COMPACT SUPPORT COHOMOLOGY OF PICARD MODULAR SURFACES 6 1 JUKKAKERANEN 0 2 Abstract. We compute the cohomology with compact supports of a Picard n modularsurfaceasavirtualmoduleovertheproductoftheappropriateGalois a group and the appropriate Hecke algebra. We use the method developed by J Ihara, Langlands, and Kottwitz: comparison of the Grothendieck-Lefschetz 4 formula and the Arthur-Selberg trace formula. Our implementation of this methodtakes asitsstartingpointtheworksofLaumonandMorel. ] T N . h t Contents a m 1. Introduction 2 [ 1.1. Goal of the Project 2 1 1.2. Related work 2 v 1.3. Structure of the Argument 3 6 2. Stable Point Counting Formula 5 7 2.1. Shimura Varieties and Integral Models 5 5 0 2.2. Kottwitz’s Stable Point Counting Formula 5 0 2.3. First Major Transition 9 . 3. Geometric Side of the Trace Formula 10 1 0 3.1. Definitions 10 6 3.2. Second Major Transition 11 1 4. Spectral Side of the Trace Formula 11 : v 4.1. Definitions 12 i 4.2. Simplifications on the Spectral Side 16 X 5. Computations at Infinity 25 r a 5.1. Strategy For Computing the d’s 25 5.2. Character Identities 26 5.3. Explicit Computations 31 6. Computations at p 34 7. Stabilization of Parabolic Terms 39 7.1. Conclusion 43 References 44 2010 Mathematics Subject Classification. Primary11R39. 1 2 JUKKAKERANEN 1. Introduction 1.1. Goal of the Project. Let E be an imaginary quadratic extension of Q. We define an algebraic group GU(2,1) over Q by setting GU(2,1)(A)= g GL (E A)tg¯Jg =c(g)J,c(g) A× 3 Q { ∈ ⊗ | ∈ } for every Q-algebra A, where 0 0 1   J = 0 1 0 GL (Z). 3 ∈  1 0 0  Then G=GU(2,1) is an algebraic group that is quasi-split over Q. Let K be a compact open subgroup of G(A ), and let S (G) be the Shimura f K variety of GU(2,1) at level K. We will assume K to be neat, so that S (G) K is a smooth quasi-projective variety over its reflex field, which in this case is the imaginary quadratic field E. Any such S (G) is called a Picard modular surface. K Fix a good prime p. Assume, that is, that p is not ramified in E, and that K p is hyperspecial, so that K =G(Z ). Let p be a prime of E dividing p. p p Definition 1.1. The Hasse-Weil L-function at p of S (G) is defined by K logLp(SK(G),s)=Pm>1 Nmm(q−s)m whereO /p=F ands C,andN the numberofrationalpointsonthe special Ep q ∈ m fiber at p of S (G) over the extension of degree m of F . K q The goal of this paper is to compute the parabolic part of the trace of certain correspondences on the cohomology with compact supports of S (G) in terms of K spectral traces. In an upcoming paper [13], we will carry out the (much simpler) computationoftheellipticpartofthesaidtrace. Puttogether,thesecomputations will allow us to give a spectral automorphic expression for the number N of m rational points. Hence, we will be able to express the Hasse-Weil L-function at p of a Picard modular surface in terms of automorphic L-functions. The distinctive challenge facing us is that the Picard modular surface S (G) K is not compact. Thus, in order to compute N , we shall have to compute the m number of fixed points N(j,fp) of a suitable correspondence on the cohomology with compact supports of S (G). Our basic tool will be the non-invarianttwisted K trace formula. 1.2. Related work. The study ofL-functionsofalgebraicvarietiesis avastdisci- pline. The previous works most closely related to that of ours are due to Laumon and Morel. (1) In 1997, Laumon computed the cohomology with compact supports for the group GSp(4); see [16]. Our work is adapted from that of Laumon’s. (2) In 2010, Morel computed the intersection cohomology of the Baily-Borel compactificationforGU(p,q)forarbitrarypandq;see[18]. WewillfollowMorelin ouroverallsetupandnotation. HerworkgeneralizesthehighlyinfluentialMontreal proceedings [17] from 1992 in which the case of GU(2,1) was worked out for the firsttime.1 Weshouldemphasizethatthereisnoobviouswaytodeduceourresults from the corresponding results for the Baily-Borelcompactification. 1WewouldliketothankPaulGunnellsforremindingustoincludethisreference. COMPACT SUPPORT COHOMOLOGY OF PICARD MODULAR SURFACES 3 1.3. Structure oftheArgument. Inthissubsection,wewillgiveabriefsynopsis ofourargument. Intheinterestofbrevity,someofthestandardnotationsemployed here will only be defined in the subsequent sections. 1. Stable Point Counting Formula Let G= GU(2,1) and let H =G(U(1) U(1,1)), the only non-trivial elliptic × endoscopic group of G. In this situation, Kottwitz’s stable point counting formula specializes to give N(j,fp)=STG(fG)+ 1STH(fH) e 2 e G H for suitable test functions f and f . Our basic project is to give a sequence of different expressions for this quantity, finally to express it in terms of spectral traces. It will be necessary to consider the following twisted sets and twisted groups: G=R G ⋊θ R G ⋊ θ =G˜, E/Q E E/Q E ⊂ h i H =R H ⋊θ R H ⋊ θ =H˜, E/Q E E/Q E ⊂ h i whereθ is anautomorphismonGandHinducedby the non-trivialelementofthe Galois group Gal(E/Q). We have the following equalities: STG(fG)=k TG(φG) e G e STH(fH)=k TH(φH) e H e where the test functions φG and φH are associated to the test functions fG and H f in a sense to be recalled, and k and k are constants to be determined. The G H G H firststep in the computation will be to replace the distributions ST and ST on e e G and H, respectively, by the distributions TG and TH on G and H, respectively. e e 2. Geometric Side of the Trace Formula Under suitable assumptions on the test functions φG and φH, we have TG(φG)=TG (φG) e geom and TH(φH)=TH (φH), e geom where TG and TH are the geometric sides of the non-invariant twisted trace geom geom formula for G and H, respectively, and the test functions φG and φH are again G H associatedwith the testfunctions f andf ,respectively. The secondstepinour computationis to replacethe distributions TG and TH with the geometric sides of e e the trace formulas for G and H, respectively. The point of the first step is that it makes this second step available to us. 3. Spectral Side of the Trace Formula Sofar,uptocertaincoefficientsweshallignoreinthisintroduction,wewillhave rewritten the stable point counting formula as N(j,fp)=TG (φG)+ 1TH (φH). geom 2 geom The third main step in our computation is to apply the trace formula individually to each term above, so as to arrive at N(j,fp)=TG (φG)+ 1TH (φH). spec 2 spec 4 JUKKAKERANEN We can break up each term in this expression according to Levi subsets: N(j,fp)=JG(φG)+JG(φG)+ 1JH(φH)+ 1JH(φH), G T 2 H 2 T whereT is the diagonalLevisubsetinGandH. The restofthe argumentconsists of rewriting the distributions J in successively more explicit forms. 4. Stabilization of the Parabolic Part of the Trace Once written out explicitly, we will compare the parabolic terms JG(φG) and T JH(φH). The expressions we will be working with are fairly complicated, and our T projectdependscruciallyonfindingcancellationsbetweencertaintermsinJG(φG) T and the corresponding terms in JH(φH). After a lengthy computation, we are left T witharelativelysimpleexpression,whichwewillidentifyasthetraceonasuitable virtual module for Gal(E¯/E) C∞(G(A )//K). This identification will allow us × c f to remove the assumptions on the test functions we had placed earlier. This is the core result of our computation. 5. The Elliptic Part of the Trace We will rewrite the elliptic terms JG(φG) and JH(φH) as traces on a suitable G H module for Gal(E¯/E) C∞(G(A )//K). This is the most routine part of our × c f computation. 6. The L-Function By using the expressions from steps 4. and 5., we will be able to express the L-function of G in terms of automorphic L-functions associated to G and H. It is perhaps worth emphasizing that the final result will, therefore, express the L- function ofthe Shimura varietyof a certainunitary groupin terms of automorphic L-functions of certain general linear groups. This is a fundamental feature of the suite of techniques employed here.2 We will carry out steps 5. and 6. in a forthcoming paper [13]. Since the choice of the test functions plays a large role in the actual execution of the argument outlined above, we pause here to explain the basic idea. First, the choice of the test functions on G is, of course, dictated by the trace we have resolved to compute in the first place, and the choice of the test functions on H is thendictatedbytherequirementthattheappropriateinstancesoftheFundamental Lemma should hold and, in particular, that the Kottwitz point-counting formula shouldhold. Inaddition,Laumon’stechnique requiresthatcertainassumptionsbe made about these functions, particularly at infinity. Second, the test functions on G and H are chosen so that we can pass from the distributions ST on the unitary groups G and H to distributions T on the linear groups G˜ and H˜, respectively. Again,inthisnewcontext,wecanandwillassumethattheadditionalassumptions hold. We willthenadaptthe bulk ofLaumon’smethodnotforthe originalunitary groups but rather for the linear groups. Finally, we can discharge the additional assumptions on the test functions on G˜ and H˜ in the same way as Laumon does and hence, finally, we can discharge the corresponding assumptions on the original groups G and H. 2Inasimilarfashion,MorelexpressestheL-functionoftheintersectioncomplexontheBaily- BorelcompactificationofaunitarygroupShimuravarietyintermsofautomorphicL-functionsof generallineargroups. See[18,p.139]. COMPACT SUPPORT COHOMOLOGY OF PICARD MODULAR SURFACES 5 2. Stable Point Counting Formula 2.1. Shimura Varieties and Integral Models. In this paper, we will follow all theassumptionsregardingShimuravarietiesadoptedbyMorelin[18]. Inparticular, the integral models of our Shimura varieties will be supplied by the work of Lan [15]. Wewillrecallthebasicdefinitionsandnotationsfromchapter1of[18]below; for more details, see [18, pp.1-6]. Let S=R G . Identify S(C)=(C C)× by using the morphism C/R m,C R ⊗ a 1+b i (a+ib,a ib), ⊗ ⊗ 7→ − and write µ :G S for the morphism z (z,1). 0 m,C C → 7→ Following Morel, the definition of pure Shimura data that will be used here is that of [21] (3.1), up to condition (3.1.4). So a pure Shimura datum is a triple (G, ,h)whereGisaconnectedreductivelinearalgebraicgroupoverQ, isaset X X with a transitive action of G(R), and h : Hom(S,G ) is a G(R)-equivarient R X → morphism, satisfying conditions (3.1.1), (3.1.2), (3.1.3), and (3.1.5) of [21], but not necessarily the condition (3.1.4) (the group Gad may have a simple factor of compact type defined over Q). Let (G, ,h) be a Shimura datum. The field of definition F of the conjugacy X class of cocharacters h µ : G G , x , is called the reflex field of the x 0 m,C C ◦ → ∈ X datum. IfK isa compactopensubgroupofG(A ), thereis anassociatedShimura f variety MK(G, ), which is a quasi-projective variety over F satisfying X MK(G, )(C)=G(Q) ( G(A )/K). f X \ X × If moreover K is neat (see [20, 0.6]), then MK(G, ) is smooth over F. X Suppose that we are given a Shimura datum as above, and that, in addition, all the assumptions in Morel’s chapter 1, [18], are in force. Then the PEL moduli schemesfromtheworkofLan[16]providesuitableintegralmodelsforourShimura varieties; see Morel [18, p. 9]. Let S denote the moduli scheme provided by Lan, K and S (G) the generic fiber of S . K K 2.2. Kottwitz’s Stable Point Counting Formula. We now assume that G is one of the groups of unitary similitudes considered by Morel in chapter 2 of [18]; concretely, we will only need the results of this section for G = GU(2,1). Let E be the imaginary quadratic field that is the reflex field of the Shimura datum associated to G. Fix a compact open subgroup K of G(A ) and suppose that f K =K K =G(Zˆ), N max ⊂ where K = Ker(G(Zˆ)։G(Zˆ/NZˆ)), N for any integer N >3. Fix a prime p that is good with respect to K, namely K =KpK p where Kp G(Ap), ⊂ f 6 JUKKAKERANEN and K =G(Z ) G(Q ), p p p ⊂ and such that p∤N, where N is as above. Fix an algebraic closure Q¯ of Q and an embedding of Q in Q¯. Fix an algebraic closure Q¯ of Q and an embedding of Q in Q¯ . Also fix an embedding of Q¯ in p p p p Q¯ . Let F¯ be the residue field of the integral closure of Z in Q¯ ; then F¯ is an p p p p p algebraic closure of F , which is the one we fix. p Fix an algebraic closure E¯ of E and an embedding of E in E¯. For any prime ℓ, we will denote the ℓ-adic cohomology with compact supports on S (G) by K Hi(S (G) E¯,Q ). c K ⊗E ℓ We will denote the convolution algebra of compactly supported, K-bi-invariant functions f :G(A ) C by f → C (G(A )//K), c f and the Q-subspace consisting of the Q-valued functions in C (G(A )//K) by c f C (G(A )//K) . c f Q There is a continuous action of Gal(E¯/E) on Hi(S (G) E¯,Q ), and also an c K ⊗E l action of C (G(A )//K) , and the two actions commute. c f Q For any prime ℓ such that p = ℓ, we will consider the virtual Gal(E¯/E) 6 × C (G(A )//K) -module c f W = ( 1)iHi(S (G) E¯,Q ). ℓ Pi>0 − c K ⊗E ℓ Fix a prime p of E dividing p. Fix an algebraic closure E¯ of E and an em- p p bedding of E in E¯ . Also fix an embedding of E¯ in E¯ . Let O /p=F , and let p p p Ep q F¯ be the residue field of the integralclosure of O in E¯ ; then F¯ is an algebraic q Ep p q closure of F , which is the one we fix. These choices, along with the choices made q earlier, determine unique homomorphisms Gal(E¯/E) ֓ Gal(E¯ /E )։ Gal(F¯ /F ). p p q q ← For each i, we have Gal(F¯ /F ) C (G(A )//K) -equivariant isomorphisms q q c f Q × Hci(SK(G)⊗E E¯,Qℓ)∼=Hci(SK(G)⊗E E¯p,Qℓ) =Hi(S (G) F¯ ,Q ). ∼ c K ⊗OEp q ℓ We shall only have to consider the case where p splits in E, since this will be enoughto determine the L-function. Forsucha p, let Frob denote the topological p generator of Gal(F¯ /F ) that is given by q q α α1/p. 7→ Let Φ be an arbitrary,fixed lift of Frob to Gal(E¯/E). p p By the above, we have tr(Φj fp1 ,W )=tr(Frobj fp,RΓ (S F¯ ,Q )) p× Kp ℓ p× c K ⊗OEp q ℓ for every integer j >0 and every function fp C (G(A )//K) , where 1 is the ∈ c f Q Kp characteristic function of K in G(Q ); see [16, p. 271]. p p COMPACT SUPPORT COHOMOLOGY OF PICARD MODULAR SURFACES 7 2.2.1. Number of Fixed Points. If fp is the characteristic function of the double coset KpgKp in G(Ap), with g G(Ap), and if f ∈ f pj >[Kp :Kp gKpg−1], ∩ thenthefixedpointsofthecorrespondenceFrobj fpontheF -schemeS F¯ p× q K⊗OEp q are isolated; see Zink [23]. We will write N(j,fp) for the number of fixed points counted with multiplicity. By Q-linearity, we define N(j,fp) for any fp C (G(A )//K) and j sufficiently large. c f Q ∈ The following theorem of Pink’s is still often called Deligne’s Conjecture. Theorem 2.1. For every function fp C (G(Ap)//Kp) , there exists an integer ∈ c f Q j(fp) > 0 with the following property. For every integer j > j(fp), the number of fixed points N(j,fp) of the correspondence Frobj fp is well-defined, and we have p× N(j,fp)=tr(Frobj fp,RΓ (S F¯ ,Q )). p× c K ⊗OEp q ℓ Proof. This is a direct consequence of theorem 7.2.2 of [19]. (cid:3) Corollary 2.2. For every function fp C (G(Ap)//Kp) and every integer j > ∈ c f Q j(fp), we have N(j,fp)=tr(Φj fp1 ,W ). p× Kp ℓ Thatis,the numberoffixedpointsofacorrespondenceonthe geometricspecial fiber is equal to the trace of the corresponding operator on the l-adic cohomology of S (G). We now recall Kottwitz’s well-known expression for this quantity. K Theorem 2.3. There is an equality N(j,fp)= ι(G,H)STH∗(fH) P(H,s,η0) e where the sum is taken over the elliptic endoscopic triples of G, and for each H, H G the function f is a transfer of f . We will take the basic concepts and results from the theory of endoscopy for granted, and we will adopt the various normalizations thereof from Morel [18]. In particular, see [18, p.88] for the notation and the references for theorem 2.3. Specializing to the case we are concerned with in this paper, we have the following Corollary 2.4. For G=GU(2,1) and H=G(U(1) U(1,1)), we have × N(j,fp)=STG(fG)+ 1STH(fH). e 2 e Proof. Inthiscase,GandHaretheonlyellipticendoscopicgroups. Further,inthis case,only(G,H)-regularorbitswillcontribute;see[18,p.89]. Thus,therestriction to such orbits in theorem 2.3, indicated with the asterisk, can be ignored. (cid:3) In particular,we have the following expressionfor the number of rationalpoints N in the definition of the L-function. j Corollary 2.5. When fp =1 , we have Kp N =N(j,fp)=STG(fG)+ 1STH(fH). j e 2 e 8 JUKKAKERANEN 2.2.2. Choice of Test Functions. We now focus on the groups G = GU(2,1) and H=G(U(1) U(1,1)). The test function fG on G can be chosen as follows: × fG =fG bG(ϕ ) fp, R × j j × where (1) fG = 1(f +f +f ), the sum of pseudocoefficients of the discrete series R 3 π1 π2 π3 L-packetΠ= π ,π ,π of G(R) associated to the trivial representation of G. 1 2 3 { } (2) ϕ C (G(Q )//K ) is the characteristic function of the double coset j ∈ c pj pj K µ(p)K , with µ the cocharacter associated to the Shimura datum, [18, p.33]; pj pj bG :C (G(Q )//K ) C (G(Q )//K ) j c pj pj → c p p the change of base map, with Q the unramified extension of Q of degree j pj p contained in Q¯ . p (3) Pick a prime q = p such that Kp = Kp,qG(Z ) and fp = fp,q1 , where fp,q C∞(G(Ap,q)//K6 p,q) is for now arbitrary. q Kq ∈ c f H The test function f is then chosen as follows: fH =fH bH(ϕ ) hp, R × j j × where H (1) fR =fρ+ +fρ− −fρ0, the (weighted) sum of pseudocoefficients of the three discrete series L-packets of H(R) that transfer to the given L-packet Π of G(R). (2) bH :C (G(Q )//K ) C (H(Q )//KH) is defined in the same way as in j c pj pj → c p p Laumon, see [16, p. 288]; (3) hp is an arbitrary transfer of fp. WewillalsoneedtestfunctionsonG=R G ⋊θ andH =R H ⋊θ that E/Q E E/Q E are associated to the given functions on G and H at all places in the sense of [14, 3.2]. Forthe details ofthese choicesandthe underlyingnormalizations,we refer to Morel [18, p. 138]. Briefly, the test function φG and φH are chosen as follows: φG =φG φG φG,p, R × j × and φH =φH φH φH,p, R × j × where,for anyfield F andanytest function φ0 onG0(F), weassociateto φ0 a test function φ on G(F)=G0(F)⋊θ by setting φ(x⋊θ):=φ0(x) for x G0(F), and similarly for H; see [14, p. 98]. The connected components of ∈ φG and φH are then given as follows (where we will ignore the superscript): (1) φG is a pseudocoefficient of the θ-discrete representationon G0(R) that cor- R respondstothediscreteseriesL-packetΠofG(R)above;see[18,p.124]. Similarly, φHR = φρ+ +φρ− −φρ0, the sum of pseudocoefficients of θ-discrete representations of H0(R) that correspond to the three L-packets of H(R) indicated above. (2) φG and φH are chosen as on p.138 of [18]. j j (3) φG,p is associatedat every place to fp, and φH,p is associatedat every place to hp. COMPACT SUPPORT COHOMOLOGY OF PICARD MODULAR SURFACES 9 2.3. FirstMajorTransition. NowletGbeaconnectedcomponentofareductive group G˜ over Q. Again, the cases we are interested in this paper are G=R G ⋊θ R G ⋊ θ =G˜, E/Q E E/Q E ⊂ h i H =R H ⋊θ R H ⋊ θ =H˜, E/Q E E/Q E ⊂ h i whereθ is anautomorphismonGandHinducedby the non-trivialelementofthe Galois group Gal(E/Q). Let T be a torus of G0 such that T (R) is a maximal torus of the set of fixed e R e points of a Cartan involution of G0(R) that commutes with θ; see [18, p. 121]. Set d(G)= Ker(H1(R,T ) H1(R,G0)). e | → | In our situation, G0 comes from a complex group by restriction of scalars, and hence H1(R,G0) = 1 and d(G) = H1(R,T ). For a quasi-split unitary group e { } | | G(U∗(n ) ... U∗(n )), with n:=n +...+n , we have T =G(U(1)n). Thus, 1 r 1 r e × × d(G)=2n−1; see [18, p. 123]. Theorem 2.6. We have TG(φG)=C d(G)STG(fG) e Gτ(G) e and TH(φH)=C d(H)STH(fH), e Hτ(H) e where the test functions are as chosen in subsection 2.2.2 above. Proof. This is a special case of proposition 8.3.1 in [18, pp. 130-1]. (cid:3) Lemma 2.7. In the situation of the theorem, we have (1) τ(G)=1 and τ(H)=2; (2) d(G)=d(H)=4; (3) C =C =1/4. G H Proof. Part(1)followsfromlemma2.3.3in[18,p.40]. Part(2)followsdirectlyfrom theremarksmadebeforetheorem2.6. Forpart(3),bycomparingMorel’sstatement of theorem 2.6 (proposition 8.3.1 in [18]) with her source for this theorem, namely Labesse’s theorem 4.3.4 in [14], we deduce that C = τ(G0)JZ(θ), G 2kd(G) where k = dim a , and J (θ) = det(1 θ a /a ); see [14, p. 97]. Note that G Z G0 G | − | | our choices concerning the spaces of test functions differ from those of Morel, and hence, the two simplifications she notes at the top of page 131 of [18] do not apply in our situation. Instead, we compute directly that J (θ) = 4 and dim a = 2, Z G while τ(G0)=1 by lemma 2.3.3 of [18]. Thus, we have C = 1×4 = 1. G 4×4 4 A similar computation shows that C =1/4. (cid:3) H 10 JUKKAKERANEN Remark. The test functions φG and φH are associated with the test functions R R G H d(G)f and d(H)f by [18, p. 124]. Thus, the appearance of the factors d(G) R R and d(H) in the foregoing identities is due to our choice of test functions on G(R) and H(R), and likewise for their reciprocals. Thus, the constants C and C are, G H essentially, equal to 1. Puttingtogethereverythingwehavedonethusfar,wecannowwriteourcentral object of interest, the number of fixed points N(j,fp) as follows. Theorem 2.8. In the situation of theorem 2.6, we have N(j,fp)=TG(φG)+TH(φH). e e Proof. Thisisadirectconsequenceofcorollary2.4,theorem2.6,andlemma2.7. (cid:3) The next step in our project is to write the distributions TG(φG) and TH(φH) e e as explicitly as we can. 3. Geometric Side of the Trace Formula In this section, we will take G to be a reductive algebraic group, not necessarily connected, over Q. Let be the set of Levi subsets of G defined over R; we will R L recall the definition of Levi subset in subsection 4.1 below. 3.1. Definitions. Afunctionf (A (R)0 G(R))iscalledcuspidal ifforevery R G ∈H \ M , M =G, and for every P (M), we have R ∈L 6 ∈P trπ(f )=0 R,P for every irreducible tempered representation π of A (R)0 M(R), where M \ f (m):=δ (m)1/2 f (k−1mnk)dndk. R,P P(R) RKmax,RRNP(R) R Further, a function f (A (R)0 G(R)) is called stable cuspidal if it is R G ∈ H \ cuspidal and, in addition, we have trπ(f )=0 R for every irreducible tempered representation π of A (R)0 M(R) that is not M \ square-integrable,and trπ (f )=trπ (f ) R,1 R R,2 R for any two square integrable representations π and π of A (R)0 G(R) that R,1 R,2 G \ belong to the same L-packet. Afunctionf (A (R)0 G(R))iscalledvery cuspidal ifitisinvariantunder R G ∈H \ conjugation by K and if for every M and every P (M), we have max,R R ∈L ∈P f (m)=δ (m)1/2 f (mn)dn=0 R,P P(R) RNP(R) R for all m in M(R). A function f that is very cuspidal is also cuspidal, but need not be stable R cuspidal. Let C R . A function f′ C∞(G(A )) is called C-regular (resp. strongly ∈ + ∈ c f C-regular) if for every M , M = G, every P (M), and m M(A ) such f ∈ L 6 ∈ P ∈ that