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DEPTH-ZERO BASE CHANGE FOR UNRAMIFIED U(2,1) 6 0 JEFFREYD.ADLERANDJOSHUAM.LANSKY 0 2 Abstract. We give an explicit description of L-packets and quadratic base n changefordepth-zerorepresentationsofunramifiedunitarygroupsintwoand a three variables. We show that this base change is compatible with unrefined J minimalK-types. 8 2 ] T R 1. Introduction h. GivenafiniteGaloisextensionE/F offinite,local,orglobalfieldsandareductive t algebraic F-group G, “base change” is, roughly, a (sometimes only conjectural) a mapping from representations of G = G(F) to those of G(E). When F is finite, m or when F is local and G = GL(n), then this mapping is the Shintani lifting (as [ introduced in [25] and extended in [17], [16], and [13] for finite groups). 1 Correspondenceslike base change that areassociatedto the Langlands program v can be difficult to describe explicitly, even in cases where they are known to exist. 5 Bushnell and Henniart [5, 6, 8, 7] are remedying this situation for base change 9 for GL(n) over local fields. Analogously, Silberger and Zink [26] have made the 6 1 AbstractMatching Theorem[11,22,2]explicitfordepth-zerodiscrete seriesrepre- 0 sentations. 6 SupposethatF isap-adicfieldofoddresiduecharacteristic. IfE/F isquadratic, 0 and G is a unitary group in three variables defined with respect to E/F, then / h Rogawski[23] has shown that a base change lifting exists, and has derived some of t itsproperties. Ourgoalinthispaperistodescribebasechangeexplicitlyfordepth- a m zero representations in the case where E/F is unramified. Depth-zero base change is particularly interesting because it should be closely related to base change for : v finitegroups. See[19]foranexplorationofanotherspecialcaseofthisphenomenon. i X In order to apply a technical lemma (Cor. 2.6), we will assume that the order q ofthe residue fieldk ofF isatleast59. Fromthe lemma,characteridentities can r F a be verified by evaluation at “very regular” elements. At such elements, character values are particularly easy to compute. Without the lemma, the verification of these identities involves evaluation at more general elements. Character values at these elements can be computed, but are far more complicated. Note that we assume that F has characteristic zero only so that we can apply resultsofRogawski[23]. Our calculationsapplyequally wellif F is a functionfield of odd residue characteristic. 1991 Mathematics Subject Classification. Primary22E50. Secondary20G05, 20G25. Key words and phrases. p-adicgroup,basechange, Shintanilift,L-packet, Langlands functo- riality,unitarygroup. Thefirst-namedauthorwaspartiallysupportedbytheNationalSecurityAgency(#MDA904- 02-1-0020). 1 2 JEFFREYD.ADLERANDJOSHUAM.LANSKY Conjecturally, one should be able to determine the depth of the representations in an L-packet from the associated Langlands parameter. Thus, liftings that arise from the Langlands correspondence should preserve depth, if depth is normalized correctly. In particular, depth-zero representations should go to depth-zero representations. We assume this throughout for base change and for endoscopic lifting from U(1,1) U(1) to U(2,1). × In 2, we present our notation, review the general notion of Shintani lifting, § describehowitappliestotherepresentationsofcertainfinitereductivesubquotients of G, and list all of the representations of G of depth zero. In 4, we give an § explicit description of the depth-zero L-packets and A-packets for G. In 5, we § determine the base change lift of each of these packets. In 6 we examine the § relationship between base change and K-types, as defined by Bushnell-Kutzko [9] and as described by Moy-Prasad [21] or Morris [20]. Recall that a (minimal) K- type (or simply a “type”) of depth zero is a pair (G ,σ), where G is a parahoric x x subgroupofG,andσ istheinflationtoG ofanirreduciblecuspidalrepresentation x of the finite reductive quotient G of G . Since all of the data in this definition x x can be lifted in a natural way to similar data for G = G(E), we have a natural notion of base change for depth-zero types. Under the above assumption on the residuecharacteristicofthep-adicfieldF,weshowtehatbasechangefordepth-zero types is compatible with base change for representations (actually, A-packets of representations): Theorem 1.1. Suppose Π is a depth-zero A-packet for G, let π˜ denote the basechange lift of Π, and let π Π. Suppose (G ,infl(σ)) is a type contained in π. x ∈ Then π˜ contains (G ,infl(σ˜)), where σ˜ is the base-change lift of σ from G to G . x x x Note that the peair (Gx,infl(σ˜)) contains a type upon restriction to some paera- horicsubgroupofG . Thus,either it is itselfa type, orit carriesmoreinformation x than a type. e In 7, we stateea formula for the character of an induced representation. The § formula itself is not new, but we need to assert that it holds for representations of groups that are not necessarily connected. In order to describe explicit base change for all representations of U(2,1) (not just of depth zero), one needs to understand depth-zero base change not just for U(2,1)but for unitary groupsin two variablesas well. We dealwith this briefly in 3. § We thank Robert Kottwitz, Jonathan Rogawski, A. Raghuram, David Pollack, Stephen DeBacker, and a referee for helpful communications. 2. Preliminaries 2.1. General notation and facts. For any nonarchimedeanlocalfieldF, let F O denote its ring of integers,p the prime ideal in , and k = /p the residue F F F F F O O field. For any abelian extension E/F, let ω denote the character of F× arising E/F via local class field theory. Wewilluseunderlinedletterstodenotealgebraicgroupsandwilldroptheunder- lining to indicate the corresponding groups of rational points. Given an algebraic F-group G and a finite extension E/F, let G = R (G), where R denotes E/F E/F e anliftTa astes.Tble A-packet∗ Base change lift ∗risk().oobtainL-pDepth-ze1. {{{IIψSnnt}ddGBGB((oψλλn)}}e-((dIIinnmddeGBGBnλλsiiiorrnrreeaddl)uucciibbllee aanndd IInnddGBGBeeeeλλ˜˜ irrerdeudcuicbilbel)e) ((((§§§4444....1111)))) IISψñntddGBGP(eeeeψ(cid:16)λ˜˜)(λ1λ˜2|·|E∓1/2◦detGL(2))⊗λ˜2(cid:17) ((((§§§5555....1111)))) DEPTH-ZERO ackroA {π G(λ),π}(λ) (§4.1) IndGGλ˜ (§5.1) BA ets,omitthe-packetsforU {I{(nππds12GG(dλ(yeI)σns,c,dπrσGB2sib(λcλeud)=}b}iinπc1cP(uλrso)pp⊕i.dπ4a.2l4()λ)) ((§§§44..12)) IInnddeBGZGPeeeeeGSeytHσ˜(cid:16)(λ1λ˜2|·|E∓1/2◦detGL(2))⊗λ˜2(cid:17) (Prop(.§§55..26)) SECHANGEFO re( e e R pre2,1 {πn(λ),πs(λ)(∗)} (§4.1) IndGP (λ1λ˜2|·|E∓1/2◦detGL(2))⊗λ˜2 (Prop. 5.6) UN sen),a (πs described in Prop. 4.4) e(cid:16) (cid:17) RA tationsmarkndtheirbase (cid:8) IIIInnnnddddGGGGGGGGyzzziiiinnnnflflflflGGGGGGGGyzzyzzzz((((−−−−RRRRCCGCGGCG′′zzy′zϕϕϕϕ1213⊗⊗⊗⊗ϕϕϕϕ2321⊗⊗⊗⊗ϕϕϕϕ3132)))),,, (Prop. 4.5) IndGZeeeGeyinflGZeeyGey(−RCGeeyϕ˜1⊗ϕ˜2⊗ϕ˜3) (Prop. 5.7) U,MIFIED(21) edch (ϕi distinct) (cid:9) wa itng he 3 4 JEFFREYD.ADLERANDJOSHUAM.LANSKY restrictionofscalars. Similarly,ifGisak -group,GwilldenoteR (G). When- F kE/kF ever we use this notation, the extension E/F will either be specifed, or it will be understood from the context. e For every nonarchimedean local field F and every reductive algebraic F-group G, one has an associated extended affine building (G,F), as defined by Bruhat B and Tits [3, 4]. As a G-set, (G,F) is a direct product of an affine space (on B which G acts via translation) and the reduced building red(G,F), which depends B only on G/Z, where Z is the center of G. Note that Z fixes red(G,F). For any B extensionE/F offiniteresiduedegree, (G,F)alwayshasanaturalembeddinginto B (G,F) = (G,E). To every point x (G,F), there is an associated parahoric B B ∈ B subgroupG ofG. ThestabilizerofxinGcontainsG withfinite index. The pro- x x p-readical of G is denoted G , and the quotient G /G is the group of rational x x+ x x+ points of a connected reductive k -group G . These objects depend only on the F x image of x in red(G,F). Thus, in the case of a torus T, we may write T , T , 0 0+ and T insteadBof T , T , and T , since these do not depend on the choice of x. x x+ x More generally, G will denote the set of topologically unipotent elements in G. 0+ We now present an elementary fact about the building that we will use several times throughout this paper. Lemma 2.1. Let Z denote the center of G, and let y,z (G,F) have distinct ∈ B images in red(G,F). Suppose G is a maximal parahoric subgroup, γ G , and y y the image γB¯ of γ in G is regular elliptic (i.e., γ¯ belongs to no proper k∈-parabolic y F subgroup of G ). Then γ ZG . y 6∈ z Proof. First, suppose γ ZG rG . From [10, Lemma 4.2.1], γ does not fix z. z z ∈ Therefore,γ mustactonsomelinecontainingz viaanontrivialtranslation. By[10, Cor. 3.1.5], γ cannot fix y, a contradiction. Now suppose γ G . Then γ G for all x lying on the geodesic between y z x ∈ ∈ and z. For such an x that is close to but not equal to y, G is a subgroup of G , x y and the image of G in G is the group of k -fixed points of a proper parabolic x y F subgroup. Thus γ G , a contradiction, and the lemma follows. (cid:3) x 6∈ If G is a connected reductive group over a finite field, T is a maximal torus in G, and θ is a (complex) character of T, then let RGθ denote the corresponding T Deligne-Lusztig virtual character of G [12]. For any reductive algebraic group G defined over a local or finite field, we have the following notation. 1 will denote the trivial representation of G. G • St will denote the Steinberg representation of G. G • For any character ψ of G, St (ψ) will denote St ψ. G G • · For any representation σ of a subgroup H of G, indGσ will denote the • H representation of G obtained from σ via normalized compact induction. If Z is the center of G and ω is a character of Z, then let C(G,ω) denote • the space of complex-valued, locally constant functions f on G such that the support of f is compactmodulo Z, and f(gz)=f(g)ω(z) for all g G ∈ and z Z. ∈ Greg denotes the set of regular semisimple elements of G. • For any admissible, finite-length representation π of G, let θ denote the π • character of π, considered either as a function on the set of elements or DEPTH-ZERO BASE CHANGE FOR UNRAMIFIED U(2,1) 5 conjugacy classes of G (of Greg in the local-field case), or as a distribution on an appropriate function space on G. Suppose ε is an automorphism of G. Then ε acts in a natural way on • the set of equivalence classes of irreducible, admissible representations of G. Suppose π is such a representation and π = πε. Let π(ε) denote an ∼ intertwining operator from π to πε. If ε has order ℓ, then we can and will normalize π(ε) by requiring that the scalar π(ε)ℓ equal 1. Then π(ε) is well determined up to a scalar ℓth root of unity. The ε-twisted character of π is the distribution θ defined by θ (f) = trace(π(f)π(ε)) for f π,ε π,ε ∈ C∞(G). As with the character, the twisted character can be represented c by a function (again denoted θ ) on G (Greg in the local-field case). We π,ε may regard θ as a function on the set of ε-twisted conjugacy classes. π,ε Note that θ still makes sense when π is an admissible, finite-length π,ε representation. For any maximal torus T of G, let W(T,G) denote the quotient of T in • its normalizer in G, and let W (T,G) denote the group of F-points of the F absolute Weyl group N (T)/T. G 2.2. Shintani lifting. Suppose that E/F is a finite, cyclic extension of local or finite fields, Γ=Gal(E/F), andGis aconnectedreductive algebraicF-group. Let ε denote a generator of Γ, and let ℓ denote the order of Γ. Then one can define a norm mapping from G to G by x x ε(x) εℓ−1(x). e e 7→ · · ··· · IfxisdefinedoverF then,ingeneral,themostthatonecansayabouttheimageof xisthatitsconjugacyclassinGisdefinedoverF. IfF islocalandGhasasimply connected derived group, then such a conjugacy class must have F-points [23]. Thus, an F-point x G determines a stable conjugacy class in G. Any stable, ∈ ε-twisted conjugate of x determines the same stable conjugacy class in G. Thus, we have a map G freom the set of stable, ε-twisted conjugacy classes of G to NE/F the set of stable conjugacy classes in G. If x commutes with its Galois conjugates, then we may and will define G (x) G via the formula above. e NE/F ∈ Call g G ε-regular if (g) is regular. Let Gε-reg denote the set of ε-regular ∈ N elements. IfΠandΠearefinitesetsofrepresentationsofGeandG,respectively,wesaythat Π is the Shintani lift (or base change) of Π if e e Θ (g)=Θ ( (g)) e Π,ε Π N for all g G (all g Gε-reg in tehe local-field case), where Θ and Θ are non- ∈ ∈ Π Π,ε trivial stable (resp. ε-stable) linear combinations of the characters (resp. ε-twisted characters)eof the elemeents of Π (resp. Π). e If T is an F-torus then for any character λ of T, define the character λ˜ of T by λ˜ =λ T . e ◦NE/F e 2.3. Notation related to unitary groups. From now on, fix a nonarchimedean local field F of characteristic zero with finite residue field k of odd order q. Let F E be the unramifiedquadraticextensionofF. LetE1 (resp.k1) denote the kernel E 6 JEFFREYD.ADLERANDJOSHUAM.LANSKY of the norm from E to F (resp. k to k ). Let ε denote the nontrivial element of E F the Galois group Γ=Gal(E/F). Let G denote a unitary group in three variables defined with respect to E/F. Then G is uniquely determined up to isomorphism, and we can and will assume that G is the unitary group defined by the Hermitian matrix 0 0 1 Φ= 0−1 0 . 1 0 0 (cid:16) (cid:17) Then G=G(F)= g GL(3,E):gΦ t(ε(g))=Φ ∈ andG=GL(3,E). LetG denot(cid:8)e the correspondingalgebraicgr(cid:9)oupoverk . Then F G=GL(3,k ). E Leet Z denote the center of G. So, following our notationalconventions,Z is the ceenter of G. Let = (G,F) and = (G,F) = (G,E). Note that ε acts on ,eand we B B B B B B may and weill identify the set of fixed points ε with . B B Since G is F-quasisplite, it conetains F-Borel subgroups. In particulare, must B contain some ε-invariant apartment with meore than one ε-fixed point. Choose A anε-fixedpointy inanε-invariantminimalfacetin ,andanε-invariantalecove A F in , suchthat the closureof contaeinsy. (Let denote the set of ε-fixedpoints A F F of .) Then these choices determine an F-Borel subegroup B together with a Levei F facteor M of B. Note that M ies isomorphic to E× E1. We may assume that our × choeices of y and allow us to realize B explicitly as the groupof upper triangular F matrices in G, and M as the group of diagonal matrices. The boundaryeof contains two points: the previously chosen point y, and F another point that we will denote z. Note that is the direct product of a one- F dimensional affine space and an ε-invariant equilateral triangle ∆ in the reduced building ofG (whichwe will identify with a subseteof ), y the ε-fixedvertexof∆, B and z is the midpoint of the wall of ∆ that is opposite y. In , y and z are both B vertices, buteonly y is hyperspecial. e ] Consider the map λ: U(1) G given by t diag(1,t,1). Since U(1)=GL(1), → 7→ ∼ we actually havea one-parametersubgroupof G. In the usualway,λ determines a parabolicF-subgroup P =P of G, together with a Levi decomposition of P. Let λ H denotethecorrespondingLevifactor. ThenHe isthegroupofinvertiblematrices of the form e e e e e ∗0∗ e 0∗0 . ∗0∗ (cid:16) (cid:17) This subgroup arises via restriction of scalars from a subgroup H of G. Note that H is an E-Levi, but not F-Levi, subgroup of G. It is an endoscopic group for G, isomorphic to U(1,1) U(1). × Similarly, we can define a subgroup H of G and a parabolic k -subgroup P of G F with Levi factor H. Note that G =G and G =H. y ∼ z ∼ Uptoconjugacy,H containstwoF-torithatareisomorphictoU(1) U(1) eU(1)e. × × ThegroupofF-peointsofoneofthesetorifixesahyperspecialvertex,andthegroup of F-points of the other fixes a non-hyperspecial vertex. Pick such a torus whose F-points fix y (resp. z) and call it C (resp. C′). Given the right choices, we can DEPTH-ZERO BASE CHANGE FOR UNRAMIFIED U(2,1) 7 and will realize C as the set of matrices of the form γ1+γ3 0 γ1−γ3 2 2 γ = 0 γ 0 2   γ1−γ3 0 γ1+γ3 2 2   whereγ U(1). WedefinethetorusC Gsimilarly. WeidentifyC (andsimilarly i C) with U∈(1) U(1) U(1) via the m⊂ap γ (γ ,γ ,γ ). We will realize C′ as 1 2 3 × × 7→ νCν−1, where 1/√̟ 0 0 F ν = 0 1 0 .   0 0 √̟ F   (The ambiguity in the choice of square root of ̟ has no effect.) F Let B (resp. B ) denote the Borel subgroup of G (resp. G ) determined by . y z y z F ForanyF-groupL,let L = L . WhenL=G,wesimplywrite . Similarly, N NE/F N for any k -group L, let L = L . When L=G, we simply write ¯. F N NkE/kF N For any subgroup S G, let det denote the restriction of the determinant to S ⊂ S. We will omit the subscript when it is clear from the context. Similar notation holds for subgroups of G. e e 2.4. CartansubgroupsofG. ForaquadraticextensionL/K,denotebyU(1,L/K) theunitarygroupinonevariableoverK definedwithrespecttoL/K. Uptostable conjugacy, there are four kinds of Cartan subgroup of G. In the notation of [23], they are isomorphic to: (2.4–0) R (GL(1)) U(1,E/F), E/F × (2.4–1) U(1,E/F) U(1,E/F) U(1,E/F), × × (2.4–2) R (U(1,EK/K)) U(1,E/F) for K a ramified quadratic extension of E/F × F, (2.4–3) R (U(1,EL/L)) for L a cubic extension of F. L/F 2.5. RepresentationtheoryofG,H,andC. Areferenceformuchofthissection is [27]. Representations of G. LetB denote a BorelsubgroupofG with Levi factor M, and let θ be a character of M. Then the induced representation indGθ is irreducible B exceptwhenθ extends toa characterofH. Inthis case,the inducedrepresentation is a sum of two irreducible components. If θ extends to a character θ of G, then 0 these components are θ0 and StG(θ0). Let L denote a cubic unramified extension of E. Then G contains a torus S that is isomorphic to the kernel of the norm map from k to k . Let T be either EL L S or C. For any character θ of T with trivial stabilizer in W (T,G), we have a kF Deligne-Lusztig cuspidal representation whose character is RGθ. For T = S, we T − will call such representations “cubic cuspidal representations.” The other irreducible representations of G have the form τ ψ, where τ is the · cuspidal unipotent representation and ψ is a character. 8 JEFFREYD.ADLERANDJOSHUAM.LANSKY Representations of H. As above let B be a Borelsubgroup of G with Levi factor M andletθ beacharacterofM. TheinducedrepresentationindH θisirreducibleex- B∩H ceptwhenθ extendstoacharacterθ ofH. Inthiscase,theinducedrepresentation 1 is the sum of θ1 and StH(θ1). The remaining representations of H are the Deligne-Lusztig cuspidal representations, whose characters are of the form RGθ for θ Hom(C,C×) in general C position with respect to the action of W (C−,H). ∈ kF Representations of C. We will need a technical result on linear combinations of characters of C. Let A be a finite abelian group of order n, and let χ ,...,χ be 1 n the irreducible characters of A. The following three lemmas concern characters of products of copies of A. Lemma 2.2. Let n f = a χ , i i i=1 X where a C. Suppose that f vanishes off of a subset of A of size 2. Then either i ∈ f =0, or the number of i such that a =0 is at least n/2. i 6 Proof. Let a,b be the above subset of A. We have { } na =n f,χ =f(a)χ¯ (a)+f(b)χ¯ (b). i i i i ·h i Assume f = 0. If f(b)= 0, then for all i, a = f(a)χ¯ (a)/n = 0. If f(b)= 0, then i i 6 6 6 a = 0 unless χ¯ (ba−1) = f(a)/f(b). Since ba−1 = 1, this equality holds for at i i mo6st n/2 values of i. − 6 (cid:3) Lemma 2.3. Let N be the subset of A A consisting of all elements (a,b) such that a=b. Suppose that for some a C×, ij 6 ∈ f = a χ χ ij i j ⊗ i,j X vanishes on N. Then either f =0 or at least n of the a are nonzero. ij Proof. Assume f = 0. Fix a A. Evaluating f at (a,b) for b = a, we obtain that 6 ∈ 6 the function a χ (a) χ ij i j ! j i X X onA vanishes on A a . It follows easily that either this function vanishes on A, −{ } or for all j, the coefficient a χ (a) is nonzero. The former case cannot happen i ij i since f = 0. In the latter case, it follows that for all j, at least one coefficient a ij must be6 nonzero. Hence atPleast n of the a must be nonzero. (cid:3) ij Lemma 2.4. Let N′ be the subset of A A A consisting of all elements (a,b,c) such that a, b, and c are distinct. Suppo×se th×at for some a C, ijk ∈ f = a χ χ χ ijk i j k ⊗ ⊗ i,j,k X vanishes on N′. Then either f vanishes on A A A or at least n/2 of the a ijk × × are nonzero. DEPTH-ZERO BASE CHANGE FOR UNRAMIFIED U(2,1) 9 Proof. Assume f =0. Fix a=b in A. Then the function 6 6 a χ (a)χ (b) χ ijk i j k k (cid:18)i,j (cid:19) X X on A vanishes off of a,b . Hence, by Lemma 2.2, either this function vanishes on { } A, or for at least n/2 values of k, the coefficient a χ (a)χ (b) is nonzero. In i,j ijk i j thelattercase,foreachsuchk,atleastonecoefficienta mustbenonzero. Hence ijk P at least n/2 of the a are nonzero. ijk We may therefore assume that the former case holds for all pairs a=b. By the 6 linear independence of characters, the coefficient a χ (a)χ (b) must vanish i,j ijk i j for all k and all pairs a = b. Since f = 0, ai′j′k′ = 0 for some i′,j′,k′. Thus the 6 6 6P function i,jaijk′χi ⊗χj on A×A vanishes on the set N of Lemma 2.3, but it does not vanish on A A since ai′j′k′ =0. Hence Lemma 2.3 implies that at least n of the cPoefficients ai×jk′ must be nonz6ero. (cid:3) Corollary 2.5. Suppose that a χ χ χ HoXm(C,C×) ∈ vanishes on C Greg, where a C. Then either this linear combination vanishes χ on C or at leas∩t (q+1)/2 of the∈a are nonzero. (cid:3) χ Corollary 2.6. Suppose that q > 59, and let f = a χ be a linear combination χ of at most 30 characters of C. If f vanishes on C Greg, then f vanishes on C. (cid:3) ∩P 2.6. Shintani lifting for G and H. According to [27], the irreducible characters ofGareoftheform RGθ,whereListhe connectedcentralizerofsomesemisimple L element of G, and θ ±is the twist of a unipotent characterof L by a one-dimensional character in general position. Moreover, one obtains a cuspidal character of G precisely when L is an elliptic torus or when L = G and θ is a twist of the unique cuspidal unipotent character of G. By [17], our assumption that k has odd characteristicguaranteesthe existence F of Shintani descent from G to G. In [14], Digne gives a general proof that Shintani descent is compatible with Deligne-Lusztig induction. In particular, if σ is an irreducible representationeof G with character RLGθ (θ a character of L), then the ± character of the Shintani lift σ˜ of σ from G to G is of the form RGθ˜, where θ˜is ± L the Shintani lift of θ. Now L is a Levi factor of a parabolic subgrouep of G unless L is isomorphic to the torus S defined in 2.5. Heence σ˜ is a paraboleically induced § representation unless L = Seor L = G. In the former case, S is an ellipteic torus ∼ isomorphic to k× and σ˜ is cuspidal. In the latter case, σ is a one-dimensional EL representation ϕ detG, a twist StG(ϕ detG) of the Steinbergerepresentation, or a ◦ ◦ twist τ(ϕ detG) of the cuspidal unipotent representation. One shows easily that ◦ the Shintani lift σ˜ is, respectively, ϕ˜ det , St (ϕ˜ det ), or τ˜(ϕ˜ det ), where τ˜ ◦ G G ◦ G ◦ G is the unipotent representation of G not equivalent to 1 or St . The remaining e e eG G e representations of G are those whose characters are of the form RGθ. By [14], the H ShintaniliftsofsuchrepresentationsearerepresentationsineducedferomP. Hencethe cubic cuspidal representations of G are exactly those irreducible representations of G whose Shintani lifts are cuspidal. e 10 JEFFREYD.ADLERANDJOSHUAM.LANSKY We nowconsiderShintaniliftingforirreduciblerepresentationsofH. From 2.5, mostsuchrepresentationshavecharactersofthe form RHθ. FromDigne [14]§, the T ± Shintani lift of such a representation has character RHθ. ± T The remaining representations of H are the one-dimensionalrepresentations ϕ e ◦ detH and the Steinberg representations StH(ϕ detH).eeIt is easy to see that the ◦ respective Shintani lifts of these representations are ϕ detH and StH(ϕ detH). ◦ ◦ e e 2.7. Depth-zero representations of G. e e Principal series of G. For λ Hom(M,C×), there exist unique characters λ 1 Hom(E×,C×) and λ Hom(∈E1,C×) such that ∈ 2 ∈ α 0 0 (2.7.1) λ 0 β 0 =λ1(α)λ2(αα¯−1β), (cid:18)(cid:18)0 0 α¯−1(cid:19)(cid:19) where α E×, β E1. By [18], indGλ is irreducible except for in the following ∈ ∈ B cases: (2.7PS–1) λ = ±1 1 |·|E (2.7PS–2) λ1|F× =ωE/F |·|±F1 (2.7PS–3) λ1 is nontrivial and λ1 F× is trivial. | In case (2.7PS–1), indGλ has two constituents: the one-dimensional representa- B tion ψ =λ det, and the square-integrableSteinberg representation St (ψ). 2 G ◦ In case (2.7PS–2), indGλ also has two constituents: a square-integrable repre- B sentation π2(λ) and a non-tempered unitary representation πn(λ). In case (2.7PS–3), indGλ decomposes into a direct sum π (λ) π (λ). B 1 ⊕ 2 By [21], indGλ has depth zero if and only if λ has depth zero. B Other representations of G. Since G has no non-minimal proper parabolic subgroups, the remaining irreducible representations are all supercuspidal. From either [21] or [20], we know that all suchrepresentations havea unique expressionof the form indG σ, where x = y or z, and σ is the inflation to G of an irreducible cuspidalreprGesxentationσ ofG . The representationsσ areclassixfiedin 2.5. Based x § on this classification, we have the following kinds of supercuspidal representation of depth zero. (2.7SC–1) indG σ, where σ¯ is a cubic cuspidal representation of G =G. Gy y ∼ (2.7SC–2) indG σ,whereσ¯isacuspidalrepresentationofG withcharacter RGyϕ Gy y − C and ϕ = ϕ ϕ ϕ is a regular character of C (with respect to 1 2 3 W (C,G )). ⊗ ⊗ kF y (2.7SC–3) indG σ, where σ¯ is the twist τ (η det) of the cuspidal unipotent rep- Gy · ◦ resentation τ of G , and η Hom(k1,C×). y ∈ E (2.7SC–4) indG σ, where σ¯ is a cuspidal representation of G = H with character Gz z ∼ RGzϕ and ϕ=ϕ ϕ ϕ is a regular character of C′ (with respect C′ 1 2 3 − ⊗ ⊗ to W (C′,G )). Recall that, according to our notational conventions, kF z C′ is the finite reductive quotient of the (unique) parahoricsubgroup of C′.