orthogonal group for the HC-descent \[\delta' = --++++---+.\] The first rows of \({\mathcal{O}}(\delta)\) and \({\mathcal{O}}(\varphi, \chi)\) are the same, and the HC-descent \(\delta'\) and the L-descent \(\chi' = +--+-++-++\) (as computed in Subsection 4.3.2) are related by the same formula as in Section 5.4.

7 Proof of the Main Theorem↩︎

Combining the results in previous sections, we prove the following main result of this paper.

From the proof of Theorem 28, we obtain the following consequence. For any \({\mathcal{O}}\in{\mathcal{N}}_F({\mathfrak {g}})_\circ\), let \((\underline{p},\{V_{p_i}\})\) be the sesquilinear Young tableau associated with \({\mathcal{O}}\) via the correspondence given in [73] as in Section 2.1. Set \(p({\mathcal{O}}):=\underline{p}\), the partition in \((\underline{p},\{V_{p_i}\})\) of \({\mathcal{O}}\). If \(\Omega\) is a subset of \({\mathcal{N}}_F({\mathfrak {g}})_\circ\), we define \(p(\Omega)\) accordingly as a set of partitions.

It is clear that \(p(\mathrm{WF}_\square(\pi)^{s-\mathrm{max}})=p(\mathrm{WF}_\square(\pi)^{r-\mathrm{max}})\) for \(\mathrm{WF}_\square(\pi) = \mathrm{WF}_{\mathrm{ari}}(\pi)\) or \(\mathrm{WF}_{\mathrm{tr}}(\pi)\) over \({\mathbb{R}}\) considered in this paper, which is expected to hold for \(\mathrm{WF}_\mathrm{wm}(\pi)\) as well.

References↩︎

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1. The work of D.J. is partially supported by the Simons Grant: SFI-MPS-TSM-00013449. The work of D.L. is partially supported by National Natural Science Foundation of China No. 12171421. The work of L.Z. is supported by AcRF Tier 1 grant A-8002960-00-00 and Tier 2 grant T2EP20225-0047 of National University of Singapore.↩︎

2. A corrigendum: in [1], when \(p_i\) is odd the sign \((-1)^{\frac{p_i-1}{2}}\) in (2.10) should be placed in the right hand side of (2.9) instead. Then the isometry (2.11) and all the subsequent calculations remain correct.↩︎