Tempered Representations Research Articles

TWISTED GAN–GROSS–PRASAD CONJECTURE FOR CERTAIN TEMPERED L-PACKETS

Abstract In this paper, we investigate the twisted GGP conjecture for certain tempered representations using the theta correspondence and establish some special cases, namely when the L-parameter of the unitary group is the sum of conjugate-dual characters of the appropriate sign.

On representations induced from the Zelevinsky segment and a tempered representation in the half-integral case

Let [Formula: see text] denote either the group [Formula: see text] or [Formula: see text] over a non-archimedean local field of characteristic zero. We determine the reducibility criteria for a parabolically induced representation of the form [Formula: see text], where [Formula: see text] denotes the Zelevinsky segment representation of the general linear group attached to the segment [Formula: see text], with [Formula: see text] half-integral, and [Formula: see text] denotes an irreducible tempered representation of [Formula: see text].

Theta lifting for tempered representations of real unitary groups

We study the theta lifting for real unitary groups and completely determine the theta lifts of tempered representations. In particular, we show that the theta lifts of (limits of) discrete series representations can be expressed as cohomologically induced representations in the weakly fair range. This extends a result of J.-S. Li in the case of discrete series representations with sufficiently regular infinitesimal character, whose theta lifts can be expressed as cohomologically induced representations in the good range.

Mackey analogy as deformation of $${\mathcal {D}}$$-modules

Given a real reductive linear Lie group \(G_{\mathbb {R}}\), the Mackey analogy is a bijection between the set of irreducible tempered representations of \(G_{\mathbb {R}}\) and the set of irreducible unitary representations of its Cartan motion group, established by Higson and Afgoustidis. We show that this bijection arises naturally from families of twisted \({\mathcal {D}}\)-modules over the ag variety of \(G_{\mathbb {R}}\).

Continuity of the Mackey–Higson bijection

When $G$ is a real reductive group and $G_0$ is its Cartan motion group, the Mackey-Higson bijection is a natural one-to-one correspondence between all irreducible tempered representations of $G$ and all irreducible unitary representations of $G_0$. In this short note, we collect some known facts about the topology of the tempered dual $\widetilde{G}$ and that of the unitary dual $\widehat{G_0}$, then verify that the Mackey-Higson bijection $\widetilde{G} \to \widehat{G_0}$ is continuous.

A remark on a trace Paley–Wiener theorem

In this paper we prove a version of a trace Paley--Wiener theorem for tempered representations of a reductive $p$--adic group. This is applied to complete certain investigation of Shahidi on the proof that a Plancherel measure is invariant of a $L$--packet of discrete series.

A geometric realisation of tempered representations restricted to maximal compact subgroups

Let G be a connected, linear, real reductive Lie group with compact centre. Let $$K<G$$ be maximal compact. For a tempered representation $$\pi $$ of G, we realise the restriction $$\pi |_K$$ as the K-equivariant index of a Dirac operator on a homogeneous space of the form G/H, for a Cartan subgroup $$H<G$$ . (The result in fact applies to every standard representation.) Such a space can be identified with a coadjoint orbit of G, so that we obtain an explicit version of Kirillov’s orbit method for $$\pi |_K$$ . In a companion paper, we use this realisation of $$\pi |_K$$ to give a geometric expression for the multiplicities of the K-types of $$\pi $$ , in the spirit of the quantisation commutes with reduction principle. This generalises work by Paradan for the discrete series to arbitrary tempered representations.

On the analogy between real reductive groups and Cartan motion groups: Contraction of irreducible tempered representations

Attached to any reductive Lie group $G$ is a "Cartan motion group" $G_0$ $-$ a Lie group with the same dimension as $G$, but a simpler group structure. A natural one-to-one correspondence between the irreducible tempered representations of $G$ and the unitary irreducible representations of $G_0$, whose existence had been suggested by Mackey in the 1970s, has recently been described by the author. In the present notes, we use the existence of a family of groups interpolating between $G$ and $G_0$ to realize the bijection as a deformation: for every irreducible tempered representation $\pi$ of G, we build, in an appropriate Fr\'echet space, a family of subspaces and evolution operators that contract $\pi$ onto the corresponding representation of $G_0$.

On the non-vanishing of theta liftings of tempered representations of U(p,q)

In this paper, we give an explicit determination of the non-vanishing of the theta liftings of tempered representations for unitary dual pairs (U(p,q),U(r,s)) for arbitrary non-negative integers p,q,r,s. For discrete series representations, in terms of Harish-Chandra parameters, we give a complete criterion when the theta lifts are nonzero. For tempered representations, we determine the non-vanishing in terms of the local Langlands correspondence assuming the local Gan–Gross–Prasad conjecture.

Endoscopic character identities for metaplectic groups

Abstract In this paper, we prove the conjectural endoscopic character identities for tempered representations of metaplectic groups Mp ⁢ ( 2 ⁢ n ) {\mathrm{Mp}({2n})} based on the formalism of endoscopy theory by J. Adams, D. Renard and W. W. Li.

A geometric formula for multiplicities of 𝐾-types of tempered representations

Let G G be a connected, linear, real reductive Lie group with compact centre. Let K &gt; G K&gt;G be compact. Under a condition on K K , which holds in particular if K K is maximal compact, we give a geometric expression for the multiplicities of the K K -types of any tempered representation (in fact, any standard representation) π \pi of G G . This expression is in the spirit of Kirillov’s orbit method and the quantisation commutes with reduction principle. It is based on the geometric realisation of π | K \pi |_K obtained in an earlier paper. This expression was obtained for the discrete series by Paradan, and for tempered representations with regular parameters by Duflo and Vergne. We obtain consequences for the support of the multiplicity function, and a criterion for multiplicity-free restrictions that applies to general admissible representations. As examples, we show that admissible representations of SU ( p , 1 ) \textrm {SU}(p,1) , SO 0 ( p , 1 ) \textrm {SO}_0(p,1) , and SO 0 ( 2 , 2 ) \textrm {SO}_0(2,2) restrict multiplicity freely to maximal compact subgroups.

ON -DISTINGUISHED REPRESENTATIONS OF THE QUASI-SPLIT UNITARY GROUPS

AbstractWe study$\text{Sp}_{2n}(F)$-distinction for representations of the quasi-split unitary group$U_{2n}(E/F)$in$2n$variables with respect to a quadratic extension$E/F$of$p$-adic fields. A conjecture of Dijols and Prasad predicts that no tempered representation is distinguished. We verify this for a large family of representations in terms of the Mœglin–Tadić classification of the discrete series. We further study distinction for some families of non-tempered representations. In particular, we exhibit$L$-packets with no distinguished members that transfer under base change to$\text{Sp}_{2n}(E)$-distinguished representations of$\text{GL}_{2n}(E)$.

The local Ginzburg–Rallis model for generic representations

We consider the local Ginzburg–Rallis model for generic representations. We prove that the summation of the multiplicities is always equal to 1 over every generic L-packet for the p-adic case and real case. This is a sequel work of [Wan15], [Wan16a] and [Wan16b] in which we considered the p-adic case and the real case for tempered representations, and considered the complex case for generic representations.

Shelstad's character identity from the point of view of index theory

Shelstad's character identity is an equality between sums of characters of tempered representations in corresponding L-packets of two real, semisimple, linear, algebraic groups that are inner forms to each other. We reconstruct this character identity in the case of the discrete series, using index theory of elliptic operators in the framework of K-theory. Our geometric proof of the character identity is evidence that index theory can play a role in the classification of group representations via the Langlands program.

Langlands classification for L-parameters

Let F be a non-archimedean local field and G=G(F) the group of F-rational points of a connected reductive F-group. The Langlands classification of complex irreducible admissible representations of G gives a parametrization of the set of all such representations in terms of triples (P,σ,ν), where P⊂G is a standard F-parabolic subgroup, σ is an irreducible tempered representation of the standard Levi-group MP and ν∈R⊗X⁎(MP) is regular and positive with respect to P. In this paper we consider Langlands' L-parameters [ϕ] which conjecturally will serve as a system of parameters for the representations π and which are equivalence classes of L-homomorphisms ϕ of the absolute Galois group Γ=Gal(F‾|F) with image in Langlands' L-group GL. Our goal is to establish a classification of L-parameters [ϕ] in terms of triples (P,[ϕb],ν) such that (P,ν) is as before and [ϕb] is a bounded L-parameter of MP.In an Appendix we also deal with the archimedean case F=R,C. This case is easier because the Weil group WR has a natural polar decomposition WR=R+××WR0.

On multiplicity in restriction of tempered representations of p-adic groups

We establish an equality between two multiplicities: one in the restriction of tempered representations of a p-adic group to its closed subgroup with the same derived group; and one occurring in their corresponding component groups in Langlands dual sides, so-called $$\mathcal {S}$$ -groups, under working hypotheses about the tempered local Langlands conjecture and the internal structure of tempered L-packets. This provides a formula of the multiplicity for p-adic groups by means of dimensions of irreducible representations of their $$\mathcal {S}$$ -groups.

Wave front sets of reductive Lie group representations II

In this paper it is shown that the wave front set of a direct integral of singular, irreducible representations of a real, reductive algebraic group is contained in the singular set. Combining this result with the results of the first paper in this series, the author obtains asymptotic results on the occurrence of tempered representations in induction and restriction problems for real, reductive algebraic groups.

The local Ginzburg–Rallis model over the complex field

We consider the local Ginzburg-Rallis model over complex field. We show that the multiplicity is always 1 for a majority of the generic representations. We also have partial results on the real case for general generic representations. This is a sequel work of [Wan15] and [Wan16] on which we considered the p-adic case and the real case for tempered representations.

Jacquet modules and irrreducibility of induced representations for classical p-adic groups

Let G be a classical p-adic group. If T is an irreducible tempered representation of such a group and $$\rho $$ an irreducible unitary supercuspidal representation of a general linear group, we can form the parabolically induced representation $$\text{ Ind }_P^G (|det|^y \rho \otimes T)$$ . The main result in this paper is the determination for which $$y \in {\mathbb R}$$ the induced representation is reducible. The key technical result in establishing this is the determination of a certain Jacquet module subquotient.

Local theta correspondence of tempered representations and Langlands parameters

In this paper, we give an explicit determination of the theta lifting for symplectic-orthogonal and unitary dual pairs over a nonarchimedean field $F$ of characteristic $0$. We determine when theta lifts of tempered representations are nonzero, and determine the theta lifts in terms of the local Langlands correspondence.