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On the irreducibility of locally analytic principal series representations 7 Sascha Orlik and Matthias Strauch 0 0 2 Abstract. Let G be a p-adic connected reductive group with Lie algebra g. For a v parabolic subgroup P ⊂ G and a finite-dimensional locally analytic representation V of o N P, we study the induced locally analytic G-representation W = IndGP(V). Our result is the following criterion concerning the topological irreducibility of W. If the Verma 0 module U(g)⊗ V′ associated to the dual representation V′ is irreducible then W is 2 U(p) topologically irreducible as well. ] T Contents R . h 1. Introduction 1 t a 2. Distribution algebras and locally analytic representations 3 m 2.1. Distribution algebras 3 [ 2.2. Norms and completions of distribution algebras 4 2 2.3. The closure of the enveloping algebra 9 v 2.4. Locally analytic representations 10 9 3. Representations induced from a parabolic subgroup 11 0 8 3.1. The setting and statement of the main result 11 2 3.2. The structure of the proof 12 1 6 3.3. Parahoric subgroups and their distribution algebras 15 0 3.4. Modules for the completed distribution algebras 21 / h 3.5. The main result 27 t a 4. Examples 29 m 4.1. Irreducibility of Verma modules 29 : 4.2. Restriction of Scalars 30 v i References 31 X r a 1. Introduction One of the principal methods for constructing representations of reductive groups is to induce representations of parabolic subgroups which come by inflation from representations of Levi factors. This applies for example to the theory of algebraic representations as well as to the theory of smooth representations of p-adic reductive groups. In this paper we consider parabolically induced representations in the theory of locally analytic representations of p-adic reductive groups. A systematic framework to study locally analytic representations of p-adic groups was developed in the recent years mainly by P. Schneider and J. Teitelbaum, cf. [ST1], [ST2]. 1 2 On theirreducibility of locally analytic principal series representations Algebraic representations and smooth representations, as well as tensor products of these, provide first examples of such locally analytic representations, but there are much more. For instance, the representations which are locally analytically induced from representations of parabolic subgroups. Locally analytic principal series representations for SL (L), L a finite extension of Q , were already defined and studied 2 p by Y. Morita in [Mo]. They were later reconsidered for the group GL in [ST1] (for 2 L = Q ) and in [KS] (for arbitrary L). p In this paper we prove a general criterion for the irreducibility of parabolically inducedlocallyanalyticrepresentations. InhisthesisH.Frommer[Fr]studiedlocally analytic representations of G = G(Q ), where G is a split reductive group over Q , p p which areinduced fromfinite-dimensional representations V of aparabolic subgroup P ⊂ G. Here V is defined over some field extension K of Q . His main theorem is p G a criterion for the (topological) irreducibility of the induced representation Ind (V) P in terms of a canonically associated Verma module. The crucial idea is to compare the structure of the dual space IndG(V)′ as a module over the distribution algebra P D(G,K), where G ⊂ G is a maximal compact subgroup, with the structure of an associated Verma module over the universal enveloping algebra U(g). In order to pass from D(G,K) to U(g) one needs a technical result about their relation and Frommer only showed this for L = Q . This is the reason for the restriction to the p base field Q . Later J. Kohlhaase [K] proved this technical result for arbitrary finite p extensions L over Q , which we use here to generalize Frommer’s theorem to the p case of a not necessarily split reductive group over an arbitrary extension of Q . p In order to state our main result let G be a connected reductive group over L, and let P ⊂ G be a parabolic subgroup. We consider a locally analytic representation V of the group P = P(L) which comes by inflation from a Levi subgroup and put G = G(L). Here V is a vector space over a discretely valued extension K which contains L and whose absolute value induces the given absolute value on L. Then we have: Main result: Suppose dim (V) < ∞. Then the induced locally analytic represen- K tation IndG(V) is topologically irreducible if U(g)⊗ V′ is irreducible as a module P U(p) over the universal enveloping algebra U(g). Our overall strategy of proof follows basically Frommer’s treatment. However, we found that an essential argument in [Fr], stating that certain distribution algebras areintegraldomains, is notobvious, asit isclaimed there. Wereprove thisinsection 3.3. But beside the fact that it is desirable to have such a criterion for the irreducibility ingeneral, a motivationwas provided bythe concrete example ofcertain Q -analytic p principal series representations of GL (Q ) (regarded as a group over Q ). The in- 2 p2 p terest in these representations comes from conjectural relations to two-dimensional S. Orlik and M. Strauch 3 crystalline representations of Gal(Q |Q ). Such induced locally analytic represen- p p2 tations play an important role in the the p-adic Langlands program, cf. [BS]. In the last section we consider a particular example, namely the case where G comes by restriction of scalars from a group which is split over L. Acknowledgements. We would like to thank Tobias Schmidt and Jan Kohlhaase for helpful discussions on distribution algebras. We thank the SFB 478 “Geometrische Strukturen in der Mathematik” at Mu¨nster for financial support of travel expenses. Notation. We let p be a prime number and denote by L a finite extension of Q . p The normalized p-adic (logarithmic) valuation is denoted by v (i.e. v (p) = 1). p p We let K be a complete discretely valued field extension of L, whose absolute value restricts to the absolute value of L. The rings of integers are denoted by o and o , L K respectively. For notions and notation in the context of non-archimedean functional analysis we refer to [S]. 2. Distribution algebras and locally analytic representations 2.1. Distribution algebras. In this section we recall some definitions and results about algebras of distributions attached to locally analytic groups, cf. [ST1], [ST2]. We consider a locally L-analytic group H and denote by Can(H,K) = Can(H,K) L the locally convex K-vector space of locally L-analytic functions on H as defined in [ST1]. The strong dual D(H,K) = D (H,K) := (Can(H,K))′ L L b is the topological algebra of K-valued distributions on H. If furthermore H is compact, then D(H,K) has the structure of a Fréchet algebra. The multiplication δ ∗δ of distributions δ ,δ ∈ D(H,K) is defined by 1 2 1 2 δ ∗δ (f) = (δ ⊗ˆ δ )((h ,h ) 7→ f(h h )), 1 2 1 2 1 2 1 2 where the distribution δ ⊗ˆ δ ∈ D(H × H,K) has the property that for functions 1 2 f ,f ∈ Can(H,K), one has 1 2 (δ ⊗ˆ δ )((h ,h ) 7→ f (h )f (h )) = δ (f )δ (f ). 1 2 1 2 1 1 2 2 1 1 2 2 The universal enveloping algebra U(h) of the Lie algebra h = Lie(H) of H acts naturally on Can(H,K). On elements x ∈ h, this action is given by d (xf)(h) = (t 7→ f(exp(−tx)h))| . t=0 dt 4 On theirreducibility of locally analytic principal series representations This gives rise to an embedding of U(h) := U(h)⊗ K into D(H,K): K L U(h) ֒→ D(H,K), z 7→ (f 7→ (z˙f)(1)). K Here z 7→ z˙ is the unique K-linear anti-automorphism of U(h) which induces K multiplication by −1 on h. 2.2. Norms and completions of distribution algebras. 2.2.1. p-valuations and global charts. Let H be a compact locally Q -analytic p group. Recall that a map 1 ω : H −{1} −→ ( ,∞) ⊂ R p−1 is called a p-valuation (cf. [L], III.2.1.2) if the following conditions hold for all g,h ∈ H: i) ω(gh−1) ≥ min{ω(g),ω(h)}, ii) ω(g−1h−1gh) ≥ ω(g)+ω(h), iii) ω(gp) = ω(g)+1. As usual one puts ω(1) = ∞ and interprets the above inequalities in the obvious sense, if a term ω(1) occurs. Let ω be a p-valuation on H. The above conditions imply that for any ν > 0 the sets H = {h ∈ H|ω(h) ≥ ν} and H = {g ∈ H|ω(g) > ν} ν ν+ are normal subgroups of H. We put gr(H) = H /H . ν ν+ ν>0 M The commutator induces a Lie bracket on gr(H) which gives gr(H) the structure of a Lie algebra over F . The map ǫ defined by p ǫ : gr(H) → gr(H), ǫ(gH ) = gpH ν+ (ν+1)+ is an F -linear map on gr(H), which gives gr(H) the structure of a graded Lie alge- p bra over F [ǫ], cf. [L], III.2.1.1. It is a free F [ǫ]-module, whose rank is equal to the p p dimension of H as a Q -analytic group, cf. loc. cit., III.3.1.3/7/9. p S. Orlik and M. Strauch 5 If(h H ,...,h H )isabasisofgr(H)overF [ǫ],thentheelementsh ,...,h 1 ω(h1)+ d ω(hd)+ p 1 d form a topological generating system of H, and the following map Zd → H, (a ,...,a ) 7→ ha1 ·...·had p 1 d 1 d is well-defined and a homeomorphism. Moreover, ω(ha1 ·...·had) = min{ω(h )+v (a )|i = 1,...,d}. 1 d i p i The sequence (h ,...,h ) is called a p-basis of H. 1 d 2.2.2. Uniform pro-p groups. We recall some definitions and results about pro-p groups, cf. [DDMS], ch. 3, 4. In this section H will be a pro-p group which is equipped with its topology as a pro-finite group. H is called powerful if p is odd (resp. p = 2) andH/Hp (resp. H/H4 ifp = 2)is abelian. Here, Hp (resp. H4) is the closure of the subgroup generated by the p-th (resp. fourth) powers of its elements. If H is topologically finitely generated one can show that the subgroups Hp (resp. H4) are open and hence automatically closed. The lower p-series (P (H)) of an i i≥1 arbitrary pro-p group H is defined inductively by P (H) = H, P (H) = P (H)p[P (H),H]. 1 i+1 i i If H is topologically finitely generated then the groups P (H) are all open in H and i form a fundamental system of neighborhoods of 1, cf. Prop. 1.16 in loc.cit. A pro-p group H is called uniform if it is topologically finitely generated, powerful and its lower p-series satisfies (H : P (H)) = (P (H) : P (H)) 2 i i+1 for all i ≥ 1. If H is a topologically finitely generated powerful pro-p group then P (H) is a uniform pro-p group for all sufficiently large i, cf. loc.cit. 4.2. Moreover, i any compact Q -analytic group contains an open normal uniform pro-p subgroup, p cf. loc.cit. 8.34. 2.2.3. The canonical p-valuation on uniform groups. Let H be a uniform pro-p group. It carries a distinguished p-valuation ωcan which is associated to the lower p-series and which we call the canonical p-valuation. In order to define it, we let ε = 1 if p = 2 and ε = 0 for odd p. For h 6= 1, we then put p p ωcan(h) = ε +max{i ≥ 1|h ∈ P (H)}. p i To verify that this gives indeed a p-valuation one makes use of the fact that [P (H),P (H)] ⊂ P (H) for all i,j ≥ 1, cf. Prop. 1.16 in [DDMS]. Also prop- i j i+j erty (iii) follows from Prop. 2.7 in loc.cit. For p = 2 one has to use the stronger 6 On theirreducibility of locally analytic principal series representations statement that [P (H),P (H)] ⊂ P (H) for all i,j ≥ 1, cf. [Sch], proof of Prop. i j i+j+1 2.1. 2.2.4. Norms induced by p-valuations. In this section we let H be a compact L- analytic group. The distribution algebra D(H,K) is then a Fréchet-Stein algebra in the sense of [ST2], sec. 3. This means in particular that there exists a family of norms k k , 1 < r < 1, on D(H,K) such that, if D (H,K) denotes the completion r p r of D(H,K) with respect to k k , the convolution product ∗ on D(H,K) extends by r continuity to a product on D (H,K). We recall briefly the construction of such a r family of norms. This is done in three steps, cf. [ST2] (proof of Thm. 5.1). Step 1. We denote by RL H the group H when considered as a locally Q -analytic Qp p group, and we put d = dim(RL H). Let H ⊂ RL H be an open normal subgroup Qp 0 Qp which is equipped with a p-valuation ω. For instance, by [DDMS], 8.34, we may choose H to be an open normal uniform pro-p subgroup, and we may take for ω the 0 canonical valuation associated to its lower p-series (cf. 2.2.3). Then we consider the graded group gr(H ) (which depends on ω). A choice of a basis of gr(H ) gives rise 0 0 to a homeomorphism ψ : Zd → H , cf. 2.2.1, which in turn induces an isomorphism p 0 of locally convex K-vector spaces ψ∗ : Can(H ,K) −≃→ Can(Zd,K) 0 p as well as an isomorphism of K-Banach spaces of continuous functions ψ∗ : C(H ,K) −≃→ C(Zd,K). 0 p Using Mahler expansions ([L], III.1.2.4) we can express elements of C(Zd,K) as p series x f(x) = c , n n∈Nd (cid:18)n(cid:19) X 0 where c ∈ K and n x x x 1 d := ··· (cid:18)n(cid:19) (cid:18)n (cid:19) (cid:18)n (cid:19) 1 d for the multi-indices x = (x ,...,x ) and n = (n ,...,n ) ∈ Nd. Further, we have 1 d 1 d 0 |c | → 0 as |n| = n +...+n → ∞. A continuous function f ∈ C(Zd,K) is locally n 1 d p analytic if and only if |c |r|n| → 0 for some r > 1. Consider the group algebra K[H ] n 0 of H . By identifying elements of H with Dirac distributions, we get an embedding 0 0 K[H ] ֒→ D(H ,K). 0 0 S. Orlik and M. Strauch 7 Put b := h −1 ∈ K[H ] and set for n ∈ Nd, i i 0 0 bn = bn1 ···bnd . 1 d Then we have bn(f) = c for any continuous function f ∈ C(H ,K), where the c n 0 n are the Mahler coefficients of ψ∗(f) ∈ C(Zd,K). It follows that every distribution p λ ∈ D(H ,K) has the shape 0 λ = d bn n n∈Nd X 0 where {d r|n||n ∈ Nd} is a bounded set for all 0 < r < 1. The norm k · k on n 0 r D(H ,K) is then defined by 0 k d bnk = sup{|d |rτ(n)|n ∈ Nd}. n∈Nd n r n 0 X 0 Here, τ(n) is given by τ(n) = n ω(h ). The Banach algebra D (H ,K) is defined i i i r 0 to be the completion of D(H0P,K) with respect to k·kr. Thus we obtain D (H ,K) = { d bn ∈ K[[b ,...,b ]]| lim |d |rτ(n) = 0}. r 0 n 1 d n Xn∈Nd0 |n|→∞ Furthermore, for 1 < r < 1, the norm k · k is multiplicative, cf. [ST2] Thm. 4.5 p r and does not depend on the chosen basis (loc. cit., before Thm. 4.11). We obtain a projective system of noetherian Banach algebras such that D(H ,K) = lim D (H ,K). 0 ←−r r 0 Moreover, the transition maps D (H ,K) → D (H ,K) r′ 0 r 0 are flat for 1 < r ≤ r′ < 1. p Step 2. We extend the norm k · k on D(H ,K) to a norm on D(RL H,K) as r 0 Qp follows. Let η ,...,η be a system of coset representatives of H/H . Then the 1 s 0 Dirac distributions δ ,...,δ form a basis of D(RL H,K) over D(H ,K). Writing η1 ηs Qp 0 µ ∈ D(RL H,K) as µ = λ δ +...+λ δ , we put Qp 1 η1 s ηs q (µ) = max{kλ k |1 ≤ i ≤ s}. r i r Again, we obtain a system of sub-multiplicative norms q , 1 < r < 1, and a pro- r p jective system D (RL H,K) of K-Banach algebras fulfilling the conditions above, r Qp and we have D(RL H,K) = limD (RL H,K). The norms q do not depend on the Qp ←− r Qp r chosen representatives. It is worth to remark at this point that the norms q are r 8 On theirreducibility of locally analytic principal series representations in general not multiplicative, because H may contain non-trivial elements of finite order which causes the group ring, as well as the rings of distributions, to have non-trivial zero divisors1. Step 3. LocallyL-analyticfunctionsonH areobviouslylocallyQ -analyticfunctions p on RL H, and hence there is a canonical map Qp Can(H,K) ֒→ Can(RL H,K) L Qp Qp which is a closed embedding (cf. the proof of Thm. 5.1 in [ST2]). This map induces by duality a continuous surjection D (RL H,K) → D (H,K) Qp Qp L on the distribution algebras. We denote the induced residue norm on D (H,K) L respectively on the completion D (H,K) by q¯ . Once again, we have D (H,K) = r r L limD (H,K). ←− r Althoughwewillmostlyconsiderthecasewheretheparameterr liesstrictlybetween 1 and 1, we point out that the norms q¯ are defined, as norms of K-vector spaces, p r for any r ∈ (0,1). This is used later in the proof of 3.4.7, cf. 3.4.6. Remark 2.2.5. We keep the notation from the preceding paragraph. In the recipe for the norm q¯ on D (H,K) given above, we could have carried out steps two r L and three in reverse order with the same resulting norm. To be precise, let k·k r be the quotient norm on D (H ,K) induced by the surjection D (RL H ,K) ։ L 0 Qp Qp 0 D (H ,K). We have again L 0 s D (H,K) = D (H ,K)∗δ L L 0 ηi Mi=1 and can hence consider the maximum norm qˆ on D (H,K) induced from this r L decomposition and the norm k·k on D (H ,K). Then it is not difficult to check r L 0 that qˆ coincides with the norm q¯ defined above, cf. [Sch], Lemma 4.4. r r 1In [Fr] it is mistakenly stated that these norms were multiplicative so that the corresponding completions would be integral domains. As a consequence, some completed distribution algebras were claimed to be obviously integral domains. S. Orlik and M. Strauch 9 2.3. The closure of the enveloping algebra. 2.3.1. Definition of U (h,H ). Let H be a compact locally L-analytic group and r 0 let H ⊂ H be an open normal subgroup, which is equipped with a p-valuation ω. 0 Associated to ω there is a norm q¯ on D(H,K) as defined in 2.2.4. We write r U (h,H ) ⊂ D (H ,K) r 0 r 0 for the topological closure of U(h) ⊗ K in D (H ,K). The following theorem by L r 0 Kohlhaase generalizes a result of Frommer who considered the case of Q -analytic p groups. It is this basic technical result which we use to generalize Frommer’s irreducibility criterion. Theorem 2.3.2. ([K], 1.3.5, 1.4.2.) Let H be a compact locally L-analytic group of dimension d with Lie algebra h = Lie(H) and r ∈ pQ with 1 < r < 1. p (i) There exists an open L-analytic subgroup H of H, and a Z -lattice Λ ⊂ Lie(H) 0 p with the following properties: (1) there is an L-basis X = (X ,...,X ) of h and a Z -basis ξ ,...,ξ of o 1 d p 1 m L such that (ξ X ,...,ξ X ) is a Z -basis of Λ; 1 1 m d p (2) the corresponding canonical coordinates of the second kind2 give an isomorphism Λ → RL H of locally Q -analytic manifolds; Qp 0 p (3) H is a uniform pro-p group. 0 (ii) Let H ⊂ H be an open subgroup as in (i) which we assume furthermore to be 0 normal. Then, if we equip RL H with its canonical valuation (2.2.3), D (H ,K) is Qp 0 r 0 a free, finitely generated module over the noetherian subalgebra U (h,H ). r 0 (iii) Let the normal subgroup H ⊂ H be as in (i), and let X = (X ,...,X ) be the 0 1 d L-basis of h as above. Then there is a norm ν on U (h,H ) which is equivalent to r r 0 q¯ , such that U (h,H ) consists of exactly those series r r 0 d Xn n n∈Nd X 0 for which lim |d |ν (Xn) = 0, n r |n|→∞ and ν ( d Xn) = sup |d |ν (Xn). r n n n n r P 2cf. [Bou], Ch. III, §4.3. 10 On theirreducibility of locally analytic principal series representations Corollary 2.3.3. Let H be a compact locally L-analytic group, and let H ⊂ H 0 be an open normal subgroup as in 2.3.2 (ii). Suppose that D (H,K) is an integral r domain. Then any non-zero left ideal I of D (H,K) has non-zero intersection with r U (h,H ). r 0 Proof. Compare [Fr], Corollary 5. Since H is compact, D (H,K) is a finite r D (H ,K)-module, and so D (H,K) is finite over U (h,H ). Therefore, there exists r 0 r r 0 for any F ∈ D (H,K), a polynomial P = N a Xi, a ∈ U (h,H ) with P(F) = 0. r i=0 i i r 0 Because we assumed Dr(H,K) to be an inPtegral domain, we may find a polynomial with a 6= 0. So, if F ∈ I then 0 N 0 6= a = − a Fi ∈ I ∩U (h,H ). 0 i r 0 Xi=1 (cid:3) 2.4. Locally analytic representations. We conclude this section by recalling some facts of locally analytic representations. Let H be a locally L-analytic group, V a locally convex K-vector space, and ρ : H → GL (V) a homomorphism. Then K ρ (or the pair (V,ρ)) is called a locally analytic representation of H if the topological K-vector space V is barrelled, the homomorphism ρ is continuous, and the orbit maps ρ : H → V, h 7→ ρ(h)(v), are locally analytic maps for all v ∈ V, cf. v [ST1], sec. 3. If V is of compact type, i.e., a compact inductive limit of Banach spaces, the strong dual V′ is a nuclear Fréchet space and a separately continuous b left D(H,K)-module. The module structure is given as follows: D(H,K)⊗ V′ → V′, δ ⊗ϕ 7→ (v 7→ δ(g 7→ ϕ(ρ(g−1)v))). K b b This functor gives an equivalence of categories locally analytic H-represen- separately continuous D(H,K)-  tations on K-vector spaces   modules on nuclear Fréchet spa-  −→  of compact type with   ces with continuous D(H,K)-      continuous linear H-maps module maps             In particular, V is topologically irreducible if V′ is a simple D(H,K)-module. b For any closed subgroup H′ of H and any locally analytic representation V of H′, we denote by IndH (V) the induced locally analytic representation. We recall the H′ definition: IndH (V) := f ∈ Can(H,V)|f(h′ ·h) = h′ ·f(h) ∀h′ ∈ H′,∀h ∈ H . H′ n o