Restrictions, L-parameters, and local coefficients for genuine representations
aa r X i v : . [ m a t h . R T ] F e b RESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTSFOR GENUINE REPRESENTATIONS
FAN GAO, FREYDOON SHAHIDI, AND DANI SZPRUCH
Abstract.
We consider the restriction and induction of representations between a cov-ering group and its derived subgroup, both on the representation-theoretic side and theL-parameter side. In particular, restriction of a genuine principal series is analyzed indetail. We also discuss a metaplectic tensor product construction for covers of the sym-plectic similitudes groups, and remark on the generality of such a construction for othergroups. Furthermore, working with an arbitrary irreducible constituent of a unitary un-ramified principal series, we prove a multiplicity formula for its restriction to the derivedsubgroup in terms of three associated R-groups. Later in the paper, we study an unram-ified L-packet on how the parametrization of elements inside such a packet varies alongwith different choices of hyperspecial maximal compact subgroups and their splittings.We also investigate the genericity of elements inside such an L-packet with respect tovarying Whittaker datum. Pertaining to the above two problems, covers of the symplec-tic similitudes groups are discussed in detail in the last part of the paper.
Contents
1. Introduction 21.1. Main results 51.2. Several remarks 141.3. Acknowledgement 152. Restriction and induction of genuine representations 152.1. Covering groups and L-groups 152.2. The Clifford–Mackey theory 182.3. Two special pictures 222.4. Covers of GL r , GSp r and GSpin r +1 r K, s K ) for covers 825.3. Whittaker datum varied 87 Mathematics Subject Classification.
Primary 11F70; Secondary 22E50.
Key words and phrases. covering groups, L-group, parameters, Whittaker functionals, local coefficientsmatrix, R-group, unramified representation, L-packet, metaplectic tensor product. r r r Introduction
Let F be a p -adic field. Let G = G ( F ) be the group of F -rational points of a connectedsplit reductive group G over F . Let H ⊂ G be a closed subgroup, which might be the F -rational points of a subgroup H ⊂ G over F but not always so. Let π ∈ Irr( G ) bean irreducible admissible representation. It is an important question to determine therestriction π | H . As the literature on this problem is vast, we only mention three represen-tative examples. The first one originated from the theory of Gelfand pair [GK75, Gro91],and it includes the spherical theory when H ⊂ G is a maximal compact subgroup, andthe Whittaker theory when H is the unipotent radical of a Borel subgroup. The secondexample concerns the Gan–Gross–Prasad conjecture, see [GP92, GGP12] and [Gan14] in-cluding references therein. The third example arises from the special case where H isclosely related to the derived subgroup G der of G .If π | H is of finite length, then we have the semisimplification( π | H ) ss = X i m i π i ∈ R ( H ) , where π i ∈ Irr( H ) and R ( H ) denotes the Grothendieck group of Irr( H ). Part of theproblem is then on determining the multiplicity m i , which usually manifests in terms ofsome representation-theoretic and arithmetic invariants associated with the pair ( π, π i ) ∈ Irr( G ) × Irr( H ). A special case is the last example mentioned above. Now we elaborateon this, as its covering analogue is the main focus of our paper.It was shown by Silberger [Sil79] that for every π ∈ Irr( G ), the restriction π | G der issemisimple of finite length and thus π | G der = k X i =1 m i · π i with π i ∈ Irr( G der ) and m i >
1. This is a weaker form of a conjecture by Borel [Bor79]that all the constituents π i belong to the same L-packet associated with the parameter f ◦ φ π : WD F → L G → L G der , where f is the natural quotient map. In fact, the conjecture of Borel concerns slightlymore general group homomorphism H → G where H may not be G der . A special caseof this is when G der ⊂ H ⊂ G , and for such H Adler and Prasad proposed a unifiedformula for the multiplicity m i , see [AP19]. The original conjecture by Borel was alsogeneralized in another direction using the enchanced L-parameters. By an approach viathe Hecke algebras, such generalized version was proved for principal series and unipotentrepresentations of G , see the work of Aubert–Baum–Plymen–Solleveld [ABPS17] andSolleveld [Sol20]. We note that there are other works on various families of groups andrepresentations for the restriction problem from G to G der , see [Key87, Nev15, BCG18,Cho19].The starting point of this paper is motivated from the above (third) example in thesetting of covering groups. More precisely, assume that F contains the full group µ n of ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 3 n -th roots of unity, and consider a central cover µ n G G of G by µ n arising from the Brylisnki–Deligne framework [BD01]. Denote byIrr gen ( G )the set of isomorphism classes of irreducible genuine representations of G , where µ n alwaysacts by a fixed embedding µ n ֒ → C × . The derived subgroup of G is equal to the cover G der of G der obtained from the pull-back of G via the canonical inclusion G der ⊂ G . While therestriction problem for representations from G is G der is already a nontrivial one involvingdata from the L-parameter side, this is even more the case for central covers.Indeed, it is already instructive to consider the restriction of the principal series rep-resentations of G and G to G der and G der respectively. Let I ( χ ) be a principal series of G , where χ is a character of the torus T . It is well-known that I ( χ ) | G der = I ( χ ) , where χ = χ | T with T ⊂ G der being the split torus. However, if I ( χ ) is a genuineprincipal series of G , where χ is a genuine character of the center Z ( T ) ⊂ T , then therestriction I ( χ ) | G der , which is still semisimple, may not be an isotypic sum of genuineprincipal series of G der . In fact, this phenomenon arises from its counterpart for thegenuine representations of covering tori. That is, the decomposition of an irreduciblegenuine representation of T , when restricted to T , is more delicate than the linear case,which is deceptively simple. Such delicacy is a consequence of the fact that Z ( T ) * Z ( T )in general.Thus, we expect multiple layers of subtleties arising from the multiplicity formula forthe restriction π | G der for π ∈ Irr gen ( G ). This also renders many new phenomena regardingthe local coefficients associated with π and certain intertwining operators. To illustrate,we consider first the linear case. Let π be a ψ -generic representation of G , where ψ : U → C × is a nondegenerate character of the unipotent radical U of a Borel subgroup B = T U .We have the canonical identificationWh ψ ( π ) = Wh ψ ( π | G der ) . It follows from the multiplicity-one property of Whittaker functionals that one hasWh ψ ( π ) = Wh ψ ( π )for a unique constituent π ⊂ π | G der . Thus, the arithmetic information encoded in thelocal coefficients of π for G is completely elucidated from this unique π . The simplestsuch example is when π = I ( χ ) is a principal series of G as above. This is in contrastwith the situation for covers. Indeed, although for π ∈ Irr gen ( G ) we still haveWh ψ ( π ) = Wh ψ ( π | G der ) , the two sides might be of high dimensions. In fact, it is possible that dim Wh ψ ( π i ) > π i ⊂ π | G der . Hence, it is necessary to study every space Wh ψ ( π i ),as there may not be any distinguished one. Already as a preliminary step, it is essentialto study the multiplicity formula in the decomposition π | G der .In fact, the relation between the Whittaker spaces and restriction problems can bedeepened if one considers the more “restrictive” Whittaker space. More precisely, the FAN GAO, FREYDOON SHAHIDI, AND DANI SZPRUCH ψ -Whittaker space Wh ψ ( π ) of π is naturally a genuine Z ( G )-module, where Z ( G ) isa Heisenberg type group and thus its genuine irreducible representations are finite di-mensional with the same dimension. If the uniqueness of ψ -Whittaker functionals for π holds, then Z ( G ) is necessarily abelian. The converse may not hold. However, assumethat Z ( G ) is abelian and µ a genuine character of it, and let ψ × µ be the correspondingrepresentation of U × Z ( G ), then one can ask for the dimension of the µ -isotypic subspaceWh ψ × µ ( π ) ⊂ Wh ψ ( π ) . For the two-fold Kazhdan–Patterson covering GL (2)2 with twisting parameter c = 0 (inthe notation of [KP84]), it was shown by Gelbart, Howe, and Piatetski-Shapiro [GHPS79]that dim Wh ψ × µ ( π ) π ∈ Irr gen (GL (2)2 ). This fact was used in [GPS80] tostudy distinguished theta representations. Such uniqueness results for Wh ψ × µ ( π ) do nothold for higher degree cover of GL , nor for covers of GL r in general. On the otherhand, similar results were obtained for certain covers of GSp r (see [Szp13b, Theorem5.1]), which follow from analyzing the restriction of π ∈ Irr gen (GSp (2)2 r ) to Sp (2)2 r and theuniqueness of Whittaker models for the metaplectic group Sp r as shown in [Szp07].We also mention that some restriction and induction results between the above dou-ble cover GL (2)2 and the derived subgroup SL (2)2 were explicated in the work [GPS83] byGelbart and Piatetski-Shapiro, for both local and global groups. For untwisted (i.e., c = 0 as above) Kazhdan–Patterson covers of GL r , similar results were obtained byAdams [Ada03]. In a more recent work [PPP16], Patel and Prasad refined the multiplic-ity formula for the restriction problem for GL (2)2 , by utilizing the Shimura–Waldspurgercorrespondence.It is also worthwhile to point out that in the tame setting when p ∤ n , even if I ( χ )is a ( K, s K )-unramified representation of G , where K ⊂ G is a hyperspecial maximalcompact subgroup of G and s K : K ֒ → G a fixed splitting (assuming its existence), there might exist constituents in I ( χ ) | G der whichare not unramified with respect to K := K ∩ G der and its inherited splitting into G der from s K . This already occurs for even-fold cover of G = GL and was discussed in some depth in [Szp19, GSS].Our paper is thus partly motivated from all the above and we investigate several aspectsof the representation theory of covering groups centered around such restriction problem,mainly on the following themes:(A1) the restriction and induction functor between G and G der on the representationside,(A2) natural speculations and results on the L-parameter side corresponding to therestriction and induction in (A1),(A3) the behaviour of the Whittaker spaces and the local coefficients matrices for rep-resentations of G and G der with respect to restriction.Since at this stage we do not have a full understanding of the ψ -Whittaker space (oreven its dimension) of an arbitrary genuine representation of a covering group, in orderto obtain precise results, at various places we focus only on genuine principal series, oreven on the unramified ones. Even with such confinement on the class of representationsunder investigation, exhibited are many new and interesting results which do not seemto exist for linear algebraic groups. ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 5
Main results.
We give a brief summary on the content and main results proved inthis paper. While some notations are standard, we refer the reader to the actual theoremswhere full elaboration is given not only on the notations and also on their content. Thethree themes in (A1)–(A3) are not studied in a strictly linear order and are interwovenwith the results proved in various sections.1.1.1. In §
2, we study the restriction and induction on the representation side between G and a certain normal subgroup H of G . For this purpose, we first recall the generalClifford–Mackey theory. There are essentially two families of such H considered in thispaper, which are of different nature.(i) In the first case, H ⊂ G is such that every π ∈ Irr gen ( G ) is H -concentrated (seeDefinition 2.1). This implies that the support of the Harish-Chandra characterdistribution Θ π lies in H . Such normal subgroup H will be the focus of discussionin § r , asan analogue to that of GL r studied in [Kab01, Mez04, Tak16, Tak17, Cai19].(ii) For the second type, we take H = Z ( G ) · G der , where Z ( G ) ⊂ G denotes the center of G . Here Z ( G ) and G der are of dual-pairalike inside G . Every irreducible representation of Z ( G ) · G der is of the form τ ⊠ ρ , where τ and ρ agree on the intersection of Z ( G ) and G der . Since Z ( G )is a Heisenberg-type group, τ ∈ Irr gen ( Z ( G )) is of finite dimension and deter-mined by its central character. The study of restriction and induction betweenrepresentations of G der and G essentially amounts to those between H and G .In fact, the consideration in (ii) above applies to the Levi subgroups M of G , andconcerns the remaining discussion in §
2. For convenience, we write henceforth M := M der for every Levi subgroup M ⊂ G , and denote M † := Z ( G ) · M ⊂ Z ( M ) · M . Thus, G denotes G der , and G † = Z ( G ) · G . Note that we use Z ( G ) (instead of Z ( M )) inthe definition of M † for all Levi subgroup M ⊂ G . In § MM † ( τ ⊠ ρ )to be irreducible. The idea is simple as we explain now. The group Q † := M /M † acts naturally on Q M := Z ( M † ) /Z ( M )by conjugation and thus on its Pontryagin dual d Q M . If the action of Q † on d Q M is free,then the Mackey’s theory gives the irreducibility of Ind MM † ( τ ⊠ ρ ). Similarly, we can define Q Z and Q M associated with Z ( G ) and M which gives a Q † -equivariant embedding d Q M c Q Z × d Q M . We have
Theorem 1.1 (Theorem 2.11) . Let M ⊂ G be a covering Levi subgroup. FAN GAO, FREYDOON SHAHIDI, AND DANI SZPRUCH (i)
Suppose the action of Q † on either c Q Z or d Q M is free. Then Ind MM † ( τ ⊠ ρ ) isirreducible for every τ ⊠ ρ ∈ Irr gen ( M † ) . (ii) Specializing to the case M = T , if the action of Q † on both c Q Z and d Q T are trivial,then for every π ∈ Irr gen ( T ) we have π | T † = ( τ ⊠ ρ ) ⊕ e for some e ∈ N . On the other hand, if Z ( T ) ⊂ Z ( Z ( G )) · Z ( T ) , then Ind TT † ( τ ⊠ ρ ) is an isotypic sum of an irreducible genuine representation of T . In Corollary 2.12, we give several explicit and equivalent conditions for the restriction π | T to be an isotypic sum. Following an idea from [Szp13b], we study the irreducibilityof parabolic induction for G and G , see Theorem 2.15.Several families of covers are worked out in detail in § r , GSp r and GSpin r +1 , and apply Theorem 2.11 and Corollary 2.12 to obtain concrete results,see Propositions 2.16, 2.20 and 2.22. We note that the example of covers of GSpin r +1 was already discussed by Kaplan in [Kap17]. The scheme of analysis in § (2)2 were generalized in[Szp13b] for GSp (2)2 r , where the latter also includes a discussion on the analogous problemof restriction and induction on covering Levi subgroups of GSp (2)2 r . It should be notedthat a common feature of all the covers studied in [Ada03, GPS83, Szp13b] is that Z ( G )is abelian. However, in this paper (and also the previous study [GSS]), we do not imposesuch constraints and treat covers in generality.In the last part of §
2, we work out explicitly in the tame case (i.e., when p ∤ n ) thedecomposition of a genuine principal series I ( χ ) of G when restricted to G . Recall thatin the tame case, if G splits over the hyperspecial maximal compact subgroup K ⊂ G generated by T ( O ) and e α ( O ), then we could fix such a splitting s K : K ֒ → G and considera ( K, s K )-unramified genuine principal series I ( χ ). Such a ( K, s K )-unramified genuineprincipal series is parabolically induced from i ( χ ) ∈ Irr gen ( T ). We set K = K ∩ G der . Theorem 1.2 (Theorem 2.24) . Let i ( χ ) := Ind TA ( ˜ χ ) ∈ Irr gen ( T ) be with central character χ . Let I ( χ ) be the associated genuine principal series of G . (i) One has i ( χ ) | T = (cid:12)(cid:12) X c Q,n (cid:12)(cid:12) · M γ ∈ X Γ Q,n / X c Q,n M ω γ,j ∈ E ( χ, γ ˜ χ O ; Z ( T )) i ( ω γ,j ) , where every i ( ω γ,j ) ∈ Irr gen ( T ) appears with multiplicity one in the double sumof the right hand side. (ii) Consequently, I ( χ ) | G = (cid:12)(cid:12) X c Q,n (cid:12)(cid:12) · M γ ∈ X Γ Q,n / X c Q,n M ω γ,j ∈ E ( χ, γ ˜ χ O ; Z ( T )) I ( ω γ,j ) for the restriction of the genuine principal series I ( χ ) . Here generically, the I ( ω γ,j ) ’s appear multiplicity-free in the double sum of the right hand side. (iii) Assume that I ( χ ) is an ( K, s K ) -unramified genuine principal series. Then I ( ω γ,j ) is ( K , s K ) -unramified if and only if ω γ,j belongs to the set E ( χ, ˜ χ O ; Z ( T )) , i.e.,with γ = 0 ∈ X Γ Q,n / X c Q,n being the trivial class.
ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 7
In the above theorem, the finite abelian group X Γ Q,n is a natural quotient of X Q,n ,the latter of which is the “moduli space” of the Whittaker space Wh ψ ( I ( χ )) when I ( χ )is unramified, as discussed in detail in [GSS]. We highlight that from Theorem 1.2 (iii)above, the genuine principal series I ( ω γ,j ) of G with γ = 0 is not ( K , s K )-unramified. Infact, such I ( ω γ,j ) , γ = 0 is unramified with respect to ( K ′ , s K ′ ) inherited from a differentpair ( K ′ , s K ′ ) for G .It is easy to work out from Theorem 1.2 when the restriction I ( χ ) | G is an isotypic sumof a genuine principal series: this is exactly when ( G, G ) is an isotypic pair in the senseof Definition 2.23, i.e., when Z ( T ) ⊂ Z ( T ) , or equivalently by Corollary 2.12, Y ∩ Y Q,n = Y ,Q,n .1.1.2. In §
3, we study the dual side of the restriction and induction discussed on therepresentation side in §
2. Some anomalies in the covering setting were already discussedin [GG18] and part of the work here may be considered as a generalization and furtherelaboration of some speculations there. To motivate the general setup, it is instructive toconsider the Kazhdan–Patterson double cover G = GL (2)2 , which by restriction gives theclassical metaplectic double cover G = SL (2)2 . Regarding the dual groups, one has G ∨ = GL and G ∨ = SL . Clearly, there is no natural nontrivial homomorphism of algebraic groups from G ∨ to G ∨ ,which supposedly encodes from the dual side the restriction functor from G to G . Onthe other hand, both G ∨ and G ∨ map naturally to PGL , which thus serves as the bridgebetween the two dual groups.For general G , there is a natural linear algebraic group H and a cover H whose dualgroup H ∨ is endowed with two natural maps: f G,H : G ∨ −→ H ∨ and f G ,H : G ∨ −→ H ∨ . These two maps can be extended to homormophisms of the corresponding L-groups.Thus, we expect a natural commutative diagram L Z ( G ) L Z ( G )WD F L G L G L H f c,z φ π φ τ φ ρ f G,z f G,H f Go,H indicating some natural speculations on the pertinent parameters for representationsinvolved in the restriction and induction between G and G . This we explain below.Denote by φ : WD F −→ L G an L-parameter of G . Let L ( φ ) be the associated hypothetical L-packet. We assume thestrong form of the local Langlands correspondence that every φ ∈ L ( φ ) is associated withan enhanced parameter ( φ, θ ) ∈ Φ en ( L G ) , where θ is an irreducible representation of the component group S φ := S ( φ ) of φ . Denote φ ♦ = f G,H ◦ φ, FAN GAO, FREYDOON SHAHIDI, AND DANI SZPRUCH and consider the naturally induced map S φ ֒ → S φ ♦ , which is an embedding (see [Sol20, Proposition 5.4]). Let Φ( L G ; φ ) be the set of param-eters φ : WD F → L G such that f G ,H ◦ φ = φ ♦ . For every φ ∈ Φ( L G ; φ ), we have anembedding S φ ֒ → S φ ♦ as well. Conjecture 1.3 (Conjecture 3.2) . Let π ∈ Irr gen ( G ) be with central character ω π and as-sociated enhanced parameter ( φ π , θ π ) ∈ Φ en ( L G ) . Let τ ⊠ ρ be an irreducible representationof G † = Z ( G ) · G occurring in the restriction of π . (i) The L-parameters φ ω π , φ τ and φ ρ are such that the following hold: • f G,z ◦ φ π = f c,z ◦ φ τ and is equal to the parameter φ ω π associated with thecentral character ω π of π ; • φ ♦ π = f G ,H ◦ φ ρ . (ii) There exists e ( π ) ∈ N such that for every τ ∈ Irr gen ( G ) with enhanced parameter ( φ τ , θ τ ) ∈ Φ en ( L G ) , one has dim Hom G ( π, τ ) = if φ τ / ∈ Φ( L G ; φ π ) ,e ( π ) · D Ind S φ ♦ S φπ θ π , Ind S φ ♦ S φτ θ τ E S φ ♦ if φ τ ∈ Φ( L G ; φ π ) , where h− , −i denotes the pairing of two representations of S φ ♦ ; in particular, if ( G, G ) is an isotypic-pair, then π | G = e ( π ) · M τ ∈L ( φ ♦ π ) (cid:10) θ τ | S φπ , θ π (cid:11) S φπ · τ, where ( φ ♦ π , θ τ ) ∈ Φ en ( L G ) is the enhanced parameter associated with τ ∈ L ( φ ♦ π ) . (iii) Let τ ∈ Irr gen ( Z ( G )) and ρ ∈ Irr gen ( G ) be compatible representations. Then everyirreducible constituent π ∈ Irr gen ( G ) of Ind GG † ( τ ⊠ ρ ) has a parameter φ π which fitsinto a commutative diagram (3.6) . Part (ii) above is clearly motivated from the case of linear algebraic groups, see [AP19]and references therein. Conjecture 1.3 is not a statement that can be tackled at the mo-ment for general π ∈ Irr gen ( G ), as it relies on an established local Langlands correspon-dence (LLC). For example, it would be interesting to see if the results as in [PPP16] forGL (2)2 fit into Conjecture 1.3, if one has an established LLC for GL (2)2 . For general coveringtorus for which LLC has been proved (see [KP84, McN12, Wei09, Wei16b, Wei18a, GG18]),we verify this in § §
3, we study the metaplectic tensor product and a metaplecticrestriction (the latter construction as a counterpart of the former) for covers GSp r . ForKazhdan–Patterson covers GL r , this problem was studied by P. Mezo [Mez04] and alsoin several works following it, as mentioned in § M r = GL r × ... × GL r k , the covering subgroups GL r i and GL r j of GL r do not commute in general, and thus therepresentation theory of M r is not obtained from the naive tensor product of a genuinerepresentation of each GL r i . However, there is still a surjective map (see [Mez04])˜ ⊗ : (Irr gen (GL r ) × ... × Irr gen (GL r k ) × Irr gen ( Z (GL r ))) ♥ Irr gen ( M r ) , ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 9 where the superscript ( − ) ♥ indicates the subset of( π , ..., π r , ω )satisfying certain compatible relation. The map ˜ ⊗ is called the metaplectic tensor productconstruction. The functorial interpretation of ˜ ⊗ on the dual side was given by Gan[Gan17].As an analogue of GL r above, in § r . Let r =( r , r , ..., r k ; r ) be a partition of r . Let M r = GL r × GL r × ... × GL r k × GSp r be the associated Levi subgroup. We show the following. Theorem 1.4 (Theorem 3.8) . Let
GSp r be a similitudes-splitting n -fold cover of GSp r with n odd. There is a well-defined bijective map ˜ ⊗ i : ( Q ki =1 Irr gen (GL r i ) × Irr gen (GSp r ) × Irr gen ( Z (GSp r ))) ♥ Irr gen ( M r ) , where the superscript ( − ) ♥ indicates the subset of ( π , ..., π k , π , ω ) satisfying a certainequality (3.15) . The inverse of ˜ ⊗ i is given by a natural and explicit “metaplectic restric-tion” map ˜R . One has a conjectural dual side description of the two maps ˜ ⊗ i and ˜R given in thetheorem above, on the relations among the parameters of the representations involved.This is Conjecture 3.10, and is clearly motivated from that in [Gan17] for Kazhdan–Patterson covers; in fact, it is a simpler situation dealing with GSp r here. For genuineprincipal series, we verify Conjecture 3.10 in Proposition 3.11. We also explain theobstacles to realizing a similar metaplectic tensor product construction for even-fold coverof GSp r .In fact, such metaplectic tensor product construction is ubiquitous for covering Levisubgroups. Indeed, it essentially amounts to the existence of a certain subgroup H ⊂ M satisfying two conditions which counteract against each other. On the one hand, wewould like that every π ∈ Irr gen ( M ) is H -concentrated, and thus H should be taken aslarge as possible. On the other hand, suppose M i ⊂ M are the covering blocks, then itis a desired property that H ∩ M i commutes with each other: this is satisfied only when H is as small as possible. Whenever a subgroup H ⊂ M satisfying these two conditionsexist, one could have a metaplectic tensor product. For M r of the Kazhdan–Pattersoncovers, we take H to be essentially the subgroup of elements with determinants lying in F × n ; while for GSp ( n )2 r with n odd, we take the subgroup with similitudes in F × n . Formore details, see § §
4, we study how the local coefficients matrices associated with certain inter-twining operators behave with respect to the restriction. More precisely, suppose π | G der = M i m i · π i , with π i ∈ Irr gen ( G der ). Then one hasWh ψ ( π ) = M i m i · Wh ψ ( π i ) . Let M ⊂ P ⊂ G be a Levi subgroup and σ ∈ Irr gen ( M ). Let T ( w, σ ) : I GP ( σ ) −→ I GP ′ ( w σ ) be the standard intertwining operator, where P ′ is a parabolic subgroup associated with P by w . It is shown in [GSS, § T ( w, σ ) ∗ : Wh ψ ( I GP ( σ )) −→ Wh ψ ( I GP ( σ )) . With the choice of a basis B for Wh ψ ( I GP ( σ )), one obtains a local coefficients matrix M B ( w, σ )representing T ( w, σ ) ∗ .For every σ ∈ Irr gen ( M ), one has an analogous map T ( w, σ ) ∗ : Wh ψ ( I G P ( σ )) −→ Wh ψ ( I G P ( σ )) . We show that T ( w, σ ) ∗ = M σ i ⊂ σ | M T ( w, σ i ) ∗ . This implies that there is a local coefficients matrix M B ( w, σ ) taking the form M σ i ⊂ σ | M M B i ( w, σ i ) , where M B i ( w, σ i ) is a local coefficients matrix associated with the triple ( P , w, σ i ) anda basis B i of Wh ψ ( I G P ( σ i )), see Proposition 4.1. In particular, the arithmetic invariants(trace and determinant, for example) of M B ( w, σ ) are completely determined by thoseof M B i ( w, σ i ).Since the Whittaker space Wh ψ ( π ) is best understood for a genuine principal series, wespecialize to this case and investigate the relation between Wh ψ ( π ) and Wh ψ ( π i ) , π i ⊂ π | G der , where π ∈ JH( I ( χ )) is an irreducible constituent of a ( K, s K )-unramified genuineprincipal series. In fact, we consider the two special cases:– when I ( χ ) is a regular unramified principal series in § I ( χ ) is a unitary unramified principal series in § I ( χ ) is a regular unramified principal series. Let Φ( χ ) be the set ofrank-one reducibility points. Then every irreducible constituent of I ( χ ) is uniquelyparametrized by a subset S ⊂ Φ( χ ), which we denote by π χ,S or π ( χ ) S . If ( G, G ) isan isotypic pair, then one has Φ( χ ) = Φ( ω ) where ω = χ | Z ( T ) ; in this case,( π χ,S ) | G = (cid:12)(cid:12) X Γ Q,n (cid:12)(cid:12) · π ω,S , which immediately yields the equality(1.1) dim Wh ψ ( π χ,S ) = (cid:12)(cid:12) X Γ Q,n (cid:12)(cid:12) · dim Wh ψ ( π ω,S ) . Assume Φ( χ ) ⊂ ∆ , i.e., it is a subset of simple roots. Then we have a formula for dim Wh ψ ( π χ,S ) anddim Wh ψ ( π ω,S ) in terms of certain permutation representations σ X and σ X of W re-spectively. It is expected that the equality (1.1) of Whittaker dimensions actually orig-inates from a relation between σ X and σ X , see Conjecture 4.6. If ( G, G ) is not anisotypic pair, then the relation between dim Wh ψ ( π χ,S ) and dim Wh ψ ( π ω,S ) for an arbi-trary I ( ω ) ⊂ I ( χ ) | G is quite complicated, since I ( ω ) may not be ( K , s K )-unramified.This already occurs for even-fold cover of GL (odd-fold cover of GL gives an isotypicpair). However, for GL , by using the work in [GSS] we can show the following: ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 11
Theorem 1.5 (Theorem 4.9) . Consider the Kazhdan-Patterson n -fold cover GL . Let χ : Z ( T ) → C × be an unramified genuine character such that Φ( χ ) = ∆ . Assume n = 2 m is even and f ( ψ ) = O F . We always have dim Wh ψ ( π ( ω γ i ,j ) ∅ ) + dim Wh ψ ( π ( ω γ i ,j ) ∆ ) = dim Wh ψ ( I ( ω γ i ,j )) = m. Moreover, we have an explicit formula for every dim Wh ψ ( π ( ω γ i ,j ) ∆ ) , and thus also dim Wh ψ ( π ( ω γ i ,j ) ∅ . Second, assume I ( χ ) is a unitary ( K, s K )-unramified principal series. Then we have adecomposition I ( χ ) = M τ ∈ Irr( S χ ) π τ of I ( χ ) into irreducible constituents, where S χ := S φ χ is the component group of the centralizer of Im( φ χ ) ⊂ L G . If I ( ω ♭ ) ⊂ I ( χ ) | G is a( K , s K )-unramified constituent as in Theorem 1.2, then one has analogously I ( ω ♭ ) = M ρ ∈ Irr( S ω♭ ) π ρ . Recall the map φ ♦ := f G,H ◦ φ χ = f G ,H ◦ φ ω ♭ and the associated component group S ♦ = S φ ⋄ . We have the inclusions S χ S ω ♭ S ♦ . The main result of § G ( π ρ , π τ ). Theorem 1.6 (Theorem 4.12) . Let I ( χ ) be a unitary ( K, s K ) -unramified genuine prin-cipal series of G and I ( ω ♭ ) a ( K , s K ) -unramified constituent of I ( χ ) | G . Then (1.2) dim Hom G ( π ρ , π τ ) = (cid:12)(cid:12) X c Q,n (cid:12)(cid:12) · D Ind S ♦ S ω♭ ρ, Ind S ♦ S χ τ E S ♦ for every τ ∈ Irr( S χ ) and ρ ∈ Irr( S ω ♭ ) . In fact, even if I ( ω ♭ ) ⊂ I ( χ ) | G is not ( K , s K )-unramified, the equality holds as well,see Corollary 4.13. In particular, this verifies Conjecture 1.3 (ii) for unitary unramifiedprincipal series of G with e ( I ( χ )) = (cid:12)(cid:12) X c Q,n (cid:12)(cid:12) . If n = 1 and G = G is a split linear group, then the above formula recovers that in[Key87], the proof of which in fact motivates that for Theorem 1.6 above.1.1.4. In §
5, we concentrate on unramified L-packets and consider the following twoproblems:(i) the variation of the parametrization of elements inside an unramified L-packetwith respect to the change of hyperspecial maximal compact subgroup K of G and different choices of splittings,(ii) the variation of the Whittaker dimensions of elements inside an unramified L-packet with respect to changing orbits of the Whittaker datum. Let χ : Z ( T ) → C × be a genuine central character. Fix a splitting s T : T ( O ) ֒ → T , andassume that i ( χ ) is ( T ( O ) , s T )-unramified. By the local Langlands correspondence forcovering tori, one has a parameter φ χ : WD F −→ L T ֒ → L G. It is postulated that the L-packet L ( φ χ ) associated with φ χ consists of exactly the sub-quotients of the principal series I ( χ ), which are ( K, s K )-unramified with respect to ahyperspecial maximal compact subgroup K ⊂ G and a splitting s K : K ֒ → G such that s K restricts to s T on T ( O ). It is also expected that there is a bijection between L ( φ χ ) ←→ Irr( S φ χ )which we denote by π ( φ χ , ρ ) ↔ ρ. Thus, we want to investigate how the (
K, s K )-unramifiedness of an element π ( φ χ , ρ ) ∈L ( φ χ ) is reflected from its parameter ρ .For this purpose, we briefly recall the linear algebraic case. Let K = { G x : x is a hyperspecial point in B ( G ) } / ∼ be the set of conjugacy classes of hyperspecial maximal compact subgroups of G , where B ( G ) is the Bruhat–Tits building associated with G . It is known that K is a torsor over d Γ tor G := Hom(( X/X sc ) tor , Q / Z ) , and there is a natural surjective map f Γ : d Γ tor G ։ Irr( S φ χ ) , which gives a natural action of d Γ tor G on Irr( S φ χ ) given by f Γ ( y ) ⊗ ρ . It follows from[Mis, Theorem 1] that for every K ∈ K and y ∈ d Γ tor G , the representation π ( φ χ , ρ ) ∈ L ( φ χ )is K -unramified if and only if π ( φ χ , f Γ ( y ) ⊗ ρ ) is y · K -unramified.To generalize the above results to covering groups, we note the following obstacles: • first, not every splitting of K into G is compatible with s T , and thus it is necessaryto filter only those compatible ones, • second, although K stays as a torsor over d Γ tor G , there is no longer any obviousnatural map from d Γ tor G to Irr( S φ χ ).For the second obstacle above, since S φ χ is the component group of the image of φ χ in L G , there is a natural finite abelian group [ Γ tor G Q,n arising from the principal endoscopicgroup G Q,n of G and a surjection ˜ f Γ : [ Γ tor G Q,n ։ Irr( S φ χ ) . The map ˜ f Γ is defined analogous to f Γ in the linear case. With this in view, it is importantto bridge the two groups d Γ tor G and [ Γ tor G Q,n . A substantial part of § B = T U and vary the non-degenerate character ψ : U → C × . Let T ad ⊂ G ad be the split torus of G ad . The set of T -orbits of non-degenerate characters of U is a torsor over T ad /T under the conjugationaction. There is a natural quotient T ad /T ։ Irr( S φ χ ) , and thus an action of t ∈ T ad /T on ρ ∈ Irr( S φ χ ), which we denote by t · ρ . In this case,it was shown in [Mis16] that π ( φ χ , ρ ) is ψ -generic if and only if π ( φ χ , t · ρ ) is t ψ -generic. ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 13
Now for a covering group G , there is no obvious map from T ad /T to Irr( S φ χ ). Similar tothe above discussion on the unramified-ness, there is a natural surjection T Q,n,ad /T Q,n ։ Irr( S φ χ ) , where T Q,n is the torus of G Q,n and T Q,n,ad is the torus for its adjoint group G Q,n,ad .Again, it is important to inspect any natural relation between T Q,n,ad /T Q,n and T ad /T , orat the level of character lattices, to relate X Q,n /X scQ,n and X/X sc . We discuss this in § ψ -Whittaker dimension of π ( φ χ , ρ ) with respect tothe action of t on the character ψ .The following is the main result of §
5, amalgamated from the discussions in § Theorem 1.7.
Conjecture 5.4 and Conjecture 5.7 hold for unitary unramified principalseries of covers of Sp r , SO , and also for Kazhdan–Patterson covers of GL r . §
6, we consider exclusively covers of GSp r and its unitary unramified principalseries I ( χ ). The discussion is oriented towards Conjecture 5.4 and Conjecture 5.7. Inparticular, we • determine the component group S φ χ and the pair ( K, s K ) with respect to which π ( φ χ , ρ ) is unramified; • determine the ψ -Whittaker dimension of each constituent π ( φ χ , ρ ), when f ( ψ ) = p F or O F ; • investigate the restriction of each π ( φ χ , ρ ) to the derived subgroup G = Sp r andthe pair ( K , s K ) with respect to which an irreducible constituent of π ( φ χ , ρ ) | Sp r is unramified, and also study the ψ -Whittaker dimension of such a constituent.We summarize (in a compressed form) the main results for § K denotes the standard hyperspecial maximal compact subgroupand K := K ∩ Sp r . Also, { K , K ′ } denotes the set of conjugacy classes of hyperspecialmaximal compact subgroup of Sp r , and s , s ′ denote the unique splittings of K and K ′ into Sp r respectively. Let ψ and t ∈ T ad /T be such that f ( ψ ) = O F and f ( t ψ ) = p F . We also write n ( r ) := n/ gcd( n, r ) . Theorem 1.8 (Theorems 6.1, 6.2 and 6.4) . Consider the n -fold cover GSp r of similitudes-splitting type. Let I ( χ ) be a unitary ( K, s K ) -unramified genuine principal series of GSp r . (i) If n is odd, then I ( χ ) | G = (cid:12)(cid:12) X Γ Q,n (cid:12)(cid:12) · I ( ω ) , where ω = χ | Z ( T ) . If I ( ω ) = π ( φ ω , ) ⊕ π ( φ ω , ε ) is reducible, and we assume that π ( φ ω , ) is ( K , s ) -unramified, then π ( φ ω , ε ) is ( K ′ , s ′ ) -unramified. (ii) If n is even and r is odd, then I ( χ ) is always irreducible and I ( χ ) | G = n ( r ) · ( I ( ω , ) ⊕ I ( ω , )) M n ( r ) · ( I ( ω e , ) ⊕ I ( ω e , )) , where every I ( ω γ,j ) is irreducible. Moreover, I ( ω ,j ) , j = 0 , is ( K , s ) -unramifiedand I ( ω e ,j ) , j = 0 , is ( K ′ , s ′ ) -unramified. (iii) If n is even and r is even, then I ( χ ) | G = n ( r ) · ( I ( ω , ) ⊕ I ( ω , )) M n ( r ) · ( I ( ω e , ) ⊕ I ( ω e , )) , where I ( ω ,j ) is ( K , s ) -unramified and I ( ω e ,j ) is ( K ′ , s ′ ) -unramified. Here every I ( ω γ,j ) is irreducible. If I ( χ ) = π ( φ χ , ) ⊕ π ( φ χ , ε ) is reducible, then π ( φ χ , ) | G ≃ π ( φ χ , ε ) | G ≃ n ( r ) · I ( ω , ) M n ( r ) · I ( ω e , ) ≃ n ( r ) · I ( ω , ) M n ( r ) · I ( ω e , ) . In all cases above, dim Wh ψ ( π ( φ ω γ,j , ρ )) and dim Wh t ψ ( π ( φ ω γ,j , ρ )) are determined explic-itly for every ρ ∈ S ( φ ω γ,j ) . The above theorem generalizes part of the results in [Szp15].1.2.
Several remarks.
Some comments are in order pertaining to the results provedabove.(i) In this paper, we mostly concentrate on genuine principal series of G or evenrestrict to the unramified ones. It is desirable to work out for other classes ofrepresentations for G in view of the conjectural formulas studied in § ψ -Whittaker space of π ( φ χ , ρ ) for varying ψ , it seems to be helpful, or perhaps even essential, to have an explicit descriptionof the local coefficients matrix in the tame case. We leave the investigation oftame local coefficients matrix to a future work.(iii) To the best of our knowledge, in the literature there has not been any computationof the archimedean local coefficients matrix or scattering matrix for double covers G R of G ( R ) in the real case, except for the metaplectic double cover Mp r ofSp r ( R ) considered in [Szp09, Szp13a]. In general, the ψ -Whittaker space of agenuine principal series of G R may also be of high dimension. Besides Mp r ,the first examples to consider are the double covers SL , R and Spin , R . It isexpected that analogous results regarding the invariants of such a local coefficientsmatrix hold in this archimedean setting, by employing the methods of partial zetaintegrals as developed in [Szp19] and already used in [GSS].(iv) When the representation is not generic, an analogue of the local coefficients forcertain classical groups was studied by Friedberg and Goldberg [FG99] by consid-ering the generalized Bessel models. The simplest family of genuine representa-tions for covering groups concerns the theta representations Θ( χ ). If the degreeof covering is “small” relative to the rank of the group, then Θ( χ ) is not generic.It is interesting to investigate the Bessel models for these theta representations.On the other hand, the leading nilpotent orbit in the Harish-Chandra characterexpansion governs the generalized or degenerate Whittaker functionals for Θ( χ ),see [MW87, Var17, PP15]. It is interesting to see further how such generalizedBessel models and generalized Whittaker functionals are related to each other, inpursuit of the representation-theoretic invariants. ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 15 (v) Especially lacking from the dual side is an interpretation of the high dimensionof Whittaker models. Recall that for linear algebraic groups, it was a conjectureof the second-named author [Sha90] that there is a bijection between L-packetsand tempered parameters, which now has been proved in many cases, see [Sha11]and references therein. Moreover, it follows from Gross–Reeder [GR10] that theadjoint L -function of a generic discrete series is necessarily holomorphic at s =0. For covering groups, it is expected that every L-packet associated with atempered parameter also contains a generic representation; however, the conversealready fails for theta representations mentioned in (iv) above. The L-parameterfor the theta representation Θ( χ ) is never tempered, although we could havedim Wh ψ (Θ( χ )) > n ≫ r . We do not yet know an interpretation from thedual side of the quantity dim Wh ψ (Θ( χ )), which was shown to be equal to thenumber of certain Weyl orbits. In fact, we are still very lacking any evidence fora general formula for dim Wh ψ ( π ) , π ∈ Irr gen ( G ).It is clear that several speculations and results proved in this paper are motivated fromits linear algebraic counterpart, while some others are new only in the covering setting,for example the metaplectic tensor product. We hope that the paper could provide amotivation to a further study on various aspects of the restriction problem for covers andthe theory of local coefficients beyond the spectrum of genuine principal series. Again,as in [GSS], we strive to work with the most general setup if possible, and assume onlythe minimal requirement. Hopefully, this could provide some convenience for any furtherstudy on the topic.1.3. Acknowledgement.
We would like to thank Wee Teck Gan, Eyal Kaplan, andDipendra Prasad for some very helpful communications on certain topics discussed inthis paper. The second author was partially supported by NSF grant DMS-1801273.2.
Restriction and induction of genuine representations
Covering groups and L-groups.
We follow the exposition in [GSS, §
2] to sum-marize some important results on the structure of covering groups and the construc-tion of their dual groups and L-groups. For more details, the reader is referred to[BD01, McN12, Wei11, Wei14, Wei18a, Wei18b, GG18].2.1.1.
Covering groups.
Let G be a split connected linear reductive group over a p -adicfield F with root datum ( X, Φ , ∆; Y, Φ ∨ , ∆ ∨ ) . Here we fix a maximal split torus T ⊂ G , and X (resp. Y ) is the character (resp.cocharacter) lattice of T . Choose a set of simple roots ∆ from the set Φ of roots. Onehas the corresponding simple coroots ∆ ∨ . Let B = TU be the Borel subgroup associatedwith ∆. Let W = N ( T ) / T be the Weyl group of ( G , T ), which we identify with the Weyl group of the coroot systemgenerated by the simply reflections w α , α ∈ ∆. We also fix a Chevalley–Steinberg systemof pinnings, i.e., a compatible system { e α : G a → U α } α ∈ Φ , where U α ⊂ G is the root subgroup associated with α . Denote by G, B, T the F -rationalpoints of G , B , T respectively.Let D : Y × Y −→ Z be a bilinear form such that(2.1) Q ( y ) = D ( y, y )is a Weyl-invariant bilinear form of Y . Here D may not be symmetric. If Q is an integralWeyl-invariant bilinear form, then it is known (see [Wei14]) that there exists D such that(2.1) holds. Let B Q : Y × Y −→ Z be the Weyl-invariant bilinear form given by B Q ( y, z ) = D ( y, z ) + D ( z, y ) , which actually only depends on Q and thus justifies the notation used here. Furthermore,let η : Y sc −→ F × be a homomorphism of the coroot lattice Y sc ⊂ Y into F × . Denote by η n : Y sc −→ F × −→ F × /F × n the composite of η with the obvious quotient.Every couple ( D, η ) gives rise to a K -extension G of G . Assuming F × contains thethe full group of n -th roots of unity, denoted by µ n , one has an n -fold cover µ n G G p obtained from the n -th Hilbert symbol( − , − ) n : K ( F ) −→ µ n . A representation ( π, V π ) of G is called genuine if µ n acts on V π by a fixed embedding µ n ֒ → C × . In this paper, we denote by Irr gen ( G )the set of isomorphism classes of irreducible genuine representations of G .The covering group G splits over unipotent subgroups canonically and G -equivariantly,where the action of G on G is by conjugation. Namely, if U ⊂ G is a unipotent subgroup,then there is a unique splitting s U : U −→ G satisfying s U ( gug − ) = gs U ( u ) g − , where the right hand side is independent of the choice of lifting g ∈ G of g ∈ G .Denote by e α ( F ) the splitting of e α ( F ) in G . For any α ∈ Φ and x ∈ F × , define(2.2) w α ( x ) := e α ( x ) e − α ( − x − ) e α ( x )and(2.3) h α ( x ) := w α ( x ) w α ( − . For any α ∈ Φ, for simplicity we write w α := w α (1) . When we consider the case n = 1 (i.e., G = G ), we will use the notations e α , w α , h α for e α , w α , h α respectively.The group G is generated by the union of the sets µ n , { e α ( F ) } α ∈ Φ and { y ( F × ) } y ∈ Y (see [BLS99, Theorem 3]). Relations among the generators include the following:(A) e α ( x ) is additive in x . ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 17 (B) If α and β are roots with α + β = 0, then the commutator[ e α ( x ) , e β ( y )] = Y e iα + jβ ( c i,j x i y j ) , where i and j are positive integers and c i,j ’s are certain integers.(B)’ For any α ∈ Φ and x ∈ F × : w α ( x ) e α ( u ) w α ( x ) − = e − α ( − x u ) . (C) There exists a section s of T over T such that s ( y ( a )) · s ( y ( b )) = s ( y ( a ) · y ( b )) · ( a, b ) D ( y ,y ) n for any y , y ∈ Y and a, b ∈ F × . For any α ∈ ∆ and x ∈ F × , one has h α ( x ) = s ( α ∨ ( x )) · ( η ( α ∨ ) , x ) Q ( α ∨ ) n . (D) For t ∈ T whose image in T is denoted by t , one has(2.4) w α · t · w − α = t · h α ( α ( t ) − ) . By (C) above, the commutator [ − , − ] : T × T −→ µ n is given by [ y ( a ) , y ( b )] = ( a, b ) B Q ( y ,y ) n , where y i ∈ Y and a, b ∈ F × . Fix a uniformizer ̟ ∈ F . For any y ∈ Y , we write ε := ( − , ̟ ) n ∈ µ n , s y := s ( y ( ̟ )) ∈ T .
For every n, r ∈ N , we also denote in this paper n ( r ) = n/ gcd( n, r ) . It is clear that n ( r ) and r ( n ) are coprime.2.1.2. Dual group.
For a cover ( G , n ) associated to ( D, η ), with Q and B Q arising from D , we define Y Q,n := Y ∩ nY ∗ , where Y ∗ ⊂ Y ⊗ Q is the dual lattice of Y with respect to B Q ; more explicitly,(2.5) Y Q,n = { y ∈ Y : B Q ( y, y ′ ) ∈ n Z for all y ′ ∈ Y } ⊂ Y. For every α ∨ ∈ Φ ∨ , denote n α := n ( Q ( α ∨ )) = n gcd( n, Q ( α ∨ ))and α ∨ Q,n = n α α ∨ , α Q,n = αn α . Let Y scQ,n ⊂ Y Q,n be the sublattice generated by Φ ∨ Q,n = { α ∨ Q,n : α ∨ ∈ Φ ∨ } . Denote X Q,n = Hom Z ( Y Q,n , Z )and Φ Q,n = { α Q,n : α ∈ Φ } . We also write∆ ∨ Q,n = { α ∨ Q,n : α ∨ ∈ ∆ ∨ } and ∆ Q,n = { α Q,n : α ∈ ∆ } . Then (cid:0) Y Q,n , Φ ∨ Q,n , ∆ ∨ Q,n ; X Q,n , Φ ∨ Q,n , ∆ Q,n (cid:1) forms a root datum with a choice of simple roots ∆
Q,n . It gives a unique (up to uniqueisomorphism) pinned reductive group G ∨ over Z , called the dual group of ( G , n ). Inparticular, Y Q,n is the character lattice for G ∨ and ∆ ∨ Q,n the set of simple roots. Let G ∨ := G ∨ ( C )be the associated complex dual group. The center of G ∨ is Z ( G ∨ ) = Hom( Y Q,n /Y scQ,n , C × ) . Let G Q,n be the pinned split reductive group over F such that G ∨ Q,n ≃ G ∨ , where G ∨ Q,n is the Langlands dual group of G Q,n . Then G Q,n is the principal endoscopicgroup of ( G , n ), and clearly G Q,n = G if n = 1.2.1.3. L-group.
In [Wei14, Wei18a], Weissman constructed the global L-group as well asthe local L-group extension G ∨ Q,n L G WD F , which is compatible with the global L-group. Here WD F is the Weil–Deligne group of F .His construction of L-group is functorial, and in particular it behaves well with respectto the restriction of G to parabolic subgroups. More precisely, let M ⊂ G be a Levisubgroup. By restriction, one has the n -fold cover M of M . Then the L-groups L M and L G are compatible, i.e., there are natural homomorphisms of extensions: G ∨ Q,n L G WD F M ∨ Q,n L M WD F . For details on the construction and some properties regarding the L-group, we refer thereader to [Wei14, Wei18a, GG18].If η n = , then there exists a so-called distinguished genuine character χ ψ : Z ( T ) −→ C × , depending on a nontrivial additive character ψ of F , such that the following propertieshold: • the character χ ψ takes values in µ ⊆ C × and is Weyl-invariant, i.e., χ ψ ( w − · t · w ) = χ ψ ( t ) for all t ∈ Z ( T ) and w ; • χ ψ gives rise to a splitting of L G over W F , with respect to which one has anisomorphism L G ≃ χ ψ G ∨ × WD F . The Clifford–Mackey theory.
We recall some results by Gelbart and Knapp[GK82, GK81] (see also [Tad92]) on the Clifford–Mackey theory of restriction and in-duction of representations in the setting of l -adic groups (see [BZ76]). In particular, itapplies to the µ n -cover of a linear algebraic group G discussed in § § ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 19
Let M be a totally disconnected group, i.e., a separable locally compact topologicalgroup whose open compact subgroups form a neighborhood base at the identity. For anadmissible representation π and e ∈ N , we will use e · π = π ⊕ e interchangeably to denote the isotypic sum of π with multiplicity e . Let H ⊂ M be an open normal subgroup such that M/H is a finite abelian group. One of two operations we consider in this paper is the restrictionof representations from M to H , while the other one is the induction from H to M . Forevery admissible representation ( ρ, V ρ ) of H and g ∈ M , there is the representation g ρ ( h ) := ρ ( g − hg )of H afforded by the same vector space V ρ but with the action twisted. We denote S ρ := Stab( ρ, M ) = { g ∈ M : g ρ ≃ ρ } , whenever the ambient group M is clear from the context. Denote by ω ρ : Z ( H ) −→ C × the central character of ρ . Also, for every subgroup K ⊂ M and g ∈ K , we define a map ϕ K,g : K −→ K by ϕ g ( x ) := [ x, g ] K , where [ g, g ′ ] K := gg ′ g − g ′− denotes the commutator of K . Definition 2.1.
For H ⊂ K ⊂ M , a representation σ ∈ Irr( K ) is called H -concentratedif (cid:8) g ∈ K : ϕ − K,g (cid:0) Z ( K ) − Ker( ω σ | Z ( K ) ) (cid:1) = ∅ (cid:9) ⊆ H. For every σ ∈ Irr( K ), denote by Θ σ the Harish-Chandra character distribution of σ ,which is conjugation invariant and can be represented by a locally-integrable function.The above definition is motivated from the following: Lemma 2.2.
Assume that σ ∈ Irr( K ) is H -concentrated. Then supp(Θ σ ) ⊂ H. Proof.
Let g ∈ K − H . Since σ is H -concentrated by assumption, there exists x ∈ K such that [ x, g ] ∈ Z ( K ) − Ker( ω σ | Z ( K ) ) . One has Θ σ ( g ) = Θ σ ([ x, g ] · g ) = ω σ ([ x, g ]) · Θ σ ( g ) , which shows that Θ σ ( g ) = 0 since ω σ ([ x, g ]) = 1. Thus, supp(Θ σ ) ⊂ H . (cid:3) Restriction.
Let π ∈ Irr( M ) be an irreducible admissible representation of M . Onehas (see [GK82, Lemma 2.1]) π | H = M g i ∈ M/S ρ ( g i ρ ) ⊕ e , where– ρ is an irreducible representation of H that occurs in π | H ;– g ρ is the twist of ρ by g ∈ M , and elements in { g i ρ : g i ∈ M/S ρ } are mutuallynon-isomorphic representations of H ;– e ∈ N is a certain ramification index (i.e., multiplicity) of g ρ , and in particular ρ ⊕ e is an isotypic component of ρ in π | H .One has S ρ = S ρ ⊕ e , and there is a natural irreducible representation of S ρ on ρ ⊕ e . Moreover, π = Ind MS ρ ( ρ ⊕ e ) . Note that the irreducible representation of S ρ on ρ ⊕ e depends on the π one started with.Regarding the restriction, there are two extreme cases:– if S ρ = M , then π | H is an isotypic sum of ρ , i.e., π | H = ρ ⊕ e ;– if S ρ = H , then e = 1 and in this case π = Ind MH ( ρ ).2.2.2. Induction.
Let ( ρ, V ρ ) be an irreducible admissible representation of H . The in-duced representation of ρ to M might not be irreducible. To obtain an irreducible repre-sentation of M from ρ , one approach is as follows. Lemma 2.3.
One can extend ρ to a representation ρ ♭ of a subgroup H ♭ ⊂ M (still actingon the same vector space V ρ ) such that S ρ ♭ = H ♭ . In this case, Ind MH ♭ ( ρ ♭ ) is irreducible.Proof. The proof is the same as in [Mez04, Page 89-90] almost word for word, by notingthat
M/H is an abelian group of finite order by assumption. (cid:3)
Using notations from the above lemma, one has(2.6) Ind H ♭ H ( ρ ) = M χ ∈ Hom( H ♭ /H, C × ) χ ⊗ ρ ♭ , where (cid:8) χ ⊗ ρ ♭ : χ ∈ Hom( H ♭ /H, C × ) (cid:9) gives all the possible extensions of ρ to H ♭ . For a general χ ∈ Hom( H ♭ /H, C × ), one canextend it to a character χ ′ : M/H −→ C × . Then one has Ind MH ♭ ( χ ⊗ ρ ♭ ) ≃ χ ′ · Ind MH ♭ ( ρ ♭ ) , which is an irreducible representation of M . Therefore, we have a decompositionInd MH ( ρ ) = M χ ∈ Hom( H ♭ /H, C × ) Ind MH ♭ ( χ ⊗ ρ ♭ )of Ind MH ( ρ ) into irreducible representations of M . ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 21
Proposition 2.4.
Assume that every irreducible representation of M is H -concentrated.Let ρ ∈ Irr( H ) . Then the representation Ind MH ( ρ ) = π ⊕ | H ♭ /H | is an isotypic sum with π := Ind HH ♭ ( ρ ♭ ) ∈ Irr( M ) given as in Lemma 2.3.Proof. Again, the argument is essentially that of [Mez04, Lemma 4.2] which deals withcovers of GL r ; the desired result in that context is given explicitly in [Tak16, Proposition4.6]. We sketch the argument for completeness.Denote σ := Ind MH ♭ ( χ ⊗ ρ ♭ ) for an arbitrary χ ∈ Hom( H ♭ /H, C × ), we want to showΘ σ = Θ π . First, applying Lemma 2.2 to K = M shows that it suffices to prove Θ σ ( x ) =Θ π ( x ) for x ∈ H . Since Θ σ ( x ) = X γ ∈ M/H ♭ ( χ · Θ ρ ♭ )( γ − xγ ) , it gives that for x ∈ H , Θ σ ( x ) = X γ ∈ M/H ♭ Θ ρ ♭ ( γ − xγ ) = Θ π ( x ) . This shows Θ σ = Θ π and thus σ ≃ π, see [BZ76, Corollary 2.20]. Since χ is arbitrary, the proof is completed. (cid:3) If every irreducible representation of M is H -concentrated, then for given ρ ∈ Irr( H )we denote by π [ ρ ] ∈ Irr( M )the unique irreducible representation appearing in Ind MH ( ρ ), as in Proposition 2.4. Since π [ ρ ] ∈ Irr( M ) is irreducible, by the discussion in § ρ ∈ Irr( S ρ ) such that(2.7) π [ ρ ] = Ind MS ρ ( ˜ ρ ) , where ˜ ρ | H = ρ ⊕ e for some e ∈ N . The relations among the two constructions are depictedin the following diagram(2.8) ( M, π [ ρ ])( S ρ , ˜ ρ ) ( H ♭ , ρ ♭ )( H, ρ ) . Note that here ˜ ρ arises from the restriction of the irreducible representation π [ ρ ]. Theleft two arrows follow from the Clifford–Mackey theory. On the other hand, the righttwo arrows in (2.8) together give an analogue of the Stone–von Neumann construction ofirreducible representations of Heisenberg groups, as illustrated by Proposition 2.4. Example 2.5.
Consider the covering torus µ n T T, which is a Heisenberg type group. The center Z ( T ) is of finite index in T . Consider thepair ( M, H ) = (
T , Z ( T )). Every irreducible representation of T is Z ( T )-concentrated. One has the diagram (2.8). To describe the right arrows of (2.8), let χ : Z ( T ) −→ C × be a genuine central character. We choose a maximal abelian subgroup A ⊂ T and anextension ˜ χ : A −→ C × of χ . Then π [ χ ] = Ind TA ( ˜ χ ) is irreducible and independent of thechoice of A and ˜ χ ; also, one hasdim π [ χ ] = q [ T : Z ( T )] , see [Wei09, Wei16b]. For the left arrows in (2.8), we have S χ = T and thus π [ χ ] | Z ( T ) = χ ⊕ e , where e = q [ T : Z ( T )]. We also use the notation i ( χ ) := π [ χ ]throughout the paper.2.3. Two special pictures.
Henceforth, we consider covers arising from the Brylinski–Deligne framework exclusively. Recall that T = Hom( X, F × ) = Y ⊗ F × . Definition 2.6.
A subgroup H ⊂ T is said to be associated with a sublattice L ⊂ Y if H = p − (Im( ι )) , where ι : L ⊗ F × → T is the map induced from the inclusion L ⊂ Y .Write L ( H ) for the unique sublattice of Y associated with H , whenever it exists.Assume that Z ( G ) is connected i.e., X/X sc is a free Z -module, or equivalently, the exactsequence X sc X X/X sc of Z -modules split. We thus have a W -equivariant embeddingHom( X/X sc , Z ) Hom( X, Z ) ≃ Y, the image of which we denote by Y c ⊂ Y . In particular, Z ( G ) is associated with thesublattice Y c ⊂ Y , see Definition 2.6. Let i Q,n : Y Q,n ⊗ F × −→ T = Y ⊗ F × be the isogeny induced from the inclusion Y Q,n ֒ → Y . It is known that (see [Wei09]) Z ( T ) = p − (Im( i Q,n ));that is, Y Q,n = L ( Z ( T )). Now we define Y z := Y Q,n ∩ Y c and let i zQ,n : Y z ⊗ F × −→ T be the map induced from the inclusion Y z ⊂ Y .The following has been observed and used in many places in the literature (for examplesee [KP84, CO13, Gan17]) Lemma 2.7.
Keeping notations as above, we have Z ( G ) = Z ( G ) ∩ Z ( T ) and Z ( G ) = p − (Im( i zQ,n )) , that is, Y z = L ( Z ( G )) . ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 23
Proof.
It suffices to prove the first equality. The inclusion Z ( G ) ⊂ Z ( G ) ∩ Z ( T )is clear. On the other hand, since G = T · G der , we see that if g ∈ Z ( G ) ∩ Z ( T ), then itcentralizes G der as well as T . Therefore Z ( G ) ∩ Z ( T ) ⊂ Z ( G ). (cid:3) We want to give a (hopefully proper) subgroup H ⊂ G such that every π ∈ Irr gen ( G )is H -concentrated, and therefore supp(Θ π ) ⊂ H. For this purpose, we assume that– there exists e s ∈ Y such that G = G der ⋊ e s ( F × );– one has Y c = Z e c for some e c ∈ Y .We denote n s = n/ gcd( n, B Q ( e c , e s )) . Proposition 2.8.
Keep notations as above and denote G n s := G der ⋊ e s ( F × n s ) . Thenevery irreducible genuine representation π of G is G n s -concentrated, and thus supp(Θ π ) ⊂ G n s . Consequently, Ind GG ns ( σ ) = m · π is an isotypic sum for every σ ∈ Irr gen ( G n s ) .Proof. Given with e s ( b ) ∈ e s ( F × ) ⊂ G , we claim that if [ e c ( a ) , e s ( b )] = 1 for all a ∈ F × ,then b ∈ F × n s . Noting that [ e c ( a ) , e s ( b )] = ( a, b ) B ( e c ,e s ) n , the claim follows from the non-degeneracy of the Hilbert symbol. Therefore, if b / ∈ F × n s ,then there exists e c ( a ) such that 1 = [ e c ( a ) , e s ( b )] ∈ µ n . Thus, if g / ∈ G der ⋊ e s ( F × n s ), then we have g = g · e s ( b ) with g ∈ G der and b / ∈ F × n s . Itthen gives[ g, e c ( a )] = [ g · e s ( b ) , e c ( a )] = [ e c ( a ) , e s ( b )] ∈ ϕ − G,g (cid:16) Z ( G ) − Ker( ω π | Z ( G ) ) (cid:17) , and this shows that π is G n s -concentrated. The rest is clear in view of Proposition 2.4. (cid:3) Remark 2.9.
Proposition 2.8 certainly does not give the sharpest bound for the supportof all Θ π . For example, for Kazhdan–Patterson covers of G = GL r , it is shown in[KP84, Proposition 0.1.4] that supp(Θ π ) ⊂ Z ( G ) · ( G ) n for every π ∈ Irr gen ( G ). This fact plays a crucial role in the metaplectic tensor productconstruction by Mezo [Mez04], Takeda [Tak16, Tak17] and the functorial interpretationby Gan [Gan17], when blocks in a covering Levi subgroup do not commute. In § r , relying on Proposition 2.8.In this paper, we will consider restrictions and inductions from two types of normalsubgroups of G . Each of these two types has some advantage and we place focus on oneof them in different contexts regarding different problems. To elaborate, in the first casewe let H ⊂ G be a normal subgroup such that G/H is a finite abelian group, and that every genuinerepresentation of G is H -concentrated. In the second case, we consider the subgroup Z · G ⊂ G, which is normal of finite index, satisfying Z ⊂ Z ( G ) , G der ⊂ G and that Z commuteswith G , as depicted from the following diagram Z ( G ) G Z G der . We have H ⊂ G ⊃ Z · G , and in general none of the two groups H and Z · G is contained in the other. Thesetwo types of groups play different roles in our paper as follows.– (First picture) Most often, H is not equal to the F -rational points of an algebraicsubgroup of G , and thus the construction of dual group and L-group of H doesnot apply directly. However, on the representation side, since H contains thesupport of an arbitrary irreducible genuine representation of G , it follows fromProposition 2.4 that the representation Ind GH ( ρ ) is an isotypic sum of a certain π [ ρ ] ∈ Irr gen ( G ). In this case, the induction functor from H to G is simple.As mentioned in Remark 2.9, some application of this includes the “metaplectictensor product” construction of representations of the covering Levi subgroups ofGL r as in [Mez04, Tak16, Tak17, Cai19, Gan17].– (Second picture) The group Z · G is almost a direct product of the commutingpair Z and G , as the intersection is a small finite group. For example, the twogroups Z ( G ) and G der form a commuting pair inside G . We consider both therestriction and induction of representations between Irr gen ( G ) and Irr gen ( Z · G ).One important aspect of the restriction of representations from G is how a localcoefficients matrix for G behaves with respect to the restriction to the subgroup G . For the groups G = GL , G = G der and Z = Z ( G ) , we carried out an extensive study in [GSS]. For the induction, the representationInd GZ ( G ) · G der ( ρ ) is not an isotypic sum in general; in fact, it is useful already todetermine when it is irreducible. For double covers of GSp r , this was investigatedin [Szp13b, Szp15].2.3.1. An irreducibility criterion.
Let G be an n -fold cover of G . We continue to assumethat Z ( G ) is associated with a sublattice Y G,c = Z e c ⊂ Y . Henceforth, we will denote G := G der , whenever there is no possibility of confusion. Let M ⊂ G be a covering Levi subgroup.Define M := M ∩ G der , which is then a Levi subgroup of G der . We have M der ⊂ M ⊂ M ,
ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 25 where the first inclusion however may not be an equality in general. We consider M † := Z ( G ) · M . There are inclusions of normal subgroups M † ⊂ Z ( M ) · M ⊂ M with M /M † a finite abelian group. The following result is straightforward. Lemma 2.10.
For every Levi subgroup M ⊂ G , one has M /M † ≃ T /T † ; in particular,the quotient is independent of the Levi subgroup. From now on, we write Q † := T /T † . We are interested in both the induction and restriction of representations between M † and M . It is natural to expect that any answer will depend on the specific representa-tion chosen. Nonetheless, for special covering groups, it is possible to obtain a uniformdescription of the induction and restriction process, to describe the irreducbility of aninduction and the decomposition from a restriction. We briefly explain the main ideafollowing [Szp13b].The two groups Z ( G ) and M have a finite intersection Z ( G ) ∩ M which lies in thecenter of both groups. Thus, M † is almost a direct product of Z ( G ) and M . Everyrepresentation of M † is of the form τ ⊠ ρ, where τ ∈ Irr gen ( Z ( G )) and ρ ∈ Irr gen ( M ) are compatible, i.e., they agree on the inter-section Z ( G ) ∩ M . Thus, τ ⊠ ρ ≃ τ ′ ⊠ ρ ′ if and only if τ ≃ τ ′ and ρ ≃ ρ ′ . Denote by ω τ the central character of τ , and similarly ω ρ for ρ . Then we can use thecentral characters ω τ or ω ρ to distinguish representations. More precisely, if ω τ = ω τ ′ or ω ρ = ω ρ ′ , then τ ⊠ ρ and τ ′ ⊠ ρ ′ are not isomorphic. In fact, since Z ( G ) is a Heisenberg-typegroup, one has τ ≃ τ ′ if and only if ω τ = ω τ ′ . We denote Q M := Z ( M † ) /Z ( M )and Q Z := Z ( Z ( G )) /Z ( G ) , Q M := Z ( M ) / ( M ∩ Z ( M )) . Note that here M ∩ Z ( M ) = Z ( M ) ∩ Z ( M ). The natural surjective map Q Z × Q M Q M has a finite kernel. Let d Q M := Hom( Q M , C × ) , c Q Z := Hom( Q Z , C × ) and d Q M := Hom( Q M , C × )be the Pontryagin duals. One has an embedding(2.9) d Q M c Q Z × d Q M . There is a natural conjugation action of M on Q Z , Q M and Q M , and thus also ontheir Pontryagin duals c Q Z , d Q M and d Q M respectively. Since M † acts trivially on all Q Z , Q M and Q M , we have a well-defined action of Q † on the three Pontryagin duals.The embedding in (2.9) is Q † -equivariant with respect to this natural action.It is clear that the action of Q † on c Q Z (resp. d Q M ) is trivial if and only if Z ( Z ( G )) ⊂ Z ( T ) (resp. Z ( M ) ⊂ Z ( T )). If Z ( M ) is associated with a sublattice Y M ,c ⊂ Y , thenit follows from Lemma 2.7 that these two inclusions are equivalent to Y G,c,Q,n ⊂ Y Q,n and ( Y M ,c ∩ Y ,Q,n ) ⊂ Y Q,n respectively. Here Y G,c,Q,n ⊂ Y G,c is the sublattice given by (2.5) with respect to therestriction of B Q on Y G,c , and Y = Y sc is the cocharacter lattice of G = G der . Theorem 2.11.
Keep notations as above. (i)
Suppose the action of Q † on either c Q Z or d Q M is free. Then Ind MM † ( τ ⊠ ρ ) isirreducible for every τ ⊠ ρ ∈ Irr gen ( M † ) . (ii) Specializing to the case M = T , if the action of Q † on both c Q Z and c Q T are trivial,then for every π ∈ Irr gen ( T ) we have π | T † = ( τ ⊠ ρ ) ⊕ e for some e ∈ N . On the other hand, if Z ( T ) ⊂ Z ( Z ( G )) · Z ( T ) , then Ind TT † ( τ ⊠ ρ ) is an isotypic sum of an irreducible representation of T .Proof. For (i), in view of (2.9), the assumption implies that the action of Q † on d Q M isfree. Thus, the assertion is a direct consequence of Mackey’s irreducibility criterion forinduced representations.For the first assertion in (ii), let ( τ ⊠ ρ ) ∈ Irr gen ( T † ) be a constituent of π | T † . Theassumption implies that S τ ⊠ ρ = T and thus by the discussion in § π | T † is an isotypic sum. We give analternate argument as follows. The assumption implies that(2.10) Z ( Z ( G )) · Z ( T ) ⊂ Z ( T ) . The representation π is determined by its central character ω π . Every constituent τ ⊠ ρ ⊂ π | T † is also determined by ω τ and ω ρ . However, the inclusion (2.10) implies that ω τ and ω ρ are uniquely determined by ω π . That is, π | T † is an isotypic sum in this case.For the second assertion in (ii), it is exactly the same argument as the above with theinclusion in (2.10) reversed. Indeed, every constituent π of Ind TT † ( τ ⊠ ρ ) is determined byits central character ω π . Since Z ( T ) ⊂ Z ( Z ( G )) · Z ( T ), we have ω π = ( ω τ · ω ρ ) | Z ( T ) , which is uniquely determined by τ ⊠ ρ . Thus, in this case Ind TT † ( τ ⊠ ρ ) = π ⊕ e for some e ∈ N . (cid:3) Corollary 2.12.
The following three statements are equivalent: – Q † acts trivially on d Q T ; – the inclusion Z ( T ) ⊂ Z ( T ) holds; – the equality Y ∩ Y Q,n = Y ,Q,n holds.If one (and thus every) of the above holds, then for every π ∈ Irr gen ( T ) the restriction π | T = ρ ⊕ e is an isotypic sum of a certain ρ ∈ Irr gen ( T ) . ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 27
Proof.
The equivalence between the first two assertions follows from the definition of Q † and d Q T . Note Y Q,n = L ( Z ( T )) and Y ,Q,n = L ( Z ( T )). One always has Y ∩ Y Q,n ⊂ Y ,Q,n ,which corresponds to the inclusion T ∩ Z ( T ) ⊂ Z ( T ) . This gives the equivalence between the second and the third assertions. The rest followsfrom the same argument as in Theorem 2.11 by noting that every irreducible genuinerepresentation of T or T is determined by its central character. (cid:3) Now we consider two Levi subgroups M ⊂ M ′ ⊂ G . It is easy to check that there arenatural maps Z ( M ′ ) /Z ( M ′ ) ∩ Z ( M ) Z ( M ) /Z ( M ) ∩ Z ( M ) = Q M Q M ′ = Z ( M ′ ) /Z ( M ′ ) ∩ Z ( M ′ ) . Thus, we have a Q † -equivaraint map d Q M −→ d Q M ′ . This immediately gives
Corollary 2.13.
For a Levi subgroup M ′ ⊂ G , if Q † acts freely on d Q M ′ , then: – Q † acts freely on d Q M for every Levi subgroup M ⊂ M ′ ⊂ G ; – hence, Ind MM † ( τ ⊠ ρ ) is always irreducible. Reducibility of parabolic inductions.
Let ( σ, V σ ) ∈ Irr gen ( M ), where V σ is a spaceof realization of σ .Let P = MN ⊂ G be a parabolic subgroup associated with θ ⊂ ∆. One has theparabolic subgroup P ⊂ G by restriction from G to P . Consider the normalized inducedrepresentation I GP ( σ ) := Ind GP ( σ )of G . Let P ′ = M ′ N ′ be another parabolic subgroup corresponding to θ ′ ⊂ ∆. Define W θ,θ ′ = { w ∈ W : w ( θ ) = θ ′ } . Call P and P ′ associated if W θ,θ ′ = ∅ . Let w ∈ W θ,θ ′ . We always take the representatives w = w α ...w α l ∈ G of w = w α ...w α l with w α given in § w M w − = M ′ and thus a representation w σ of M ′ given by w σ ( m ′ )( v ) := σ ( w − mw )( v )for any m ′ ∈ M ′ and v ∈ V σ . In particular, for the underlying vector spaces, we have V w σ = V σ .Consider the intertwining operator T ( w, σ ) : Ind GP ( σ ) −→ Ind GP ′ ( w σ )given by the meromorphic continuation of the integral(2.11) T ( w, σ )( f )( g ) = Z N w f ( w − ng ) dn where N w = U ∩ ( wN − w − ) with N − the unipotent opposite N . Recall that we write G = G der whenever there is no confusion. There is a similar intertwining operator T ( w, σ ) : Ind G P ( σ ) −→ Ind G P ′ ( w σ )defined for every σ ∈ Irr gen ( M ). The following is well-known (see [BJ04, Page 409] forexample). Lemma 2.14.
Let σ ∈ Irr gen ( M ) . We have (Ind GP ( σ )) | G = M σ ⊂ σ | M Ind G P ( σ ) . On the other hand,
Ind GP Ind MM † ( τ ⊠ σ ) = Ind GG † (cid:0) τ ⊠ Ind G P ( σ ) (cid:1) , where τ ∈ Irr gen ( Z ( G )) and σ ∈ Irr gen ( M ) . Moreover, the above two equalities arecompatible with the standard intertwining operators T ( w, σ ) and T ( w, σ ) for every σ ⊂ σ | M . Theorem 2.15.
Let P = M N ⊂ G be a parabolic subgroup. Assume Q † acts freely oneither c Q Z or d Q M . Let σ ∈ Irr gen ( M ) , and let σ ⊂ σ | M be an irreducible constituent. (i) If Q † acts freely on either c Q Z or d Q G , then I GP ( σ ) is irreducible if and only if I G P ( σ ) is irreducible. (ii) Assume σ is supercuspidal. If Q † acts trivially on c Q Z and does not act freely on d Q G , then I GP ( σ ) is irreducible if and only if I G P ( σ ) is irreducible and ( W · σ ) ∩ (cid:8) t ( σ ) : t ∈ Q † (cid:9) = { σ } holds, i.e., σ is not Weyl-conjugate to any representation of the form t σ with = t ∈ Q † .Proof. Since Q † acts freely on either c Q Z or d Q M , we see that σ = I MM † ( τ ⊠ σ )for some τ ∈ Irr gen ( Z ( G )). Lemma 2.14 gives(2.12) I GP ( σ ) = I GG † ( τ ⊠ I G P ( σ )) . Thus it is clear that if I GP ( σ ) is irreducible, then I G P ( σ ) is irreducible.For (i), it suffices to prove the if part. If I G P ( σ ) is irreducible and Q † acts freely on c Q Z or d Q G , then I GP ( σ ) is irreducible by Theorem 2.11.For (ii), in view of Proposition 2.12, I GP ( σ ) is irreducible if and only if τ ⊠ I G P ( σ ) isnot isomorphic to ( t τ ) ⊠ t ( I G P ( σ )) for any 1 = t ∈ Q † ; since t ( I G P ( σ )) ≃ I G P ( t σ ))and the assumption implies that t τ ≃ τ , this condition is equivalent to that I G P ( σ ) is notequivalent to I G P ( t σ ) for any 1 = t ∈ Q † . Again, as σ is assumed to be supercuspidal, itfollows from [BZ77, Theorem 2.9] that this amounts to that σ is not in the W -orbit of t σ for any 1 = t ∈ Q † . The proof is thus completed. (cid:3) ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 29
Covers of GL r , GSp r and GSpin r +1 . In this subsection, we work out explicitlycovers of GL r , GSp r and GSpin r +1 in view of the above discussion . Some results thusgeneralize earlier work as in [Szp13b, Szp15], and they will also be used in later sectionsof the paper.2.4.1. Covers of GL r . Every Brylinski–Deligne cover GL r arises from two parameters p , q ∈ Z such that B Q ( e i , e j ) = ( p if i = j, q otherwise.Here { e i : 1 i r } is a standard basis for the cocharacter lattice Y of GL r such that α ∨ i := e i − e i +1 , i r − ❡ ❡ ❡ ❡ ❡❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ α ∨ α ∨ α ∨ r − α ∨ r − α ∨ r One has Q ( α ∨ ) = 2 p − q for every coroot α ∨ . There are two special families of Brylinski–Deligne covers we high-light below (see [Gao18b, GW19]): • (Kazhdan–Patterson covers [KP84]) A Kazhdan–Patterson covering GL r, KP is aBrylinski–Deligne cover GL r such that2 p − q = − . Here p equals the twisting parameter c in [KP84]. We have n α = n for every root α , and also the inclusion nY ⊂ Y Q,n , which however may not be an equality in general. • (Savin covers [Sav]) Consider the Brylinski–Deligne covers GL ( n ) r , which are asso-ciated with the pairs ( p , q ) such that Q ( α ∨ ) = 2 p − q = − . In particular, the cover parametrized by ( p , q ) = ( − ,
0) was first studied by Savin[Sav], and we denote it by GL r, Sav . In fact, GL r, Sav arises from the restriction ofthe n -fold cover of Sp r to its Siegel Levi subgroup GL r . The fact that q = 0accounts for the block commutativity of the covering Levi subgroups of GL r, Sav .In this case, we have n α = n gcd(2 , n )and Y Q,n = n α · Y .Let r = ( r , r , ..., r k ) be a partition of r . Let M := M r = GL r × ... × GL r k ⊂ GL r be the associated Levi subgroup of GL r . One has M := M r , = n ( g i ) : g i ∈ GL r i and Y det( g i ) = 1 o ⊂ M r . In order to determine Q † , Q Z and Q M , we compute the sublattice L ⊂ Y associatedwith the following subgroups of T : T † , Z ( Z ( G )) , Z ( G ) , Z ( M ) and M ∩ Z ( M ) , whenever they exist.First, we have Y c = Z e c with e c = r X i =1 e i . It is clear that L ( T † ) = Y c + Y and thus Q † ≃ e ( F × ) /e ( F × r ) . An easy computation gives that Q ( e c ) = r p + r ( r − q , Q ( e c ) = r · B ( e c , e i ) for every i , where B ( e c , e i ) = 2 p + r ( r − q = (2 p + 1) r − . It follows that L ( Z ( Z ( G ))) = Y c,Q,n = Z · ne c gcd( n, Q ( e c )) =: Z ( n e c ) , and also L ( Z ( G )) = Y c ∩ Y Q,n = Z · ne c gcd( n, B ( e c , e )) =: Z ( n e c ) . We have n = n · gcd( n, r ) , since r and B ( e c , e i ) are coprime. Hence, Q Z ≃ e c ( F × n ) /e c ( F × n ) . Recall that the action of g ∈ Q † on χ ∈ c Q Z is given by the multiplication χ · χ g , where χ g : Q Z −→ C × with χ g ( h ) = [ g − , h ] . In the case of GL r , we take g = e ( a ) , h = e c ( b n ) , a, b ∈ F × to obtain(2.13) χ e ( a ) ( e c ( b n )) = ( b, a ) n · B Q ( e c ,e ) n . Two special cases to note:– if n and r are coprime, then Z ( Z ( G )) = Z ( G ). In particular, Q † acts trivially on c Q Z in this case.– if r | n , then it is easy to see from (2.13) that χ e ( a ) is trivial on Q Z if and only if a ∈ F × r , i.e., e ( a ) = 1 ∈ Q † . Thus, Q † acts freely on c Q Z . However, since in thiscase (cid:12)(cid:12) Q † (cid:12)(cid:12) = | Q Z | , the group c O Z is a torsor over O † .On the other hand, to compute d Q M , we note that Y M = Y = Y sc . Recalling that n α = n/ gcd( n, Q ( α ∨ )), if Z ( M ) is associated with a sublattice Y M ,c of Y , then we have Y M ,c = Y ∩ Y M,c , where Y M,c is the lattice associated with Z ( M ). In this case, it is easy to obtain L ( Z ( M )) = Y M ,c ∩ Y ,Q,n = ( y , ..., y ; y , ..., y ; ... ; y k , ..., y k ) ∈ ⊕ ri =1 Z e i : • n α | ( y i − y i +1 ) for every i, • P ki =1 r i y i = 0 . , ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 31 where y i appears with multiplicity r i for every i . On the other hand, we have L ( M ∩ Z ( M )) = Y M ,c ∩ Y Q,n = ( y , ..., y ; y , ..., y ; ... ; y k , ..., y k ) ∈ ⊕ ri =1 Z e i : • n α | y i for every i, • P ki =1 r i y i = 0 . , For simplicity, we specialize to the case M = T and obtain L ( Z ( T )) = Y ,Q,n = ( y , y , ..., y r ) ∈ ⊕ ri =1 Z e i : • n α | ( y i − y j ) for every i, j, • P ri =1 y i = 0 . and L ( T ∩ Z ( T )) = Y ∩ Y Q,n = nY . Setting v i = e i − e r for 1 i r − , and v r − = ( e + ... + e r − + e r ) − r · e r , it is clear that Y ,Q,n has a basis { n α v i : 1 i r } ∪ (cid:8) n α, ( r ) v r − (cid:9) where n α, ( r ) = n α / gcd( n α , r ), and Y Q,n has a basis { nv i : 1 i r − } . It thus follows that Q T = v r − ( F × n α, ( r ) ) /v r − ( F × n α ) . The action of g ∈ Q † on χ ∈ d Q T is given by χ · χ g , where χ g : Q T −→ C × takes the form χ g ( h ) = [ g − , h ] . We take g = e ( a ) , h = v r − ( b n α, ( r ) ) , a, b ∈ F × to obtain(2.14) χ e ( a ) ( v r − ( b n α, ( r ) )) = ( b, a ) Q ( α ∨ ) · n α, ( r ) n . Again, there are two special cases:– if gcd( n α , r ) = 1, then Q † acts trivially on d Q T .– if gcd( n α , r ) = r (i.e., r | n α ), then d Q T is a torsor over Q † .We give a partial summary of some results for covers of GL r , concentrating on the specialcases when the induced representation Ind MM † ( τ ⊠ ρ ) is irreducible, and when the restriction π | T is an isotypic sum for π ∈ Irr gen ( T ). Proposition 2.16.
Let GL r be a Brylinski–Deligne cover associated with p , q ∈ Z . (i) If r | n , then every Ind MM † ( τ ⊠ ρ ) is irreducible for τ ⊠ ρ ∈ Irr gen ( M † ) . (ii) If gcd( n α , r ) = 1 , then for every π ∈ Irr gen ( T ) , the restriction π | T is an isotypicsum. If we assume the stronger equality gcd( n, r ) = 1 , then π | T † is already anisotypic sum.Proof. The assertion (i) follows from Theorem 2.11. The first part of (ii) follows fromCorollary 2.12, while the second part of (ii) is an immediate consequence of second state-ment (ii) of Theorem 2.11. (cid:3)
The case of n -fold Kazhdan–Patterson covers of GL was already discussed extensivelyin [GSS, § to SL . Covers of
GSp r . Let GSp r be the group of similitudes of symplectic type, andlet ( X, ∆ , Y, ∆ ∨ ) be its root data given as follows. The character lattice X ≃ Z r +1 has astandard basis { e ∗ i : 1 i r } ∪ { e ∗ } , where the simple roots are∆ = { e ∗ i − e ∗ i +1 : 1 i r − } ∪ { e ∗ r − e ∗ } . The cocharacter lattice Y ≃ Z r +1 is given with a basis { e i : 1 i r } ∪ { e } , such that the paring between X and Y is (cid:10) e i , e ∗ j (cid:11) = δ ij . The simple coroots are∆ ∨ = { e i − e i +1 : 1 i r − } ∪ { e r } . Write α i = e ∗ i − e ∗ i +1 , α ∨ i = e i − e i +1 for 1 i r −
1, and also α r = 2 e ∗ r − e ∗ , α ∨ r = e r . The Dynkin diagram for the simple coroots is as follows: ❡ ❡ ❡ ❡ ❡♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ > α ∨ α ∨ α ∨ r − α ∨ r − α ∨ r We have Y c = Z e c , where e c := 2 e + X i r e i . Consider the covering GSp r incarnated by ( D, ). The following relations are easy tocheck: Q ( e i ) = − B ( e i , e ) for every i ; Q ( e c ) = B Q ( e c , e ) = 4 Q ( e ) − rQ ( e i ) for any i. Here B ( e i , e ) dictates the (non-)commutativity of blocks in a covering Levi subgroup ofGSp r . More precisely, if a Levi subgroup M ⊂ GSp r contains blocks GL k ’s and GSp m ,then GL k and GL k ′ always commute; however, GL k and GSp m commute if and only if B ( e i , e ) = 0 ∈ Z /n Z . We are interested in those GSp r whose restriction to Sp r is the one with Q ( α ∨ r ) = Q ( e i ) = − i . Thus, we assume Q ( α ∨ i ) = − i r − , and Q ( α ∨ r ) = − . In this case, Q ( e c ) = 4 Q ( e ) + r. Since ∆ ∨ ∪ { e } gives a basis for Y , to determine Q it suffices to specify Q ( e ). Thereare two special families of GSp r (depending on the choice of Q ( e )) we want to highlightbelow. ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 33 • (Type I: similitudes-splitting type) If Q ( e ) = 0, then the similitude factor F × corresponding to the cocharacter e splits into GSp r , and we haveGSp r ≃ Sp r ⋊ F × . For n = 2, this recovers the classical double cover GSp (2)2 r ≃ Sp (2)2 r ⋊ F × as discussedin [Szp15]. We note that it suffices to take Q ( e ) = 0 ∈ Z /n Z to ensure thesplitting of GSp r over e ( F × ). In any case, if Q ( e ) = 0, then Q ( e c ) = r. • (Type II: Kazhdan–Patterson type) The second family of covers of GSp r arisesfrom restricting the Kazhdan–Patterson covers of GL r . To avoid confusion, we let { v i : 1 i r } ⊂ Y be the standard basis of GL r . The one has a natural embeddingGSp r ⊂ GL r such that α ∨ r = v r − v r +1 , α ∨ i = v i − v i +1 − v r +1 − i + v r − i and e = X r +1 i r v i . Let GL r be a Brylinski–Deligne cover associated with p , q ∈ Z such that Q ( v i ) = p , B Q ( v i , v j ) = q , i = j , as in § Q ( α ∨ r ) = 2 p − q , Q ( α ∨ i ) = 2 Q ( α ∨ r ) for 1 i < r and Q ( e ) = r · (2 p + 1) r − , Q ( e c ) = r · (2(2 p + 1) r − . In particular, the n -fold cover GSp r obtained from such restriction may not splitover the similitude factor F × ⊂ GSp r .We will carry out the computation mainly for type (I) and (II) covers of GSp r as above.Let r = ( r , r , ..., r k , m ) be an ordered partition of r . Let M r = GL r × ... × GL r k × GSp m ⊂ GSp r be the associated Levi subgroup. We have G = Sp r , and M = GL r × ... × GL r k × Sp m ⊂ Sp r is the Levi subgroup of Sp r . Every element in T ⊂ GSp r is of the form( a ; λ ) := ( a , ..., a r ; λ ) := (cid:16) r Y i =1 e i ( a i ) (cid:17) · e ( λ ) with a i , λ ∈ F × . Here ( a ; λ ) ∈ T if and only if λ = 1. Lemma 2.17.
For ( a ; λ ) , ( b ; δ ) ∈ T , one has the commutator [( a ; λ ) , ( b ; δ )] = ( λ, det( b )) − Q ( α ∨ r ) n · (det( a ) , δ ) − Q ( α ∨ r ) n · ( λ, δ ) Q ( e ) n · Y i ( a i , b i ) Q ( α ∨ r ) n . Proof.
By abuse of notation, we will write [ a, b ] for [( a ; 1) , ( b ; 1)], and [ λ, b ] for [(1; λ ) , ( b ; 1)].One has [( a ; λ ) , ( b ; δ )] = [ λ, b ] · [ a, b ] · [ λ, δ ] · [ a, δ ] ∈ µ n . Now, it is easy to obtain[ λ, b ] = ( λ, det( b )) B ( e ,e ) n = ( λ, det( b )) − Q ( α ∨ r ) n , [ a, b ] = Y i ( a i , b i ) Q ( e i ) n = Y i ( a i , b i ) Q ( α ∨ r ) n , [ λ, δ ] = ( λ, δ ) Q ( e ) n , [ a, δ ] = (det( a ) , δ ) B ( e ,e ) n = (det( a ) , δ ) − Q ( α ∨ r ) n . The result immediately follows from combining these equalities. (cid:3)
For GSp r of type (I) or (II), we have Q ( α ∨ r ) = − a ; λ ) , ( b ; δ )] = ( λ, det( b )) n · (det( a ) , δ ) n · ( λ, δ ) Q ( e ) n · Y i ( a i , b i ) − n . Now, we want to explicate the three groups Q † , Q G and Q M . Equivalently, we want todetermine the sublattices of Y associated with the following groups T † , Z ( Z ( G )) , Z ( G ) , Z ( M ) and M ∩ Z ( M ) . As L ( T † ) = Y c + Y , we see that Q † ≃ e ( F × ) /e ( F × ) , where representatives are taken from e ( F × ). We have L ( Z ( Z ( G ))) = Y c,Q,n = Z · ne c gcd( n, Q ( e c )) =: Z ( n e c )and L ( Z ( G )) = Y c ∩ Y Q,n = Z · ne c gcd( n, Q ( e c )) =: Z ( n e c ) . Thus, Q Z = e c ( F × n ) /e c ( F × n ) . For every natural number n , we let ℘ ( n ) ∈ N > be the 2-exponent in n such that n = 2 ℘ ( n ) · n ′ with n ′ odd. Lemma 2.18.
For
GSp r of type (I) or (II), we have: (i) if ℘ ( n ) ℘ ( r ) , then Q Z = { } ; (ii) if ℘ ( n ) > ℘ ( r ) , then c Q Z is a torsor over Q † .Proof. One has Q ( e c ) = ( r if GSp r is of type (I) ,r · (2(2 p + 1) r −
1) if GSp r is of type (II) . If ℘ ( n ) ℘ ( r ), the assertion is clear. Now we assume ℘ ( n ) > ℘ ( r ). The action of e ( a ) ∈ Q † on c Q Z is given by the multiplication of the character χ e ( a ) : Q Z −→ C × given by χ e ( a ) ( e c ( b n )) = ( b, a ) n · B ( e ,e c ) n = ( b, a ) n · Q ( e c ) n . ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 35 If χ e ( a ) ∈ c Q Z is the trivial character, then a ∈ F × , i.e., e ( a ) = 1 ∈ Q † . That is, Q † acts freely on c Q Z . Since in this case n = 2 n , the two groups Q † ≃ F × /F × and Q Z ≃ F × n /F × n have the same size. Hence, c Q Z is a torsor over Q † . (cid:3) To determine Z ( M ), we first note that for every I ⊂ ( N ∩ [1 , r ]), one has Q (cid:16) X i ∈ I e i (cid:17) = | I | . Let e c,i ∈ Y be such that Y GL ri ,c = Z e c,i , i k . We have L ( Z ( M )) = Y M ,c ∩ Y ,Q,n = k M i =1 Z · ne c,i gcd( n, Q ( e i )) = k M i =1 Z · ne c,i gcd( n, Q ( α ∨ ))= ( k X i =1 y i e c,i : n α | ( y i ) for every i ) . It is not hard to see that L ( M ∩ Z ( M )) = Y M ,c ∩ Y Q,n = P ki =1 y i e c,i ∈ Y M ,c : • n α | y i for every i, • n α r | ( P ki =1 y i r i ) . . Lemma 2.19.
Let M ⊂ GSp r be the Levi subgroup associated with the partition r =( r , r , ..., r k ; m ) of r . There are three cases: (i) if n α = n α r , then Q M = { } ; (ii) if n α r = 2 · n α and r i is even for every i k , then Q M = { } ; (iii) if n α r = 2 · n α and r i is odd for some i , then d Q M is a torsor over Q † .Proof. First, (i) and (ii) are clear since in this case n α r | ( k X i =1 y i r i )holds whenever n α | y i for all 1 i k . Now for (iii), the assumption implies that L ( M ∩ Z ( M )) is of index two inside L ( Z ( M )), and representatives of Q M can be chosenfrom e c,i ( b n α ) , b ∈ F × . The action of e ( a ) ∈ Q † on d Q M is given by the multiplicationwith the character χ e ( a ) : Q M −→ C × which takes the form χ e ( a )( e c,i ( b nα )) = ( b, a ) r i n α · B ( e ,e i ) n = ( b, a ) − r i n α Q ( α ∨ r ) n . If χ e ( a ) ∈ d Q M is trivial, then a ∈ F × , i.e., e ( a ) = 1 ∈ Q † . On the other hand, Q † and Q M have the same size. Hence, d Q M is a torsor over Q † . (cid:3) For n = 2, this agrees with the results in [Szp13b], as expected. Note that if ( a , a , ..., a k )is a partition of an odd a ∈ N , then one of the a j ’s is odd. It follows that if the assertion(iii) in Lemma 2.19 holds for M , then it holds for every Levi subgroup M ′ ⊂ M ⊂ G .This is consistent with Corollary 2.13.Similar as Proposition 2.16, we give a partial summary for the above discussion onGSp r . Combining Theorem 2.11, Lemma 2.18 and Lemma 2.19, we have the following. Proposition 2.20.
Let
GSp r be a cover of GSp r either of type (I) or (II). (i) For a Levi subgroup M associated with ( r , ..., r k , m ) , if – either ℘ ( n ) > ℘ ( r ) , or – n is even and r i is odd for some i ,then Ind MM † ( τ ⊠ ρ ) is irreducible for every τ ⊠ ρ ∈ Irr gen ( M † ) . (ii) If n is odd, then for every π ∈ Irr gen ( T ) , its restriction to T † is an isotypic sum;hence the restriction π | T is also an isotypic sum. Covers of
GSpin r +1 . We consider the simply-connected algebraic group Spin r +1 which sits in the exact sequence Z Spin r +1 SO r +1 , where Z = { , z } is the center of Spin r +1 . The Dynkin diagram for the simple corootsof the group Spin r +1 is as follows: ❡ ❡ ❡ ❡ ❡♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ < α ∨ α ∨ α ∨ r − α ∨ r − α ∨ r Pushing out the above exact sequence by the embedding
Z ֒ → GL with z sent to − GSpin r +1 SO r +1 , where by definition GSpin r +1 = GL × Spin r +1 { (1 , , ( − , z ) } . In [AS06], the group GSpin r is also discussed. One of the advantegeous properties ofGSpin r +1 is that its Levi subgroups are associated with partitions ( r , ..., r k , m ) of r andtake the form of GL r × ... × GL r k × GSpin m +1 ; similarly for GSpin r . However, the Levisubgroups of Spin r +1 are complicated to describe explicitly, as explained in [Asg02] anddiscussed with details in [Mat09].The Langlands dual group of GSpin r +1 is GSp r , and thus the root datum( X, ∆ , Y, ∆ ∨ )of the former is obtained from inverting that of the latter, which is given in § X ≃ Z r +1 is given with a basis { e ∗ i : 1 i r } ∪ { e ∗ } . The simple roots are ∆ = { e ∗ i − e ∗ i +1 : 1 i r − } ∪ { e ∗ r } . The cocharacter group Y ≃ Z r +1 has a standard basis { e i : 1 i r } ∪ { e } , where the simple coroots are∆ ∨ = { e i − e i +1 : 1 i r − } ∪ { e r − e } . Write α i = e ∗ i − e ∗ i +1 , α ∨ i = e i − e i +1 for 1 i r −
1, and also α r = e ∗ r , α ∨ r = 2 e r − e . Consider the covering GSpin r +1 incarnated by ( D, ). We are interested in those GSpin r whose restriction to Spin r +1 is the one with Q ( α ∨ ) = 1. That is, we assume Q ( α ∨ i ) = 1 for 1 i r − , and Q ( α ∨ r ) = 2 . ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 37
We have Y c = Z e and thus 0 = B Q ( α ∨ r , e ) = 2 B Q ( e r , e ) − Q ( e ) . Hence,(2.15) B Q ( e i , e ) = Q ( e )for every i . It also follows from the two equalities Q ( e i ) = Q ( e j ) for every i, j and Q (2 e r − e ) = 2 Q ( e i − e i +1 )that(2.16) Q ( e ) = 2 B ( e i , e j ) for every i = j. On the other hand, we have Q ( α ∨ i ) = 2 Q ( e i ) − B ( e i , e i +1 )for 1 i r −
1. Thus, the cover GSp r +1 we are interested in is determined by the twonumbers p := Q ( e i ) and q := B ( e i , e j ) for i = j constrained by 2 p − q = 1 . Again, we want to determine the three groups Q † , Q Z and Q M and the structure ofthe Pontryagin duals of latter two under the action of the first. For Q † and Q Z , we wantto determine the sublattices of Y associated with the following groups T † , Z ( Z ( G )) , Z ( G ) . To avoid technical difficulties while dealing with general M (see [Mat09]), for Q M weonly illustrate the minimal parabolic case when M = T , and thus consider Z ( T ) and T ∩ Z ( T ) . First, L ( T † ) = Y c + Y with Y / ( Y c + Y ) = { , e r } . Hence, Q † ≃ e r ( F × ) /e r ( F × ) , where representatives are taken from e r ( F × ). We have L ( Z ( Z ( G ))) = Y c,Q,n = Z · ne gcd( n, Q ( e )) =: Z ( n e )and L ( Z ( G )) = Y c ∩ Y Q,n = Z · ne gcd( n, Q ( e )) =: Z ( n e ) . This gives that Q Z = e ( F × n ) /e ( F × n ) . Lemma 2.21.
We have two cases for the action of Q † on c Q Z : (i) if ℘ ( n ) = 0 , or equivalently n = n , then Q Z = { } ; (ii) if ℘ ( n ) > , or equivalently n = 2 n , then c Q Z is a torsor over Q † . Proof.
The assertion (i) is clear. For (ii), we have Q Z ≃ F × n /F × n , which has the samesize as Q † ≃ F × /F × . It suffices to show that the action of Q † on c Q Z is free. Considerthe character χ e r ( a ) ∈ c Q Z given by χ e r ( a ) ( e ( b n )) = ( b, a ) n · B ( e r ,e ) n = ( b, a ) n · Q ( e ) n . If χ e r ( a ) is the trivial character of Q Z , then a ∈ F × , i.e., e r ( a ) = 1 ∈ Q † . This showsthat the action of Q † on c Q Z is free and completes the proof. (cid:3) To compute Q T , we have L ( Z ( T )) = Y ,Q,n and L ( T ∩ Z ( T )) = Y ∩ Y Q,n . A simple computation gives that Y ∩ Y Q,n = Y scQ,n = r M i =1 Z ( n α i α ∨ i ) , where n α i = n/ gcd( n, Q ( α ∨ i )). Note that this shows that the cover GSpin r +1 is alwayssaturated in the sense of [Gao20, Definition 2.1]. Regarding the lattice Y ,Q,n , we havetwo cases:– if n is odd or 4 | (2 r − n ), then Y ,Q,n = Y scQ,n , and we have the dual groupSpin ∨ r +1 = ( PGSp r if n is odd , SO r +1 if 4 | (2 r − n ) . – if 4 ∤ (2 r − n ) with n = 2 m , for each 1 i r defining v i := m (2 e i − e ) and v := v + v + ... + v r , then one can check easily that { v i : 1 i r − } ∪ { v } constitutes a basis of Y ,Q,n . In this case,Spin ∨ r +1 = Spin r +1 . It is clear that in the first case above ( n is odd or 4 | (2 r − n )), one has Q T = { } . On the other hand, if 4 ∤ (2 r − n ), then Q T = v ( F × ) /v ( F × ) ≃ F × /F × , where more explicitly v = m ( e + e + ... + e r ) − ( mr · e ) /
2. The action of e r ( a ) ∈ Q † on d Q T is given multiplication by the character χ e r ( a ) : Q T −→ C × , which takes the form χ e r ( a ) ( v ( b )) = ( b, a ) B Q ( e r ,v ) n = ( b, a ) mQ ( α ∨ ) n = ( b, a ) . From this it follows that d Q T is a Q † -torsor in this case.We give a summary of the results for GSpin r +1 below. Proposition 2.22.
Let
GSpin r +1 be a Brylinski–Deligne cover associated with a qua-dratic form Q such that Q ( α ∨ r ) = 2 . ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 39 (i) If ℘ ( n ) > , then Ind MM † τ ⊠ ρ is irreducible for every τ ⊠ ρ ∈ Irr gen ( M † ) . Moreover,if ∤ (2 r − n ) with n even, then Ind TT † τ ⊠ ρ is irreducible for τ ⊠ ρ ∈ Irr gen ( T † ) . (ii) If n is odd or | (2 r − n ) , then the restriction of every π ∈ Irr gen ( T ) to T is anisotypic sum. Moreover, if ℘ ( n ) = 2 and | r , then the restriction π | T † is alreadyan isotypic sum. Restriction of genuine principal series.
In this subsection, we consider exclu-sively the restriction of genuine principal series (especially in the tame case) by using adirect analysis from the perspective of Mackey’s theory.The notations are the same as before, G is a cover of G and G = G der the derivedsubgroup of G . Let T ⊂ G be the covering torus.Every genuine principal series is parabolically induced from an irreducible representa-tion of T of the form Ind TA ( ˜ χ )where A ⊂ T is a maximal abelian subgroup, necessarily containing Z ( T ), and˜ χ : A −→ C × is an extension of the central character χ : Z ( T ) −→ C × . For every γ ∈ T \ T /A , denote A γ = ( γAγ − ) ∩ T . Define the character γ ˜ χ : A γ −→ C × by γ ˜ χ ( g ) = ˜ χ ( γ − gγ ) for g ∈ A γ . The Mackey theory gives a decomposition (cid:16)
Ind TA ( ˜ χ ) (cid:17) | T = M γ ∈ T \ T /A
Ind T A γ ( γ ˜ χ ) . We have the following inclusions between various groups
A T T A γ Z ( T ) Z ( T ) ∩ T Z ( T ) . Here the dashed arrow indicates that there may not be an inclusion Z ( T ) ⊂ A γ ; inparticular, A γ may not be a maximal abelian subgroup of T . Definition 2.23.
A pair (
G, G ) is called an isotypic-pair if any of the following equivalentconditions (see Corollary 2.12) is satisfied:(i) the equality Y ∩ Y Q,n = Y ,Q,n holds;(ii) the inclusion Z ( T ) ⊂ Z ( T ) holds.To continue the analysis, we specialize to the tame case when p ∤ n . In this case, wefix a splitting of G over K := G ( O ) denoted by s K : K −→ G. This gives inherited splittings of G ( O ) , T ( O ) and T ( O ) in the respective coveringgroups. If no confusion arises, we will omit s K and simply write K ⊂ G . The group A = Z ( T ) · T ( O )is a maximal abelian subgroup of T . One has A γ = ( Z ( T ) ∩ T ) · ( γ T ( O ) ∩ T ) = ( Z ( T ) ∩ T ) · γ T ( O ) ⊂ Z ( T ) · T ( O ) , where γ T ( O ) = γ T ( O ) γ − for every γ . Note that A := Z ( T ) · T ( O )is a maximal abelian subgroup of T . Defining A ♮ = ( Z ( T ) ∩ T ) · T ( O ) , We write every ˜ χ : A −→ C × as ˜ χ = χ ⊠ ˜ χ O , where χ is the central character mentioned above, and˜ χ O : T ( O ) −→ C × is a character such that χ and ˜ χ O agree on the intersection Z ( T ) ∩ T ( O ). Thus thecharacter γ ˜ χ of A γ is just γ ˜ χ = ( γ χ ) ⊠ ( γ ˜ χ O ) = χ ⊠ ( γ ˜ χ O ) , where γ ˜ χ O : γ T ( O ) ∩ T = γ T ( O ) −→ C × is given by γ ˜ χ O ( γ k ) = ˜ χ O ( k )for every k ∈ T ( O ). In view of the identification A γ = A ♮ given by z · γ k ( z · [ γ, k ]) · k , where z ∈ Z ( T ) ∩ T and k ∈ T ( O ), we can identify γ ˜ χ = χ ⊠ γ ˜ χ O : A γ −→ C × with a character γ ˜ χ = χ ⊠ ([ γ, − ] − ˜ χ O ) : A ♮ −→ C × . Thus, we have (cid:16)
Ind TA ( ˜ χ ) (cid:17) | T = M γ ∈ T / ( T A ) Ind T A ♮ ( γ ˜ χ ) , where(2.17) T / ( T A ) ≃ Y / ( Y + Y Q,n ) =: X Γ Q,n . Now we analyze the decomposition of Ind T A ♮ ( γ ˜ χ ). By induction in stages, one hasInd T A ♮ ( γ ˜ χ ) = Ind T A Ind A A ♮ ( γ ˜ χ ) . Denote by E ( γ ˜ χ ; A ♮ , A ) ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 41 the set of all the possible extensions of γ ˜ χ to A ; then it is exactly the set of irreducibleconstituents of Ind A A ♮ ( γ ˜ χ ). We thus haveInd T A ♮ ( γ ˜ χ ) = M ( γ ˜ χ ) ′ ∈ E ( γ ˜ χ ; A ♮ ,A ) Ind T A (( γ ˜ χ ) ′ ) . Recall that T is a Heisenberg-type group and thus irreducible genuine representation isdetermined by the central character. In particular, the isomorphism class of Ind T A (( γ ˜ χ ) ′ )is in fact determined by the restriction of ( γ ˜ χ ) ′ to Z ( T ). We thus consider E ( χ, γ ˜ χ O ; Z ( T )) = ω γ,j : Z ( T ) −→ C × : • ω γ,j extends χ | Z ( T ) ∩ T , • ω γ,j extends ([ γ, · ] − ˜ χ O ) | T ( O ) ∩ Z ( T ) . . We note that χ and [ γ, · ] − ˜ χ O are compatible on the intersection of Z ( T ) ∩ T and T ( O ) ∩ Z ( T ). It is easy to see that there is a natural bijection E ( χ, γ ˜ χ O ; Z ( T )) −→ E ( γ χ ; A ♮ , A )with size (cid:12)(cid:12) E ( χ, γ ˜ χ O ; Z ( T )) (cid:12)(cid:12) = | Y ,Q,n / ( Y ∩ Y Q,n ) | . This gives the bounds for the index j in ω γ,j as1 j | Y ,Q,n / ( Y ∩ Y Q,n ) | . We also have(2.18) Ind T A ♮ ( γ ˜ χ ) = M ω γ,j ∈ E ( χ, γ ˜ χ O ; Z ( T )) Ind T A (˜ ω γ,j ) = M ω γ,j ∈ E ( χ, γ ˜ χ O ; Z ( T )) i ( ω γ,j ) , where in the middle term ˜ ω γ,j : A −→ C × is any extension of ω γ,j to A . Here i ( ω γ,j ) denotes the isomorphism class of Ind T A (˜ ω γ,j ),which only depends on the central character ω γ,j . Note that the decomposition in (2.18)is multiplicity-free. We thus get(2.19) (cid:16) Ind TA ( ˜ χ ) (cid:17) | T = M γ ∈ X Γ Q,n M ω γ,j ∈ E ( χ, γ ˜ χ O ; Z ( T )) i ( ω γ,j ) , where we recall that X Γ Q,n = Y / ( Y + Y Q,n )by definition in (2.17).To simplify the formula (2.19), we consider the group homomorphism(2.20) c : X Γ Q,n
Hom( T ( O ) ∩ Z ( T ) , µ n )given by c ( y )( k ) := [ y ( ̟ ) , k ] ∈ µ n . Define Y c = { y ∈ Y : B Q ( y, y ′ ) ∈ n Z for all y ′ ∈ Y ,Q,n } ⊂ Y. Note that T ( O ) ∩ Z ( T ) = Y ,Q,n ( O ), and thus c ( y )( y ′ ( u )) = ( ̟, u ) B Q ( y,y ′ ) n for every y ′ ∈ Y ,Q,n . Hence, c ( y ) = if and only if y ∈ Y c . It is easy to see that Y + Y Q,n ⊂ Y c and thus we have the well-defined group(2.21) X c Q,n := Y c / ( Y Q,n + Y ) ⊂ X Γ Q,n . We summarize the above discussion to give the following:
Theorem 2.24.
Assume p ∤ n . Let i ( χ ) = Ind TA ( ˜ χ ) ∈ Irr gen ( T ) be with central character χ . Let I ( χ ) be the associated genuine principal series of G . (i) One has
Ker( c ) = X c Q,n and thus i ( χ ) | T = (cid:12)(cid:12) X c Q,n (cid:12)(cid:12) · M γ ∈ X Γ Q,n / X c Q,n M ω γ,j ∈ E ( χ, γ ˜ χ O ; Z ( T )) i ( ω γ,j ) , where i ( ω γ,j ) appears with multiplicity one in the double sum of the right handside. (ii) Consequently, I ( χ ) | G = (cid:12)(cid:12) X c Q,n (cid:12)(cid:12) · M γ ∈ X Γ Q,n / X c Q,n M ω γ,j ∈ E ( χ, γ ˜ χ O ; Z ( T )) I ( ω γ,j ) for the restriction of genuine principal series I ( χ ) . Here generically, the I ( ω γ,j ) ’sappear multiplicity-free in the above decompostion. (iii) Assume that I ( χ ) is a ( K, s K ) -unramified genuine principal series. Then I ( ω γ,j ) is ( K , s K ) -unramified if and only if ω γ,j belongs to the set E ( χ, ˜ χ O ; Z ( T )) , i.e.,with γ = 0 ∈ X Γ Q,n / X c Q,n being the trivial class.
Corollary 2.25.
Assume p ∤ n . If ( G, G ) is an isotypic-pair, then X Γ Q,n = X c Q,n ; in this case i ( χ ) | T = i ( ω ) ⊕ | X Γ Q,n | and I ( χ ) | G = I ( ω ) ⊕ | X Γ Q,n | , are both isotypic sums with multiplicity (cid:12)(cid:12) X Γ Q,n (cid:12)(cid:12) , where ω = χ | Z ( T ) is the unique characterin E ( χ, ˜ χ O ; Z ( T )) .Proof. Under the assumption, we have Y c = Y and thus X c Q,n = X Γ Q,n . The rest is clearin view of Theorem 2.24. (cid:3)
Remark 2.26. If I ( χ ) is ( K, s K )-unramified, then it is ( γ · K, γ · s K )-unramified as well.Here γ · K = γKγ − and γ · s K is a splitting of G over γ · K given by( γ · s K )( γ · k ) := γ · s K ( k ) · γ − . Then every I ( ω γ,j ) is in fact ( γ · K , γ · s K )-unramified. Note that γ · K and K may notbe in the same G -conjugacy classes of maximal compact subgroup of G . See § § § L-parameters and functoriality
We continue to assume in this section that Z ( G ) is connected as in § ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 43
Relations among several L-groups.
Recall that from Lemma 2.7 the group Z ( G )is associated with the sublattice Y z := Y Q,n ∩ Y c . We have the L-group extension Z ( G ) ∨ L Z ( G ) WD F where Z ( G ) ∨ := Hom( Y z , C × ) denotes the dual group of Z ( G ). Similarly, we have anL-group for Z ( G ) as an extension Z ( G ) ∨ L Z ( G ) WD F . Here Z ( G ) ∨ := Hom( Y c,Q,n , C × ). It is easy to see that(3.1) Y z ⊂ Y c,Q,n and thus there is a natural homormorphism(3.2) f c,z : L Z ( G ) −→ L Z ( G )such that f c,z | Z ( G ) ∨ : Z ( G ) ∨ −→ Z ( G ) ∨ is just the restriction map Hom( Y c,Q,n , C × ) −→ Hom( Y z , C × )induced from (3.1).On the other hand, the dual group for G has root datum ( Y Q,n , Φ ∨ Q,n ; X Q,n , Φ Q,n ), where Y Q,n is the character lattice for G ∨ . We see that there is a natural homomorphism f G,z : G ∨ −→ Z ( G ) ∨ which extends to a homomorphism of L-groups(3.3) f G,z : L G −→ L Z ( G ) , depending on the choice of a distinguished character χ ψ . We give some details of themap f G,z as follows. Let G Q,n be the split linear algebraic group over F such that theLanglands dual group of G Q,n = G Q,n ( F ) is equal to G ∨ , i.e., G ∨ Q,n = G ∨ . One can restrict the Brylisnki–Deligne data (
D, η ) to the cocharacter lattice Y Q,n of G Q,n and to the coroot lattice Y scQ,n respectively to obtain an n -fold cover G Q,n . As therestriction of B Q to Y Q,n is trivial modulo n , the cover G Q,n is almost a trivial cover over G Q,n . In particular, the covering torus T Q,n ⊂ G Q,n is abelian. In fact, we have G ∨ Q,n = G ∨ Q,n = G ∨ , where G ∨ Q,n is the dual group for the n -fold cover G Q,n . One also has Z ( G Q,n ) = Z ( G Q,n ) ∩ Z ( T Q,n ) = Z ( G Q,n ) . Recall the isogeny i Q,n : T Q,n −→ T induced from the inclusion Y Q,n ⊂ Y . By pull-back, one naturally has the map (by abuseof notation) denoted by i Q,n : T Q,n −→ T .
Lemma 3.1.
Keeping notations as above, we have (3.4) i Q,n ( Z ( G Q,n )) = Z ( G ) . Hence, Z ( G ) ∨ = Z ( G Q,n ) ∨ and the map f G,z : G ∨ −→ Z ( G ) ∨ is just f G,z : G ∨ −→ G ∨ / [ G ∨ , G ∨ ] , where [ G ∨ , G ∨ ] ⊂ G ∨ is the derived subgroup.Proof. For simplicity, writing i for i Q,n . One has a commutative diagram
X/X sc XX Q,n /X scQ,n X Q,n , which gives the commutative diagram Z ( G ) TZ ( G Q,n ) T Q,n . i i It follows that i ( Z ( G Q,n )) ⊂ Z ( G ). To show the other inclusion, we consider the exactsequence Z ( G Q,n ) T Q,n
Hom( X scQ,n , F × ) Ext( X Q,n /X scQ,n , F × ) Y z ⊗ F × , ri z where i z is the map induced from the inclusion Y z ⊂ Y Q,n . It suffices to show that r ◦ i z = 0. For every y ⊗ a ∈ Y z ⊗ F × and x ∈ X scQ,n , one has r ◦ i z ( y ⊗ a )( x ) = a h x,y i , where h− , −i : X Q,n × Y Q,n −→ Z denotes the canonical pairing. However, since x ∈ X scQ,n and y ∈ Y c ∩ Y Q,n , we have h x, y i = 0. This shows that r ◦ i z = 0 and completes the proofof (3.4). It follows that Z ( G ) ∨ = Z ( G Q,n ) ∨ ; as G ∨ = G ∨ Q,n , the rest is clear. (cid:3)
To extend f G,z to be an L-homomorphism. We choose a distinguished character χ ψ : Z ( T ) −→ C × which gives an isomorphism L G ≃ G ∨ × WD F , see § T Q,n ⊂ G Q,n and further restriction to Z ( G Q,n ), one has a distinguished genuine characterof Z ( G Q,n ), with respect to which we also have L Z ( G ) = L Z ( G Q,n ) = Z ( G ) ∨ × WD F . From this, we have an L-map as in (3.3).Next, we want to understand the relation between L G and L G . Our exposition herefollows closely that of [GG18, § Y be the cocharacter lattice of G . One has Y sc ⊂ Y ⊂ Y. The root datum of G is ( X , Φ; Y , Φ ∨ ) , ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 45 where X ⊃ X sc is the character lattice. By pull-back, one has a cover G of G . Thendual group G ∨ of G has root datum( Y ,Q,n , Φ ∨ Q,n ; X ,Q,n , Φ Q,n ) . Unlike the linear algebraic case, there might be no group homomorphism from G ∨ to G ∨ in general. However, these two groups are related as follows.First, we have Y ∩ Y Q,n ⊂ Y ,Q,n . Let H be the split algebraic group over F whose Langlands dual group H ∨ has rootdatum ( Y ∩ Y Q,n , Φ ∨ Q,n ; dual lattice of Y ∩ Y Q,n , Φ Q,n ) . We obtain two homomorphisms of algebraic groups valued in H ∨ (3.5) G ∨ −→ H ∨ ←− G ∨ . By restricting the Brylinski–Deligne data D and η to Y ∩ Y Q,n and Y scQ,n respectively, oneobtains an n -fold cover H of H such that H ∨ ≃ H ∨ ≃ G ∨ . Suppose there exist distinguished genuine characters for G and G , then we can extendthe homomorphism in (3.5) to obtain two homomorphisms of L-groups, f G,H : L G −→ L H and f G ,H : L G −→ L H. Note that if (
G, G ) is an isotypic-pair, then G ∨ = H ∨ . In particular, this is true when G is a saturated cover, i.e., when G ∨ is of adjoint type;in this case, f G ,H is an isomorphism.3.2. L-parameters and some speculations.
To every parameter φ : WD F −→ L G we associate the group of connected components of the centralizer of Im( φ ): S φ = S ( φ ) := π (cid:16) Cent G ∨ (Im( φ ) /Z ( G ∨ )) (cid:17) . Let Φ( L G ) be the set of G ∨ -conjugacy classes of L-parameters of WD F valued in L G . LetΦ en ( L G ) be the set of enhanced parameters of the form( φ, θ ) with φ ∈ Φ( L G ) and θ ∈ Irr( S φ ) . We assume that there is a local Langlands correspondenceIrr gen ( G ) Φ en ( L G ) , assigning to every π ∈ Irr gen ( G ) a pair ( φ π , θ π ) ∈ Φ en ( L G ). For every φ ∈ Φ( L G ), denoteby L ( φ ) := (cid:8) π ∈ Irr gen ( G ) : φ π = φ (cid:9) ⊂ Irr gen ( G )the hypothetical L-packet associated with φ . Besides the one for linear algebraic groupsgiven in [Bor79], there are several additional desiderata for such a correspondence asgiven in [GG18, § We expect on the parameter side relations among various L-groups, and also the com-patibility of certain L-parameters, as depicted in the following diagram.(3.6) L Z ( G ) L Z ( G )WD F L G L G L H. f c,z φ π φ τ φ ρ f G,z f G,H f Go,H
To explain this, for every φ ∈ Φ( L G ), we denote φ ♦ = f G,H ◦ φ and set Φ( L G ; φ ) = (cid:8) φ ∈ Φ( L G ) : f G ,H ◦ φ = φ ♦ (cid:9) . Consider the induced homomorphism S φ ֒ → S φ ♦ , which is actually injective (see [Sol20, Proposition 5.4]). For every φ ∈ Φ( L G ; φ ), thereis also an embedding S φ ֒ → S φ ♦ . The following are some (refined) speculations from [GG18, § § Conjecture 3.2.
Let π ∈ Irr gen ( G ) be with central character ω π and associated enhancedparameter ( φ π , θ π ) ∈ Φ en ( L G ) . Let τ ⊠ ρ be an irreducible representation of G † = Z ( G ) · G that occurs in the restriction of π . (i) The L-parameters φ ω π , φ τ and φ ρ satisfy the following: • f G,z ◦ φ π = f c,z ◦ φ τ and is equal to the parameter φ ω π associated with thecentral character ω π of π ; • φ ♦ π = f G ,H ◦ φ ρ . (ii) There exists e ( π ) ∈ N such that for every τ ∈ Irr gen ( G ) , one has dim Hom G ( π, τ ) = if φ τ / ∈ Φ( L G ; φ π ) ,e ( π ) · D Ind S φ ♦ S φπ θ π , Ind S φ ♦ S φτ θ τ E S φ ♦ if φ τ ∈ Φ( L G ; φ π ) , where h− , −i denotes the pairing of two representations of S φ ⋄ . In particular, if ( G, G ) is an isotypic-pair, then π | G = e ( π ) · M τ ∈L ( φ ♦ π ) (cid:10) θ τ | S φπ , θ π (cid:11) S φπ · τ, where ( φ ♦ π , θ τ ) ∈ Φ en ( L G ) is the enhanced parameter associated with τ ∈ L ( φ ♦ π ) . (iii) Let τ ∈ Irr gen ( Z ( G )) and ρ ∈ Irr gen ( G ) be compatible representations. Then everyirreducible constituent π ∈ Irr gen ( G ) of Ind GG † ( τ ⊠ ρ ) has a parameter φ π which fitsinto a commutative diagram (3.6) . Part (ii) is clearly motivated from the case of linear algebraic groups, especially thework of Silberger [Sil79], Keys [Key87], Gelbart–Knapp [GK82, GK81], Adler–Prasad[AP19]; see also that of Ban, Choiy and Goldberg [BCG18, Cho19].
ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 47
Functoriality for genuine principal series.
We first briefly recall the local Lang-lands correspondence for covering tori. The description here follows closely [Wei18a, § L T of T and establish the local Langlands corre-spondence for T along the same line of the construction.Let T = Y ⊗ F × be a linear torus and T an n -fold cover associated with a quadraticform Q : Y → Z . Let i Q,n : T Q,n → T be the isogeny induced from the inclusion Y Q,n ⊂ Y . Recall that the preimage of T † := Im( i Q,n ) ⊂ T inside T is just the center Z ( T ). Using the fixed embedding ε : µ n ֒ → C × we obtain thepush-out ε ∗ ( Z ( T )) of Z ( T ). At the same time, any genuine character χ of Z ( T ) givesrise to a splitting s χ : ε ∗ ( Z ( T )) C × given by s χ : [( z, t )] z · χ − ( t ) , where [( z, t )] denote the class of ( z, t ) ∈ C × × Z ( T )in ε ∗ ( Z ( T )). We illustrate this by using the following diagram µ n Z ( T ) T † C × ε ∗ ( Z ( T )) T † . ε χs χ Clearly s χ entails a splitting φ χ as in C × ε ∗ ( Z ( T )) T † φ χ given by φ χ : t [( χ ( t ) , t )] , where t ∈ Z ( T ) is any lifting of t ∈ T † .By definition T ∨ = X Q,n ⊗ C × , where X Q,n = Hom( Y Q,n , Z ) is the lattice dual to Y Q,n . By abuse of notation, we still use i Q,n to denote the naturally induced map i Q,n : X Q,n ⊗ T Q,n −→ X Q,n ⊗ T † . Consider the composite m : WD F F × X Q,n ⊗ T Q,n ≃ Hom( Y Q,n , T
Q,n ) , rec f where the first map is the reciprocity map of class field theory sending a geometricFrobenius to the uniformizer ̟ ∈ F × and trivial on SL ( C ) ⊂ WD F , and the secondmap is given by f ( a )( y ) = y ⊗ rec( a ) , y ∈ Y Q,n . The L-group L T is defined to be the pull-back of X Q,n ⊗ ε ∗ ( Z ( T )) via i Q,n ◦ m :(3.7) T ∨ X Q,n ⊗ ε ∗ ( Z ( T )) X Q,n ⊗ T † T ∨ L T WD F . φ χ i Q,n ◦ m φ χ Here, the bottom splitting s χ of L T over WD F is the one inherited from the splitting ofthe top extension.The above construction Irr gen ( T ) Φ( L T )given by i ( χ ) φ χ is the local Langlands correspondence (LLC) for covering torus, which is an injectivemap.We want to obtain functoriality of LLC with respect to the restriction to a subgroupof T , which is associated with a W -stable lattice J ⊂ Y . Note that J Q,n may not be asublattice of Y Q,n in general, and we have J Q,n ⊃ ( J ∩ Y Q,n ) ⊂ Y Q,n . Let T J be the torus associated with J and T J,♮ the torus associated with J ∩ Y Q,n . Wehave a relation Z ( T J ) Z ( T J,♮ ) = T J,♮ Z ( T ) . Lemma 3.3.
Let i ( χ J ) ∈ Irr gen ( T J ) and i ( χ ) ∈ Irr gen ( T ) . Then i ( χ J ) ⊂ i ( χ ) | T J if andonly if χ J and χ agree on T J,♮ , or equivalently, the following diagram Z ( T J ) T J,♮ Z ( T ) C × χ J χ commutes.Proof. The only if part is clear, and it suffices to prove the if part. Assume χ J and χ agree on T J,♮ , then we get a genuine character χ J ⊗ χ : Z ( T ) · Z ( T ) −→ C × . Let A ⊂ T be a maximal abelian subgroup of T . We extend χ J ⊗ χ to a character χ ′ ⊗ χ : A · Z ( T ) −→ C × , where χ ′ : A −→ C × ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 49 is an extension of χ J . We can find a maximal abelian subgroup A ⊂ T containing A · Z ( T ), and let χ ′ : A −→ C × be extending χ ′ ⊗ χ . Note that up to isomorphism class, we have i ( χ J ) ≃ Ind T A ( χ ′ ) and i ( χ ) ≃ Ind TA ( χ ′ ) . Now Frobenius reciprocity gives thatHom T ( i ( χ J ) , i ( χ )) = Hom A ( χ ′ , Ind TA ( χ ′ )) = 0 , as χ ′ extends χ ′ ⊗ χ . This shows that i ( χ J ) ⊂ i ( χ ) | T . (cid:3) If χ J and χ agree on T J,♮ , then we denote χ J,♮ = χ | T J,♮ . By the construction of L-groups,we have L T J L T J,♮ L
T . If χ and χ J are compatible, then by the LLC for covering torus, the two parameters φ χ J and φ χ are compatible, i.e., the following diagram commutes:(3.8) L T J L T J,♮ L T WD F , φ χJ φ χ where the middle vertical arrow is the parameter associated with χ J,♮ . Proposition 3.4.
Let π := I ( χ ) and ρ := I ( χ ) be irreducible genuine principal seriesof G and G respectively. Let τ ∈ Irr gen ( Z ( G )) . (i) If τ ⊠ ρ ∈ Irr( G † ) occurs in I ( χ ) | G † , then the L-parameters φ τ and φ ρ are suchthat the diagram in (3.6) commutes. Moreover, if L G = L H , then I ( χ ) | G is anisotypic sum of a genuine principal series of G . (ii) The parameter φ π of every irreducible constituent π of Ind GG † ( τ ⊠ ρ ) satisfies acommutative diagram (3.6) involving φ τ and φ ρ .Proof. For (i), we first consider φ π and φ τ . Note that we have a natural map L T −→ L G ,see § J = Y c , we thus have a commutative diagram from (3.8):(3.9) L Z ( G )WD F L Z ( G ) L T L G. φ τ φ χ φ ωπ f G,z
Similarly, taking J = Y coupled with (3.8) give another diagram:(3.10) L T L G WD F L T Y ,♮ L H L T L G , f G,H φ χ φ χJ φ χ f G ,H where we have φ χ := φ χ J . The above two diagrams show the commutativity of (3.6). If L G = L H , then φ χ is uniquely determined by φ χ , and thus i ( χ ) | T = e · i ( χ )is an isotypic sum of i ( χ ) ∈ T . This shows that I ( χ ) | G = e · I ( χ ) is an isotypic sum.The proof of (i) is completed.For (ii), assume π = I ( χ ) ⊂ Ind GG † ( τ ⊠ I ( χ )), it is easy to see that (3.9) commutes. Onthe other hand, using Frobenius reciprocity, one has I ( χ ) ⊂ I ( χ ) | G and thus i ( χ ) ⊂ i ( χ ). The commutativity of (3.10) then follows from Lemma 3.3. This concludes theproof. (cid:3) Metaplectic tensor product for
GSp r . In this subsection, we investigate forGSp r an analogue of the metaplectic tensor product construction for Kazhdan–Pattersoncovers studied in [Mez04, Tak16, Tak17, Cai19]. The parameter side interpretation for sucha construction is given in [Gan17] using the formalism of L-groups as in [Wei14, Wei18a].More precisely, consider a Levi subgroup M = GL r × GL r × ... × GL r k ⊂ GL r . It is well-known (by checking on the covering tori of GL r i ’s for example) that the blocksGL r i ⊂ M do not commute. Thus, the representation of M can not be simply reducedto that of each GL r i . However, given with π i ∈ Irr gen (GL r i ) satisfying certain conditionconstrained by the central character of Z ( G ), Mezo [Mez04] gave a natural constructionof a representation of M . Coarsely, the construction goes through several steps as follows:(1) For every i and n ∈ N , we denoteGL h n i r i = (cid:8) g ∈ GL r i : det( g ) ∈ F × n (cid:9) . One can check that for i = j , the two covering groups GL h n i r i and GL h n i r j commutewith each other. Hence one can define M h n i := GL h n i r × µ n × ... × µ n GL h n i r k . For each π i ∈ GL r i , let σ i ∈ Irr gen (GL h n i r i ) be an irreducible summand in therestriction of π i . We have a representation σ := σ ⊠ ... ⊠ σ k of M h n i ⊂ M .(2) Pick an irreducible genuine character ω : Z (GL r ) −→ C × such that(3.11) ω = σ on Z (GL r ) ∩ M h n i . This gives an irreducible representation ω ⊠ σ of Z (GL r ) · M h n i . ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 51 (3) One extends ω ⊠ σ as much as possible to a representation ( ω ⊠ σ ) ′ of a subgroup M ′ such that the Mackey’s irreducibility criteria are satisfied. We thus obtain˜ ⊗ i π i := Ind MM ′ (( ω ⊠ σ ) ′ ) , which is an irreducible representation of M .It is shown in [Mez04] that the representation ˜ ⊗ i π i depends only on π i ’s and ω , and isindependent of the intermediate choices σ i , M ′ and ( ω ⊠ σ ) ′ . We highlight that suchindependence relies on the crucial property that for every π ∈ Irr gen ( M ), one hassupp(Θ π ) ⊂ Z (GL r ) · M h n i ;thus, it follows from (the proof of) Proposition 2.4 that Ind MM h n i ( ω ⊠ σ ) is an isotypic sumof an irreducible representation of M , which is exactly ˜ ⊗ i π i in (3) above. This observationwas explicated in [Tak16, Proposition 4.6]. In view of this, we could replace (3) above bythe following:(3)’ One has Ind MM h n i ( ω ⊠ σ ) = m · π for a certain π ∈ Irr gen ( M ) . Now the representation ˜ ⊗ i π i := π ∈ Irr gen ( M ) is the one we seek.The above construction gives a well-defined surjective map (for the surjectivity, see[Tak16, Lemma 4.4])˜ ⊗ : (Irr gen (GL r ) × ... × Irr gen (GL r k ) × Irr gen ( Z (GL r ))) ♥ Irr gen ( M ) . The superscript ( − ) ♥ indicates the subset of( π , ..., π r , ω )satisfying the relation (3.11). For every character χ i of F × which is trivial on F × n ,replacing π by π ⊗ ( χ i ◦ det) gives the same representation on Irr gen ( M ), and this accountsfor the non-injectivity of the map ˜ ⊗ , see [Mez04, Lemma 5.1]. This metaplectic tensorproduct construction was further analyzed and refined in [Tak16, Tak17] especially in theglobal context, see also [Cai19].To proceed, in this subsection, we assume GSp r is the type I (similitudes-splitting) n -fold cover associated with Q ( α ∨ r ) = − Q ( e ) = 0 , see § B ( e , e i ) = 1 for every i , and B Q ( e c , e ) = Q ( e c ) = r . Also,(3.12) Y Q,n = P ri =0 y i e i ∈ Y : • n | ( − y i + y ) for every i, • n | ( y + y + ... + y r ) . . On the other hand, Y scQ,n is spanned by (cid:8) α ∨ i,Q,n = n · α ∨ i : 1 i r − (cid:9) ∪ (cid:8) α ∨ r,Q,n = nα ∨ r (cid:9) . We consider a partition r = ( r , r , ..., r k ; r )of r with associated Levi subgroup M r = GL r × GL r × ... × GL r k × GSp r . For every j ∈ N , let GSp h j i r = (cid:8) g ∈ GSp r : e ∗ ( g ) ∈ F × j (cid:9) be the subgroup of GSp r with similitudes lying in F × j , where e ∗ : GSp r −→ F × is the similitude map. Also define M h j i r = GL r × GL r × ... × GL r k × GSp h j i r . Accordingly, we have the covering subgroups GSp h j i r ⊂ GSp r and M h j i r ⊂ M r . Thecovering blocks GL r i in M r commute with each other; however, they may not commutewith GSp h j i r for general j . Lemma 3.5. (i)
The covering group
GSp h j i r commutes with every GL r i , i k ifand only if n | j . (ii) For every genuine representation π of M r , one has supp(Θ π ) ⊂ M h n ( r ) i r for the character Θ π of π .Proof. For (i), we note that every element in GSp r can be written as g · e ( a ) with g ∈ Sp r and a ∈ F × . We have block commutativity among the blocks GL r i and Sp r ,thus g commutes with every element in GL r i . On the other hand, e ( a ) commutes withevery e α i ( a ) , i r −
1, thus e ( a ) commutes with GL r i if and only if[ e ( a ) , e i ( x )] = ( a, x ) B Q ( e ,e i ) n for every e i ∈ Y GL ri and x ∈ F × . Since B Q ( e , e i ) = 1, the above equality amounts to a ∈ F × n . For (ii), the proof is the same as Proposition 2.8 by using the above fact that g commutes with GL r i for every i . (cid:3) Let M r ⊂ GSp r be a Levi subgroup as above. We assume in the rest of this subsectionthat gcd( n, r ) = 1which implies supp(Θ π ) ⊂ GSp h n i r for every π ∈ Irr gen (GSp r ), by Lemma 3.5. It isalso easy to check that(3.13) L ( Z (GSp r ) ∩ M h n i r ) = ( Z ( ne c ) if 2 ∤ gcd( n, r ) , Z ( me c ) if 2 | gcd( n, r ) . Also,(3.14) Z (GSp r ) · M h n i r = M h n ( r ) i r if 2 ∤ gcd( n, r ) ,M h n ( r ) i r if 2 | gcd( n, r ) . We will discuss the two cases separately: (1) n is odd, (2) n is even. ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 53
The case of odd n . For n odd, it is easy to obtain from (3.12) that { ne i : 1 i r } ∪ (cid:8) n ( r ) · e c (cid:9) constitutes a Z -basis for Y Q,n . This shows thatGSp ∨ r = (cid:8) ( g, a ) ∈ GSpin r +1 × GL : λ ( g ) = a gcd( n,r ) (cid:9) , where λ : GSpin r +1 −→ GL is the similitude map of GSpin r +1 associated with ne c ∈ Y Q,n . Since we have assumedgcd( n, r ) = 1, it gives GSp ∨ r = GSpin r +1 . Now we describe a construction analogous to the metaplectic tensor product for GL r . Inparallel, there are three steps as follows.(S1) Let π i ∈ Irr gen (GL r i ) and π ∈ Irr gen (GSp r ). Let σ ⊂ π be an irreduciblesummand in the restriction of π to GSp h n i r . The representation σ := π ⊠ ... ⊠ π k ⊠ σ is then a representation of M h n i r .(S2) Let ω : Z (GSp r ) −→ C × be a central character such that(3.15) ω = σ on Z (GSp r ) ∩ M h n i r . We have ω ⊠ σ ∈ Irr( Z (GSp r ) · M h n i r ).(S3) One extends ω ⊠ σ to a representation ( ω ⊠ σ ) ′ of a subgroup M ′ r ⊂ M r such thatthe representation ˜ ⊗ i π i := Ind MM ′ ( ω ⊠ σ ) ′ is irreducible. This representation ˜ ⊗ i π i is the sought metaplectic-tensor productof the π i ’s.Similar to the case of Kazhdan–Patterson covers GL r discussed earlier, (S3) can be re-placed by an equivalent statement as follows.(S3’) It follows from (ii) of Lemma 3.5, the equality (3.14) and Proposition 2.8 thatInd M r Irr( Z (GSp r ) · M h n i r ) ( ω ⊠ σ ) = m · π is an isotypic sum of π ∈ Irr gen ( M r ). The representation π is just ˜ ⊗ i π i describedin (S3) above. In particular, fixing ω ⊠ σ , the representation ˜ ⊗ i π i is independentof the choice of M ′ r and the extension ( ω ⊠ σ ) ′ .We have to justify the above construction (S1)-(S3) (or (S1), (S2) and (S3’)) as follows. Lemma 3.6.
The representation ˜ ⊗ i π i is independent of the choice of σ ⊂ π , the choiceof M ′ r and the extension ( ω ⊠ σ ) ′ .Proof. Again, the argument is essentially the same as in [Mez04]. By the equivalencebetween (S3) and (S3’), it suffices to show the independence on the choice of constituent σ in (S1). Every summand in the restriction of π to GSp r is of the form g π , where g ∈ GSp r / GSp h n i r ≃ e ( F × ) /e ( F × n ). Taking g = e ( a ) , a ∈ F × , since Θ π i ⊂ GL h n i r i forevery π i ∈ Irr gen (GL r i ) by Proposition 2.8, we see that g Θ π i = Θ π i and thus g π i ≃ π i . It follows that π ⊠ ... ⊠ π k ⊠ g σ = g σ. If ω and σ agrees on Z (GSp r ) ∩ M h n i r , then ω = g σ on this intersection subgroup as well.Now we have Ind M r Z (GSp r ) · M h n i r ( ω ⊠ g σ ) = Ind M r Z (GSp r ) · M h n i r g ( ω ⊠ σ )= g Ind M r Z (GSp r ) · M h n i r ( ω ⊠ σ )= Ind M r Z (GSp r ) · M h n i r ( ω ⊠ σ )= e · ˜ ⊗ i π i , where the last equality follows from Proposition 2.8, see also (S3) and (S3’) above. Thisshows the independence on g σ . (cid:3) We also have a reverse construction from Irr gen ( M r ) to Q ki =1 Irr gen (GL r i ) × Irr gen (GSp r )as follows.(RS1) Let Π ∈ Irr gen ( M r ) be an irreducible genuine representation. We pick a summandΠ ⊂ Π in the restriction of Π to M h n i r , it takes the formΠ = π ⊠ π ⊠ ... ⊠ π k ⊠ σ with π i ∈ Irr gen (GL r i ) and σ ∈ Irr gen (GSp h n i r ). Every summand of Π | M h n i r is ofthe form g (Π ) = g π ⊠ ... ⊠ g π k ⊠ g σ for some g ∈ M r /M h n i r = e ( F × ) /e ( F × n ). Taking g = e ( a ) , a ∈ F × , since wehave supp(Θ π i ) ⊂ Z (GSp r ) · GL h n i r i by Proposition 2.8, it follows that g Θ π i = Θ π i as g commutes with every elements in GL h n i r i . This shows that g π i ≃ π i . Thus,every constituent Π ′ ⊂ Π is of the formΠ ′ = π ⊠ π ⊠ ... ⊠ π k ⊠ g σ for some g ∈ e ( F × ). In particular, we have π i ∈ Irr gen (GL r i ) uniquely determinedby Π for each 1 i k .(RS2) Extend σ as much as possible to a representation σ ′ of a subgroup GSp ′ r ⊂ GSp r , such that the induced representation π := Ind GSp r GSp ′ r ( σ ′ )is irreducible. This gives the desired representation π ∈ Irr gen (GSp r ). Usingthe crucial fact that Θ π ⊂ GSp h n i r for every π ∈ Irr gen (GSp r ), it follows fromProposition 2.8 that Ind GSp r GSp h n i r ( σ ) = m · π , as an isotypic sum of π ∈ Irr gen (GSp r ). Thus, π is the desired representation. Proposition 3.7.
Keep notations as above. (i)
One has ˜ ⊗ i π i ≃ ˜ ⊗ i π ′ i if and only if π i ≃ π ′ i for every i k . (ii) Every representation Π ∈ Irr gen ( M r ) is a metaplectic tensor product of the form ˜ ⊗ i π i , where the π i ’s are uniquely determined by (RS1)-(RS2) . ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 55
Proof.
For (i), it suffices to prove the “only if” part. Consider the restriction of ˜ ⊗ i π i ≃ ˜ ⊗ i π ′ i to M h n i r , we see that there is g ∈ e ( F × ) and σ ⊂ π , σ ′ ⊂ π ′ such that g ( π ⊠ ... ⊠ π k ⊠ σ ) = π ′ ⊠ ... ⊠ π ′ k ⊠ σ ′ , where the left hand side is in fact π ⊠ ... ⊠ π k ⊠ g σ . It follows that π i ≃ π ′ i for every 1 i k , and also σ ′ = g σ . On the other hand, we haveInd GSp r GSp h n i r ( σ ′ ) = e ′ · π ′ and Ind GSp r GSp h n i r ( σ ) = e · π for some e, e ′ ∈ N . SinceInd GSp r GSp h n i r ( σ ′ ) = Ind GSp r GSp h n i r ( g σ ) = g Ind
GSp r GSp h n i r ( σ ) = Ind GSp r GSp h n i r ( σ ) , it follows that e = e ′ and π ≃ π ′ .It is clear from the above argument that every Π ∈ Irr gen ( M r ) is a metaplectic tensorproduct with π i determined as above. This gives (ii) and completes the proof. (cid:3) The construction in (S1)-(S3) gives a well-defined map˜ ⊗ i : ( Q ki =1 Irr gen (GL r i ) × Irr gen (GSp r ) × Irr gen ( Z (GSp r ))) ♥ Irr gen ( M r ) . The superscript ( − ) ♥ indicates the subset of( π , ..., π k , π , ω )satisfying the equality (3.15) in (S2). On the other hand, when restricted to Z (GSp r ) ⊂ M r , the representation Π acts by a character ω Π : Z (GSp r ) → C × . Thus, the construction in (RS1)-(RS2) gives a “metaplectic restriction”˜R : Irr gen ( M r ) −→ (cid:16) k Y i =1 Irr gen (GL r i ) × Irr gen (GSp r ) × Irr gen ( Z (GSp r )) (cid:17) ♥ given by ˜R(Π) = ( π , ..., π k , π ; ω Π ) . The following is immediate from Proposition 3.7.
Theorem 3.8.
Assume n is odd and gcd( n, r ) = 1 . Then the map ˜ ⊗ i : (cid:16) k Y i =1 Irr gen (GL r i ) × Irr gen (GSp r ) × Irr gen ( Z (GSp r )) (cid:17) ♥ −→ Irr gen ( M r ) is a bijection with the inverse given by ˜R , i.e. ˜R ◦ ˜ ⊗ i = id = ˜ ⊗ i ◦ ˜R . It is expected that for the parameters associated with the representations involved inthe constructions ˜ ⊗ and ˜R, one should have a corresponding identification. To proceed,we note that the dual group of M r is M ∨ r = ( ( g , g , ..., g k , g ; z ) : λ ( g ) · k Y i =1 det( g i ) = z g ( n,r ) ) ⊂ k Y i =1 GL r i × GSpin r +1 × GL ( C ) . Assuming that we have a local Langlands correspondence for the covers, then associatedwith π i ’s and ω we have φ i : WD F −→ GL r i for 1 i k, φ : WD F −→ GSpin r +1 parametrizing π i ’s for 0 i k , and φ ω : WD F −→ GL ( C )parametrizing ω . This gives a parameter( φ × ... × φ k × φ ) × φ ω : WD F −→ k Y i =1 GL r i × GSpin r +1 ! × C × . Lemma 3.9.
Assume the local Langlands correspondence with desiderata. Then the com-patibility condition in (3.15) is equivalent to k Y i =1 det( φ i ) · λ ( φ ) = φ gcd( n,r ) ω . Thus, the parameter ( φ × ... × φ k × φ ) × φ ω factors through M ∨ r .Proof. It follows from (3.13) that Z (GSp r ) ∩ M r = Z (GSp r ) n , where Z (GSp r ) = Z (GSp r ) n ( r ) . If we denote by e i,c ∈ Y GL ri the natural generator of Y GL ri ,c , then Z (GL r i ) = Z (GL r i ) n and Z (GSp r ) = Z (GSp r ) n . From the desiderata (i) of Conjecture (3.2), the center character ω i : Z (GL r i ) −→ C × of π i has a parameterdet( φ i ) : WD F GL ∨ r i GL ∨ r i / [GL ∨ r i , GL ∨ r i ] ≃ C × , φ π det where the second map is the determinant map of GL ∨ r i ≃ GL r i . The center character ω : Z (GSp r ) −→ C × of π has a parameter λ ( φ ) : WD F GSp ∨ r GSp ∨ r / [GSp ∨ r , GSp ∨ r ] ≃ C × , φ π λ where the second map is the similitude map of GSp ∨ r ≃ GSpin r +1 . The identity is thusclear. (cid:3) Let i M : M ∨ r Q ki =1 GL ∨ r i × GSp ∨ r × C × be the natural inclusion. Assuming the local Langlands correspondence, we expect thatthe following diagram(3.16) WD F Q ki =1 GL ∨ r i × GSp ∨ r × C × M ∨ r . φ × ... × φ k × φ × φ ω φ Π i M ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 57 commutes, when φ Π is related to φ i ’s and φ ω via the two metaplectic constructions ˜ ⊗ i and ˜R. The following is in analogue with [Gan17, § Conjecture 3.10.
Keep notations as above. (i)
The metaplectic tensor product ˜ ⊗ i π i defined above has an associated L -parameterexactly ( φ × ... × φ k × φ ) × φ ω , which is a parameter factoring through i M andthus is valued in M ∨ r . (ii) The metaplectic restriction ˜R(Π) above has L -parameter exactly i M ◦ φ Π . Proposition 3.11.
Conjecture 3.10 holds when the π i ’s and Π are genuine principalseries.Proof. We denote by T r i ⊂ GL r i and T r ⊂ GSp r the covering tori of the blocks GL r i and GSp r respectively. Every genuine principalseries π i = I ( i ( χ i )) ∈ Irr gen (GL r i ) is associated with χ i : Z ( T r i ) −→ C × , a central genuine character; similarly, a genuine principal series π i = I ( i ( χ )) ∈ Irr gen (GSp r )is associated with χ : Z (GSp r ) −→ C × . Denote by Y r i the character lattice of GL r i , also by Y r that of GSp r .We have L ( Z ( T r i )) = nY r i , i k and L ( Z ( T r )) = nY scr ⊕ Z ( ne c ) , where Y scr ⊂ Y r is the coroot lattice of GSp r . Since L ( Z (GSp r )) = Z n ( r ) e c and L ( Z ( T )) = nY sc ⊕ Z · n ( r ) e c , it follows that L ( Z ( T )) = Z n ( r ) e c + k X i =1 L ( Z ( T r i )) + L ( Z ( T r )) ! . Thus giving a genuine character χ : Z ( T ) −→ C × is equivalent to giving( χ r , ..., χ r k , χ r ; ω ) ∈ Irr gen ( Z ( T r )) × ... × Irr gen ( Z ( T r k )) × Irr gen ( Z ( T r )) × Irr gen ( Z (GSp r ))such that χ r ⊠ ...χ r k ⊠ χ r and ω agree on (cid:0) Z ( T r ) × µ n ... × Z ( T r k ) × µ n Z ( T r ) (cid:1) ∩ Z (GSp r ) . In particular, χ r i , i k and ω are obtained from restricting χ to the correspondingsubgroups of Z ( T ). The equality between the parameter χ Π of Π := Π( χ ) and φ × ... × φ k × φ × φ ω then easily follows from functoriality of the local Langlands correspondencefor covering tori with respect to restriction, as discussed in § (cid:3) Remarks on the even n case. The metaplectic tensor product ˜ ⊗ and metaplecticrestriction ˜R for odd fold cover of GSp r do not generalize naively in the even fold covercase. Now, assuming 2 | n , we highlight the obstacles to the constructions in ˜ ⊗ and ˜R.Recall that we assume gcd( n, r ) = 1.Let n = 2 m . It follows from Proposition 2.8 that supp(Θ π i ) ⊂ GL h m i r i . However, theremay exist π i such that supp(Θ π i ) * GL h n i r i . For such π i , we may have g π i = π i . This gives an obstruction to validating the construc-tions both in (S1)-(S3) and (RS1)-(RS2), as the seen from proof of Lemma 3.6. In fact,if 2 | gcd( n, r ), then it follows from (3.14) that Z (GSp r ) · M h n i r = M h n ( r ) i r , which however is not equal to M h n ( r ) i r . In this case,(3.17) Ind M r Z (GSp r ) · M h n i r = M j ∈ F × / Ind M r M h n ( r ) i r ( ω ⊠ σ ) j = M j e j · Π j , where ( ω ⊠ σ ) j is an extension of ( ω ⊠ σ ) and F × / e ( F × n ( r ) ) /e ( F × n ( r ) ) ≃ F × /F × . The last equality in (3.17) follows from Proposition 2.8 with Π j ∈ Irr gen ( M r ). This givesanother obstruction in the case 2 | gcd( n, r ).We also remark that the dual side seems to be deceivable when n is even, and thisprovides a counterexample where the naive version of functoriality might fail for general n . We illustrate this below according to the parity of gcd( n, r ).In the first case, we assume n = 2 m with 2 ∤ gcd( n, r ). Setting v := n ( r ) · e c and v i := me i + v / , i r, It is easy to check that Y Q,n = r M i =0 Z v i with Y scQ,n spanned by { mα ∨ i : 1 i r − } ∪ { nα ∨ r } . Thus, for 2 ∤ gcd( n, r ) one hasGSp ∨ r = GSp r . It follows that (given with the partition r = ( r , ..., r k , r ) of r ) M ∨ r ≃ k Y i =1 GL r i × GSp r = k Y i =1 GL ∨ r i × GSp ∨ r . Analyzing the discussion in (S1)-(S3) and (RS1)-(RS2), one can define a correspondence between Q ki =1 Irr gen (GL r i ) × Irr gen (GSp r ) and Irr gen ( M r ). However, there is no map (instead of correspondence) from one to the other. This indicates that a naive analogueof the functoriality might fail when considering representations of M r and those of itscovering blocks inside.In the second case, we assume 2 | gcd( n, r ), then it follows that Y Q,n = Y scQ,n ⊕ Z · v , where v = n ( r ) e c as above. In this case,GSp ∨ r = PGSp r × GL ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 59 and thus one has a map k Y i =1 GL ∨ r i × GSp ∨ r −→ M ∨ r . However, this map does not seem to be associated from the representation-side by a“metaplectic tensor product” construction, the same as we remarked above for the n =2 m, ∤ gcd( n, r ) case.3.4.3. General metaplectic tensor product.
From the example of Kazhdan–Patterson cov-ers of GL r and the odd-fold cover of GSp r discussed in this subsection, we see that theformulation of the metaplectic tensor product construction could be carried out in quitegeneral setting, provided the following constraints are satisfied. To elaborate, assume forsimplicity that M = M × M × ... × M k is a Levi subgroup of G . We like to have normal subgroups M ♮j ⊂ M j , j k such that every M j /M ♮j is a finite abelian group and satisfy the following two conditions:(i) the blocks M ♮j , j k commute with each other,(ii) supp(Θ π ) ⊂ Q j M ♮j for every π ∈ Irr gen ( M ).If (i) and (ii) are satisfied, then the process described in steps (S1)–(S3) in § M ♮j is a smaller subgroup of M j ; on the other hand, for (ii) to besatisfied the group M ♮j should be closer to M j . Thus, a family n M ♮j o j satisfying both (i)and (ii) is a balancing choice, if it exists.For G = GL r and GSp r , the natural candidate for M ♮j is essentially the subgroup ofelements with determinant or similitude lying in F × n , respectively. Another reason oftaking such M ♮j is to ensure we have a natural (even conjectural) interpretation of thetensor product construction on the dual side.4. Local coefficients matrix under restriction
Local coefficients matrix.
Let B = T U be the Borel subgroup of G , where U isidentified as a subgroup of G via the canonical splitting given by e α ( x ) e α ( x ). Let ψ : F −→ C × be a nontrivial character. By abuse of notation, denote by ψ : U −→ C × the unique character such that ψ ( e α ( x )) = ψ ( x )for every α ∈ ∆ and x ∈ F . Let ( π, V π ) ∈ Irr gen ( G ) be a genuine irreducible representationof G . A functional λ : V π −→ C is called a ψ -Whittaker functional if λ ( π ( u ) v ) = ψ ( u ) · λ ( v ) for all u ∈ U and v ∈ V π . Denote by Wh ψ ( π ) the space of ψ -Whittaker functionalsfor π . It follows from the work of Patel [PP15], which generalizes [MW87] by Mœglin-Waldspurger, that dim Wh ψ ( π ) < ∞ for every π ∈ Irr gen ( G ) . See [KP84, Theorem I.5.2 (i)] also for a proof of this finite dimensionality for covers ofGL r .We briefly recall the definition of a local coefficients matrix as in [Szp19, GSS]. Let P = MN ⊂ G be a parabolic subgroup of G associated with θ ⊂ ∆. Let σ ∈ Irr gen ( M ). We have theparabolic subgroups P = M N ⊂ G and P = M N ⊂ G , see § T ( w, σ ) in (2.11), whenever holomorphic at σ ,gives a linear map between finite dimensional vector spaces: T ( w, σ ) ∗ : Wh ψ ( I GP ′ ( w σ )) Wh ψ ( I GP ( σ )) . Let ψ M be the restriction of ψ to the unipotent radical U M of the Borel subgroup T U M of M . Here ψ M and w are compatible (in the sense of [Sha10, Page 51]). We have anatural isomorphism (see [CS80, Rod73], [Sha81, §
3] or [Sha10, § J ( σ ) : Wh ψ M ( σ ) −→ Wh ψ ( I GP ( σ ))given by J ( σ )( λ )( f ) = Z N ′ λ ( f ( w − n ′ )) · ψ ( n ′ ) dn ′ , where N ′ = w N − w − with w = w l · w l,M . Here w l (resp. w l,M ) is the longest Weylelement in the Weyl group W = W ( T, G ) (resp. W M = W ( T, M )).Similarly, one has J ( w σ ) : Wh ψ M ′ ( w σ ) −→ Wh ψ ( I GP ′ ( σ )) . At the same time, since V w σ = V σ , we also have a canonical identification ι : Wh ψ M ( σ ) = Wh ψ M ′ ( w σ ) , which we briefly explained as follows. First, note that l ∈ Wh ψ M ( σ ) if and only if l ( σ ( u )( v )) = ψ M ( u ) · v for all u ∈ U M , v ∈ V σ . For every u ′ ∈ U M ′ one has w − u ′ w ∈ U M and ψ M ′ ( u ′ ) = ψ ( u ′ ) = ψ ( w − u ′ w ) = ψ M ( w − u ′ w ) . It is verified easily (cf. [GSS, § l ( w σ ( u ′ )( v )) = l ( σ ( w − u ′ w )( v )) = ψ M ( w − u ′ w ) · v = ψ M ′ ( u ′ ) · v for all u ′ ∈ U M ′ and v ∈ V w σ = V σ . This shows that l ∈ Wh ψ M ′ ( w σ ) and gives thecanonical identification ι .This gives a natural isomorphism of vector spaces C ( w, σ ) = J ( w σ ) ◦ ι ◦ J ( σ ) − : Wh ψ ( I GP ( σ )) Wh ψ ( I GP ′ ( w σ )) . We therefore obtain an endomorphism(4.1) T ( w, σ ) ∗ = T ( w, σ ) ∗ ◦ C ( w, σ ) : Wh ψ ( I GP ( σ )) Wh ψ ( I GP ( σ )) ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 61 of the finite dimensional vector space Wh ψ ( I GP ( σ )). A local coefficients matrix associatedto ( P , w, σ ) is the matrix M B ( w, σ )representing T ( w, σ ) ∗ with respect to an ordered basis B ⊂ Wh ψ ( I GP ( σ )). Proposition 4.1.
Fix σ ∈ Irr gen ( M ) . There is a local coefficients matrix M B ( w, σ ) ofthe form M σ i ⊂ σ | M M B i ( w, σ i ) , where M B i ( w, σ i ) is a local coefficients matrix associated with the triple ( P , w, σ i ) anda basis B i of Wh ψ ( I G P ( σ i )) . In particular, for the two invariants trace and determinant,one has (4.2)Tr( M B ( w, σ )) = X σ i ⊂ σ | M Tr( M B i ( w, σ i )) , det( M B ( w, σ )) = Y σ i ⊂ σ | M det( M B i ( w, σ i )) . Proof.
Clearly, it suffices to show that T ( w, σ ) ∗ decomposes as a direct sum of the oper-ators T ( w, σ i ) ∗ . First, we have an isomorphism of vector spaces(4.3) Wh ψ ( I GP ( σ )) ≃ M σ i ⊂ σ | M Wh ψ ( I G P ( σ i )) , since Wh ψ ( − ) is the dual of the twisted Jacquet module, which depends only on ψ : U → C × and is an exact functor. It follows from Lemma 2.14 that T ( w, σ ) ∗ = M σ i ⊂ σ | M T ( w, σ i ) ∗ . Now we consider the decomposition of C ( w, σ ) over the C ( w, σ i )’s with respect to thedecomposition in (4.3). For this purpose, we note first for each σ i ⊂ σ | M the isomorphism J ( σ i ) : Wh ψ M ( σ i ) −→ Wh ψ ( I G P ( σ i ))and the decomposition Wh ψ M ( σ ) = M σ i ⊂ σ | M Wh ψ M ( σ i ) . It is easy to see from the defining formula of J ( σ i ) that J ( σ ) = M σ i ⊂ σ | M J ( σ i ) . Similar decomposition for J ( w σ ) holds. Lastly, the canonical identification ι also decom-poses: ι = M σ i ⊂ σ | M ι σ i , where ι σ i : Wh ψ M ( σ i ) → Wh ψ M ′ ( w σ i ) is the canonical identification. Thus, the isomor-phism C ( w, σ ) decomposes as a direct product of the C ( w, σ i )’s. Hence we have T ( w, σ ) ∗ = M σ i ⊂ σ | M T ( w, σ i ) ∗ , which gives the desired result. (cid:3) The above proposition shows that the invariants associated to a local coefficients matrix M B ( w, σ ) is a priori determined by all the local coefficients matrices M B i ( w, σ i ) arisingfrom the restriction of σ to M . This has a striking consequence even in the case where σ = i ( χ ) is a genuine representation of T ⊂ G , where χ is a central character of Z ( T ).Indeed, in this case, the two invariantsTr( M B ( w α , σ )) and det( M B ( w α , σ ))associated with a simple reflection w α are computed in [GS16, Szp19, GSS18, GSS]. Theyare expressed in terms of the Plancherel measure and gamma or metaplectic-gammafactors associated with i ( χ ); in particular, these two invariants depend only on the centralcharacter χ .It is expected that Tr( M B i ( w α , σ i )) and det( M B i ( w α , σ i )) depend on σ i only. Note thateven if σ ∈ G is an unramified representation, it is possible to have a ramified constituent σ i ⊂ σ | T . In this case, the equalities in (4.2) entail that there is a cancellation of theramified information encoded in the ramified constituents of σ | T . A prototype of suchexamples is the pair (GL ( n )2 , SL ( n )2 ) with even n , see [GSS, § § Corollary 4.2.
Let σ ∈ Irr gen ( M ) . Assume that σ | M = σ ⊕ m is an isotypic sum of σ ∈ Irr gen ( M ) . Then dim Wh ψ M ( σ ) = m · dim Wh ψ M ( σ ) and moreover M B ( w, σ ) ⊕ m is a local coefficients matrix for T ( w, σ ) ∗ , where M B ( w, σ ) is a local coefficients matrix for T ( w, σ ) ∗ . In particular, Tr( M B ( w, σ )) = m · Tr( M B ( w, σ )) , det( M B ( w, σ )) = det( M B ( w, σ )) m . One immediate application of Corollary 4.2 is the computation of the multiplicity m byconsidering the Whittaker dimensions. Indeed, such multiplicity (or ramification index e in the discussion of § Genuine principal series.
We assume p ∤ n , and fix a splitting s K of G over K = G ( O ). We consider a ( K, s K )-unramified genuine principal series I ( χ ) = I ( i ( χ )) of G , where χ : Z ( T ) −→ C × is an unramified character, i.e., it is trivial on Z ( T ) ∩ K . In this case, as discussed in § A = Z ( T ) · T ( O ) ⊂ T is a maximal abelian subgroup. There is a unique extension also denoted by˜ χ : A −→ C × , which is trivial on T ( O ). From now on, we will take i ( χ ) = Ind TA ( ˜ χ )for this ˜ χ , which gives the unramified principal series I ( χ ) = I ( i ( χ )).It follows from Theorem 2.24 that one has I ( χ ) | G = (cid:12)(cid:12) X c Q,n (cid:12)(cid:12) · M γ ∈ X Γ Q,n / X c Q,n M ω γ,j ∈ E ( χ, γ ˜ χ O ; Z ( T )) I ( ω γ,j ) , where generically I ( ω γ,j ) is a multiplicity-free genuine principal series of G appearningin the double sum of the right hand side. We note thatdim Wh ψ ( I ( ω γ,j )) = | X ,Q,n | ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 63 for every ω γ,j ∈ E ( χ, γ ˜ χ O ; Z ( T )), where X ,Q,n := Y /Y ,Q,n . One the other hand, we have dim Wh ψ ( I ( χ )) = | X Q,n | . The relations among various quotients of lattices are illustrated as follows.(4.4) Y ,Q,n / ( Y ∩ Y Q,n ) Y /Y Y / ( Y ∩ Y Q,n ) X Q,n X Γ Q,n X ,Q,n . f If Z ( T ) ⊂ Z ( T ), or equivalently Y ,Q,n = Y ∩ Y Q,n by Corollary 2.12, then the abovediagram becomes(4.5) X ,Q,n X Q,n X Γ Q,n . f Since X Q,n is the “moduli space” of Wh ψ ( I ( χ )), i.e., there is a natural parametrizationof the Whittaker space by X Q,n (see [GSS, § X Q,n into the disjoint union of the fibers of f , and thus a corresponding decompositionof Wh ψ ( I ( χ )). Each fibre of f is a torsor over X ,Q,n and in particular of the same sizeas | X ,Q,n | = dim Wh ψ ( I ( χ | Z ( T ) )) . This decomposition is compatible with the fact that I ( χ ) | G = I ( χ | Z ( T ) ) ⊕ | X Γ Q,n | is anisotypic sum, see Corollary 2.25 and Corollary 4.2.In the remaining of this section, we assume f ( ψ ) = O F and investigate dim Wh ψ ( π ) and dim Wh ψ ( π ), where π is an irreducible constituent of aregular or unitary unramified principal series and π ⊂ π | G .4.3. Regular unramified principal series.
In this subsection, we consider the restric-tion to G of an irreducible constituent π of a regular unramified genuine principal series I ( χ ) of G . In particular, we will study how the Whittaker dimension dim Wh ψ ( π ) isdistributed into the constituents of π | G .Temporarily, we use G ♯ to denote either G or G , and T ♯ ⊂ G ♯ is the covering torus.Let χ ♯ : Z ( T ♯ ) −→ C × be a central character satisfying:– if T ♯ = T , then χ ♯ is unramified;– if T ♯ = T , then i ( χ ♯ ) ⊂ i ( χ ) | T for an unramified χ : Z ( T ) −→ C × .We highlight that in the second case above, χ ♯ may not be unramified, unless ( G, G ) isan isotypic pair; also, by Lemma 3.3 we have i ( χ ♯ ) ⊂ i ( χ ) | T if and only if χ ♯ | T ∩ Z ( T ) = χ | T ∩ Z ( T ) . Denoting Y ♯Q,n := Y ∩ Y Q,n and let T ♯Q,n be the preimage in T of Im( i ♯Q,n ) ⊂ T , where i ♯Q,n : Y ♯Q,n ⊗ F × −→ Y Q,n ⊗ F × −→ Y ⊗ F × is induced from the inclusions Y ♯Q,n ⊂ Y Q,n ⊂ Y . One has T ♯Q,n ⊂ Z ( T ) . Similarly, we have the subgroup T scQ,n ⊂ Z ( T ) associated with Y scQ,n ⊂ Y Q,n . In any case,since T ♯Q,n = T ∩ Z ( T ) = Z ( T ) ∩ Z ( T ) , we see that every χ ♯ is unramified when restricted to T ♯Q,n . In particular, the restriction χ ♯ | T scQ,n : T scQ,n −→ C × is always unramified.In this subsection, we assume that χ ♯ is regular, i.e., the stabilizer subgroup of χ ♯ inthe Weyl group is trivial. Thus the irreducible constituents of I ( χ ♯ ) are multiplicity-free.We denote its Jordan–Holder set by JH( I ( χ ♯ )). ConsiderΦ( χ ♯ ) := (cid:8) α ∈ Φ : χ ♯ ( h α ( ̟ n α )) = q − (cid:9) ⊂ Φ . Let P (Φ( χ ♯ )) be the power set of Φ( χ ♯ ), and denote by C ( X ⊗ R ; χ ♯ )the set of connected components of X ⊗ R − [ α ∈ Φ( χ ♯ ) Ker( α ∨ ) . Proposition 4.3.
There are bijections between the three sets P (Φ( χ ♯ )) ←→ JH( I ( χ ♯ )) ←→ C ( X ⊗ R ; χ ♯ ) denoted by S ↔ π S ↔ Γ S , which is given as follows. First, we have Γ S = { x ∈ X ⊗ R : h α ∨ , x i < if and only if α ∈ S } . Second, the representation π S is characterized by its Jacquet module ( π S ) U = M w ∈ W S δ / B · i ( w − χ ♯ ) , where W S = (cid:8) w ∈ W : Φ( χ ♯ ) ∨ ∩ w (Φ ∨− ) = S ∨ (cid:9) ⊂ W. Proof.
For linear algebraic group and a regular character χ of T ⊂ G (not necessarilyunramified), the result was shown in [Rod81]. For covering groups and unramified χ , itwas extended in [Gao20]. We explain how the argument in [Gao20], which is adoptedfrom [Rod81], actually works for χ ♯ here.Indeed, the key property needed in the proof in [Gao20] is the computation of thePlancherel measure µ ( w α , i ( χ ♯ )) associated with T ( w α , i ( χ ♯ )) with w α being a simplereflection. For unramified χ ♯ , this was explicitly computed in [Gao18a] and used in[Gao20, Proposition 3.5]; it shows that(4.6) µ ( w α , i ( χ ♯ )) − = 1 − q − χ ♯ ( h α ( ̟ n α ))1 − χ ♯ ( h α ( ̟ n α )) · − q − χ ♯ ( h α ( ̟ n α )) − − χ ♯ ( h α ( ̟ n α )) − . ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 65
One thus concludes that the poles of µ ( w α , i ( χ ♯ )) are achieved when α ∈ Φ( χ ♯ ) or − α ∈ Φ( χ ♯ ). This is the key ingredient used in the proof in [Rod81], the rest of which appliesin the covering setting.Now if χ ♯ is for G with i ( χ ♯ ) ⊂ i ( χ ) | T , then we can proceed in two ways in order todetermine µ ( w α , i ( χ ♯ )). First, one can invoke [GS16, Theorem 5.1] directly. Alternatively,it follows from Lemma 2.14 that µ ( w α , i ( χ ♯ )) = µ ( w α , i ( χ ))for every ω : Z ( T ) −→ C × such that i ( ω ) ⊂ i ( χ ) | T . In particular, considering theunramified ω and using the formula of µ ( w α , i ( ω )) in [Gao18a], one also obtains the sameformula (4.6) for µ ( w α , i ( χ ♯ )). Now the same argument in [Gao20, Proposition 3.5] appliesto give the desired result. (cid:3) For every S ⊂ Φ( χ ♯ ) we denote by π χ ♯ ,S = π ( χ ♯ ) S ∈ JH( I ( χ ♯ ))the associated irreducible constituent of I ( χ ♯ ). Corollary 4.4.
Let χ : Z ( T ) → C × be a regular unramified genuine character. (i) For every ω : Z ( T ) −→ C × such that i ( ω ) ⊂ i ( χ ) | T , one has Φ( χ ) = Φ( ω ) . (ii) For every S ⊂ Φ( χ ) , we have ( π χ,S ) | G = (cid:12)(cid:12) X c Q,n (cid:12)(cid:12) · M γ ∈ X Γ Q,n / X c Q,n M ω γ,j ∈ E ( χ, γ ˜ χ O ; Z ( T )) π ω γ,j ,S . Proof.
Part (i) follows from the proof of Proposition 4.3, since µ ( w, i ( χ )) = µ ( w, i ( ω )) forsuch ω and depends only on χ | T scQ,n = ω | T scQ,n . For (ii), we note that the representation π χ,S is characterized by its Jacquet module, and since computing Jacquet module commuteswith restriction to G , it suffices to determineRes TT ( π χ,S ) U = M w ∈ W S δ / B · i ( w − χ ) | T . Since i ( χ ) | T = (cid:12)(cid:12) X c Q,n (cid:12)(cid:12) · M γ ∈ X Γ Q,n / X c Q,n M ω γ,j ∈ E ( χ, γ ˜ χ O ; Z ( T )) i ( ω γ,j ) , and similar decomposition for w − χ for each w ∈ W S holds, we getRes TT ( π χ,S ) U = (cid:12)(cid:12) X c Q,n (cid:12)(cid:12) · M γ ∈ X Γ Q,n / X c Q,n M ω γ,j ∈ E ( χ, γ ˜ χ O ; Z ( T )) M w ∈ W S δ / B · i ( w − ω γ,j )= (cid:12)(cid:12) X c Q,n (cid:12)(cid:12) · M γ ∈ X Γ Q,n / X c Q,n M ω γ,j ∈ E ( χ, γ χ O ; Z ( T )) ( π ω γ,j ,S ) U . As noted, the representation π ω γ,j ,S is uniquely determined by its Jacquet module ( π ω γ,j ,S ) U ,this shows (ii) and concludes the proof. (cid:3) For a regular (
K, s K )-unramified I ( χ ), we want to determine the Whittaker spaceWh ψ ( π χ,S ) in terms of the Wh ψ ( π ω γ,j ,S )’s. As in general there is much difficulty, in theremaining part of this subsection, we will consider two special cases: – when Z ( T ) ⊂ Z ( T ), and thus π χ,S | G = (cid:12)(cid:12) X Γ Q,n (cid:12)(cid:12) · π ω,S is an isotypic sum, where ω = χ | Z ( T ) ;– when the pair is (GL ( n )2 , SL ( n )2 ) with arbitrary n .We hope to illustrate the subtleties on determining even the relation between dim Wh ψ ( π χ,S )and dim Wh ψ ( π ω γ,j ,S ).4.3.1. The isotypic case.
If (
G, G ) is an isotypic pair, then we denote by ω := χ | Z ( T ) the unramified character of Z ( T ). We have π χ,S | G = (cid:12)(cid:12) X Γ Q,n (cid:12)(cid:12) · π ω,S anddim Wh ψ ( π χ,S ) = (cid:12)(cid:12) X Γ Q,n (cid:12)(cid:12) · dim Wh ψ ( π ω,S ) . Now we want to give an interpretation of this equality from the perspective of the shortexact sequence (4.5).
Lemma 4.5.
Assume that G is a saturated cover. Then (i) G is also a saturated cover, and thus both G and G are persistent covers; (ii) one has Z ( T ) ⊂ Z ( T ) in this case.Proof. Recall that we have the inclusions Y scQ,n ⊂ ( Y sc ∩ Y Q,n ) ⊂ ( Y ∩ Y Q,n ) ⊂ Y ,Q,n . The assumption implies that Y scQ,n = Y ,Q,n . This enforces the equality ( Y ∩ Y Q,n ) = Y ,Q,n and thus (ii) holds. This also enforces another equality Y scQ,n = Y sc ∩ Y Q,n , which exactlyshows that G is saturated and gives (i). (cid:3) Let s be any splitting, if it exists, of the short exact sequence of finite abelian groups:(4.7) X ,Q,n X Q,n X Γ Q,n . fs Every such splitting gives a set s ( X Γ Q,n ) of representatives of X Γ Q,n in X Q,n . We have X Q,n = G z ′ ∈ X Γ Q,n f − ( z ′ ) = G z ∈ s ( X Γ Q,n ) ( z + X ,Q,n ) . Assume G is a saturated cover. Also assume thatΦ( χ ) ⊂ ∆ . It then follows from Lemma 4.5 and [Gao20, Theorem 6.6] thatdim Wh ψ ( π χ,S ) = h σ X , σ S i W , dim Wh ψ ( π ω,S ) = h σ X , σ S i W . Here σ S is a certain Kazhdan–Lusztig representation of W of dimension | W S | . On theother hand, σ X : W −→ Perm( X Q,n )is the permutation representation of W given by σ X ( w )( y ) := w [ y ], where w [ y ] = w ( y − ρ ) + ρ. Similarly, σ X : W −→ Perm( X ,Q,n )is also given by σ X ( w )( y ) = w [ y ] ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 67 for every y ∈ X ,Q,n . In fact, σ X is the subrepresentation of σ X restricted on X ,Q,n ⊂ X Q,n .For every coset z + X ,Q,n ⊂ X Q,n , the action w [ · ] is given by w [ z + y ] = w ( z ) + w [ y ] = z + ( w [ y ] + w ( z ) − z )for every y ∈ X ,Q,n , where w ( z ) denotes the usual action arising from reflection. As theimage of w ( z ) − z in X Γ Q,n via f is trivial, we see that w ( z ) − z ∈ X ,Q,n . Thus, w [ z + y ] lies in z + X ,Q,n ; that is, the action w [ · ] on z + X ,Q,n is well-defined.This gives a permutation representation σ z + X : W −→ Perm( z + X ,Q,n ) . On the other hand, we consider the permutation representation σ X ,z : W −→ Perm( X ,Q,n )given by σ X ,z ( w )( y ) = w h y i := w [ y ] + w ( z ) − z. It is easy to check that the action w h·i is well-defined. Thus σ X ,z is a well-definedpermutation representation of W on X ,Q,n . In fact, we have:(i) The representation σ z + X is the one obtained from transporting σ X ,z via thetranslation X ,Q,n → z + X ,Q,n given by y z + y . Thus, as representations of the Weyl group, σ z + X and σ X ,z are equivalent. In particular, we have σ X = M z ∈ s ( X Γ Q,n ) σ X ,z . (ii) Denote by ρ z : W −→ Perm( X ,Q,n )the representation given by the translation ρ z ( w )( y ) = y + w ( z ) − z . It is clearthat σ X ,z = ρ z ◦ σ X .We note that the exact sequence (4.7) is W -equivariant with respect to the usual action w ( · ). Also, the action of W on X Γ Q,n is trivial. In general, the splitting s may not be W -equivariant. However, we believe that in our setting this is indeed the case: Conjecture 4.6.
Assume that G is a saturated cover. Then there exists a W -equivariantsplitting s : X Γ Q,n → X Q,n with respect to the Weyl action w ( · ) . Consequently, with respectto such a splitting s , one has σ X ,z ≃ σ X for every z ∈ s ( X Γ Q,n ) and thus σ X = (cid:12)(cid:12) X Γ Q,n (cid:12)(cid:12) · σ X . In particular, dim Wh ψ ( π χ,S ) = (cid:12)(cid:12) X Γ Q,n (cid:12)(cid:12) · dim Wh ψ ( π χ ,S ) . If s is W -equivariant, then w ( z ) − z = 0and thus ρ z ( w ) = id for every z ∈ s ( X Γ Q,n ); hence, it is clear that σ X ,z = σ X in this case. For a fixed splitting s , consider the following three statements:(S1) σ X ,z ≃ σ X for every z ∈ s ( X Γ Q,n );(S2) h σ X ,z , σ S i W = h σ X , σ S i W for every z ∈ s ( X Γ Q,n ); (S3) dim Wh ψ ( π χ,S ) = (cid:12)(cid:12) X Γ Q,n (cid:12)(cid:12) · dim Wh ψ ( π χ ,S ).Clearly, we have the implications(S1) = ⇒ (S2) = ⇒ (S3) . Here Conjecture (4.6) asserts the strongest (S1) in our context, and is certainly compatiblewith (S3) which holds a priori.
Example 4.7.
Consider the cover G = GL ( n )2 associated with p , q ∈ Z such that 2 p − q = −
1, as discussed from § G = SL ( n )2 is saturated if and only if n isodd. Thus, we assume that n is odd. An easy computation gives that Y Q,n = { y e + y e ∈ Z e ⊕ Z e : n | (4 p + 1) y and n | ( y − y ) } . Setting d = gcd(4 p + 1 , n ), we have X ,Q,n X Q,n X Γ Q,n , with X ,Q,n = Z α ∨ / ( Z nα ∨ ) , | X Q,n | = n /d and (cid:12)(cid:12) X Γ Q,n (cid:12)(cid:12) = n/d. We claim that there is a W -equivariant splitting s given by s ( X Γ Q,n ) = { ke c : 0 k ( n/d − } ⊂ X Q,n . For this purpose, it suffices to show that ke c / ∈ Y sc + Y Q,n for every 1 k ( n/d − mα ∨ ∈ Y sc such that ke c + mα ∨ ∈ Y Q,n . This implies that( n/d ) | ( k + m ) and n | ( k + a − k + a ) . As n is odd, this implies that ( n/d ) | k , which is a contradiction. Thus, Conjecture 4.6 holdsfor this pair (GL , SL ). We remark that when d = n (for example, if n = 3 , p = − X ,Q,n = X Q,n ; in this case π χ,S | G = π χ ,S with multiplicity one. Example 4.8.
We consider the cover GSp (3)4 from § Q ( α ∨ ) = −
1. Notethat in this case, Sp is saturated, as its dual group is SO . Using notations in § Y Q,n = y e + y e + y e ∈ ⊕ i =0 Z e i : • | ( − y + y ) • | ( − y + y ) • | ( y + y + 2 y · Q ( e )) . There are two cases as follows.– If Q ( e ) = ± X ,Q,n = X Q,n in this case.Thus, Conjecture 4.6 holds trivially, and we have π χ,S | G = π χ ,S with multiplicityone.– If Q ( e ) = 0 mod 3, then Y Q,n = 3 Y . In this case, one has X ,Q,n = Y / Y . On the other hand, X Q,n = Y / Y . Thus, a W -equivariant splitting is given by s ( X Γ Q,n ) = { ke c : 0 k } . In either case, we see that Conjecture 4.6 holds.
ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 69
Covers of GL and SL . We consider the pair (GL , SL ) of n -fold covers associatedwith p and q such that Q ( α ∨ ) = 2 p − q = −
1. For odd n , this was discussed in Example4.7; thus, we assume in this subsection that n = 2 m is even. Again, we set d := gcd(4 p + 1 , n ) . It is easy to see the diagram (4.4) becomes(4.8) Z ( mα ∨ ) / Z ( nα ∨ ) Y /Y Z α ∨ / Z ( nα ∨ ) X Q,n X Γ Q,n X c Q,n Z α ∨ / Z ( mα ∨ ) X ,Q,n , where | X Q,n | = n /d and (cid:12)(cid:12) X Γ Q,n (cid:12)(cid:12) = n/d . Also, it is easy to obtain Y c = { y e + y e ∈ Y : 2 | ( y − y ) } , and thus we see that [ X Γ Q,n : X c Q,n ] = 2 . It follows that X Γ Q,n / X c Q,n = { ie : 0 i } . For every γ i ∈ X Γ Q,n / X c Q,n , we have E ( χ, γ ˜ χ O ; Z ( T )) = { ω γ,j : 0 j } . Setting γ i := ie . It follows from Theorem 2.24 (see also [GSS, Theorem 8.15]) that I ( χ ) | G = ( m/d ) · M i M j I ( ω γ i ,j ) . Also, the two representations I ( ω γ ,j ) , j = 0 , K , s K )-unramified con-stituents of I ( χ ). Again, for every S ⊂ Φ( χ ), we have(4.9) π χ,S | G = ( m/d ) · M i M j π ( ω γ i ,j ) S , where we have written π ( ω γ i ,j ) S instead of π ω γi,j ,S for notational convenience.In the rest of this subsection, we assume that Φ( χ ) = ∆ ∨ = { α ∨ } . In this case, wehave π χ, ∅ I ( χ ) π χ, ∆ , where π χ, ∅ is the covering analogue of the Steinberg representation, and π χ, ∆ is the thetarepresentation associated with χ . Since GL is always a saturated cover and thus persis-tent (see [Gao20, Lemma 2.7]), it follows from [Gao20, Theorem 6.6] that– dim Wh ψ ( π χ, ∅ ) = h σ X , i W = m ( n + 1) /d , which is equal to the number of W -orbits in X Q,n ;– dim Wh ψ ( π χ, ∆ ) = h σ X , ε W i W = m ( n − /d , which is equal to the number of free W -orbits in X Q,n .Here (resp. ε W ) denotes the trivial character (resp. sign character) of W . Theorem 4.9.
Let χ : Z ( T ) → C × be an unramified genuine character such that Φ( χ ) =∆ . We always have dim Wh ψ ( π ( ω γ i ,j ) ∅ ) + dim Wh ψ ( π ( ω γ i ,j ) ∆ ) = dim Wh ψ ( I ( ω γ i ,j )) = m. Moreover, the following fold. (i) If | n (i.e. | m ), then dim Wh ψ ( π ( ω γ i ,j ) ∆ ) = m/ for i = 0 , j ∈ { , } ,m/ for i = 1 , j = 0 , ( m − / for i = 1 , j = 1 . (ii) If n = 2 m with m odd, then dim Wh ψ ( π ( ω γ i ,j ) ∆ ) = ( m + 1) / for i = 0 , j = 0 , ( m − / for i = 0 , j = 1 , ( m − / for i = 1 , j ∈ { , } . In either case, π ( ω γ ,j ) ∆ , j = 0 , are the two ( K , s K ) -unramified theta representationsassociated with I ( ω γ ,j ) .Proof. We first consider the case m is even. In this case, the cover SL ( n )2 is persistent. As I ( ω γ ,j ) for j ∈ { , } is unramified, it thus follows from [Gao20, Proposition 6.2] that(4.10) dim Wh ψ ( π ( ω γ ,j ) ∆ ) = h σ X , ε W i W = m , which is equal to the number of free W -orbits in X ,Q,n . To deal with dim Wh ψ ( π ( ω γ ,j ) ∆ ),we consider a local coefficients matrix M ( w, i ( ω γ ,j )) associated with the map T ( w, i ( ω γ ,j )) ∗ : Wh ψ ( I ( ω γ ,j )) −→ Wh ψ ( I ( ω γ ,j )) , which arises from the intertwining operator T ( w, i ( ω γ ,j )) : I ( i ( ω γ ,j )) −→ I ( w i ( ω γ ,j )) . One has dim Wh ψ ( π ( ω γ ,j ) ∆ ) = rank of T ( w, i ( ω γ ,j )) ∗ . To have a description of M ( w, i ( ω γ ,j )), we first reconcile some notations used in [GSS].Let ξ : F × −→ C × be a character such that ξ ( x ) n = χ ( h α ( x n )) . Now we have from [GSS, § M ( w, i ( ω γ ,j )), by swapping certain rows and columns, becomes ablock-diagonal matrix with m/ W -orbit in X ,Q,n . We label these twoby two blocks as M k , k ( m/ . Every M k is a non-zero matrix, and thus in particular rank( M k ) > . – There is a special two by two block labelled by M , which takes the form M = (cid:18) γ (1 , ξ − n ) βββ ( ω γ ,j , s, ψ ) 0 (cid:19) , ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 71 where βββ ( ω γ ,j , s, ψ ) = (1 − q − / · ( ξη u ) − m γψ )(1 − q − / ξ m γψ )(1 − ( ξη u ) m γψ )(1 − ξ − m γψ ) . Here u ∈ O × is a representative of the unique nontrivial coset of O × /O × , and η u ( a ) = ( u, a ) n for a ∈ F × ; also, m γ ψ := m · γ ψ ( ̟ ) ∈ {± m } , where γ ψ is the Weil-index associated with ψ .– For every 2 k ( m/ M k ) = C × µ ( w, i ( χ )) − , where = C × means the equality holds modulo some elements in C × (in fact mod-ulo an element in {± ξ − m } as shown in the proof of [GSS, Theorem 9.13]), and µ ( w, i ( χ )) denotes the Plancherel measure.Since Φ( χ ) = ∆, we see that µ ( w, i ( χ )) − = 0 and thusrank( M k ) = 1 for 2 k ( m/ . Also, we have γ (1 , ξ − n ) = (1 − q − ξ − n ) / (1 − ξ n ) , which equals 0 in this case. Let sgn( ξ ) ∈ {± } be such that ξ ( ̟ ) m = sgn( ξ ) · q − / . Then it is not hard to see that βββ ( ω γ ,j , s, ψ ) = 0 ⇐⇒ sgn( ξ ) = η u, (2) ( ̟ ) , where η u, (2) ( x ) = ( u, x ) . Thus, we could label ω γ , to be such that sgn( ξ ) = − η u, (2) ( ̟ )and thus βββ ( ω γ , , s, ψ ) = 0; in this case, rank( M ) = 1 anddim Wh ψ ( π ( ω γ , ) ∆ ) = 1 + ( m − / m/ . On the other hand, ω γ , is such that βββ ( ω γ , , s, ψ ) = 0, and thusdim Wh ψ ( π ( ω γ , ) ∆ ) = ( m − / . This completes the proof for the case 4 | n .Now we assume that m = 2 l + 1 is odd. Again, we first consider the unramified thetarepresentation π ( ω γ ,j ) ∆ , j ∈ { , } . Note that the n -fold cover SL is not persistent in thiscase, and thus formula (4.10) does not hold. The W -orbits in X ,Q,n = Z ( α ∨ ) / Z ( mα ∨ )are {O iα ∨ : 1 i l } ∪ {O − lα ∨ } with O iα ∨ = { iα ∨ , (1 − i ) α ∨ } for 1 i l and O − lα ∨ = {− lα ∨ } . Every free orbit O iα ∨ , i l supports a one-dimensional subspace of Wh ψ ( π ( ω γ ,j ) ∆ )for both j ∈ { , } . On the other hand, it follows from [Gao17, Theorem 3.14 and § O − lα ∨ supports a one-dimensional subspace of Wh ψ ( π ( ω γ ,j ) ∆ ) for exactly one of theunramified characters in { ω γ ,j : j = 0 , } ; we assume without loss of generality thatWh ψ ( π ( ω γ , ) ∆ ) O − lα ∨ = 1 , and thus Wh ψ ( π ( ω γ , ) ∆ ) O − lα ∨ = 1 . This gives that(4.11) Wh ψ ( π ( ω γ , ) ∆ ) = l + 1 = ( m + 1) / , and Wh ψ ( π ( ω γ , ) ∆ ) O − lα ∨ = l = ( m − / . To determine Wh ψ ( π ( ω γ ,j ) ∆ ), we consider a local coefficients matrix M ′ ( w, i ( ω γ ,j )) of size m × m associated with the map T ( w, i ( ω γ ,j )) ∗ as above. Again, let ξ : F × → C × be the character such that ξ ( x ) n = χ ( h α ( x n )) for x ∈ F × . We may assume that ξ isramified and ξ is unramified. Such a matrix M ′ ( w, i ( ω γ ,j )) is computed in the proof of[GSS, Theorem 9.12] with the following properties.– The matrix M ′ ( w, i ( ω γ ,j )), by swapping certain rows and columns, becomes ablock-diagonal matrix with every block associated to a W -orbits in X ,Q,n . Thus,we have ( m + 1) / M ′ i , each associated with O iα ∨ , i l or i = − l . Also the size of M ′ i is exactly |O iα ∨ | for every such i .– For every 1 i l , the matrix M ′ i is a non zero two by two matrix withdet( M ′ i ) = C × µ ( w, i ( χ )) − . On the other hand, M ′− l = ˜ γ (1 , ξ − m , ψ ) , which is the metaplectic-gamma factor defined in [Szp11].Now, since Φ( χ ) = ∆, we see that µ ( w, i ( χ )) − = 0 and thusrank( M ′ i ) = 1 for every 1 i l. On the other hand, one has from [GS16, Theorem A.1] (see also [GSS, Theorem 4.4])that ˜ γ (1 , ξ − m , ψ ) = C × γ (1 / , ξ m , ψ ) γ (0 , ξ n , ψ ) , where ψ ( x ) = ψ (2 x ). Since ξ is ramified with ξ unramified, we see that ξ m is ramifiedand therefore γ (1 / , ξ m , ψ ) ∈ C × . Also, ξ n is unramified with ξ n ( ̟ ) = q − , and this implies that˜ γ (1 , ξ − m , ψ ) = C × − q − ξ − n ( ̟ )1 − ξ n ( ̟ ) = 0 . This shows that M ′− l = 0 for either j = 0 ,
1, and thusdim Wh ψ ( π ( ω γ ,j ) ∆ ) = m − j = 0 , π ( ω γ ,j ) ∆ , j =0 ,
1. Indeed, in this case, we may assume that ξ is unramified. Thus,˜ γ (1 , ξ − m , ψ ) = γ (1 / , ξ m , ψ ) γ (0 , ξ n , ψ ) = 1 − q / ξ m ( ̟ )1 − q − / ξ − m ( ̟ ) · − q − ξ − n ( ̟ )1 − ξ n ( ̟ ) . Recall that ξ m ( ̟ ) = sgn( ξ ) · q − / , and therefore˜ γ (1 , ξ − m , ψ ) = 0 ⇐⇒ sgn( ξ ) = − . This also gives the desired equalities for dim Wh ψ ( π ( ω γ ,j ) ∆ ) , j = 0 ,
1. In any case, theproof of the case m is odd is completed. (cid:3) Remark 4.10.
We always have the equality X i,j dim Wh ψ ( π ( ω γ i ,j ) ∆ ) = n − . For n = 4 this equality and (4.10) together enforce dim Wh ψ ( π ( ω γ ,j ) ∆ ) = 0 for both j = 0 ,
1. For n = 2, the above equality coupled with (4.11) also imply thatdim Wh ψ ( π ( ω γ ,j ) ∆ ) = 0 for j = 0 , . ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 73
However, for n large, it is necessary to have an explicit description of a local coefficientsmatrix, as used in the proof of Theorem 4.9.4.4. Unitary unramified principal series.
In this subsection, we investigate the de-composition of an irreducible constituent of a unitary (
K, s K )-unramified principal seriesunder the restriction from G to G , and also how the Whittaker dimension behaves withrespect to the restriction.We first recall some notations and results following [Gao]. Let I ( χ ) be a unitary( K, s K )-unramified genuine principal series of G . Denote by R χ ⊂ W ( χ )the R-group associated with I ( χ ), where W ( χ ) ⊂ W is the stabilizer subgroup of χ . Onehas an algebra isomorphism C [ R χ ] ≃ End G ( I ( χ )) , given by R χ ∋ w
7→ A ( w, i ( χ )) , where A ( w, i ( χ )) = γ ( w, χ ) · T ( w, i ( χ ))is the normalized intertwining operator. Here γ ( w, χ ) is the gamma factor associatedwith w ∈ W . One knows that R χ is abelian (see [Luo20] or [Gao, Theorem 4.6]) and thusthere is a multiplicity-free decomposition I ( χ ) = M τ ∈ Irr( R χ ) π τ , where Irr( R χ ) = Hom( R χ , C × ) denotes the characters of R χ . One has A ( w, i ( χ )) | π τ = τ ( w ) · idfor every w ∈ R χ .On the dual side, let φ χ : WD F −→ L T −→ L G be the parameter associated with I ( χ ). Recall the component group S χ := S φ χ of the centralizer of Im( φ χ ) modulo Z ( G ) ∨ . By using the results of Keys [Key87], it isshown in [Gao, Theorem 4.9] that(4.12) R χ ≃ S χ . Note that the same consideration applies to unitary unramified principal series of G ,and analogous results hold.Consider the decomposition from Theorem 2.24 (using slightly different notation)(4.13) I ( χ ) | G = (cid:12)(cid:12) X c Q,n (cid:12)(cid:12) · M γ i M j I ( ω γ i ,j ) , where X Γ Q,n / X c Q,n = { γ i } i . For every γ i we have E ( χ, γ i ˜ χ O ; Z ( T )) = { ω γ i ,j } j , which always has size | Y ,Q,n / ( Y ∩ Y Q,n ) | . For simplicity of notations, we may even write ω i,j := ω γ i ,j . For every constituent I ( ω i,j ) := I ( ω γ i ,j ), we have the L-parameter φ ω i,j which fits intothe following commutative diagram:(4.14) W F L G L G L H. φ χ φ ωi,j f G,H f G ,H Recall that we defined φ ♦ = f G,H ◦ φ χ = f G ,H ◦ φ ω i,j and thus have S ♦ := S φ ♦ . We denote Ω χ = (cid:8) ω i,j ∈ Hom gen ( Z ( T ) , C × ) : I ( ω i,j ) ⊂ I ( χ ) | G (cid:9) , and have from Lemma 3.3 thatΩ χ = n ω i,j ∈ Hom gen ( Z ( T ) , C × ) : ω i,j | T ♯Q,n = χ | T ♯Q,n o . We also denoteΩ un χ = E ( χ, ˜ χ O ; Z ( T ))= (cid:8) ω i,j ∈ Hom gen ( Z ( T ) , C × ) : I ( ω i,j ) ⊂ I ( χ ) | G and is ( K , s K )-unramified (cid:9) For every ω ♭ ∈ Ω un χ , setΩ un χ ( ω ♭ ) := (cid:8) w ( ω ♭ ) : w ∈ W (cid:9) ∩ Ω un χ = (cid:8) w ( ω ♭ ) : w ∈ W (cid:9) ∩ E ( χ, ˜ χ O ; Z ( T )) . For ω ♭ ∈ Ω un χ there are two natural inclusions (see § § S χ S ω ♭ S ♦ . Note that S ♦ depends only on φ χ and not on choice of ω ♭ . We write χ re := χ | T ♯Q,n = ω ♭ | T ♯Q,n , where L ( T ♯Q,n ) = Y ∩ Y Q,n . Since Y scQ,n ⊂ ( Y ∩ Y Q,n ), it follows from (4.12) and [Gao, § S ♦ / S ω ♭ ≃ W ( χ re ) /W ( ω ♭ ) . Lemma 4.11.
For every ω ♭ ∈ Ω un χ , one has Ω un χ ( ω ♭ ) = (cid:8) w ( ω ♭ ) : w ∈ W ( χ re ) (cid:9) , with Stab W ( χ re ) ( ω ♭ ) = W ( ω ♭ ) . Thus, Ω un χ ( ω ♭ ) is a torsor over S ♦ / S ω ♭ .Proof. It follows from the definition of Ω un χ ( ω ♭ ) that w ( ω ♭ ) ∈ Ω un χ ( ω ♭ ) ⇐⇒ w ( ω ♭ ) is an unramified extension of χ re ⇐⇒ w ( ω ♭ ) | Z ( T ) ∩ T = χ re = ω ♭ | Z ( T ) ∩ T ⇐⇒ w ∈ W ( χ re ) . ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 75
It is clear that the stabilizer subgroup of ω ♭ is W ( ω ♭ ) ⊂ W ( χ re ). It follows that Ω un χ ( ω ♭ )is a torsor over S ♦ / S ω ♭ . (cid:3) We have the short exact sequence S χ / ( S χ ∩ S ω ♭ ) S ♦ / S ω ♭ S ♦ / ( S χ · S ω ♭ ) . The action of S χ on Ω un χ ( ω ♭ ) factors through S χ / ( S χ ∩ S ω ♭ ), and every such orbit is infact a torsor over S χ / ( S χ ∩ S ω ♭ ) by Lemma 4.11. Thus, we have a decompositionΩ un χ ( ω ♭ ) = | S ♦ / ( S χ ·S ω♭ ) | G k =1 O ( ω ♭ ) k into a disjoint union of S χ -orbits, where every O ( ω ♭ ) k is a torsor over S χ / ( S χ ∩ S ω ♭ ).To proceed, we analyze the space Hom G ( π ρ , π τ ) with ρ ∈ Irr( S ω ♭ ) , τ ∈ Irr( S χ ) in moredetail by adapting ideas from [Key87]. To set up some notations:– f : A −→ B is a ring homomorphism;– W A is a A -module, and W B , V are both B -modules;– setting E = End B ( V ), then Hom B ( W B , V ) and Hom A ( W A , V f ) are both E -modules,where V f = V is view as a A -module via f .It is proved in [Key87, Page 53] that if W B is a direct summand of V , then the naturalhomomorphism(4.15) Hom A ( W A , ( W B ) f ) −→ Hom E (Hom B ( W B , V ) , Hom A ( W A , V f ))of vector spaces is an isomorphism. We apply the above to the following data A = C ∞ c [ G ] , B = C ∞ c [ G ] , V = I ( χ ) , W A = π ρ , W B = π τ to obtain(4.16) Hom G ( π ρ , π τ | G ) = Hom C [ S χ ] (cid:0) τ, Hom G ( π ρ , I ( χ ) | G ) (cid:1) , where we note that τ ≃ Hom G ( π τ , I ( χ ))as C [ S χ ]-module. In view of the decomposition of I ( χ ) | G in (4.13), we first observe that(see [BZ77, Theorem 2.9])Hom G ( π ρ , I ( ω i,j )) = 0 ⇐⇒ ω i,j ∈ Ω un ( ω ♭ ) . Denoting Π( ω ♭ ) := M ω i,j ∈ Ω un ( ω ♭ ) I ( ω i,j )and Π( O ( ω ♭ ) k ) := M ω i,j ∈O ( ω ♭ ) k I ( ω i,j ) , one clearly has Π( ω ♭ ) = M k Π( O ( ω ♭ ) k ) . Also, Hom G ( π ρ , I ( χ ) | G ) = (cid:12)(cid:12) X c Q,n (cid:12)(cid:12) · M k Hom G ( π ρ , Π( O ( ω ♭ ) k )) . The action of w ∈ S χ on Hom G ( π ρ , I ( χ ) | G ) is given by transporting the action of A ( w, χ ) on I ( χ ). Also, we have a decomposition A ( w, χ ) = (cid:12)(cid:12) X c Q,n (cid:12)(cid:12) · M i,j A ( w, ω i,j ) , where A ( w, ω i,j ) : I ( ω i,j ) −→ I ( w ω i,j )is the normalized intertwining operator of each I ( ω i,j ). We see that the restriction A ( w, χ ) | Π( O ( ω ♭ ) k ) is well-defined for every k . Thus, the spaceHom G ( π ρ , Π( O ( ω ♭ ) k ))affords a representation of C [ S χ ] of dimension (cid:12)(cid:12) O ( ω ♭ ) k ) (cid:12)(cid:12) , and S χ also acts on the (cid:12)(cid:12) Ω un χ ( ω ♭ ) (cid:12)(cid:12) -dimensional space Hom G ( π ρ , Π( ω ♭ )) = M k Hom G ( π ρ , Π( O ( ω ♭ ) k )) . Theorem 4.12.
Let I ( χ ) be a unitary ( K, s K ) -unramified genuine principal series of G and I ( ω ♭ ) an ( K , s K ) -unramified constituent of I ( χ ) | G . Keep notations as above. Thenas representations of S χ , (4.17) Hom G ( π ρ , Π( O ( ω ♭ ) k )) ≃ Ind S χ S χ ∩S ω♭ ( ρ ) for every k . Consequently, (4.18) dim Hom G ( π ρ , π τ ) = (cid:12)(cid:12) X c Q,n (cid:12)(cid:12) · D Ind S ♦ S ω♭ ρ, Ind S ♦ S χ τ E S ♦ for every τ ∈ Irr( S χ ) and ρ ∈ Irr( S ω ♭ ) .Proof. First, Hom G ( π ρ , Π( O ( ω ♭ ) k )) = M ω i,j ∈O ( ω ♭ ) k Hom G ( π ρ , I ( ω i,j )) . Since I ( ω i,j ) ≃ I ( ω ♭ ) for every ω i,j ∈ O ( ω ♭ ) k , it is easy to check that as representationsof S χ ∩ S ω ♭ , one has Hom G ( π ρ , I ( ω i,j ) ≃ ρ | S χ ∩S ω♭ . Moreover, as O ( ω ♭ ) k is a torsor over S χ / ( S χ ∩ S ω ♭ ) for every k , the representation of S χ on Hom G ( π ρ , Π( O ( ω ♭ ) k )) is indeed Ind S χ S χ ∩S ω♭ ρ . This gives (4.17).Now to prove (4.18), we see that as representations of S χ ,Hom G ( π ρ , I ( χ ) | G ) = (cid:12)(cid:12) X c Q,n (cid:12)(cid:12) · M k Hom G ( π ρ , Π( O ( ω ♭ ) k ))= (cid:12)(cid:12) X c Q,n (cid:12)(cid:12) · (cid:12)(cid:12) S ♦ / ( S χ · S ω ♭ ) (cid:12)(cid:12) · Ind S χ S χ ∩S ω♭ ρ. On the other hand, the Frobenius reciprocity gives D Ind S ♦ S ω♭ ρ, Ind S ♦ S χ τ E S ♦ = D Res S ♦ S χ Ind S ♦ S ω♭ ρ, τ E S χ , and by Mackey’s theory one hasRes S ♦ S χ Ind S ♦ S ω♭ ρ = M s ∈S ♦ / ( S χ ·S ω♭ ) Ind S χ S χ ∩S ω♭ ρ = (cid:12)(cid:12) S ♦ / ( S χ · S ω ♭ ) (cid:12)(cid:12) · Ind S χ S χ ∩S ω♭ ρ. This coupled with (4.16) give the desired equality (4.18). The proof is completed. (cid:3)
ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 77
Corollary 4.13.
Let I ( χ ) be a unitary ( K, s K ) -unramified genuine principal series of G and I ( ω ) ⊂ I ( χ ) | G , ω ∈ Ω( χ ) an arbitrary constituent. Then dim Hom G ( π ρ , π τ ) = (cid:12)(cid:12) X c Q,n (cid:12)(cid:12) · D Ind S ♦ S ω ρ, Ind S ♦ S χ τ E S ♦ for every τ ∈ Irr( S χ ) and ρ ∈ Irr( S ω ) . In particular, this verifies Conjecture 3.2 (ii) forunitary unramified principal series of G with e ( I ( χ )) = (cid:12)(cid:12) X c Q,n (cid:12)(cid:12) .Proof.
Assume that ω = ω γ,j in the notation of (4.13). Since I ( χ ) is ( K, s K )-unramified, itis also ( γ · K, γ · s K )-unramified. Since the genuine principal series I ( ω ) is ( γ · K , γ · s K )-unramified in this case (see Remark 2.26), we can apply Theorem 4.12 to obtain thedesired equality regarding dim Hom G ( π ρ , π τ ). The rest is clear. (cid:3) A special case is the following
Corollary 4.14.
Assume that ( G, G ) is an isotypic-pair. Let I ( χ ) be a unitary unram-ified principal series of G and let ω ♭ = χ | Z ( T ) . Then dim Hom G ( π ρ , π τ ) = (cid:12)(cid:12) X Γ Q,n (cid:12)(cid:12) · (cid:10) ρ | S χ , τ (cid:11) S χ . Proof.
For an isotypic pair (
G, G ), one has G ∨ = H ∨ and thus S ω ♭ = S ♦ . In particular, S χ ⊂ S ω ♭ . The result follows from Theorem 4.12. (cid:3) Remark 4.15.
Corollary 4.14 confirms for unitary unramified genuine principal seriespart (ii) of Conjecture 3.2. In fact, Theorem 4.12 generalizes this to the case where(
G, G ) may not be an isotypic-pair.It follows from Corollary 4.14 that for an isotypic pair ( G, G ), one has(4.19) dim Wh ψ ( π τ ) = (cid:12)(cid:12) X Γ Q,n (cid:12)(cid:12) · X ρ ∈ Irr( S ω♭ ) (cid:10) ρ | S χ , τ (cid:11) S χ · dim Wh ψ ( π ρ )for every τ ∈ Irr( S χ ) and ρ ∈ Irr( S ω ♭ ). We want to justify the equality (4.19) from[Gao, Theorem 5.6] which we briefly recall as follows:– there is a representation σ Wh0 : S ω ♭ −→ GL( C | X ,Q,n | )which is given by σ Wh0 ( w ) = A ( w, ω ♭ ) ∗ , the induced isomorphism of Wh ψ ( I ( ω ♭ ))of dimension | X ,Q,n | . One hasdim Wh ψ ( π ρ ) = (cid:10) ρ, σ Wh0 (cid:11) S ω♭ for every ρ ∈ Irr( S ω ♭ ).Note that one also has a representation(4.20) σ Wh : S χ −→ GL( C | X Q,n | )given by σ Wh ( w ) := A ( w, χ ) ∗ for w ∈ S χ , and similarly(4.21) dim Wh ψ ( π τ ) = (cid:10) τ, σ Wh (cid:11) S χ for every τ ∈ Irr( S χ ). The decompositionWh ψ ( I ( χ )) = | X Γ Q,n | M i =1 Wh ψ ( I ( ω ♭ )) , which is compatible with the decomposition of A ( w, χ ) = L i A ( w, ω ♭ ), shows that(4.22) σ Wh ≃ (cid:12)(cid:12) X Γ Q,n (cid:12)(cid:12) · ( σ Wh0 ) | S χ . It is easy to see that the three equalities (4.20), (4.21) and (4.22) together imply (4.19)as well.Now we explain relations among (4.19), Conjecture 4.6 and [Gao, Conjecture 5.3].Thus, for the rest of this subsubsection, we assume that G is simply-connected and G is a saturated cover. We have– ([Gao, Conjecture 5.3]) the equality dim Wh ψ ( π ρ ) = h ρ, σ X i S ω♭ holds for every ρ ∈ Irr( S ω ♭ ).Here σ X : S ω ♭ ֒ → W −→ Perm( X ,Q,n )is the permutation representation given by σ X ( w ) = w [ · ]. The following is straightfor-ward. Proposition 4.16.
Consider a pair ( G, G ) with G being simply-connected and G asaturated cover. (i) Assume [Gao, Conjecture 5.3] , then dim Wh ψ ( π τ ) = (cid:12)(cid:12) X Γ Q,n (cid:12)(cid:12) · (cid:10) τ, ( σ X ) | S χ (cid:11) S χ for every τ ∈ Irr( S χ ) . (ii) Assume both [Gao, Conjecture 5.3] and Conjecture 4.6, then dim Wh ψ ( π τ ) = h τ, σ X i S χ for every τ ∈ Irr( S χ ) . Note that if I ( χ ) is irreducible, then the equalities in Proposition 4.16 hold triviallyand unconditionally. We give two examples below in the non isotypic-pair case. Example 4.17.
Let G = GL ( n )2 be associated with p = 0 , q = 1, and n = 2 m is even.One has Y ,Q,n = Z ( mα ∨ ) , Y Q,n = nY, Y ∩ Y Q,n = nY . Thus, G ∨ = GL , G ∨ = SL and H ∨ = PGL . For every (
K, s K )-unramified I ( χ ), one hasΩ un χ = { ω , , ω , } . Note that we always have S χ = { } and S ω ,j = { } for j = 0 ,
1. Regarding S ♦ , thereare two cases:– if χ ( h α ( ̟ n )) = −
1, then S ♦ = { } and thus Ω un χ ( ω ,j ) = { ω ,j } for each j .– if χ ( h α ( ̟ n )) = −
1, then S ♦ = W and we have Ω un χ ( ω ,j ) = Ω un χ for j = 0 , ω ,j ) (in thesense of [GG18] if m is odd) for ω ,j issat( ω ,j ) = √− j +1 √− − (2 j +1) ! ∈ G ∨ . This example is easily generalizable to the pair (GL ( n ) r , SL ( n ) r ) with r | n . For such a pair,one always has S χ = S ω ♭ = { } . However, the group S ♦ will be more complicated. ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 79
Example 4.18.
Consider the pair of double covers (
G, G ) = (GSp (2)2 r , Sp (2)2 r ) as from § § G ∨ = ( GSp r ( C ) , if r is odd;PGSp r ( C ) × GL ( C ) , if r is even.On the other hand, we have G ∨ = Sp r , H ∨ = PGSp r . Assume r is even. It follows from [Key82, Page 399] that the only nontrivial S χ = R χ is { , w } ≃ Z / Z which is generated by w := w α w α ...w α r − , with the character χ satisfying χ ( h α i ( ̟ n αi )) = − i = 2 k − , k r/ . For such χ , one has Ω un χ = { ω , , ω , } ; also, S χ = S ♦ = { , w } and S ω ♭ = { } for every ω ♭ ∈ Ω un χ . In this case, for any j ∈ { , } ,Ω un χ ( ω ,j ) = Ω un χ , which is a torsor over S ♦ . It follows from Theorem 4.12 thatdim G ( I ( ω ,j ) , π τ ) = (cid:12)(cid:12) X c Q,n (cid:12)(cid:12) · h τ, C [ S χ ] i S χ = (cid:12)(cid:12) X c Q,n (cid:12)(cid:12) = 1for every τ ∈ Irr( S χ ) and j = 0 , Unramified L-packets
In this section, we consider the internal structure of an unramified L-packets, the linearalgebraic analogue has been investigated in [Mis]. The following two topics are pertainingto our discussion:(i) the parametrization of elements inside an L-packet with respect to changing thehyperspecial maximal compact subgroups of G ,(ii) the variation of the Whittaker dimension of elements inside an unramified L-packetwith respect to different orbits of the Whittaker datum.Regarding (ii), some earlier work include [Kuo02, Kuo10, Mis16]. In this section, we adoptsome ideas from these works in the covering setting. The fact that the three groups G , its n -fold cover G and the principal endoscopic group G Q,n all play a role here necessitatesa careful analysis of certain relations among them, which allows us to postulate somenatural problems and prove certain results for covers. For this purpose, we first brieflyrecall the results for (split) linear algebraic groups.5.1.
Unramified L-packets for linear groups.
Let G be the F -rational points of alinear algebraic group G , and T the F -points of T . For a character χ : T −→ C × which is trivial on T := T ( O ), one obtains on the dual side an unramified parameter φ χ : WD F −→ L T which is trivial on I F ⊂ W F and SL ( C ) ⊂ WD F . We call φ χ an unramified L-parametervalued in L T . Post-composing φ χ with the natural inclusion L T −→ L G gives a parameter which is still denoted by φ χ : W F −→ L G. The local Langlands correspondence postulates that the L-packet L ( φ χ ) associated with φ χ consists exactly of the subquotients of the principal series I ( χ ), which are K -unramifiedwith respect to a hyperspecial maximal compact subgroup K of G , see [Bor79, § L ( φ χ ) ←→ Irr( S ( φ χ )) , where S ( φ χ ) is the component group of the image of the parameter φ χ in L G .If we conveniently index the packet as L ( φ χ ) = n π ( φ χ , ρ ) : ρ ∈ Irr( S ( φ χ )) o , then as remarked each π ( φ χ , ρ ) is K -unramified for some hyperspecial subgroup K ⊂ G .It is natural to ask how the indexing by ρ varies with respect to changing the hyperspecialmaximal compact subgroup K . Following [Mis], we describe the answer to this questionwhich embodies part of the internal structure of L ( φ χ ).Consider the root datum ( X, Φ , ∆; Y, Φ ∨ , ∆ ∨ )of the group G , and also recall the root lattice X sc = Z [∆] and coroot lattice Y sc = Z [∆ ∨ ].One has the standard affine Weyl group W a = X sc ⋊ W acting on the vector space X sc ⊗ Z R . On the other hand, there is also a natural actionof the extended affine Weyl group ˜ W a = X ⋊ W on X ⊗ Z R . Defining Γ := ˜ W a /W a , one has a split short exact sequence W a ˜ W a Γwith a splitting given by s (Γ) = n x ∈ ˜ W a : x · C = C o , where C ⊂ X ⊗ R is an alcove for the action of ˜ W a on X ⊗ R . That is, we have˜ W a = W a ⋊ Γwith Γ ≃ X/X sc . Note that Z ( G ) = Hom(Γ , F × ). For every lattice L , we write L Q := L ⊗ Q for the Q -vector space. It is easy to see thatΓ tor = ( X ∩ X sc Q ) /X sc , ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 81 where Γ tor ⊂ Γ denotes the torsion-subgroup. Denote by Y ad the cocharacter group of G ad , the adjoint quotient group of G . Then Y ad ⊂ Q [∆ ∨ ] is the lattice of coweights,which is Z -dual to Z [∆]. Consider b Γ † := Coker( Y −→ Y ad ) , we see that there is an embedding b Γ † ֒ → b Γ := Hom(Γ , Q / Z ) . In fact, the composite b Γ † b Γ d Γ tor is an isomorphism, see [Mis, Lemma 11].Consider the set K = { G x : x is a hyperspecial point in B ( G ) } / ∼ of conjugacy classes of hyperspecial maximal compact subgroup of G , where B ( G ) isthe Bruhat-Tits building associated with G . Here K is a torsor over b Γ † with the action y · G x , y ∈ b Γ † inherited from the conjugation action of G ad := G ad ( F ) on G . One has y · G x = G y · x ∈ K , where y · x is the action of G ad on B ( G ), see [Tit79, § b Γ † to S ( φ χ ) given as follows, byreducing to the case of unitary χ first (see [Mis, § φ := φ χ for simplicity, it gives a commuting pair φ o , φ + : W F −→ L T ⊂ L G of parameters such that φ o is tempered and φ ( a ) = φ o ( a ) · φ + ( a )for every a ∈ W F . Let χ + : T −→ C × be the character attached to φ + . Then the set∆ M := n α ∈ ∆ : (cid:12)(cid:12)(cid:12) χ + ( α ∨ ( ̟ )) (cid:12)(cid:12)(cid:12) = 1 o ⊂ ∆gives a Levi subgroup M ⊂ G such that Im( φ ) ⊂ M ∨ and S M ∨ ( φ o ) = S G ∨ ( φ ) . Moreover, it is shown (see [Mis, Lemma 15]) that one has two embeddings S M ∨ ( φ o ) Γ tor M Γ tor G . This shows in particular that the group S ( φ ) is abelian. By applying Hom( − , Q / Z ) weobtain a surjection (writing b Γ † = b Γ † G and omitting G ∨ in S G ∨ ( φ ))(5.1) f Γ : b Γ † Irr( S ( φ )) . Theorem 5.1 ([Mis, Theorem 1]) . For every conjugacy class K ∈ K and every y ∈ b Γ † ,the representation π ( φ χ , ρ ) ∈ L ( φ χ ) is K-unramified if and only if π ( φ χ , f Γ ( y ) ⊗ ρ ) is y · K -unramified. If we choose K ∈ K , and require that π ( φ χ , ) ∈ JH( I ( χ )) is the unique K -unramifiedconstituent of I ( χ ), then the above theorem implies that π ( φ χ , ρ ) , ρ ∈ Irr( S ( φ χ )) is theunique y · K -unramified constituent in I ( χ ), where y ∈ f − ( ρ ) is any lifting of ρ . Variation of ( K, s K ) for covers. Splittings over K . Now we consider an n -fold cover G of G , and still denote by K the set of conjugacy classes of hyperspecial maximal compact subgroup of G . Lemma 5.2.
With the assumption that gcd( n, p ) = 1 and η n : Y sc → F × /F × n is trivial,the group G splits over every hyperspecial maximal compact subgroup K ∈ K .Proof. We presents two arguments. First, since K is hyperspecial, one has K = G O ( O )for a smooth group scheme G O over Spec( O ). By [BD01, Construction 12.11], one knowsthat K ⊂ G is the pull-back of a residual extension µ n G O ( κ ) G O ( κ ) . The proof of [GG18, Proposition 4.1], which was written for G ( O ), also applies for such K and gives that the above extension splits, if we assume the triviality of η n . Thus, wehave a splitting of K over K . Note that the existence of such a splitting also follow from[Wei16a, Theorem 4.3].Alternatively, we note that G splits over K ♮ = G ( O ) by [GG18, Theorem 4.2], and thuswe let s : K ♮ ֒ → G be a fixed splitting. The conjugation action of G ad on G extends to G (see [BD01]), and the group G ad acts transitively on K . We see that for t · K ♮ ∈ K , t ∈ G ad ,one has a splitting t · s : t · K ♮ ֒ → G given by ( t · s )( t · k ) := t · s ( k ) · t − , k ∈ K ♮ . This is a well-defined splitting of G over the hyperspecial maximal compact subgroup y · K ♮ ⊂ G . (cid:3) We impose the same assumption in the Lemma 5.2 throughout the remaining part ofthis section. For every K ∈ K , the set Spl( G, K )of splittings of G over K is a torsor over Hom( K, µ n ). Since G = T · G der and thus K = ( T ∩ K ) · ( G der ∩ K ), we see that every homomorphism h ∈ Hom(
K, µ n ) must betrivial on K ∩ G der . Hence, the restriction mapHom( K, µ n ) Hom( T ∩ K, µ n )is an injection, where T ∩ K = T ( O ). As the restriction of h to T der ⊂ T is also trivial,and the above injection factors through the inclusionHom( T ( O ) / T der ( O ) , µ n ) ֒ → Hom( T ( O ) , µ n )we have ι K : Hom( K, µ n ) Hom( T ( O ) / T der ( O ) , µ n ) Hom( T ( O ) , µ n ) . Note that we do not consider the whole set Hom(
K, µ n ), but only a subset of “admis-sible” elements constrained as follows. Let ( K, s K ) be such that K ∈ K and s K ∈ Spl(
G, K ). If a representation π is ( K, s K )-unramified, then it is also ( K, h ⊗ s K )-unramified if ι K ( h ) is trivial on T Q,n ( O ) ⊂ T , where by abuse of notation we use T Q,n ( O )to denote the image of T Q,n ( O ) in T ( O ) with respect to the natural map T Q,n → T . Forthis reason, we defineHom( K, µ n ) ♮ := ι − K (Hom( T ( O ) / T Q,n ( O ) , µ n )) . ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 83
We thus have natural injections (still denoted by ι K ) ι K : Hom( K, µ n ) ♮ Hom( T ( O ) / T der ( O ) · T Q,n ( O ) , µ n ) Hom( T ( O ) / T Q,n ( O ) , µ n ) . Since A := C T ( T ( O )) = T ( O ) · Z ( T ) is a maximal abelian subgroup of T , the commu-tator map gives a perfect pairing[ − , − ] : T /A × T ( O ) / T Q,n ( O ) −→ µ n , which gives a group isomorphismHom( T ( O ) / T Q,n ( O ) , µ n ) −→ Y /Y
Q,n given by [ y ( ̟ ) , − ] ↔ y. By composing with ι K , we obtain a natural injection(5.2) ι ♮K : Hom( K, µ n ) ♮ Y /Y
Q,n . We denote (
Y /Y
Q,n ) ♮K := Im( i ♮K ) ⊂ Y /Y
Q,n and also for z ∈ ( Y /Y
Q,n ) ♮K , we write f z ∈ Hom(
K, µ ) ♮ for the unique element such that i ♮K ( f z ) = z .If we define(5.3) Y ♮K = { y ∈ Y : B Q ( y, z ) ∈ n Z for all z ∈ Y } , which clearly contains Y Q,n , then one has the inclusions(
Y /Y
Q,n ) ♮K ֒ → Y ♮K /Y Q,n ֒ → X Q,n . The first inclusion is often an identity. It is also easy to see that there is a commutativediagram(5.4) Y ,Q,n /Y ∩ Y Q,n ( Y /Y
Q,n ) ♮K X c Q,n Y /Y ∩ Y Q,n X Q,n X Γ Q,n X ,Q,n Y /Y ,Q,n , p Γ p Γ where the map p Γ : ( Y /Y
Q,n ) ♮K → X c Q,n obtained from restriction is well-defined in viewof (2.20) and (2.21), and the middle horizontal and left vertical maps are short exactsequences.5.2.2.
A relation between d Γ tor G and [ Γ tor G Q,n . For G we have the following commutative dia-gram: X Q,n /X X
Q,n /X scG X/X scG Γ G ( X Q,n ∩ X scG, Q ) / ( X ∩ X scG, Q ) ( X Q,n ∩ X scG, Q ) /X scG ( X ∩ X scG, Q ) /X scG Γ tor G . Here X Q,n = Hom( Y Q,n , Z ) and we have used the subscript G in X scG, Q = X scG ⊗ Q (forexample) to emphasize the underlying group involved. Similarly, one has for a Levisubgroup M the commutative diagram X Q,n /X X
Q,n /X scG X/X scM Γ M ( X Q,n ∩ X scM, Q ) / ( X ∩ X scM, Q ) ( X Q,n ∩ X scM, Q ) /X scM ( X ∩ X scM, Q ) /X scM Γ tor M . The above two diagrams are compatible, and for the bottom two exact sequences this isillustrated from the following diagram X Q,n /X ( X Q,n ∩ X scG, Q ) / ( X ∩ X scG, Q ) ( X Q,n ∩ X scG, Q ) /X scG Γ tor G ( X Q,n ∩ X scM, Q ) / ( X ∩ X scM, Q ) ( X Q,n ∩ X scM, Q ) /X scM Γ tor M . Denote by P ( L ) ⊂ Q [∆ ∨ ]the Z -dual of the lattice L ⊂ Q [∆]. Applying Hom( − , Q / Z ) to the above diagram, weobtain Y /Y
Q,n P ( X ∩ X scG, Q ) /P ( X Q,n ∩ X scG, Q ) P ( X scG ) /P ( X Q,n ∩ X scG, Q ) d Γ tor G . For the principal endoscopic group G Q,n of G , one has the group Γ G Q,n and Γ tor G Q,n defined analogously as for Γ G and Γ scG . Thus,Γ G Q,n = X Q,n /X scQ,n and Γ tor G Q,n = ( X Q,n ∩ X scG, Q ) /X scQ,n , where the last equality follows from the fact that X scG Q,n , Q = X scG, Q . This gives that [ Γ tor G Q,n = P ( X scQ,n ) /P ( X Q,n ∩ X scG, Q ) . The discussion in § G Q,n . In particular, the set of conjugacy classes ofhyperspecial maximal compact subgroup of G Q,n is a torsor over [ Γ tor G Q,n . Moreover, forevery genuine character χ : Z ( T ) −→ C × , which is s Q,n ( T ( O ))-unramified with respectto a splitting s Q,n of T ( O ), there is the associated parameter φ χ : W F −→ L T ֒ → L G. One has a surjection [ Γ tor G Q,n ։ Irr( S ( φ χ )) . To relate the two group d Γ tor G and [ Γ tor G Q,n , we note that there is a natural map ϕ : P ( X scG ) /P ( X Q,n ∩ X scG, Q ) −→ [ Γ tor G Q,n
ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 85 given as follows. Recall that X scQ,n is generated by { α/n α : α ∈ ∆ } , and thus there is agroup homomorphism ϕ : X scQ,n −→ X scG uniquely determined by ϕ ( α/n α ) = α for every α ∈ ∆ . This gives a well-defined map ϕ as in(5.5)( X Q,n ∩ X scG, Q ) / ( X ∩ X scG, Q ) ( X Q,n ∩ X scG, Q ) /X scG ( X ∩ X scG, Q ) /X scG Γ tor G Q,n ( X Q,n ∩ X scG, Q ) /X scQ,n . Γ tor Gϕ By taking applying Hom( − , Q / Z ), we obtain the desired homomorphism ϕ . In fact, let { ω α i } ⊂ Q [∆ ∨ ]be the set of fundamental coweights such that h ω α i , α j i = δ ij for every α j ∈ ∆. We have P ( X scG ) = Z [ ω α : α ∈ ∆] . Similarly, P ( X scQ,n ) = Z [ n α ω α : α ∈ ∆]Then ϕ is induced from the map P ( X scG ) −→ P ( X scQ,n )given by ω α n α w α for every α ∈ ∆. In particular, ϕ is surjective.As a summary of the above discussion, we have the following diagram(5.6) Y /Y
Q,n P ( X ∩ X scG, Q ) /P ( X Q,n ∩ X scG, Q ) P ( X scG ) /P ( X Q,n ∩ X scG, Q ) d Γ tor G H ϕ P ( X scQ,n ) /P ( X Q,n ∩ X scG, Q ) [ Γ tor G Q,n /H ϕ [ Γ tor G Q,n . hϕ ϕ ϕ Here by definition H ϕ is the image of P ( X ∩ X scG, Q ) /P ( X Q,n ∩ X scG, Q ) via ϕ . The followingcompressed diagram will play a key role in our subsequent discussion:(5.7) Hom( K, µ n ) ♮ Y /Y
Q,n d Γ tor G H ϕ [ Γ tor G Q,n [ Γ tor G Q,n /H ϕ Irr( S ( φ χ )) . ι ♮K ϕ ◦ h ϕq χ s ϕ ? The packet L ( φ χ ) versus unramifiedness. For every K ∈ K and y ∈ d Γ tor G the action y · K is realized by the conjugacy action of y ( ̟ ) ∈ T ad on K . For every f ∈ Hom(
K, µ n ),by transport of structure, we define y · f : y · K −→ µ n by ( y · f )( y ( ̟ ) · k · y ( ̟ ) − ) := f ( k ) . Recall that G ad and thus T ad acts on G . As used in Lemma 5.2, if s K : K ֒ → G is asplitting, then we obtain a splitting y · s K : y · K −→ G of y · K ∈ K given by( y · s K )( y ( ̟ ) · k · y ( ̟ ) − ) := y ( ̟ ) · s K ( k ) · y ( ̟ ) − . Definition 5.3.
For
K, K ′ ∈ K , two splittings s K : K ֒ → G and s K ′ : K ′ ֒ → G are calledassociated if they agree on T Q,n ( O ).In particular, if K = K ′ , then s K and s ′ K = f ⊗ s K , f ∈ Hom(
K, µ n ) are associatedonly if f ∈ Hom(
K, µ n ) ♮ . Conjecture 5.4.
There is a canonical splitting s ϕ of [ Γ tor G Q,n over [ Γ tor G Q,n /H ϕ , as depictedin (5.7) . Moreover, there is a bijection L ( φ χ ) Irr( S ( φ χ )) which we denote conveniently by π ( φ χ , ρ ) ↔ ρ , such that – for every K ∈ K and splitting s K : K ֒ → G with s K = s Q,n on T ( O ) , – for every f z ∈ Hom(
K, µ n ) ♮ , – for every y ∈ d Γ tor G with y · s K being associated with s K ,one has that π ( φ χ , ρ ) is ( K, s K ) -unramified if and only if π ( φ χ , γ z,y · ρ ) is ( y · K, ( y · f z ) ⊗ ( y · s K )) -unramified, where γ z,y = (cid:0) ϕ ◦ h ◦ i ♮K ( f z ) (cid:1) · (cid:0) s ϕ ◦ ϕ ( y ) (cid:1) ∈ [ Γ tor G Q,n . Note that y · f z ∈ Hom( y · K, µ n ) ♮ and in fact y · f z ⊗ y · s K = y · ( f z ⊗ s K ). Remark 5.5.
There is an action of d Γ tor G on the set˜ K = (cid:8) ( K, s K ) : K ∈ K and s K is a splitting of K into G (cid:9) / ∼ , where ∼ means modulo the conjugation action of G . The action is still free, but willbe not be transitive in general. This implies that the notion of unramifiedness is moredelicate for covering groups. ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 87
Different models of I ( χ ) . To facilitate later computations, we provide some furtheranalysis of a (
K, s K )-unramified genuine principal series I ( χ ) of G , especially on itsdifferent concrete realizations. Recall that χ : Z ( T ) → C × is a central character and˜ χ : A −→ C × an extension to a maximal abelian subgroup A ⊂ T . The isomorphism class i ( χ ) onlydepends on χ , and not on the chosen A and the extension ˜ χ . The commutator pairing[ − , − ] : T /Z ( T ) × T /Z ( T ) −→ C × is non-degenerate, and thus every character of T /Z ( T ) is of the form [ − , t ] for some t ∈ T .In particular, if we fix A and consider two extensions ˜ χ, ˜ χ ′ of χ to A , we get a character˜ χ/ ˜ χ ′ : A/Z ( T ) −→ C × , then there exists t ∈ T such that ˜ χ ′ = t ˜ χ, where t ˜ χ ( x ) = ˜ χ ( t − xt ) = [ x, t ] · ˜ χ ( x ) . For any fixed lifting t ∈ T of t , we have an isomorphism φ t : Ind TA ( ˜ χ ) −→ Ind TA ( t ˜ χ )given by φ t ( f )( x ) = f ( t − x ) . Hence, there is an induced isomorphism(5.8) Φ t : Ind GAU ( ˜ χ ) −→ Ind
GAU ( t ˜ χ )given by Φ t ( f )( g ) = f ( t − g ) for g ∈ G, where we note that the two sides are different models for the same isomorphism class I ( χ ).5.3. Whittaker datum varied.
We consider in this subsection how the Whittaker di-mension of an element inside an unramified L-packet varies with respect to differentWhittaker data.5.3.1.
Linear case.
First we recall the results in the linear case, see [Kuo02, Kuo10,GGP12, Kal13, Mis16]. Let w = ( B, U, ψ )be a fixed Whittaker datum such that ψ : U −→ C × is a non-degenerate character. Thegroup T = T ( F ) acts on the set of such non-degenerate characters ψ and gives finiteorbits. On the other hand, from the short exact sequence Z ( G ) ֒ → G ։ G ad we obtain by Galois cohomology Z ( G ) G G ad H ( F, Z ( G )) ..., where G and G ad denote the F -points of G and G ad respectively. One has an embedding G ad /G ֒ → H ( F, Z ( G ))and also an isomorphism T ad /T ≃ G ad /G, which acts simply and transitively on the set of T -orbits of non-degenerate characters of U . We denote the action by t w for every t ∈ T ad , or simply t ψ , since the action of t isonly on ψ .On the other hand, for every unramified χ , there is an embedding S ( φ χ ) ֒ → X/X sc , and by applying Hom( − , Q / Z ) we obtain the surjection Y ad /Y = P ( X sc ) /Y ։ Irr( S ( φ χ )) , where Y ad = P ( X sc ) is the cocharacter lattice of G ad . Note that the valuation map v F : F × → Z induces a well-defined map v F : T ad /T −→ Y ad /Y. We thus have a surjection(5.9) T ad /T Irr( S ( φ χ )) . In fact, by using Tate duality, one can define a natural map (see [Mis16, § H ( F, Z ( G )) −→ Irr( S ( φ χ )) , the composite of which with the inclusion T ad /T ֒ → H ( F, Z ( G )) gives rise to (5.9).It is shown in [Mis16, Theorem 12] that for an unramified L-packet L ( φ χ ) associatedwith an unramified χ , the representation π ( φ χ , ρ ) is ψ -generic if and only if π ( φ χ , t · ρ ) is t ψ -generic, for every t ∈ T ad . Here ψ -genericity of π ∈ Irr( G ) means thatdim Wh ψ ( π ) = 1 . Moreover, π ( φ χ , t · ρ ) is the unique t ψ -generic element in L ( φ χ ) for the non-degeneratecharacter t ψ , see [Var17, Ato17] and references therein.5.3.2. Speculations for coverings.
It is natural to consider the analogous question forcovering groups and an unramified L-packet L ( φ χ ), by viewing U as a subgroup of G via the canonical splitting. However, similar to the earlier discussion in the precedingsubsection, we encounter the problem that on the one hand the Whittaker datum has anatural action by T ad /T , depending on G ; on the other hand, the S -group pertains to thedual group G ∨ and thus to the principal endoscopic group G Q,n . It is therefore crucial torelate the two aspects.We denote by G Q,n,ad and T Q,n,ad the F -rational points of the quotient adjoint group G Q,n,ad of G Q,n and the split torus T Q,n,ad inside G Q,n,ad , respectively. We ask when isthere a natural map T ad /T −→ T Q,n,ad /T Q,n ?In fact, for our purpose, it suffices to consider if there is a natural map X Q,n /X scQ,n −→ X/X sc , or equivalently, a map Z ( G ) −→ Z ( G Q,n ) . For simplicity of discussion, we assume that G is a semsimple group through the restof this subsection. In this case, one has from (5.5) the following diagram(5.10) X Q,n /X X
Q,n /X sc X/X sc X Q,n /X scQ,n j ϕ ˜ ϕ ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 89
Taking ϕ to be the natural candidate, the question is whether it factors through j . Inthe special case of G being almost simple and simply-connected, the answer is as follows. Proposition 5.6.
Assume G is almost simple and simply-connected. If ( G, G ∨ ) = (Spin r +1 , SO r +1 ) , where the equality holds exactly when r is odd and n α ≡ for every short coroot α ∨ of Spin r +1 , then the map ϕ factors through j , i.e., ϕ ( X Q,n /X scQ,n ) ⊂ X/X sc . Proof.
For every x = P α ∈ ∆ c α α ∈ X Q,n , c α ∈ Q , we want to show that ϕ ( x ) = X α ∈ ∆ c α n α α lies in X , or equivalently, h ϕ ( x ) , y i = X α ∈ ∆ c α n α h α, y i lies in Z for every y ∈ Y . Our argument is essentially a case by case discussion. First,assume G is simply-laced. Then α n α is constant on ∆ and we have Y scQ,n = n α · Y ⊂ Y Q,n . In this case, h ϕ ( x ) , y i = h x, n α y i ∈ Z , since x ∈ X Q,n and n α y ∈ Y Q,n . Thus, we are left with G being of type B r , C r . Note thatthough F and G have two lengths for the simple roots, we always have X Q,n = X scQ,n for such groups, and thus there is nothing to check. The two nontrivial cases will be for B r , C r and when [ X Q,n : X scQ,n ] = 2, i.e., when the dual group G ∨ of G is of adjoint type.In this case Y Q,n = Y scQ,n .First, if G = Sp r and G ∨ is adjoint, then G Q,n = Sp r as well. In particular, G Q,n hasthe same root system type as G , and thus α n α is constant on ∆. In this case, Y scQ,n = n α · Y sc = Y. Hence h ϕ ( x ) , y i ∈ Z for all x ∈ X Q,n and y ∈ Y , and this gives ϕ ( x ) ∈ X .Second, if G = Spin r +1 and G ∨ is adjoint, then there are exactly two possibilities asfollows. Assume that α ∨ r is unique long simple coroot in ∆ ∨ , and α ∨ i ∈ ∆ ∨ is short for1 i r − G ∨ = ( PGSp r if n α is odd , SO r +1 if r is odd and n α ≡ . Now if G ∨ = PGSp r , then G Q,n = Spin r +1 and thus α n α is constant on ∆. Inthis case, the same argument as above gives that ϕ ( x ) ∈ X for every x ∈ X Q,n . Now if G ∨ = SO r +1 , then we get G Q,n = Sp r with X scQ,n = Z [ α /n α ] ⊕ Z [ α /n α ] ⊕ ... ⊕ Z [ α r /n α r ]where n α r = n α /
2, and also X scQ,n = Z [ α /n α ] ⊕ Z [ α /n α ] ⊕ ... ⊕ Z [ α r /n α ]Thus, ϕ ( α r /n α ) = α r / / ∈ X . This completes the proof. (cid:3) To proceed with our discussion, we assume ϕ factors through j and thus one has awell-defined map ˜ ϕ : X Q,n /X scQ,n −→ X/X scG . Note that since we assumed G is semisimple, one has Γ G = Γ tor , and similarly for G Q,n .For every unramified genuine character χ : Z ( T ) → C × , this gives a commutative diagram(5.11) c Γ G P ( X sc ) /Y T ad /T H ( F, Z ( G )) [ Γ G Q,n P ( X scQ,n ) /Y Q,n T Q,n,ad /T Q,n H ( F, Z ( G Q,n ))Irr( S ( φ χ )) , ϕ v F q χ v F where ϕ is induced from ϕ and the right vertical map arises from ˜ ϕ ∗ : Z ( G ) −→ Z ( G Q,n )and the functoriality of Galois cohomology. By our assumption on ϕ , the map ϕ : P ( X sc ) /Y Q,n ։ P ( X scQ,n ) /Y Q,n as in (5.6) factors through the quotient ϕ : P ( X sc ) /Y Q,n ։ c Γ G , giving rise to the map(still denoted by) ϕ in (5.11). For convenience, we denote by ζ χ := q χ ◦ ϕ ◦ v F : T ad /T ։ Irr( S ( φ χ ))for the composed surjection. Conjecture 5.7.
Assume that ˜ ϕ : X Q,n /X scQ,n −→ X/X scG is well-defined. Then for everyWhittaker data w = ( B, U, ψ ) and ( K, s K ) -unramified genuine principal series I ( χ ) , onehas dim Wh ψ ( π ( φ χ , ρ )) = dim Wh t ψ ( π ( φ χ , ζ χ ( t ) ⊗ ρ )) for every π ( φ χ , ρ ) ∈ L ( φ χ ) and t ∈ T ad /T . Remark 5.8.
It seems to us that the map ϕ : X Q,n /X scQ,n → X Q,n /X sc considered here isthe most natural one, though its image may not be in X/X sc , as shown in Proposition 5.6even for G almost simple and simply-connected. On the one hand, one could definitelyconsider any other “natural” map ϕ ′ : X Q,n /X scQ,n → X Q,n /X sc whose image lies in X/X sc , for example, x nx . However, such a map may not be injective and thus theinduced map ϕ ′ : P ( X sc ) /Y → P ( X scQ,n ) /Y Q,n may not be surjective. It is expected that the analogue of Conjecture 5.7 may not holdfor such ϕ ′ . Indeed, if ϕ ′ is the trivial map, then the action of t ∈ T ad /T on Irr( S ( φ χ ))via ϕ ′ is trivial; however, dim Wh t ψ ( π ( φ χ , ρ )) may not equal to dim Wh ψ ( π ( φ χ , ρ )). Onthe other hand, we expect that a cover G such that G Q,n has simply-connected derivedgroup but of a different root system type is quite unusual and exhibits many interestingproperties. The pair (
G, G ∨ ) = (Spin r +1 , SO r +1 )in Proposition 5.6 is such an example. Another example is( G, G ∨ ) = (GSp r , PGSp r × GL )for even fold cover of GSp r with r even, see § § ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 91
Scattering matrix for f ( ψ ) = p F . To facilitate some computations later, we recallbriefly the scattering matrix associated with T ( w, i ( χ )) when f ( ψ ) = p F . We use freely the notations in [GSS] and those in earlier sections of this paper. There isa square matrix [ τ ( w, χ, γ, γ ′ )] γ,γ ′ ∈ T /A of size | Y /Y
Q,n | representing the operator T ( w, χ ; r un w ) ∗ : Wh ψ ( I ( w χ )) −→ Wh ψ ( I ( χ ))such that T ( w, χ ; r un w ) ∗ ( λ w χγ ) = X γ ′ ∈ T /A τ ( w, χ, γ, γ ′ ) · λ χγ ′ . In particular, one has a local scattering matrix S R ( w, χ ) := [ τ ( w, χ, s y , s y )] y ,y ∈ R indexed by R for every ordered set R ⊂ Y of representatives Y /Y
Q,n . Some immediateproperties are: • For w ∈ W and z, z ′ ∈ A , the identity holds: τ ( w, χ, γ · z, γ ′ · z ′ ) = ( w χ ) − ( z ) · τ ( w, χ, γ, γ ′ ) · χ ( z ′ ) . • For w , w ∈ W such that l ( w w ) = l ( w ) + l ( w ), one has τ ( w w , χ, γ, γ ′ ) = X γ ′′ ∈ T /A τ ( w , w χ, γ, γ ′′ ) · τ ( w , χ, γ ′′ , γ ′ ) , which is referred to as the cocycle relation.To explicate the matrix for a simple reflection w = w α associated with α ∈ ∆, we fix theHaar measure µ of F such that µ ( O F ) = 1; thus, µ ( O × F ) = 1 − /q . The Gauss sum isdefined by G ψ ( a, b ) = Z O × F ( u, ̟ ) an · ψ ( ̟ b u ) µ ( u ) , a, b ∈ Z . It satisfies (compared with [GSS, § ψ has conductor p F ) G ψ ( a, b ) = b < , − /q if n | a, b > , n ∤ a, b > , − /q if n | a, b = 0 ,G ψ ( a,
0) with | G ψ ( a, | = q − / if n ∤ a, b = 0 . One has G ψ ( a, b ) = ( − , ̟ ) an · G ψ ( − a, b ). For every k ∈ Z , we write g ψ ( k ) := G ψ ( k, . Proposition 5.9.
Suppose γ = s y is represented by y and γ ′ = s y by y . Then we canwrite τ ( w α , χ, γ, γ ′ ) = τ ( w α , χ, γ, γ ′ ) + τ ( w α , χ, γ, γ ′ ) with the following properties: • τ i ( w α , χ, γ · z, γ ′ · z ′ ) = ( w α χ ) − ( z ) · τ i ( w α , χ, γ, γ ′ ) · χ ( z ′ ) for every z, z ′ ∈ A , • τ ( w α , χ, γ, γ ′ ) = 0 unless y ≡ y mod Y Q,n , • τ ( w α , χ, γ, γ ′ ) = 0 unless y ≡ w α ( y ) mod Y Q,n .Moreover, • if y = y , then τ ( w α , χ, γ, γ ′ ) = (1 − q − ) χ ( h α ( ̟ n α )) k y,α − χ ( h α ( ̟ n α )) , where k y,α = (cid:24) h y, α i n α (cid:25) ; • if y = w α ( y ) , then τ ( w α , χ, γ, γ ′ ) = ( − , ̟ ) h y,α i· D ( y,α ∨ ) n · g ψ − ( h y, α i Q ( α ∨ )) . Proof.
The computation in [McN16] (with refinement given in [Gao17]) applies here mu-tatis mutantis, by noting that ψ has conductor p F instead of O F as in the referencesmentioned. We omit the details of the computation. (cid:3) Some simple examples.
We illustrate on the previous discussion, especially onConjectures 5.4 and Conjecture 5.7 by considering some simple examples. These includecovers of SL , SO and GL r . We focus only on unitary ( K, s K )-unramified principal seriesof these groups.5.4.1. Covers of SL . Let SL be the n -fold cover of SL associated with Q ( α ∨ ) = − . Let K = SL ( O ) and K ′ = e · K = e ( ̟ ) · K · e ( ̟ ) − , where e := α ∨ / ∈ Y ad is a generator for the cocharacter lattice for PGL . Here e ∈ d Γ tor G is the nontrivial element and K = { K, K ′ } . Consider the two unique splittings s K and s K ′ of K and K ′ respectively, one has(5.12) s ′ K = [ e ( ̟ ) , − ] ⊗ s K on T ( O ), where(5.13) [ e ( ̟ ) , α ∨ ( a k )] = ( ̟, a ) − kn for all a ∈ F × and k ∈ Z , see [BD01, § n .First, if n is odd, then it is easy to see Y Q,n = Y scQ,n = Z [ nα ∨ ] , X Q,n = Z [ α/ n ] , X scQ,n = Z [ α/n ] . This immediately gives that the diagram in (5.6) becomes Z [ α ∨ ] / Z [ nα ∨ ] Z [ α ∨ ] / Z [ nα ∨ ] Z [ α ∨ / / Z [ nα ∨ ] Z [ α ∨ / / Z [ α ∨ ] d Γ tor G { } Z [ nα ∨ / / Z [ nα ∨ ] Z [ nα ∨ / / Z [ nα ∨ ] [ Γ tor G Q,n
Irr( S ( φ χ )) , ≃ ϕ ϕ ϕ ≃ where ϕ ( x ) = n · x for middle vertical map. The existence of the splitting s ϕ is clear. Asmentioned, we concentrate on the case where χ is unramified and unitary. ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 93 If n | k , or equivalently kα ∨ ∈ Y Q,n and thus kα ∨ ( a ) ∈ Z ( T ), then[ e ( ̟ ) , α ∨ ( a k )] = 1 . This shows that s K and s ′ K agree on T Q,n ( O ), and thus are associated in the sense ofDefinition 5.3. It follows that in this case I ( χ ) is both ( K, s K ) and ( K ′ , s K ′ )-unramified.If I ( χ ) is unitary unramified and irreducible, then L ( φ χ ) = { I ( χ ) } . It is easy to see that Conjecture 5.4 holds. On the other hand, if I ( χ ) is a unitary( K, s K )-unramified and reducible, then it occurs exactly when χ α := χ ( h α ( ̟ n α )) = − R χ = W . In particular, w α χ = χ . Note that we have an isomorphism R χ ≃S ( φ χ ) (see [Gao]) and thus we always make such an identification. In this case, L ( φ χ ) = JH( I ( χ )) = { π ( φ χ , ) , π ( φ χ , ε W ) } , where ε W is the nontrivial character of S ( φ χ ). Proposition 5.10.
There is exactly one π ( φ χ , ρ ) which is ( K, s K ) -unramified, while theother is ( K ′ , s K ′ ) -unramified. In particular, this verified Conjecture 5.4.Proof. The proof uses the same idea as in the linear case, that is, to compute the coefficientof the intertwining operator applied to the unramified vector. More precisely, recall thatthe normalized intertwining operator A ( w α , χ ) = γ ( w α , χ ) · T ( w α , χ ; r un w α )acts on π ( φ χ , ρ ) ⊂ I ( χ ) by ρ ( w α ) · id for each ρ ∈ Irr( S ( φ χ )) = Irr( W ).Note that as character of A , we have e ( ̟ ) ˜ χ = [ − , e ( ̟ )] ⊗ ˜ χ. However, the problem is that e ( ̟ ) / ∈ T . Setting t := ̟ ( n +1) α ∨ / ∈ T , it is easy to see t ˜ χ = e ( ̟ ) ˜ χ, since [ − , t ] = [ − , e ( ̟ )] as a function on T . Let ˜ χ = χ ⊠ ˜ χ O be the unique extension of χ such that ˜ χ O ◦ s K = on T ( O ). We see that t ˜ χ ◦ s K ′ | T ( O ) = and(5.14) s K ′ = [ t, − ] ⊗ s K on T ( O ). Here ˜ χ and t ˜ χ are both extensions of χ and give rise to two modelsInd GAU ( ˜ χ ) , Ind
GAU ( t ˜ χ )of the principal series I ( χ ). For any fixed lifting t of t , we have the isomorphismΦ t : Ind GAU ( ˜ χ ) −→ Ind
GAU ( t ˜ χ )given by Φ t ( f )( g ) = f ( t − g ), see (5.8).Without loss of generality, we assume that π ( φ χ , ) is ( K, s K )-unramified and thus A ( w α , χ )( f K ) = f K , where f K ∈ Ind
GAU ( ˜ χ ) is the normalized ( K, s K )-unramified vector. We want to computethe scalar arising from A ( w α , χ ) applied to the ( K ′ , s K ′ )-unramified vector. Note that inthe model Ind GAU ( t ˜ χ ) the ( K ′ , s K ′ )-unramified vector is given by f K ′ ( b · s ′ ( k ′ )) = ( δ / B ( t ) · ˜ χ ( a ) if b = au with a ∈ A, u ∈ U and k ′ ∈ K ′ , − t ( f K ′ ) ∈ Ind
GAU ( ˜ χ ) is the ( K ′ , s K ′ )-unramified vector. We want to determine thescalar c ∈ C such that T ( w α , χ ; r un w α )(Φ − t ( f K ′ )) = c · Φ − t ( f K ′ ) . We see that c = T ( w α , χ ; r un w α )(Φ − t ( f K ′ ))( t − )= Z U Φ − t ( f K ′ )( w − α ut − ) du = Z U f K ′ ( t · w − α u · t − ) du = Z U f K ′ ( t · w − α t − w α · w − α · tut − ) du = Z U f K ′ ( h α ( ̟ n ) · h α ( ̟ ) · w − α · tut − ) du, where the last equality follows from (2.4). We have c = q · χ α · Z U f K ′ ( h α ( ̟ ) · w − α · u ) du. Now to compute the integral, we split U into p F and U − p F . If u ∈ p F , it is easy to seethat h α ( ̟ ) w − α u ∈ s K ′ ( K ′ ) and thus Z p F f K ′ ( h α ( ̟ ) · w − α · u ) du = d ( p ) = q − . On the other hand, Z U − p F f K ′ ( h α ( ̟ ) · w − α · e α ( u )) du = Z U − p f K ′ ( h α ( ̟ ) · h α ( u − ) · e α ( − u ) · e − α ( − u − )) du = X k Z u ∈ O × f K ′ ( h α ( ̟ ) · h α ( ̟ − k u − ) · e α ( − u )) d ( ̟ k u )= X k (cid:12)(cid:12) ̟ k (cid:12)(cid:12) Z u ∈ O × f K ′ ( h α ( ̟ ) · h α ( ̟ − k ) h α ( u − ) · ( u, ̟ ) kQ ( α ∨ ) n ) d ( u )= X k (cid:12)(cid:12) ̟ k (cid:12)(cid:12) Z u ∈ O × f K ′ ( h α ( ̟ ) · h α ( ̟ − k ) s K ( h α ( u − )) · ( u, ̟ ) kQ ( α ∨ ) n ) d ( u ) . Now by (5.14) we have s K ′ ( h α ( u − )) = s K ( h α ( u − )) · ( u, ̟ ) Q ( α ∨ ) n . ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 95
This gives that the above integral over U − p F equals X k (cid:12)(cid:12) ̟ k (cid:12)(cid:12) Z u ∈ O × f K ′ ( h α ( ̟ ) · h α ( ̟ − k ) s K ′ ( h α ( u − )) · ( u, ̟ ) ( k − Q ( α ∨ ) n ) d ( u )= X k ,n | ( k − (cid:12)(cid:12) ̟ k (cid:12)(cid:12) Z u ∈ O × f K ′ ( h α ( ̟ − k )) d ( u )= q − χ α (1 − q − )1 − χ α . Combining the above we get that c = χ α · c gk ( w α , χ ) = ( −
1) 1 − q − χ α − χ α and therefore in our setting A ( w α , χ )( f K ′ ) = ( − · f K ′ . This concludes the proof. (cid:3)
Second, we consider the case n = 2 m is even, which gives then Y Q,n = Z [ mα ∨ ] , Y scQ,n = Z [ nα ∨ ] , X Q,n = X scQ,n = Z [ α/n ] . In this case, diagram (5.6) becomes Z [ α ∨ ] / Z [ mα ∨ ] Z [ α ∨ ] / Z [ mα ∨ ] Z [ α ∨ / / Z [ mα ∨ ] Z [ α ∨ / / Z [ α ∨ ] d Γ tor G { } Z [ mα ∨ ] / Z [ mα ∨ ] { } [ Γ tor G Q,n
Irr( S ( φ χ )) , ≃ ϕ ϕ ϕ As [ Γ tor G Q,n = { } , every L ( φ χ ) is a singleton and if χ is unitary, then L ( φ χ ) = { I ( χ ) } . Note that in this case, if I ( χ ) is ( K, s K )-unramified, then it is not ( K ′ , s K ′ )-unramified.Indeed, for every z = imα ∨ ∈ Y Q,n and u ∈ O × , it follows from (5.12) and (5.13) that s K ′ ( z ⊗ u ) = ( ̟, u ) − min · s K ( z ⊗ u ) , where ( ̟, u ) min = ( ̟, u ) i may not be trivial in general. Thus, the two splittings s K and s K ′ = y · s K are not associated. This agrees with Conjecture 5.4.Regarding the Whittaker dimension with respect to varying Whittaker datum, weassume that for the conductor, f ( ψ ) = O F . For simplicity of notation, we denote e ψ := e ( ̟ ) ψ where e ( ̟ ) ∈ T ad /T such that f ( e ψ ) = p F . Consider the n -fold cover SL such that n is odd and I ( χ ) = π ( φ χ , ) ⊕ π ( φ χ , ε W ) , where the labelling is such that π ( φ χ , ) is the unique ( K, s K )-unramified constituent,and thus π ( φ χ , ε W ) is ( K ′ , s K ′ )-unramified by Proposition 5.10. Proposition 5.11.
Keep notations and conventions as above, one has dim Wh ψ ( π ( φ χ , )) = dim Wh e ψ ( π ( φ χ , ε W )) = ( n + 1) / and dim Wh ψ ( π ( φ χ , ε W )) = dim Wh e ψ ( π ( φ χ , )) = ( n − / . In particular, Conjecture 5.7 is verified in this case.Proof.
We first note that for any non-degenerate character ψ : U −→ C × , there is arepresentation (see [Gao, § σ Wh ψ : R χ −→ GL(Wh ψ ( I ( χ )))given by σ Wh ψ ( w ) = A ( w, i ( χ )) ∗ , the induced map from A ( w, i ( χ )) = γ ( w, χ ) · T ( w, χ ; r un w ), where γ ( w, χ ) is the gamma-factor associated with w and i ( χ ). It is proved in [Gao, Theorem 5.6] that one has(5.15) dim Wh ψ ( π ( φ χ , ρ )) = (cid:10) ρ, σ Wh ψ (cid:11) R χ . If f ( ψ ) = O F and π ( φ χ , ) is ( K, s K )-unramified, as is assumed here, then it is shown in[Gao, Theorem 7.1] that σ Wh ψ = n + 12 M n − ε W , which immediately gives thatdim Wh ψ ( π ( φ χ , )) = n + 12 and dim Wh ψ ( π ( φ χ , ε W )) = n − . The proof relies on an explicit form of the scattering matrix for f ( ψ ) = O , as describedin [Gao17] and [GSS].Now to compute σ Wh eψ , we use Proposition 5.9 instead, since f ( e ψ ) = p F . We take theset R = { iα ∨ : − ( n − / i ( n − / } ⊂ Y of representatives of Y /Y
Q,n . In view of Proposition 5.9, by permuting the set R properly,the scattering matrix [ τ ( w α , χ, s z , s y )] z,y ∈ R is equal to a block-diagonal matrix with ( n + 1) / M i : 0 i ( n − / i = 0 we get the size-one matrix S R ( w α , χ ) = [ τ ( w α , χ, s , s )] = χ α − q − − χ α = − q − , where the second equality follows from χ α = − ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 97 – for each 1 i ( n − /
2, we have the two by two matrix S R ( w α , χ ) iα ∨ =[ τ ( w α , χ, s z , s y )] with z, y ∈ {± iα ∨ } , which by Proposition 5.9 is equal to(5.16) S R ( w α , χ ) iα ∨ = (cid:18) χ α (1 − q − ) / (1 − χ α ) g t ψ − (2 i )) g t ψ − ( − i ) (1 − q − ) / (1 − χ α ) (cid:19) = (cid:18) − (1 − q − ) / g t ψ − (2 i )) g t ψ − ( − i ) (1 − q − ) / (cid:19) . Here the left-upper and right-lower entry is τ ( w α , χ, s iα ∨ , s iα ∨ ) and τ ( w α , χ, s − iα ∨ , s − iα ∨ )respectively.Note that in this case γ ( w α , χ ) = 1 − χ α − q − χ α = 21 + q − . This coupled with the above equalities for the S R ( w α , χ ) iα ∨ ’s gives that σ Wh tψ = n + 12 ε W M n − , and it thus follows from (5.15) thatdim Wh e ψ ( π ( φ χ , )) = n −
12 and dim Wh e ψ ( π ( φ χ , ε W )) = n + 12 , as desired. This completes the proof. (cid:3) Remark 5.12.
Results in this subsection generalize to covers of Sp r without too muchdifficulty. Indeed, for even-fold cover every unitary ( K, s K )-unramified I ( χ ) is irreducible.If n is odd, then I ( χ ) could be reducible with R χ = { , w α r } , where α r is the unique longsimple root. The same proof as Proposition 5.10 applies to Sp r to give the same result.Write I ( χ ) = π ( φ χ , ) ⊕ π ( φ χ , ε W ), assuming π ( φ χ , ) is ( K, s K )-unramified. Regardingthe Whittaker dimension, one hasdim Wh ψ ( π ( φ χ , )) = dim Wh e ψ ( π ( φ χ , ε W )) = n r − ( n + 1)2and dim Wh ψ ( π ( φ χ , ε W )) = dim Wh e ψ ( π ( φ χ , )) = n r − ( n − . For these equalities, one can either apply Rodier’s heredity coupled with Proposition 5.11,or argue as in the proof of Proposition 5.11 directly.5.4.2.
Covers of SO . Let Y = Z · e be the cocharacter lattice of SO with α ∨ = 2 e generating the co-root lattice Y sc . Let Q : Y → Z be the Weyl-invariant quadratic form such that Q ( e ) = 1. Thus, Q ( α ∨ ) = 4. We get α ∨ Q,n = n gcd(4 , n ) α ∨ and thus Y scQ,n = n (4) α ∨ . On the other hand, Y Q,n = Z n gcd(2 , n ) e = Z [ n (2) e ] . Let χ be a unitary unramified genuine character of Z ( T ). The diagram (5.6) becomes Z [ e ] / Z [ n (2) e ] Z [ e ] / Z [ n (2) e ] Z [ e ] / Z [ n (2) e ] { } d Γ tor G Z [ n (4) e ] / Z [ n (2) e ] Z [ n (4) e ] / Z [ n (2) e ] { } [ Γ tor G Q,n
Irr( S ( φ χ )) , ≃ ≃ ϕ ϕ ϕ ≃ where one has ϕ ( e ) = n (4) · e . It is shown in [Gao, § S ( φ χ ) = W if and only if4 | n and χ α is a non-trivial quadratic character; otherwise S ( φ χ ) = { } .Note that K = PGL ( O ) = SO ( O ) ∈ K is the unique conjugacy class of hyperspecialmaximal compact subgroup. If we fix a splitting s K : K ֒ → G , then the other splitting s K ⊗ f ξ is given by the twist f ξ : PGL ( O ) −→ µ n where f ξ = ξ ◦ det : GL ( O ) −→ µ n for a quadratic character ξ : O × /O × −→ µ n . In particular, • if n is odd, then Hom( K, µ n ) = { } as ξ is always trivial, and thus there is onlyone splitting s K of K ; • if n = 2 k with k odd, then Hom( K, µ n ) = { , f ξ } , and for the unique nontrivial ξ two splittings s K ⊗ f ξ and s K are not associated; • if 4 | n , then we have Hom( K, µ n ) = { , f ξ } for the unique nontrivial ξ , and in thecase the two splittings s K ⊗ f ξ and s K are associated, i.e., f ξ : T ( O ) → µ n istrivial on T Q,n ( O ).From the perspective of § Y /Y
Q,n ) ♮ ≃ Z n (4) e/ Z n (2) e. If we consider an (
K, s K )-unramified I ( χ ), then there are two cases as follows. • First, if 4 ∤ n , then every unitary ( K, s K )-unramified I ( χ ) is irreducible and onehas L ( φ χ ) = { I ( χ ) } . then Hom( K, µ n ) ♮ = { } . In particular, if n = 2 k with k odd, then I ( χ ) is not( K, s K ⊗ f ξ )-unramified, where Hom( K, µ n ) = { , f ξ } . In this case, Conjecture5.4 is vacuously true since Hom( K, µ n ) ♮ = { } . • Second, if 4 | n , then Hom( K, µ n ) ♮ = Hom( K, µ n ) = { , f ξ } , and the two splittings s K and s K ⊗ f ξ are associated. Thus, if I ( χ ) is irreducible, then it is both ( K, s K )and ( K, s K ⊗ f ξ )-unramified; in this case, Conjecture 5.4 holds as well.In fact, for 4 | n the nontrivial element in Hom( K, µ n ) ♮ is f ξ = f ne/ : T ( O ) −→ µ n , ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 99 explicitly given by f ne/ ( y ⊗ u ) = [ ̟ ne/ , y ⊗ u ] = ( ̟, u ) nB Q ( e,y ) / n . Proposition 5.13.
Keep notations as above. For a unitary ( K, s K ) -unramified principalseries I ( χ ) , if L ( φ χ ) = JH( I ( χ )) = { π ( φ χ , ) , π ( φ χ , ε ) } , which occurs exactly for | n and that χ α is the non-trivial quadratic character, then π ( φ χ , ρ ) is ( K, s K ) -unramified for exactly one ρ ∈ Irr( S ( φ χ )) , and the other is ( K, s K ⊗ f ξ ) -unramified. In particular, Conjecture 5.4 holds in this case.Proof. For simplicity, in the proof we write s ′ K := s K ⊗ f ξ . We first remark that thesetting is different from Proposition 5.10, since here we only have one conjugacy class K of hyperspecial maximal compact subgroup, but equipped with two splittings. However,the idea of proof in Proposition 5.10 applies. That is, we will consider the eigenvalueof the normalized intertwining operator A ( w α , χ ) applied to the normalized unramifiedvectors f s K and f s ′ K .We take the model Ind TA ( ˜ χ ) for i ( χ ), where˜ χ : A −→ C × is trivial on s K ( T ( O )). In this case Ind GAU ( ˜ χ ) is ( K, s K )-unramified and the normalized( K, s K )-unramified vector f s K ∈ Ind
GAU ( ˜ χ ) is given by(5.17) f s K ( b · s K ( k )) = ( δ / B ( t ) · ˜ χ ( a ) if b = tu with t ∈ A, u ∈ U and k ∈ K, A ( w α , χ )( f s K ) = f s K . To compute A ( w α , χ )( f s ′ K ), we first note that in the model Ind GAU ( ˜ χ ) above, the normal-ized ( K, s ′ K )-unramified vector f s ′ K is not given by the formula (5.17) above.Recall that the character s ′ K /s K : T ( O ) / T Q,n ( O ) −→ µ n is given by ( s ′ K /s K )( k ) = ξ ◦ det( k ) = [ e ( ̟ n/ ) , det( k )]for every k ∈ T ( O ). Fix a lifting t := e ( ̟ n α ) ∈ T of e ( ̟ n α ) ∈ T , where n α = n/
4. We see that the model Ind
GAU ( t ˜ χ ) contains the normalized( K, s ′ K )-unramified vector f s ′ K given by f s ′ K ( b · s ′ K ( k )) = ( δ / B ( a ) · ( t ˜ χ )( a ) if b = au with a ∈ A, u ∈ U and k ∈ K, G -isomorphism (see (5.8))Φ t : Ind GAU ( ˜ χ ) −→ Ind
GAU ( t ˜ χ )we see that Φ − t ( f s ′ K ) is the ( K, s ′ K )-unramified vector in the model Ind GAU ( ˜ χ ). We wantto compute the constant c ∈ C such that T ( w α , χ ; r un w α ) (cid:0) Φ − t ( f s ′ K ) (cid:1) = c · Φ − t ( f s ′ K ) .
00 FAN GAO, FREYDOON SHAHIDI, AND DANI SZPRUCH
By noting that Φ − t ( f s ′ K )( t − ) = f s ′ K (1) = 1, one obtains c = T ( w α , χ ; r un w α )(Φ − t ( f s ′ K ))( t − ) = Z U (Φ − t ( f s ′ K ))( w − α ut − ) du = Z U f s ′ K ( t · ( w − α t − w α ) · w − α tut − ) du = Z U f s ′ K ( h α ( ̟ n α ) · w − α tut − ) du = χ α · Z U f s ′ K ( w − α u ) du. Now similar (and in fact simpler) argument as in Proposition 5.10 gives that the lastintegral is equal to c gk ( w α , χ ) and thus c = χ α · c gk ( w α , χ ) = ( − γ ( w α , χ ) − = ( −
1) 1 + q − . This shows that A ( w α , χ )( f s ′ K ) = − f s ′ K as desired and the proof is completed. (cid:3) Consider the Whittaker dimension, note that:– if 4 ∤ n , then the hypothesis of Conjecture 5.7 is satisified, or equivalently the map ϕ is trivial. However, in this case, I ( χ ) is always irreducible, and thus Conjecture5.7 holds trivially.– if 4 | n , then the cover SO does not fit into the hypothesis of Conjecture 5.7. Onthe other hand, since T ad /T = { } , we see that a priori one expects(5.18) dim Wh ψ ( π ( φ χ , ρ )) = dim Wh t ψ ( π ( φ χ , ρ ))for every ρ ∈ Irr( S ( χ χ )) and t ∈ T .For 4 | n we verify below the equality (5.18) for reducible I ( χ ), for which S ( φ χ ) = W . Weassume ψ = O F and t = e ( ̟ ) ∈ T .First, writing n = 2 k with k even, we have the ordered set R = { ie : − k/ i k/ } of representatives of Y /Y
Q,n . To apply Proposition 5.9, we consider the action w ( − ) on Y /Y
Q,n . There are two trivial orbits { } and { ke/ } in Y /Y
Q,n . The remaining elementsin
Y /Y
Q,n are formed by free W -orbits with respect to the action w ( − ). Accordingly, upto a permutation of elements in R , we see that the scattering matrix[ τ ( w α , χ, s z , s y )] z,y ∈ R is a block-diagonal matrix with each block S R ( w α , χ ) ie described as follows.– There is a size-one block S R ( w α , χ ) = τ ( w α , χ, s , s ) = χ α − q − − χ α = − q − W -orbit { } ; similarly, associated to { ke/ } one hasanother size-one block S R ( w α , χ ) ke/ = τ ( w α , χ, s ke/ , s ke/ ) = (1 − q − ) χ α − χ α − q − χ α = 1 + q − . – For every 1 i k/ −
1, one has a size-two block(5.19) S R ( w α , χ ) ie = (cid:18) − (1 − q − ) / g t ψ − ( i )) g t ψ − ( − i ) (1 − q − ) / (cid:19) , ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 101 where the left-upper entry and right-lower entry are equal to τ ( w α , χ, s ie , s ie ) and τ ( w α , χ, s − ie , s − ie ) respectively.From the above, we immediately get (writing e ψ for e ( ̟ ) ψ ) σ Wh eψ = ( k/ · ⊕ ( k/ · ε W = σ Wh ψ , where the second equality follows from [Gao, Proposition 8.3]. Hence,dim Wh ψ ( π ( φ χ , ρ )) = dim Wh e ψ ( π ( φ χ , ρ )) = n/ ρ ∈ { , ε W } = Irr( S ( φ χ ), as expected.5.4.3. Covers of GL r . It seems to be a folkloric result that a Zelevinsky-type classificationin terms of segements of reducibility points of parabolic induction from supercuspidalrepresentations holds for covers of GL r . However, the classical proof of Zelevinsky reliescrucially on the fact that the Whittaker model of the linear GL is unique, and thusthe approach is not directly adaptable in the covering settings. If one restricts to therepresentations with unique Whittaker model, then one can obtain similar results byusing the same method as in [BZ76, BZ77, Zel80].On the other hand, it is expected that every unitary unramified genuine principal seriesof GL r is irreducible. Recall that every Brylinski–Deligne cover is associated with B Q such that B ( e i , e i ) = 2 p and B ( e i , e j ) = q for i = j. One has Q ( α i ) = 2 p − q . Proposition 5.14.
Let GL r be an arbitrary Brylinski–Deligne cover of GL r such that n α · Y ⊂ Y Q,n . Then one has Γ tor G Q,n = { } . Hence, every unitary unramified genuine principal series of such cover GL r is irreducible.Proof. In view of the embedding S ( φ χ ) ֒ → Γ tor G,Q,n for every unitary unramified χ , itsuffices to prove the triviality of the latter group. Recall thatΓ tor G Q,n = ( X Q,n ∩ X scG, Q ) /X scQ,n . Assume x ∈ r − X i =1 c i · α i ∈ X Q,n ∩ X scG, Q with c i ∈ Q and α i = e ∗ i − e ∗ i +1 for each i . Since n α Y ⊂ Y Q,n by assumption, we have n α e k ∈ Y Q,n for every 1 k r −
1. Then we have h x, n α e k i ∈ Z , that is, n α c ∈ Z and n α ( c i +1 − c i ) ∈ Z for every 1 i r −
1. This gives that c i ∈ Z /n α and thus x ∈ X scQ,n . (cid:3) It is clear that Proposition 5.14 applies to the cases when n α = n or q = 0, in particularto the Kazhdan–Patterson covers and Savin covers discussed in § Remark 5.15.
It is expected that the last assertion in Proposition 5.14 holds for arbitraryBrylinski–Deligne cover of GL r , without the assumption n α Y ⊂ Y Q,n , i.e., we expect theequality S ( φ χ ) = { } for every unitary unramified χ . However, the group Γ tor G Q,n may notbe trivial in general. For example, consider the Brylinski–Deligne cover GL (4)2 associatedwith p = 1 , q = 2 and n = 4 . Then it is easy to check that n α = 1 and Γ tor G Q,n = ( Z α/ / Z α ≃ Z / Z in this case.
02 FAN GAO, FREYDOON SHAHIDI, AND DANI SZPRUCH Analysis for covers of
GSp r In this section, we analyze the unitary unramified principal series I ( χ ) of G = GSp r of type I (see § π ( φ χ , ρ ) , ρ ∈ Irr( S ( φ χ )) of an( K, s K )-unramified I ( χ ), determine the pair ( K ′ , s K ′ ) with respect to which π ( φ χ , ρ )is unramified,(ii) determining the ψ -Whittaker dimension of each constituent π ( φ χ , ρ ),(iii) investigation of the restriction of each π ( φ χ , ρ ) to Sp r and the pair ( K , s K )with respect to which a constituent is unramified, and also how the Whittakerdimension of each constituent varies with respect to different ψ -Whittaker datum.Recall that the n -fold cover GSp r of type I is associated with the quadratic form Q : Y → Z such that Q ( α ∨ r ) = − Q ( e ) = 0 , where we follow the notations in § K = GSp r ( O ) and a splitting s K : K ֒ → G . Then for every f : K −→ µ n , one obtains a splitting f ⊗ s K . The homomorphism f factors through the similitude mapsim : GSp r −→ F × and thus corresponds to a homomorphism ξ : O × /O × n −→ µ n such that f = ξ ◦ sim. We are interested in the subgroup Hom( K, µ n ) ♮ ⊂ Hom(
K, µ n ), thatis, those f z with z ∈ ( Y /Y
Q,n ) ♮K . For every z ∈ ( Y /Y
Q,n ) ♮K , as a function on T ( O ) ⊂ K ,one has f z ( y ⊗ u ) = [ y ⊗ u, z ( ̟ )] , and in particular f z ( e ( u )) = [ e ( u ) , z ( ̟ )] = ( u, ̟ ) B Q ( e ,z ) n and(6.1) 1 = f z ( α ∨ i ( u )) = [ α ∨ i ( u ) , t ] = ( u, ̟ ) B Q ( α ∨ i ,z ) n for every u ∈ O × and 1 i r . The equalities in (6.1) are equivalent to B Q ( α ∨ i , z ) ∈ n Z . for all 1 i r , or equivalently, B Q ( e i , z ) ∈ n Z for every 1 i r . We have in this case ( Y /Y
Q,n ) ♮K = Y ♮K /Y Q,n , where Y ♮K := { z ∈ Y : B Q ( z, e i ) ∈ n Z for all i } and Y Q,n is given in (3.12). Setting z = P ri =1 y i e i , it is easy to see that(6.2) Y ♮K = (cid:26) P ri =0 y i e i ∈ Y : • n | ( − y i + y ) for every i (cid:27) . Let I ( χ ) be a ( K, s K )-unramified principal series. It is also ( K, f z ⊗ s K )-unramified forevery z ∈ ( Y /Y
Q,n ) ♮K . On the other hand, for any y ∈ Y , consider K y := y ( ̟ ) · K · y ( ̟ ) − . ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 103
The representation I ( χ ) is unramified with respect to( K y , y · s K )as well. Indeed, since [ y ( ̟ ) , T Q,n ( O )] = 1, we see that the splittings y · s K : K y ֒ → G and s K : K ֒ → G agree on T Q,n ( O ). We also have( Y /Y
Q,n ) ♮K ≃ ( Y /Y
Q,n ) ♮K y for every y ∈ Y . Combining the above, we see I ( χ ) is ( K y , f z ⊗ y · s K )-unramified forevery z ∈ ( Y /Y
Q,n ) ♮K and y ∈ Y .We set K = Sp r ∩ K = Sp r ( O ) , and for any y ∈ Y denote K y, = K z ∩ Sp r = y ( ̟ ) · K · y ( ̟ ) − . In particular, we write K ′ = K e and thus K ′ = K e , . It follows that K = { K , K ′ } is the set of conjugacy classes of hyperspecial maximal compact subgroup of Sp r .For the questions in (i)–(iii), we will give a seperate discussion according to the parityof n .6.1. Odd fold cover of
GSp r . If n is odd, then it is given in § Y Q,n has a Z -basis { ne i : 1 i r } ∪ (cid:8) n ( r ) · e c (cid:9) and also that(6.3) GSp ∨ r = (cid:8) ( g, a ) ∈ GSpin r +1 × GL : λ ( g ) = a gcd( n,r ) (cid:9) . One has | X Q,n | = 2 n r · n ( r ) . Since n is odd, we have Y ,Q,n = Y scQ,n , which is then equal to Y ∩ Y Q,n . In any case, wehave for odd n the identity X Γ Q,n = X c Q,n . Moreover, it is easy to see that { e i : 1 i r } ∪ (cid:8) n ( r ) e c (cid:9) constitutes a basis for Y + Y Q,n , and therefore (cid:12)(cid:12) X Γ Q,n (cid:12)(cid:12) = 2 · n ( r ) . One has in this case( Y /Y
Q,n ) ♮K = (cid:8) i · e c : 0 i n ( r ) − (cid:9) . The diagram (5.4) becomes(6.4) (
Y /Y
Q,n ) ♮K X c Q,n X ,Q,n X Q,n X Γ Q,n . p Γ p Γ One can check that the image of (
Y /Y
Q,n ) ♮K in X c Q,n via p Γ is of index 2.
04 FAN GAO, FREYDOON SHAHIDI, AND DANI SZPRUCH
Every unitary (
K, s K )-unramified genuine principal series I ( χ ) is irreducible, in viewof (6.3). As remarked, for every y ∈ Y and z ∈ ( Y /Y
Q,n ) ♮K , the representation I ( χ ) isalso unramified with respect to ( K y , f z ⊗ y ( ̟ ) · s K ) . The pair (
G, G ) is an isotypic pair, and one has the natural map f G,G between the L -groups arising from the composites as in the following diagram G ∨ GSpin r +1 × GL G ∨ SO r +1 . f G,G By the choice of a distinguished genuine character χ ψ : Z ( T ) −→ C × , which by restrictiongives a distinguished genuine character χ ψ : Z ( T ) → C × , one can extend f G,G to be ahomomorphism of L -groups f G,G : L G −→ L G . We have I ( χ ) | G = (cid:12)(cid:12) X Γ Q,n (cid:12)(cid:12) · I ( ω ) , where ω = χ | Z ( T ) . For any non-degenerate ψ , one hasdim Wh ψ ( I ( χ )) = 2 · n r · n ( r ) and dim Wh ψ ( I ( ω )) = n r . Note that I ( ω ) might be reducible. Indeed, if χ α r = −
1, then I ( ω ) = π ( φ ω , ) ⊕ π ( φ ω , ε )where ε is the nontrivial character of S ( φ ω ) = { , w α r } . Let K = { K , K ′ } be the aboveset of conjugacy classes of hyperspecial maximal compact subgroups of Sp r , as above.Let s : K ֒ → Sp r and s ′ : K ′ ֒ → Sp r be the two unique splittings of K and K ′ respectively. Theorem 6.1.
For
GSp ( n )2 r with odd n and a unitary ( K, s K ) -unramified I ( χ ) , one has I ( χ ) | G = (cid:12)(cid:12) X Γ Q,n (cid:12)(cid:12) · I ( ω ) where ω = χ | Z ( T ) . If I ( ω ) = π ( φ ω , ) ⊕ π ( φ ω , ε ) is reducible, and we assume that π ( φ ω , ) is ( K , s ) -unramified, then π ( φ ω , ε ) is ( K ′ , s ′ ) -unramified. Regarding the Whittaker di-mension, for an additive character ψ with f ( ψ ) = O F , one has dim Wh ψ ( π ( φ ω , )) = n r + n r − , dim Wh ψ ( π ( φ ω , ε )) = n r − n r − and also dim Wh t ψ ( π ( φ ω , )) = n r − n r − , dim Wh t ψ ( π ( φ ω , ε )) = n r + n r − , where t := e ( ̟ ) .Proof. The result here is essentially the content of Remark 5.12. Indeed, the assertionthat π ( φ ω , ε ) is ( K ′ , s ′ )-unramified follows from the same argument as in Proposition5.10. The equalities regarding the ψ -Whittaker dimensions follow from [Gao], while thosefor the t ψ -Whittaker dimension follows from the same argument in Proposition 5.11, bynoting f ( e ψ ) = p F . ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 105
Alternatively, to prove the equalities on Whittaker dimensions, as noted in Remark5.12, one may reduce (essentially) to the SL ( n )2 by parabolic induction. Indeed, let M ⊂ P ⊂ Sp r be the maximal Levi subgroup associated with ∆ − { α r − } . Then one has M ≃ GL r − × µ n Sp and also ω = χ ⊠ µ. If χ α r = ω α r = −
1, equivalently µ α r = −
1, then one has a decomposition of the genuineprincipal series I ( µ ) = π ( φ µ , ) ⊕ π ( φ µ , ε )of Sp , such that π ( φ ω , ρ ) = Ind Sp r P I ( χ ) ⊠ π ( φ µ , ρ )for ρ ∈ { , ε } = Irr( S ( φ χ )). Since dim ψ ( I ( χ )) = n r = dim t ψ ( I ( χ )), we see that thedesired equalities follow directly from Rodier’s heredity and the results in Proposition5.11. This also concludes the proof. (cid:3) Regarding Conjecture 5.4 and 5.7, we see that the three groupsHom(
K, µ ) ♮ , d Γ tor G and [ Γ tor G Q,n are all trivial. Also, every unitary (
K, s K )-unramified I ( χ ) is irreducible. Hence, we seethat Conjecture 5.4 and Conjecture 5.7 hold trivially.6.2. Even fold cover of
GSp r . In this subsection, we assume n = 2 m is even. We fixa unitary ( K, s K )-unramified I ( χ ) of GSp r , it follows from Theorem 2.24 that I ( χ ) | G = (cid:12)(cid:12) X c Q,n (cid:12)(cid:12) · M γ ∈ X Γ Q,n / X c Q,n M ω γ,j ∈ E ( χ, γ ˜ χ O ; Z ( T )) I ( ω γ,j ) , where (cid:12)(cid:12) E ( χ, γ ˜ χ O ; Z ( T )) (cid:12)(cid:12) = | Y ,Q,n /Y ∩ Y Q,n | = 2 . We also have | X ,Q,n | = m r . There are two situations according to the parity of r .6.2.1. When r is odd. In this case, as discussed in § (cid:8) v := n ( r ) · e c (cid:9) ∪ { me i + v / i r } constitutes a basis for Y Q,n and GSp ∨ r = GSp r . Thus, every unitary (
K, s K )-unramified I ( χ ) is irreducible. Moreover, | Y /Y
Q,n | = 2 · m r · n ( r ) . It is also easy to see that Y + Y Q,n ⊂ Y is the Z -span of { e i : 1 i r } ∪ { v / } and thus X Γ Q,n = Y / ( Y + Y Q,n ) = (cid:8) ie : 0 i n ( r ) − (cid:9) , which has size n ( r ) . It is straightforward to compute from its definition that X c Q,n = (cid:8) ie : 2 | i and 0 i n ( r ) − (cid:9) and thus X Γ Q,n / X c Q,n = { , e } .
06 FAN GAO, FREYDOON SHAHIDI, AND DANI SZPRUCH
We have(6.5) 0 Y ,Q,n /Y ∩ Y Q,n ( Y /Y
Q,n ) ♮K X c Q,n Y /Y ∩ Y Q,n X Q,n X Γ Q,n X ,Q,n Y /Y ,Q,n , m r p Γ m r − · n ( r ) n ( r ) m r where the numbers around an arrow indicate the ratio of the orders of the two groupsas the domain and codomain of the pertaining map; for example 2 m r = | X Q,n | / (cid:12)(cid:12) X Γ Q,n (cid:12)(cid:12) .Since Z e c /Y Q,n ⊂ ( Y /Y
Q,n ) ♮K , we see that p Γ is surjective.From the above, we obtain more explicitly I ( χ ) | G = n ( r ) · ( I ( ω , ) ⊕ I ( ω , )) M n ( r ) · ( I ( ω e , ) ⊕ I ( ω e , )) . Note that I ( ω ,i ) , i = 0 , K , s )-unramified. We show that I ( ω e ,i ) , i = 0 , K ′ , s ′ )-unramified. Indeed, we have i ( ω e ,i ) = Ind T A ( ω e ,i ⊠ e ( ̟ ) ˜ χ O ) ∈ Irr( T ) , where e ( ̟ ) ˜ χ O : T ( O ) −→ C × is the twisted character. Since the character ˜ χ O restricted to s ( T ( O )) is trivial and bydefinition s ′ ( k ′ ) = e ( ̟ ) · s (cid:0) e ( ̟ ) − · k ′ · e ( ̟ ) (cid:1) · e ( ̟ ) − for every k ′ ∈ K ′ , it gives that e ( ̟ ) ˜ χ O ( s ′ ( k )) = ˜ χ O ◦ s ( k ) = 1for every k ∈ T ( O ). Thus, by the Satake isomorphism, every I ( ω e ,i ) for j = 0 , K ′ , s ′ )-unramified.Since the dual group Sp ∨ r = Sp r , we have S ( φ ω γ,j ) = { } and thus every I ( ω γ,i ) isirreducible. We give a summary of the above discussion. Theorem 6.2.
Assume n = 2 m and r is odd. Then every ( K, s K ) -unramified unitaryprincipal series I ( χ ) of GSp r is irreducible, and one has I ( χ ) | G = n ( r ) · ( I ( ω , ) ⊕ I ( ω , )) M n ( r ) · ( I ( ω e , ) ⊕ I ( ω e , )) where I ( ω ,j ) , j = 0 , is ( K , s ) -unramified and I ( ω e ,j ) , j = 0 , is ( K ′ , s ′ ) -unramified.Moreover, for every γ ∈ X Γ Q,n / X c Q,n and j ∈ { , } one has dim Wh ψ ( I ( ω γ,j )) = m r for any non-degenerate character ψ . ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 107
Similar to the odd-fold cover of GSp r , we see that if n is even with r odd, the twogroups d Γ tor G and [ Γ tor G Q,n are trivial. This shows that Conjecture 5.4 and Conjecture 5.7 holdtrivially as well.6.2.2.
When r is even. In this case, we have Y Q,n = Y scQ,n ⊕ Z v with v = n ( r ) · e c asabove. Thus, { m ( e i − e i +1 ) : 1 i r − } ∪ { ne r } ∪ { v } constitutes a basis for Y Q,n , and alsoGSp ∨ r = PGSp r × GL , see § | X Q,n | = 4 m r · n ( r ) . It is easy to see that Y + Y Q,n is the Z -span of { e i : 1 i r } ∪ (cid:8) v = n ( r ) · e c (cid:9) . Thus X Γ Q,n = (cid:8) ie c + ce : 0 i n ( r ) − c (cid:9) with (cid:12)(cid:12) X Γ Q,n (cid:12)(cid:12) = 2 · n ( r ) . In view of the map c : X Γ Q,n −→ Hom( T ( O ) ∩ Z ( T ) , µ n ) , we see that X c Q,n := Ker( c ) = (cid:8) ie c : 0 i n ( r ) − (cid:9) ⊂ X Γ Q,n . We have(6.6) 0 Y ,Q,n /Y ∩ Y Q,n ( Y /Y
Q,n ) ♮K X c Q,n Y /Y ∩ Y Q,n X Q,n X Γ Q,n X ,Q,n Y /Y ,Q,n m r p Γ m r − · n ( r )
22 2 n ( r ) m r where the map p Γ is surjective. In particular, I ( χ ) | G = n ( r ) · ( I ( ω , ) ⊕ I ( ω , )) M n ( r ) · ( I ( ω e , ) ⊕ I ( ω e , )) . Note that I ( χ ) itself might be reducible, exactly when χ satisfies that χ α i = − i = 2 k − , k r/ . In this case S ( χ χ ) = { , w } with w = w α w α ...w α r − and we write I ( χ ) = π ( φ χ , ) ⊕ π ( φ χ , ε ) , where ε ∈ Irr( S ( φ χ )) is the nontrivial character. Proposition 6.3.
Assume I ( χ ) is unitary ( K, s K ) -unramified and reducible, and assume π ( φ χ , ) ⊂ I ( χ ) is the unique ( K, s K ) -unramified constituent. Then π ( φ χ , ε ) is the unique ( K, f me r ⊗ s K ) -unramified constituent.
08 FAN GAO, FREYDOON SHAHIDI, AND DANI SZPRUCH
Proof.
First note that me r ∈ ( Y /Y
Q,n ) ♮K , in view of (6.2). The idea of the proof isthe same as Proposition 5.13, though here we are dealing with an intertwining operatorassociated with w = w α w α ...w α r − . Let ˜ χ be a genuine character of A extending χ and trivial on s K | T ( O ) . For the proof, weset t = me r ( ̟ ) and s ′ K = f me r ⊗ s K . Then t ˜ χ also extends χ and is trivial on s ′ K restricted to T ( O ).Let t be any lifting of t . One has the isomorphismΦ t : Ind GAU ( ˜ χ ) −→ Ind
GAU ( t ˜ χ )between the two models given by Φ t ( f )( g ) = f ( t − g ) . Let f s ′ K ∈ Ind
GAU ( t ˜ χ ) be the ( K, s ′ K )-unramified vector, which gives Φ − t ( f s ′ K ) ∈ Ind
GAU ( ˜ χ ).We want to compute the number c ∈ C such that T ( w, ˜ χ ; r un w α ) (cid:0) Φ − t ( f s ′ K ) (cid:1) = c · Φ − t ( f s ′ K ) . We have the following commutative diagram I ( ˜ χ ) I ( w ˜ χ ) I ( t ˜ χ ) I ( w ( t ˜ χ )) I ( t ( w ˜ χ )) , T ( w, ˜ χ )Φ t Φ wtw − Φ t T ( w, t ˜ χ ) Φ twt − w − where w ˜ χ = ˜ χ. Let c ( w, t ˜ χ ) ∈ C be the number such thatΦ t ◦ T ( w, ˜ χ ) ◦ Φ − t ( f s ′ K ) = c ( w, t ˜ χ ) · f s ′ K . Thus, we get c ( w, t ˜ χ ) = T ( w, t ˜ χ )( f s ′ K )( wtw − t − ) . Now we see that wtw − t − = w r − · me r ( ̟ ) · w − r − · me r ( ̟ ) − = h α r − ( ̟ m ) ∈ Z ( T ) . It then follows that c ( w, t ˜ χ ) = w χ ( h α r − ( ̟ m )) · c gk ( w, χ ) = ( − · c gk ( w, χ ) . This shows that the eigenvalue of A ( w, χ ) : I ( ˜ χ ) −→ I ( ˜ χ )is equal to − K, s ′ K )-normalized unramified vector Φ − t ( f s ′ k ) ∈ I ( ˜ χ ). This con-cludes the proof. (cid:3) Regarding Conjecture 5.4, we have d Γ tor G = { } and [ Γ tor G Q,n = Z / Z , where the nontrivialelement is equal to ϕ ◦ h ( me r ) with the function ϕ ◦ h given as in (5.7). Thus, we seethat Proposition 6.3 implies Conjecture 5.4 for z = me r (while y = 1). It is not hard toverify Conjecture 5.4 for any other z ′ ∈ Hom(
K, µ ) ♮ . Thus, Conjecture 5.4 holds for evenfold cover of GSp r , where r is also even. ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 109
On the other hand, the hypothesis of Conjecture 5.7 is not satisfied for such cover ofGSp r . We analyze the Whittaker dimension dim Wh ψ ( π ( φ χ , ρ )) as follows. It followsfrom the analysis in § I ( ω , ) ≃ I ( ω , )since ω , = w ω , , and that these two principal series are both ( K , s )-unramified. Onthe other hand, we have I ( ω e , ) ≃ I ( ω e , ) , which are both ( K ′ , s ′ )-unramified. Note that Theorem 4.12 implies that π ( φ χ , ) | G and π ( φ χ , ε ) | G both contain n ( r ) · I ( ω , ). Since I ( χ ) is also ( K e , e · s K )-unramified, it is easyto see that I ( ω e ,i ) is ( K ′ , s ′ )-unramified. Theorem 4.12 then also implies n ( r ) · I ( ω e , )is contained in both π ( ) and π ( ε ). Thus, as representations of G , we have π ( φ χ , ) | G ≃ π ( φ χ , ε ) | G ≃ n ( r ) · I ( ω , ) M n ( r ) · I ( ω e , ) ≃ n ( r ) · I ( ω , ) M n ( r ) · I ( ω e , ) . For every nondegenerate ψ , we havedim Wh ψ ( π ( φ χ , )) = dim Wh ψ ( π ( φ χ , ε )) = 2 m r · n ( r ) . We give a summary of this as follows.
Theorem 6.4.
For
GSp ( n )2 r with n and r both even, a unitary ( K, s K ) -unramified principalseries I ( χ ) has the decomposition I ( χ ) | G = n ( r ) · ( I ( ω , ) ⊕ I ( ω , )) M n ( r ) · ( I ( ω e , ) ⊕ I ( ω e , )) , where I ( ω ,j ) is ( K , s ) -unramified and I ( ω e ,j ) is ( K ′ , s ′ ) -unramified. Here every I ( ω γ,j ) is irreducible. If I ( χ ) = π ( φ χ , ) ⊕ π ( φ χ , ε ) is reducible, then π ( φ χ , ) | G ≃ π ( φ χ , ε ) | G ≃ n ( r ) · I ( ω , ) M n ( r ) · I ( ω e , ) ≃ n ( r ) · I ( ω , ) M n ( r ) · I ( ω e , ) . Regarding the Whittaker dimension, one has dim Wh ψ ( I ( ω γ,j )) = m r for every non-degenerate ψ . Example 6.5.
Consider GSp (4)4 , i.e., n = 4 and r = 2. In this case, n ( r ) = 2. We have X Γ Q,n = { , e c , e , e c + e } ≃ Z / Z × Z / Z . and X c Q,n = { , e c } . On the other hand, we see(
Y /Y
Q,n ) ♮K = { , e , e c , e c − e } ≃ Z / Z × Z / Z with p − (0) = { , e } and p − ( e c ) = { e c , e c − e } . When the (
K, s K )-unramified I ( χ ) is reducible, we have π ( φ χ , ) | G ≃ π ( φ χ , ε ) | G ≃ · I ( ω , ) M · I ( ω e , ) ≃ · I ( ω , ) M · I ( ω e , )with dim Wh ψ ( π ( φ χ , )) = dim Wh ψ ( π ( φ χ , ε )) = 16 . Assume π ( φ χ , ) is ( K, s K )-unramified, then it is also ( K, f e c ⊗ s K )-unramified, and that π ( φ χ , ε ) is both ( K, f e ⊗ s K ) and ( K, f e c − e ⊗ s K )-unramified.
10 FAN GAO, FREYDOON SHAHIDI, AND DANI SZPRUCH
Remark 6.6.
Here we observe again that covers satisfying G ∨ ≃ G ad seem to exhibit quite special properties. For instance, they do not satisfy the hypothesisof Conjecture 5.7. One such example here is the even fold cover of GSp r with r beingeven. A previous example, as noted in Remark 5.8, is the n -fold cover of Spin r +1 with r odd and n α ≡ α ∨ . The n -fold cover of SO studied in § | n , is also of such type. Remark 6.7.
For the double cover GSp (2)2 r of GSp r , many results in this section arealready proved in [Szp15], even without assuming the tame condition p ∤ n . Indeed,the restriction problem for general genuine principal series of GSp (2)2 r is analyzed system-atically in loc. cit. by utilizing a natural maximal abelian subgroup A ⊂ T chosenindependent of the residual characteristic p of F , see [Szp15, Lemma 2.1]. References [Ada03] Jeffrey Adams,
Characters of covering groups of
SL( n ), J. Inst. Math. Jussieu (2003), no. 1,1–21, DOI 10.1017/S147474800300001X. MR1955205[AP19] Jeffrey D. Adler and Dipendra Prasad, Multiplicity upon restriction to the derived subgroup ,Pacific J. Math. (2019), no. 1, 1–14, DOI 10.2140/pjm.2019.301.1. MR4007368[Asg02] Mahdi Asgari,
Local L -functions for split spinor groups , Canad. J. Math. (2002), no. 4,673–693, DOI 10.4153/CJM-2002-025-8. MR1913914[AS06] Mahdi Asgari and Freydoon Shahidi, Generic transfer for general spin groups , Duke Math. J. (2006), no. 1, 137–190, DOI 10.1215/S0012-7094-06-13214-3. MR2219256[Ato17] Hiraku Atobe,
On the uniqueness of generic representations in an L -packet , Int. Math. Res.Not. IMRN (2017), 7051–7068, DOI 10.1093/imrn/rnw220. MR3801418[ABPS17] Anne-Marie Aubert, Paul Baum, Roger Plymen, and Maarten Solleveld, The principal seriesof p -adic groups with disconnected center , Proc. Lond. Math. Soc. (3) (2017), no. 5, 798–854, DOI 10.1112/plms.12023. MR3653247[BCG18] Dubravka Ban, Kwangho Choiy, and David Goldberg, R -group and multiplicity in restrictionfor unitary principal series of GSpin and Spin , Geometry, algebra, number theory, and theirinformation technology applications, Springer Proc. Math. Stat., vol. 251, Springer, Cham,2018, pp. 59–69, DOI 10.1007/978-3-319-97379-1-4. MR3880383[BJ04] Dubravka Ban and Chris Jantzen, Duality and the normalization of standard intertwiningoperators , Manuscripta Math. (2004), no. 4, 401–415, DOI 10.1007/s00229-004-0504-7.MR2103658[BLS99] William D. Banks, Jason Levy, and Mark R. Sepanski,
Block-compatible metaplectic cocycles ,J. Reine Angew. Math. (1999), 131–163.[BZ76] I. N. Bernstein and A. V. Zelevinsky,
Representations of the group GL ( n, F ) , where F is a localnon-Archimedean field , Uspehi Mat. Nauk (1976), no. 3(189), 5–70 (Russian). MR0425030[BZ77] , Induced representations of reductive p -adic groups. I , Ann. Sci. ´Ecole Norm. Sup. (4) (1977), no. 4, 441–472. MR0579172[Bor79] A. Borel, Automorphic L -functions , Automorphic forms, representations and L -functions(Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Proc. Sympos. PureMath., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 27–61. MR546608[BBBF18] Ben Brubaker, Valentin Buciumas, Daniel Bump, and Solomon Friedberg, Hecke modules frommetaplectic ice , Selecta Math. (N.S.) (2018), no. 3, 2523–2570, DOI 10.1007/s00029-017-0372-0. MR3816510[BD01] Jean-Luc Brylinski and Pierre Deligne, Central extensions of reductive groups by K , Publ.Math. Inst. Hautes ´Etudes Sci. (2001), 5–85, DOI 10.1007/s10240-001-8192-2. MR1896177[Cai19] Yuanqing Cai, Fourier coefficients for theta representations on covers of general linear groups ,Trans. Amer. Math. Soc. (2019), no. 11, 7585–7626, DOI 10.1090/tran/7429. MR3955529[CS80] W. Casselman and J. Shalika,
The unramified principal series of p -adic groups. II. The Whit-taker function , Compositio Math. (1980), no. 2, 207–231. MR581582[CO13] Gautam Chinta and Omer Offen, A metaplectic Casselman-Shalika formula for GL r , Amer.J. Math. (2013), no. 2, 403–441, DOI 10.1353/ajm.2013.0013. MR3038716 ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 111 [Cho19] Kwangho Choiy,
On multiplicity in restriction of tempered representations of p -adic groups ,Math. Z. (2019), no. 1-2, 449–471, DOI 10.1007/s00209-018-2091-4. MR3936078[CR90] Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I , Wiley ClassicsLibrary, John Wiley & Sons, Inc., New York, 1990. With applications to finite groups andorders; Reprint of the 1981 original; A Wiley-Interscience Publication. MR1038525[FG99] Solomon Friedberg and David Goldberg,
On local coefficients for non-generic represen-tations of some classical groups , Compositio Math. (1999), no. 2, 133–166, DOI10.1023/A:1000723719451. MR1686785[Gan14] Wee Teck Gan,
Recent progress on the Gross-Prasad conjecture , Acta Math. Vietnam. (2014), no. 1, 11–33, DOI 10.1007/s40306-014-0047-2. MR3176460[Gan17] , The metaplectic tensor product as an instance of Langlands functoriality , L-functionsand automorphic forms, Contrib. Math. Comput. Sci., vol. 10, Springer, Cham, 2017, pp. 97–114. MR3931450[GG18] Wee Teck Gan and Fan Gao,
The Langlands-Weissman program for Brylinski-Deligne exten-sions , Ast´erisque (2018), 187–275 (English, with English and French summaries). L-groupsand the Langlands program for covering groups. MR3802419[GGP12] Wee Teck Gan, Benedict H. Gross, and Dipendra Prasad,
Symplectic local root numbers, cen-tral critical L values, and restriction problems in the representation theory of classical groups ,Ast´erisque (2012), 1–109 (English, with English and French summaries). Sur les conjec-tures de Gross et Prasad. I. MR3202556[Gao17] Fan Gao, Distinguished theta representations for certain covering groups , Pacific J. Math. (2017), no. 2, 333–379, DOI 10.2140/pjm.2017.290.333.[Gao18a] ,
The Langlands-Shahidi L-functions for Brylinski-Deligne extensions , Amer. J. Math. (2018), no. 1, 83–137, DOI 10.1353/ajm.2018.0001.[Gao18b] ,
Generalized Bump-Hoffstein conjecture for coverings of the general linear groups , J.Algebra (2018), 183–228, DOI 10.1016/j.jalgebra.2017.12.002.[Gao20] ,
Kazhdan–Lusztig representations and Whittaker space of some genuine representa-tions , Math. Ann. (2020), no. 1, 289–358, DOI 10.1007/s00208-019-01925-1.[Gao] ,
R-group and Whittaker space of some genuine representations , J. Inst. Math. Jussieu(2021, accepted), available at https://arxiv.org/abs/1912.07408.[GSS18] Fan Gao, Freydoon Shahidi, and Dani Szpruch,
On the local coefficients matrix for coveringsof SL , Geometry, algebra, number theory, and their information technology applications,Springer Proc. Math. Stat., vol. 251, Springer, Cham, 2018, pp. 207–244. MR3880389[GSS] , Local coefficients and gamma factors for principal series of covering groups , Memoirsof the AMS (2019, accepted), available at https://arxiv.org/abs/1902.02686.[GW19] Fan Gao and Martin H. Weissman,
Whittaker models for depth zero representations of cov-ering groups , Int. Math. Res. Not. IMRN (2019), 3580–3620, DOI 10.1093/imrn/rnx235.MR3961710[GK75] I. M. Gelfand and D. A. Kazhdan, Representations of the group
GL( n, K ) where K is a localfield , Lie groups and their representations (Proc. Summer School, Bolyai J´anos Math. Soc.,Budapest, 1971), Halsted, New York, 1975, pp. 95–118. MR0404534[GHPS79] Stephen Gelbart, Roger Howe, and Ilya Piatetski-Shapiro, Uniqueness and existence of Whit-taker models for the metaplectic group , Israel J. Math. (1979), no. 1-2, 21–37 (1980), DOI10.1007/BF02761822. MR571393[GK82] S. S. Gelbart and A. W. Knapp, L -indistinguishability and R groups for the special linear group ,Adv. in Math. (1982), no. 2, 101–121, DOI 10.1016/0001-8708(82)90030-5. MR644669[GK81] , Irreducible constituents of principal series of SL n ( k ), Duke Math. J. (1981), no. 2,313–326. MR620252[GPS80] Stephen Gelbart and I. I. Piatetski-Shapiro, Distinguished representations and modular formsof half-integral weight , Invent. Math. (1980), no. 2, 145–188, DOI 10.1007/BF01390042.MR577359[GPS83] , Some remarks on metaplectic cusp forms and the correspondences of Shimura andWaldspurger , Israel J. Math. (1983), no. 2, 97–126, DOI 10.1007/BF02760615. MR693355[GS16] David Goldberg and Dani Szpruch, Plancherel measures for coverings of p -adic SL ( F ), Int. J.Number Theory (2016), no. 7, 1907–1936, DOI 10.1142/S1793042116501189. MR3544420[Gro91] Benedict H. Gross, Some applications of Gel ′ fand pairs to number theory , Bull. Amer. Math.Soc. (N.S.) (1991), no. 2, 277–301, DOI 10.1090/S0273-0979-1991-16017-9. MR1074028
12 FAN GAO, FREYDOON SHAHIDI, AND DANI SZPRUCH [GP92] Benedict H. Gross and Dipendra Prasad,
On the decomposition of a representation of SO n when restricted to SO n − , Canad. J. Math. (1992), no. 5, 974–1002, DOI 10.4153/CJM-1992-060-8. MR1186476[GR10] Benedict H. Gross and Mark Reeder, Arithmetic invariants of discrete Langlands parameters ,Duke Math. J. (2010), no. 3, 431–508, DOI 10.1215/00127094-2010-043. MR2730575[Kab01] Anthony C. Kable,
The tensor product of exceptional representations on the general lin-ear group , Ann. Sci. ´Ecole Norm. Sup. (4) (2001), no. 5, 741–769, DOI 10.1016/S0012-9593(01)01075-8 (English, with English and French summaries). MR1862025[Kal13] Tasho Kaletha, Genericity and contragredience in the local Langlands correspondence , AlgebraNumber Theory (2013), no. 10, 2447–2474, DOI 10.2140/ant.2013.7.2447. MR3194648[Kap17] Eyal Kaplan, The double cover of odd general spin groups, small representations, and appli-cations , J. Inst. Math. Jussieu (2017), no. 3, 609–671, DOI 10.1017/S1474748015000250.MR3646283[Kap] , Doubling constructions and tensor product L-functions: coverings of the symplecticgroup , preprint, available at https://arxiv.org/abs/1902.00880.[KP84] D. A. Kazhdan and S. J. Patterson,
Metaplectic forms , Inst. Hautes ´Etudes Sci. Publ. Math. (1984), 35–142. MR743816[Key82] C. David Keys, Reducibility of unramified unitary principal series representations of p -adic groups and class- representations , Math. Ann. (1982), no. 4, 397–402, DOI10.1007/BF01457019. MR670188[Key87] , L -indistinguishability and R -groups for quasisplit groups: unitary groups in even di-mension , Ann. Sci. ´Ecole Norm. Sup. (4) (1987), no. 1, 31–64. MR892141[Kuo02] Wentang Kuo, Principal nilpotent orbits and reducible principal series , Represent. Theory (2002), 127–159, DOI 10.1090/S1088-4165-02-00132-2. MR1915089[Kuo10] , The Langlands correspondence on the generic irreducible constituents of principalseries , Canad. J. Math. (2010), no. 1, 94–108, DOI 10.4153/CJM-2010-006-3. MR2597025[Luo20] Caihua Luo, Knapp-Stein dimension theorem for finite central covering groups , Pacific J. Math. (2020), no. 1, 265–280, DOI 10.2140/pjm.2020.306.265. MR4109915[Mat09] Ivan Mati´c,
Levi subgroups of p -adic Spin(2 n + 1), Math. Commun. (2009), no. 2, 223–233.MR2743171[McN12] Peter J. McNamara, Principal series representations of metaplectic groups over local fields ,Multiple Dirichlet series, L-functions and automorphic forms, Progr. Math., vol. 300,Birkh¨auser/Springer, New York, 2012, pp. 299–327, DOI 10.1007/978-0-8176-8334-413.MR2963537[McN16] ,
The metaplectic Casselman-Shalika formula , Trans. Amer. Math. Soc. (2016),no. 4, 2913–2937, DOI 10.1090/tran/6597. MR3449262[Mez04] Paul Mezo,
Metaplectic tensor products for irreducible representations , Pacific J. Math. (2004), no. 1, 85–96, DOI 10.2140/pjm.2004.215.85. MR2060495[Mis16] Manish Mishra,
Generic representations in L -packets , Int. J. Number Theory (2016), no. 6,1613–1624, DOI 10.1142/S1793042116500986. MR3529884[Mis] , Structure of the unramified L-packet , preprint, available athttps://arxiv.org/abs/1212.1439.[MW87] C. Mœglin and J.-L. Waldspurger,
Mod`eles de Whittaker d´eg´en´er´es pour des groupes p -adiques ,Math. Z. (1987), no. 3, 427–452 (French). MR913667[Nev15] Monica Nevins, Restricting toral supercuspidal representations to the derived group, and appli-cations , J. Pure Appl. Algebra (2015), no. 8, 3337–3354, DOI 10.1016/j.jpaa.2014.10.018.MR3320223[PP15] Shiv Prakash Patel,
A theorem of Mœglin and Waldspurger for covering groups , Pacific J.Math. (2015), no. 1, 225–239. MR3290452[PPP16] Shiv Prakash Patel and Dipendra Prasad,
Multiplicity formula for restriction of represen-tations of g GL ( F ) to g SL ( F ), Proc. Amer. Math. Soc. (2016), no. 2, 903–908, DOI10.1090/proc12721. MR3430864[Rod73] Fran¸cois Rodier, Whittaker models for admissible representations of reductive p -adic splitgroups , Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI,Williams Coll., Williamstown, Mass., 1972), Amer. Math. Soc., Providence, R.I., 1973, pp. 425–430. MR0354942 ESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS 113 [Rod81] ,
D´ecomposition de la s´erie principale des groupes r´eductifs p -adiques , Noncommutativeharmonic analysis and Lie groups (Marseille, 1980), Lecture Notes in Math., vol. 880, Springer,Berlin-New York, 1981, pp. 408–424 (French). MR644842[Sav] Gordan Savin, A nice central extension of GL r , preprint.[Sha81] Freydoon Shahidi, On certain L -functions , Amer. J. Math. (1981), no. 2, 297–355, DOI10.2307/2374219. MR610479[Sha90] , A proof of Langlands’ conjecture on Plancherel measures; complementary series for p -adic groups , Ann. of Math. (2) (1990), no. 2, 273–330, DOI 10.2307/1971524. MR1070599[Sha10] , Eisenstein series and automorphic L -functions , American Mathematical SocietyColloquium Publications, vol. 58, American Mathematical Society, Providence, RI, 2010.MR2683009[Sha11] , Arthur packets and the Ramanujan conjecture , Kyoto J. Math. (2011), no. 1, 1–23,DOI 10.1215/0023608X-2010-018. MR2784745[Sil79] Allan J. Silberger, Isogeny restrictions of irreducible admissible representations are finite directsums of irreducible admissible representations , Proc. Amer. Math. Soc. (1979), no. 2, 263–264, DOI 10.2307/2042302. MR516475[Sol20] Maarten Solleveld, Langlands parameters, functoriality and Hecke algebras , Pacific J. Math. (2020), no. 1, 209–302, DOI 10.2140/pjm.2020.304.209. MR4053201[Szp07] Dani Szpruch,
Uniqueness of Whittaker model for the metaplectic group , Pacific J. Math. (2007), no. 2, 453–469, DOI 10.2140/pjm.2007.232.453. MR2366363[Szp09] ,
Computation of the local coefficients for principal series representations of the meta-plectic double cover of SL ( F ), J. Number Theory (2009), no. 9, 2180–2213, DOI10.1016/j.jnt.2009.01.024. MR2528059[Szp11] , On the existence of a p -adic metaplectic Tate-type ˜ γ -factor , Ramanujan J. (2011),no. 1, 45–53, DOI 10.1007/s11139-010-9277-7. MR2837718[Szp13a] , Some irreducibility theorems of parabolic induction on the metaplectic group via theLanglands-Shahidi method , Israel J. Math. (2013), no. 2, 897–971, DOI 10.1007/s11856-012-0140-y. MR3096578[Szp13b] ,
Some results in the theory of genuine representations of the metaplectic dou-ble cover of
GSp n ( F ) over p-adic fields , J. Algebra (2013), 160–193, DOI10.1016/j.jalgebra.2013.05.001. MR3061683[Szp15] , Symmetric genuine spherical Whittaker functions on
GSp n ( F ), Canad. J. Math. (2015), no. 1, 214–240, DOI 10.4153/CJM-2013-033-5. MR3292701[Szp19] , On Shahidi local coefficients matrix , Manuscripta Math. (2019), no. 1-2, 117–159,DOI 10.1007/s00229-018-1052-x. MR3936136[Tad92] Marko Tadi´c,
Notes on representations of non-Archimedean
SL( n ), Pacific J. Math. (1992), no. 2, 375–396. MR1141803[Tak16] Shuichiro Takeda, Metaplectic tensor products for automorphic representation of g GL ( r ),Canad. J. Math. (2016), no. 1, 179–240, DOI 10.4153/CJM-2014-046-2. MR3442519[Tak17] , Remarks on metaplectic tensor products for covers of GL r , Pacific J. Math. (2017), no. 1, 199–230, DOI 10.2140/pjm.2017.290.199. MR3673084[Tit79] Jacques Tits, Reductive groups over local fields , Automorphic forms, representations and L -functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Proc. Sym-pos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 29–69. MR546588[Var17] Sandeep Varma, On descent and the generic packet conjecture , Forum Math. (2017), no. 1,111–155, DOI 10.1515/forum-2015-0113. MR3592596[Wei09] Martin H. Weissman, Metaplectic tori over local fields , Pacific J. Math. (2009), no. 1,169–200, DOI 10.2140/pjm.2009.241.169. MR2485462[Wei11] ,
Managing metaplectiphobia: covering p -adic groups , Harmonic analysis on reductive, p -adic groups, Contemp. Math., vol. 543, Amer. Math. Soc., Providence, RI, 2011, pp. 237–277,DOI 10.1090/conm/543/10738. MR2798431[Wei14] , Split metaplectic groups and their L-groups , J. Reine Angew. Math. (2014), 89–141, DOI 10.1515/crelle-2012-0111. MR3276164[Wei16a] ,
Covering groups and their integral models , Trans. Amer. Math. Soc. (2016), no. 5,3695–3725, DOI 10.1090/tran/6598. MR3451891[Wei16b] ,
Covers of tori over local and global fields , Amer. J. Math. (2016), no. 6, 1533–1573, DOI 10.1353/ajm.2016.0046. MR3595494
14 FAN GAO, FREYDOON SHAHIDI, AND DANI SZPRUCH [Wei18a] ,
L-groups and parameters for covering groups , Ast´erisque (2018), 33–186 (English,with English and French summaries). L-groups and the Langlands program for covering groups.MR3802418[Wei18b] ,
A comparison of L-groups for covers of split reductive groups , Ast´erisque (2018),277–286 (English, with English and French summaries). L-groups and the Langlands programfor covering groups. MR3802420[Zel80] A. V. Zelevinsky,
Induced representations of reductive p -adic groups. II. On irreducible repre-sentations of GL( n ), Ann. Sci. ´Ecole Norm. Sup. (4) (1980), no. 2, 165–210. MR584084 Fan Gao: School of Mathematical Sciences, Yuquan Campus, Zhejiang University, 38Zheda Road, Hangzhou, China 310027
Email address : [email protected] Freydoon Shahidi: Department of Mathematics, Purdue University, 150 N. UniversityStreet, West Lafayette, IN 47907
Email address : [email protected] Dani Szpruch: Department of Mathematics and Computer Science, Open Universityof Israel, Raanana 43107, Israel
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