The Jordan-Hölder Series of the

Locally Analytic Steinberg Representation

Sascha Orlik, Benjamin Schraen

Received: November 11, 2010 Revised: March 14, 2014

Communicated by Peter Schneider

Abstract. We determine the composition factors of a Jordan-Hölder series including multiplicities of the locally analytic Steinberg representation. For this purpose, we prove the acyclicity of the evaluated locally analytic Tits complex giving rise to the Steinberg representation. Further we describe some analogue of the Jacquet functor applied to the irreducible principal series representation constructed in [OS2].

2010 Mathematics Subject Classification: 22E50, 20G05, 11S37

1. Introduction

In this paper we study the locally analytic Steinberg representation V_{B}^{G} for a given split reductive p-adic Lie group G. This type of object arises in various fields of Representation Theory, cf. [Hum1, DOR]. In the smooth representation theory of p-adic Lie groups as well as in the case of finite groups of Lie type, it is related to the (Bruhat-)Tits building and has therefore interesting applications [Car, DM, Ca1]. In the locally analytic setting it comes up so far in the p-adic Langlands program with respect to semi-stable, non-crystalline Galois representation [Br]. More precisely, V_{B}^{G} coincides with the set of locally analytic vectors in the continuous Steinberg representation which should arise in a possible local p-adic Langlands correspondence on the automorphic side. Our main result gives the composition factors including multiplicities of a Jordan-Hölder series for V_{B}^{G}. This answers a question raised by Teitelbaum in [T]. Let G be a split reductive p-adic Lie group over a finite extension L of ℚ_{p} and let B ⊂ G be a Borel subgroup. The definition of V_{B}^{G} is completely analogous to the above mentioned classical cases. It is given by the quotient

V_{B}^{G} = I_{B}^{G}/ ∑_{P ⊋B} I_{P}^{G},

where I_{P}^{G} = C^{an}(G/P, K) is the G-representation consisting of locally Lanalytic functions on the partial flag manifold G/P with coefficients in some fixed finite extension K of L. In contrast to the smooth situation or that of a finite group of Lie type, the locally analytic Steinberg representation is not irreducible. Indeed, it contains the smooth Steinberg representation v_{B}^{G} = i_{B}^{G}/ ∑_{P ⊋B} i_{P}^{G}, where i_{P}^{G} = C^{∞}(G/P, K), as a closed subspace. On the other hand, it is a consequence of the construction in [OS2] that V_{B}^{G} has a composition series of finite length and therefore the natural question of determining its Jordan-Hölder series comes up. Morita [Mo] proved that for G = GL_{2}, the topological dual of V_{B}^{G} is isomorphic to the space of K-valued sections of the canonical sheaf ω on the Drinfeld half space X = ℙ^{1} \ ℙ^{1}(L). In higher dimensions Schneider and Teitelbaum [ST1] construct an injective map from the space of K-valued sections of ω to the topological dual of V_{B}^{G}. However, the natural hope that this map is an isomorphism in general turns out not to be correct. In fact, this follows by gluing some results of [Schr] and [O] considering the Jordan-Hölder series of both representations in the case of GL_{3}. In order to determine the composition factors of V_{B}^{G} in the general case, we apply the machinery constructing locally analytic G-representations F_{P}^{G}(M, V) developed in [OS2]. Here M is an object of type P in the category O of Lie algebra representations of Lie G and V is a smooth admissible representation of a Levi factor L_{P} ⊂ P (we refer to Section 2 for a more detailed recapitulation). It is proved in loc.cit. that F_{P}^{G}(M, V) is topologically irreducible if M and V are simple objects and if furthermore P is maximal for M. On the other hand, it is shown that F_{P}^{G} is bi-exact which allows us to speak of a locally analytic BGG-resolution. This latter aspect is one main ingredient for proving the acyclicity of the evaluated locally analytic Tits complex

0 → I_{G}^{G} → ⨁_{{K⊂Δ \atop |Δ\K|=1}} I_{P_{K}}^{G} → ⨁_{{K⊂Δ \atop |Δ\K|=2}} I_{P_{K}}^{G} → ⋯ → ⨁_{{K⊂Δ \atop |K|=1}} I_{P_{K}}^{G} → I_{B}^{G} → V_{B}^{G} → 0.

Here Δ is the set of simple roots with respect to B and a choice of a maximal torus T ⊂ B. Hence the determination of the composition factors of V_{B}^{G} is reduced to the situation of an induced representation I_{P}^{G} which lies in the image of the functor F_{P}^{G}. This leads to the question when two irreducible representations of the shape F_{P}^{G}(M, V) with V a composition factor of i_{B}^{G} are isomorphic. It turns out that this holds true if and only if all ingredients are the same. Thus we arrive at the stage where Kazhdan-Lusztig theory enters the game. Now we formulate our main result. For two reflections w, w′ in the Weyl group W, let m(w′, w) ∈ ℤ_{≥0} be the multiplicity of the simple highest weight module L(w · 0) of weight w · 0 ∈ Hom_{L}(Lie T, K) in the Verma module M(w′ · 0) of weight w′ ·0. Let supp(w) ⊂ W be the subset of simple reflections which appear in w. In the following theorem we identify the set of simple reflections with the set Δ.

Theorem: For w ∈ W, let I ⊂ Δ be a subset such that the standard parabolic subgroup P_{I} attached to I is maximal for L(w · 0). Let v_{P_{J}}^{P_{I}} be the smooth generalized Steinberg representation of L_{P_{I}} with respect to a subset J ⊂ I. Then the multiplicity of the irreducible representation F_{P_{I}}^{G} (L(w · 0), v_{P_{J}}^{P_{I}}) in V_{B}^{G} is given by

∑_{{w′∈W \atop supp(w′)=J}} (−1)^{ℓ(w′)+|J|}m(w′, w),

and we obtain in this way all the Jordan-Hölder factors of V_{B}^{G}. In particular, the smooth Steinberg representation v_{B}^{G} is the only smooth subquotient of V_{B}^{G}. Moreover, the representation F_{P_{I}}^{G} (L(w · 0), v_{P_{J}}^{P_{I}}) appears with a non-zero multiplicity if and only if J ⊂ supp(w). We close this introduction by mentioning that we discuss in our paper more generally generalized locally analytic Steinberg representation V_{P}^{G}, as well as their twisted versions V_{P}^{G}(λ) involving a dominant weight λ ∈ X^{∗}(T).

Acknowledgments: The second author would like to thank the Bergische Universität Wuppertal for an invitation where some part of this work was done and to mention that a part of this work was done when he was a member of the DMA at the École normale supérieure.

Notation: We denote by p a prime number and by K ⊃ L ⊃ ℚ_{p} finite extensions of the field of p-adic integers ℚ_{p}. Let O_{L} be the ring of integers in L and fix an uniformizer π of O_{L}. We let k_{L} = O_{L}/(π) be the corresponding residue field. For a locally convex K-vector space V, we denote by V′ its strong dual, i.e., the K-vector space of continuous linear forms equipped with the strong topology of bounded convergence, cf. [S]. For an algebraic group G over L we denote by G = G(L) the p-adic Lie group of L-valued points. We use a Gothic letter 𝔤 to indicate its Lie algebra. We denote by U(𝔤) := U(𝔤 ⊗_{L} K) the universal enveloping algebra of 𝔤 after base change to K. We let O be the category O of Bernstein-Gelfand-Gelfand in the sense of [OS2].

2. The functors F_{P}^{G}

In this first section we recall the definition of the functors F_{P}^{G} constructed in [OS2]. As explained in the introduction they are crucial for the determination of a Jordan-Hölder series of the locally analytic Steinberg representation. Let G be a split reductive algebraic group over L. Let T ⊂ G be a maximal torus and fix a Borel subgroup B ⊂ G containing T. Let Δ be the set of simple roots, Φ the set of roots and Φ^{+} the set of positive roots of G with respect to T ⊂ B. In what follows we assume that our prime number p is odd, if the root system Φ has irreducible components of type B, C or F 4, and if Φ has irreducible components of type G2 we assume that p > 3.^{1} We identify the group X^{∗}(T) of algebraic characters of T via the derivative as a lattice in Hom_{L}(𝔱, K). Let P be standard parabolic subgroup (std psgp). Consider the Levi decomposition P = L_{P} · U_{P} where L_{P} is the Levi subgroup containing T and U_{P} is its unipotent radical. Let U_{P}^{−} be the unipotent radical of the opposite parabolic subgroup. Let O^{𝔭} be the full subcategory of O consisting of U(𝔤)-modules of type 𝔭 = Lie P. Its objects are U(𝔤)-modules M over the coefficient field K satisfying the following properties:

1This restriction is here to use results of [OS2].

(1) The action of 𝔲_{P} on M is locally finite.

(2) The action of 𝔩_{P} on M is semi-simple and locally finite.

(3) M is finitely generated as U(𝔤)-module. In particular, we have O = O^{𝔟}. Moreover, if Q is another parabolic subgroup of G with P ⊂ Q, then O^{𝔮} ⊂ O^{𝔭}. Let Irr(𝔩_{P})^{fd} be the set of finite-dimensional irreducible 𝔩_{P} -modules. Any object in O^{𝔭} has by property (2) a direct sum decomposition into 𝔩_{P} -modules

(2.1)  M = ⨁_{𝔞∈Irr(𝔩_{P})^{fd}} M_{𝔞}

where M_{𝔞} ⊂ M is the 𝔞-isotypic part of the finite-dimensional irreducible representation 𝔞. We let O_{alg}^{𝔭} be the full subcategory of O^{𝔭} given by objects such that all 𝔩_{P} -representations appearing in (2.1) are induced by finite-dimensional algebraic L_{P}-representations. The above inclusion O^{𝔮} ⊂ O^{𝔭} is compatible with these new subcategories, i.e. we also have O_{alg}^{𝔮} ⊂ O_{alg}^{𝔭}. In particular, the category O_{alg}^{𝔭} contains all finite-dimensional 𝔤-modules which come from algebraic G-modules. Every object in O_{alg}^{𝔭} has a Jordan-Hölder series which coincides with the Jordan-Hölder series in O. Let Rep_{K}^{∞,adm}(L_{P}) be the category of smooth admissible L_{P} -representations with coefficients over K. In [OS2] there is constructed a bi-functor

F_{P}^{G} : O_{alg}^{𝔭} × Rep_{K}^{∞,adm}(L_{P}) ⟶ Rep_{K}^{ℓa}(G),

where Rep_{K}^{ℓa}(G) denotes the category of locally analytic G-representations with coefficients in K. It is contravariant in the first and covariant in the second variable. Furthermore, F_{P}^{G} factorizes through the full subcategory of admissible representations in the sense of Schneider and Teitelbaum [ST2]. Let us recall the definition of F_{P}^{G}. Let M be an object of O_{alg}^{𝔭}. By the defining axioms (1) - (3) above there is a surjective map

(2.2)  ϕ : U(𝔤) ⊗_{U(𝔭)} W → M

for some finite-dimensional algebraic P -representation W ⊂ M. Let V be additionally a smooth admissible L_{P} -representation. We consider V via the trivial action of U_{P} as a P -representation. Further by equipping V with the finest locally convex topology it becomes a locally analytic P -representation, cf. [ST4, §2]. Hence we may consider the tensor product representation W′ ⊗_{K} V as a locally analytic P -representation. Let Ind_{P}^{G} : Rep_{K}^{ℓa}(P) → Rep_{K}^{ℓa}(G) be the locally analytic induction functor [Fe]. Then

F_{P}^{G}(M, V) = Ind_{P}^{G}(W′ ⊗_{K} V)^{𝔡}

denotes the subset of functions f ∈ Ind_{P}^{G}(W′ ⊗_{K} V) which are killed by the U(𝔤)-submodule 𝔡 = ker(ϕ) ⊂ U(𝔤) ⊗_{U(𝔭)} W for the action obtained by joining left translation for elements of U(𝔤) and evaluation in W′ for elements of W. Then F_{P}^{G}(M, V) is a well-defined G-stable closed subspace of Ind_{P}^{G}(W′ ⊗_{K} V) and has therefore a natural structure of a locally analytic G-representation. Further the above construction is even functorial. If V = 1 is the trivial L_{P} - representation, we simply write F_{P}^{G}(M) instead of F_{P}^{G}(M, 1). The functors F_{P}^{G} satisfy the following properties, cf. [OS2, Prop. 4.9, Thm. 5.8]:

• (exactness) The bi-functor F_{P}^{G} is exact in both arguments.

• (PQ-formula) Let Q ⊃ P be another parabolic subgroup. We suppose that L_{P} ⊂ L_{Q}. If V is a smooth admissible L_{P} -representation, then we denote by

i_{P}^{Q}(V) = Ind_{P}^{∞,Q}(V)

the smooth induced Q-representation. Note that as L_{Q}-representation one has the identification i_{P}^{Q}(V) = i_{L_{P} ·(U_{P} ∩L_{Q})}^{L_{Q}}(V). Let M ∈ O_{alg}^{𝔮} ⊂ O_{alg}^{𝔭}. Then there is the following identity:

F_{P}^{G}(M, V) = F_{Q}^{G}(M, i_{P}^{Q}(V)).

• (irreducibility) A std psgp P ⊂ G is called maximal for an object M ∈ O if M ∈ O^{𝔭} and if M ∉ O^{𝔮} for all 𝔮 ⊋ 𝔭. Let P be a std psgp, maximal for a simple object M ∈ O_{alg}. Further let V be an irreducible smooth admissible L_{P} -representation, then F_{P}^{G}(M, V) is (topologically) irreducible.

In [OS2] it is explained how one can deduce from the previous properties of the bi-functors F_{P}^{G} the Jordan-Hölder series of a representation F_{P}^{G}(M, V). Let us recall this procedure in the case V = 1. The smooth generalized Steinberg representation to P is the quotient

v_{P}^{G} = i_{P}^{G}/ ∑_{P ⊊Q⊂G} i_{Q}^{G}.

This is an irreducible G-representation and all irreducible subquotients of i_{P}^{G} occur in the shape v_{Q}^{G} with Q ⊃ P and with multiplicity one, cf. [BoWa, Ch. X,§4], [Ca2]. Let M be an object of the category O_{alg}^{𝔭}. Then it has a Jordan-Hölder series

M = M^{0} ⊃ M^{1} ⊃ M^{2} ⊃ … ⊃ M^{r} ⊃ M^{r+1} = (0) in the same category. By applying the functor F_{P}^{G} to it we get a sequence of surjections

F_{P}^{G}(M) →^{p_{0}} F_{P}^{G}(M^{1}) →^{p_{1}} F_{P}^{G}(M^{2}) →^{p_{2}} … →^{p_{r−1}} F_{P}^{G}(M^{r}) →^{p_{r}} (0).

For any integer i with 0 ≤ i ≤ r, we put

q_{i} := p_{i} ∘ p_{i−1} ∘ ⋯ ∘ p_{1} ∘ p_{0}

and set

F^{i} := ker(q_{i})

which is a closed subrepresentation of F_{P}^{G}(M). We obtain a filtration

(2.3)  F^{−1} = (0) ⊂ F^{0} ⊂ ⋯ ⊂ F^{r−1} ⊂ F^{r} = F_{P}^{G}(M)

by closed subspaces with

F^{i}/F^{i−1} ≃ F_{P}^{G}(M^{i}/M^{i+1}) .

Let Q_{i} = L_{i} · U_{i} ⊃ P a std psgp maximal for M^{i}/M^{i+1}. Then by the P Qformula, we get

F_{P}^{G}(M^{i}/M^{i+1}) = F_{Q_{i}}^{G}(M^{i}/M^{i+1}, i_{P}^{Q_{i}})

where i_{P}^{Q_{i}} = i_{L_{i}∩P}^{L_{i}} . We conclude that the representations

F_{Q_{i}}^{G}(M^{i}/M^{i+1}, v_{R}^{Q_{i}})

where R is a std psgp of G with Q_{i} ⊃ R ⊃ P and v_{R}^{Q_{i}} = v_{L_{i}∩R}^{L_{i}} are the topologically irreducible constituents of F_{Q_{i}}^{G}(M^{i}/M^{i+1}, i_{P}^{Q_{i}}). By refining the filtration (2.3) we get thus a Jordan-Hölder series of F_{P}^{G}(M). Finally we recall the parabolic BGG resolution of a finite-dimensional algebraic G-representation [Le]. We let ⟨ , ⟩ be the natural pairing X^{∗}(T) × X_{∗}(T) → ℤ defined by x(u(t)) = t^{⟨x,u⟩}. For each α ∈ Φ, we denote by α^{∨} ∈ X_{∗}(T) the cocharacter associated to α as in [Sp, §2.2]. Let W be the Weyl group of G and consider the dot action · of W on X^{∗}(T) given by

w · χ = w(χ + ρ) − ρ,

where ρ = \frac{1}{2} ∑_{α∈Φ^{+}} α. For a character λ ∈ X^{∗}(T), let

M(λ) = U(𝔤) ⊗_{U(𝔟)} K_{λ}

be the corresponding Verma module. Clearly M(λ) is an object of O_{alg}. We denote its irreducible quotient by L(λ). Let

X_{+} = {λ ∈ X^{∗}(T) | ⟨λ, α^{∨}⟩ ≥ 0 ∀α ∈ Δ}

be the set of dominant weights in X^{∗}(T). If λ ∈ X_{+}, then L(λ) is finitedimensional and comes from an irreducible algebraic G-representation. In this situation, we also write V (λ) for L(λ). For a subset I ⊂ Δ, let P = P_{I} ⊂ G be the attached std psgp and denote by

X_{I}^{+} = {λ ∈ X^{∗}(T) | ⟨λ, α^{∨}⟩ ≥ 0 ∀α ∈ I} be the set of L_{I}-dominant weights. Every λ ∈ X_{I}^{+} gives rise to a finitedimensional algebraic L_{I}-representation

(2.4)  V_{I}(λ) = V_{P} (λ).

We consider V_{I}(λ) as a P_{I}-module by letting act U_{I} trivially on it. The generalized (parabolic) Verma module attached to the weight λ is given by

M_{I}(λ) = U(𝔤) ⊗_{U(𝔭_{I})} V_{I}(λ).

We have a surjective map

q_{I} : M(λ) → M_{I}(λ),

where the kernel is given by the image of ⊕_{α∈I}M(s_{α} · λ) → M(λ), cf. [Le, Prop. 2.1]. If J ⊂ I, then we obtain a transition map q_{J,I} : M_{J}(λ) → M_{I}(λ) such that q_{I} = q_{J,I} ∘ q_{J}. Let W_{I} ⊂ W be the parabolic subgroup induced by I ⊂ Δ. Consider the set IW = W_{I}\W of left cosets which we identify with their representatives of shortest length in W. Let ^{I}w be the element of maximal length in ^{I}W. If λ is in X_{+} and w ∈ ^{I}W then w · λ ∈ X_{I}^{+}, cf. [Le, p. 502]. The I-parabolic BGG-resolution of V (λ), λ ∈ X_{+}, is given by the exact sequence

0 → M_{I}(^{I}w · λ) ⟶^{d_{ℓ(^{I} w)}^{I}} ⨁_{{w∈^{I} W \atop ℓ(w)=ℓ(^{I} w)−1}} M_{I}(w · λ) ⟶^{d_{ℓ(^{I} w)−1}^{I}} …

⋯ ⟶^{d_{2}^{I}} ⨁_{{w∈^{I} W \atop ℓ(w)=1}} M_{I}(w · λ) ⟶^{d_{1}^{I}} M_{I}(λ) → V (λ) → 0.

We shall specify the differentials in this complex. For each w ∈ W, fix once and for all an embedding i_{w} : M(w · λ) ↪ M(λ). If w′ ≥ w (for the Bruhat order ≤ on W), i_{w′} factors through the image of i_{w} and we can define thus a unique map i_{w′,w} ∈ Hom_{𝔤}(M(w′ · λ), M(w · λ)) such that i_{w′} = i_{w} ∘ i_{w′,w}. These maps satisfy the relations i_{w′′,w} = i_{w′,w} ∘ i_{w′′,w′} for all w′′ ≥ w′ ≥ w. If I ⊂ Δ is a subset and w′ ≥ w are elements of ^{I}W, then the previous relations and the identification of M_{I}(w · λ) with the cokernel of ⊕_{α∈I}i_{s_{α}w,w} show that i_{w′,w} factors through a map M_{I}(w′ · λ) → M_{I}(w · λ) which we will denote by the same symbol i_{w′,w}. Now recall that to each pair (w′, w) ∈ W^{2} such that w′ ≥ w and ℓ(w′) = ℓ(w) + 1, we can associate an element e(w′, w) ∈ {±1}, with e(w_{2}, w_{4})e(w_{1}, w_{2}) = −e(w_{3}, w_{4})e(w_{1}, w_{3}) once we have w_{4} < w_{2} < w_{1}, w_{4} < w_{3} < w_{1}, w_{2} ≠ w_{3} and ℓ(w_{1}) = ℓ(w_{4}) + 2 (cf. [BGG, Lemma 10.4]). Finally we define

d_{k}^{I} = ⨁_{{w′∈^{I} W \atop ℓ(w′)=k}} ∑_{{w∈^{I} W, w<w′ \atop ℓ(w′)=ℓ(w)+1}} e(w′, w)i_{w′,w}.

The exactness of the above sequence is a consequence of [Le, Thm. 4.3].

By applying our exact functor F_{P}^{G} to this exact sequence, we get another exact sequence

0 ← F_{P}^{G}(M_{I}(^{I}w · λ)) ← ⨁_{{w∈^{I} W \atop ℓ(w)=ℓ(^{I} w)−1}} F_{P}^{G}(M_{I}(w · λ)) ← …

⋯ ← ⨁_{{w∈^{I} W \atop ℓ(w)=1}} F_{P}^{G}(M_{I}(w · λ)) ← F_{P}^{G}(M_{I}(λ)) ← F_{P}^{G}(V (λ)) ← 0

which coincides by the very definition of F_{P}^{G} and the PQ-formula with

(2.5) 0 ← Ind_{P}^{G}(V_{I}(^{I}w · λ)′) ← ⨁_{{w∈^{I} W \atop ℓ(w)=ℓ(^{I} w)−1}} Ind_{P}^{G}(V_{I}(w · λ)′) ← …

⋯ ← ⨁_{{w∈^{I} W \atop ℓ(w)=1}} Ind_{P}^{G}(V_{I}(w · λ)′) ← Ind_{P}^{G}(V_{I}(λ)′) ← V (λ)′ ⊗_{K} i_{P}^{G} ← 0.

3. Isomorphism between simple objects

In this section we analyse when two simple modules of the shape F_{P_{I}}^{G} (M, v_{P_{J}}^{P_{I}}), with P_{I} maximal for M and a subset J ⊂ I, are isomorphic. This will be used in the next section for determining the multiplicities of composition factors of the locally analytic Steinberg representation. Let’s begin by recalling some additional notation of [OS2]. Let G_{0} be a split reductive group model of G over O_{L}. Let T_{0} ⊂ B_{0} ⊂ G_{0} be O_{L}-models of T and B. Fix a std psgp P_{0} and denote by U_{P,0} its unipotent radical. Let U_{P,0}^{−} be its opposite unipotent radical and denote by L_{P,0} the Levi factor containing T_{0}. Let G_{0} = G_{0}(O_{L}) ⊂ G, P_{0} = P_{0}(O_{L}) = G_{0} ∩ P ⊂ P etc. be the corresponding compact open subgroups consisting of O_{L}-valued points. We denote by I = p^{−1}(B_{0}(k_{L})) ⊂ G_{0} the standard Iwahori subgroup where p : G_{0} → G_{0}(k_{L}) is the reduction map. Consider now for an open subgroup H of G_{0} the distribution algebra D(H) := D(H, K) = C_{L}^{an}(H, K)′ which is defined by the dual of the locally convex K-vector space C_{L}^{an}(H, K) of locally L-analytic functions [ST2]. It has the structure of a Fréchet-Stein algebra. More precisely, for each \frac{1}{p} < r < 1, there is a multiplicative norm q_{r} on the Fréchet algebra D(H) such that if D(H)_{r} denotes the Banach algebra given by the completion of D(H) with respect to q_{r}, we have a topological isomorphism of algebras D(H) ≃ \underleftarrow{lim}_{r} D(H)_{r}. For the precise definition of a Fréchet-Stein algebra we refer to loc.cit. We set P_{H} = P_{0} ∩ H and let U(𝔤, P_{H}) be the subalgebra of D(H) generated by U(𝔤) and D(P_{H}). Let U(𝔤, P_{H})_{r} be the topological closure of U(𝔤, P_{H}) in D(H)_{r}. The notion of a coadmissible module on a Fréchet-Stein algebra is defined in [ST2, §3]. If M is such a coadmissible D(H)-module, it comes along with a family (q_{r,M})_{r} of seminorms such that if M_{r} denotes the completion of M with respect to q_{r,M}, we have M ≃ \underleftarrow{lim}_{r} M_{r} and each M_{r} is a finitely-generated D(H)_{r}-module.

Let M be a simple object of O_{alg}^{𝔭} As such an object it has naturally the structure of a U(𝔤, P_{0})-module. By [OS2, Section 4] there is a continuous D(G_{0})- isomorphism

D(G_{0}) ⊗_{U(𝔤,P_{0})} M ≃ F_{P}^{G}(M)′.

Let M = D(H) ⊗_{U(𝔤,P_{H})} M. Then M is a coadmissible D(H)-module in the above sense, cf. [OS2, Prop. 4.4] and M_{r} is given by M_{r} = D(H)_{r} ⊗_{U(𝔤,P_{H})}M. We denote by q_{r,M} the restriction of the seminorm q_{r,M} to M via the inclusion M ↪ M and by M_{r} the completion of M for q_{r,M}, which we may identify with U(𝔤, P_{H})_{r} ⊗_{U(𝔤,P_{H})}M. Thus we obtain a U(𝔤, P_{H})_{r}-equivariant map M_{r} → M_{r} giving rise to a D(H)_{r}-equivariant isomorphism

D(H)_{r} ⊗_{U(𝔤,P_{H})_{r}} M_{r} ⟶^{∼} M_{r}.

Hence we get a topological isomorphism

(3.1)  M ≃ \underleftarrow{lim}_{r} D(H)_{r} ⊗_{U(𝔤,P_{H})_{r}} M_{r}.

Let D be the category of topological U(𝔱)-modules M whose topology is metrizable, which are semi-simple with finite-dimensional eigenspaces and such that the topology can be defined by a family of norms (q_{r})_{r} such that

(3.2)  q_{r}(∑_{μ} x_{μ}) = sup_{μ} q_{r}(x_{μ}),

for x_{μ} ∈ M_{μ} in a decomposition M = ⨁_{μ∈Hom_{L}(𝔱,K)} M_{μ}. In this case, the completion M_{r} of M with respect to the norm q_{r} is given by

M_{r} = {∑_{μ} x_{μ} | q_{r}(x_{μ}) → 0 cofinite}.

A simple consequence of this description is the uniqueness of the expansion ∑_{μ} x_{μ} and the fact that (M_{r})_{μ} = M_{μ} for all μ ∈ Hom_{L}(𝔱, K). In particular, the U(𝔱)-eigenspaces of M_{r} are all finite-dimensional.

Lemma 3.1. If M is an object of D, each U(𝔱)-submodule (resp. quotient) of M with the induced (resp. quotient) topology is an object of D. Moreover, each U(𝔱)-module coming from O_{alg} is the image of an object of D under the functor forgetting the topology.

Proof. A U(𝔱)-submodule S of an object M in D is clearly contained in D. In particular, we have S = ⨁_{μ} S_{μ} with S_{μ} = S ∩ M_{μ}. Since the eigenspaces M_{ν} are finite-dimensional, each subspace S_{μ} is closed in M_{μ}. It follows that S is closed in M. As a consequence the quotient topology on M/S is metrizable. It is given by the induced family of quotient norms (\overline{q}_{r})_{r}. We have to check that condition (3.2) is satisfied for each such norm \overline{q}_{r}. For every ϵ > 0, there is an element y = ∑_{μ} y_{μ} ∈ S such that

\overline{q}_{r}(∑_{μ} x_{μ} + S) ≥ q_{r}(∑_{μ} x_{μ} + y) − ϵ

= sup_{μ} q_{r}(x_{μ} + y_{μ}) − ϵ

≥ sup_{μ} \overline{q}_{r}(x_{μ} + S) − ϵ.

Hence \overline{q}_{r}(∑_{μ} x_{μ} + S) ≥ sup_{μ} \overline{q}_{r}(x_{μ} + S). The other inequality is immediate since q̄_{r} is non-archimedian. As explained above, if M is an object of O_{alg}, then it has the structure of a topological metrizable U(𝔱)-module. It is a consequence of [K, Theorem 1.4.2] that each module of the form U(𝔤) ⊗_{U(𝔟)} W ≃ U(𝔲^{−}) ⊗_{K} W, with W a finitedimensional representation of B, is an object of D. But each object of O_{alg} is a quotient of some module of the shape U(𝔤) ⊗_{U(𝔟)} W, so that we deduce the last assertion. □

The following statement generalizes [OS1, Prop. 3.4.8] in the split case and is again a consequence of [Fe, 1.3.12] and the finiteness of U(𝔱)-eigenspaces in M_{r} when M is an object of D.

Proposition 3.2. Let M be an object of D. We have an inclusion preserving bijection

{closed U(𝔱)-invariant subspaces of M_{r}} ⟶^{∼} {U(𝔱)-invariant subspaces of M}.
S ⟼ S ∩ M

The inverse map is induced by taking the closure. In particular, any weight vector for the action of 𝔱 lies already in M.

Let U = U_{B} be the unipotent radical of our fixed Borel subgroup B ⊂ G and let 𝔲 = Lie U be its Lie algebra. Recall that if N is a Lie algebra representation of 𝔤, then H^{0}(𝔲, N) = {n ∈ N | 𝔵 · n = 0 ∀ 𝔵 ∈ 𝔲} denotes the subspace of vectors killed by 𝔲. This is a U(𝔱)-module which is an object of D if N ∈ O_{alg}.

Corollary 3.3. Let M be an object of O_{alg}. Then H^{0}(𝔲, M_{r}) = H^{0}(𝔲, M). In particular, H^{0}(𝔲, M_{r}) is finite-dimensional.

Proof. We clearly have H^{0}(𝔲, M_{r}) ∩ M = H^{0}(𝔲, M). As H^{0}(𝔲, M_{r}) is closed in M_{r} by the continuity of the action of 𝔤 and as H^{0}(𝔲, M) is finite-dimensional and therefore complete the statement follows by Proposition 3.2. □

Let V be additionally a smooth admissible representation of L_{P}. Below we compute the first U-homology resp. U-cohomology group of the various representations F_{P}^{G}(M, V). More precisely, we denote by \overline{H}_{0}(U, F_{P}^{G}(M, V)) the quotient of F_{P}^{G}(M, V) by the topological closure of the K-subspace generated by the elements ux − x for x ∈ F_{P}^{G}(M, V) and u ∈ U. It is the largest Hausdorff quotient of F_{P}^{G}(M, V) on which U acts trivially. On the other hand, H^{0}(𝔲, F_{P}^{G}(M, V)′) is a Fréchet space since it is a closed subspace of F_{P}^{G}(M, V)′.

Lemma 3.4. As Fréchet spaces we have H^{0}(𝔲, F_{P}^{G}(M, V)′) = H^{0}(𝔲, F_{P}^{G}(M)′) ˆ⊗_{K}V′. (Note that the tensor product topology on the right hand side is unambiguous, since both factors are Fréchet spaces).

Proof. Since the action of 𝔲 is trivial on V, there is an identification F_{P}^{G}(M, V) = F_{P}^{G}(M) ⊗_{K} V as 𝔲-modules. Now we write V = \underrightarrow{lim}_{n} V_{n} as the union of finite-dimensional K-vector spaces. Then F_{P}^{G}(M, V) = \underrightarrow{lim}_{n} F_{P}^{G}(M) ⊗_{K} V_{n} as locally convex K-vector spaces. By passing to the dual we get

F_{P}^{G}(M, V)′ = \underleftarrow{lim}_{n}(F_{P}^{G}(M) ⊗_{K} V_{n})′
= \underleftarrow{lim}_{n} F_{P}^{G}(M)′ ⊗_{K} V′_{n} = F_{P}^{G}(M)′ ˆ⊗_{K}V′.

For the first identity confer [Em, Prop. 1.1.22] resp. [S, Prop. 16.10], the second one follows as V_{n} is finite-dimensional, the third one is [Em, Prop. 1.1.29]. Now, the space H^{0}(𝔲, F_{P}^{G}(M)′) is the kernel of the map

d : F_{P}^{G}(M)′ → (F_{P}^{G}(M)′)^{dim 𝔲}

given by v ↦ (𝔵_{i}v) where (𝔵_{i}) is a basis of 𝔲. By the exactness of the tensor product − ⊗_{K} V′_{n} and the left exactness of the projective limit, the space H^{0}(𝔲, F_{P}^{G}(M)′) ˆ⊗_{K}V′ is the kernel of the map d ˆ⊗1 : F_{P}^{G}(M)′ ˆ⊗_{K}V′ → (F_{P}^{G}(M)′ ˆ⊗_{K}V′)^{dim 𝔲}. The result follows. □

Now we can prove the main result of this section.

Theorem 3.5. Let M be a simple object of O_{alg}^{𝔭} such that P is maximal for M, and let V be a smooth admissible representation of L_{P}. Let λ ∈ X^{∗}(T) be the highest weight of M, so that M ≃ L(λ). Then there are T -equivariant isomorphisms

(3.3)  H^{0}(U, F_{P}^{G}(M, V)′) = λ ⊗_{K} J_{U∩L_{P}} (V)′,

and

(3.4)  \overline{H}_{0}(U, F_{P}^{G}(M, V)) = λ′ ⊗_{K} J_{U∩L_{P}} (V),

where J_{U∩L_{P}} is the usual Jacquet functor for the unipotent subgroup U ∩ L_{P} ⊂ L_{P}.

Proof. The underlying topological space of F_{P}^{G}(M, V) is of compact type. As the category of locally convex vector spaces of compact type is stable by forming Hausdorff quotients, the underlying topological vector space of \overline{H}_{0}(U, F_{P}^{G}(M, V)) is reflexive. As H^{0}(U, F_{P}^{G}(M, V)′) is the topological dual of \overline{H}_{0}(U, F_{P}^{G}(M, V)), it is sufficient to prove the first isomorphism (3.3). Let’s begin by determining H^{0}(𝔲, F_{P}^{G}(M, V)′). The Iwahori decomposition G_{0} = ∐_{w∈W^{I}} IwP_{0} induces a decomposition

D(G_{0}) ⊗_{U(𝔤,P_{0})} M ≃ ⨁_{w∈W^{I}} D(I) ⊗_{U(𝔤,I∩wP_{0}w^{−1})} M^{w}.

Here M^{w} denotes the module M with the twisted action given by conjugation with w. For each w ∈ W^{I}, we have

H^{0}(𝔲, D(I) ⊗_{U(𝔤,I∩wP_{0}w^{−1})} M^{w}) ≃

≃ H^{0}(Ad(w^{−1})𝔲, D(w^{−1}Iw) ⊗_{U(𝔤,w^{−1}Iw∩P_{0})} ⊗M).

Now we apply the precedent discussion with H = w^{−1}Iw. Set 𝔫 = Ad(w^{−1})𝔲. First we consider the case w ≠ 1. By [OS1, Lemma 3.3.2], there is an Iwahori decomposition w^{−1}Iw = (U_{P,0}^{−} ∩ w^{−1}Iw)(P_{0} ∩ w^{−1}Iw), hence there is by [K, 1.4.2] a finite number of elements u in U_{P,0}^{−} such that D(H)_{r} = ⨁ δ_{u}·U(𝔤, P_{H})_{r}, so that

M_{r} ≃ ⨁_{u} δ_{u} ⊗ M_{r}

and the action of 𝔵 ∈ 𝔫 is given by

𝔵 · ∑ δ_{u} ⊗ m_{u} = ∑ δ_{u} ⊗ (Ad(u^{−1})𝔵)m_{u}.

Now we can find a non-trivial element 𝔵 ∈ 𝔲_{P}^{−} ∩ Ad(w^{−1})𝔲 = 𝔲_{P}^{−} ∩ 𝔫. Indeed, the set Φ^{−} ∩ w^{−1}(Φ^{+}) contains an element of Φ^{−}\Φ_{I}^{−}. For that, choose β ∈ Φ^{+}\Φ_{I}^{+} such that w^{−1}β ∈ Φ^{−}. This is possible since w ∉ W_{I}. Then we have w^{−1}β ∉ Φ_{I}^{−} since W^{I} is exactly the set of w such that w(Φ_{I}^{+}) ⊂ Φ^{+}. Now Ad(u^{−1})𝔵 ∈ 𝔲_{P}^{−} since u ∈ U_{P}^{−}. By [OS2, Corollary 8.6], elements of 𝔲_{P}^{−} act injectively on M, and arguing as in Step 1 of the proof of [OS2, Theorem 5.7], they act injectively on M_{r}, as well. We conclude that H^{0}(Ad(u^{−1})𝔫, M_{r}) = 0 and therefore H^{0}(𝔫, M_{r}) = 0. By identity (3.1), we get H^{0}(𝔫, M) = 0. Now consider the case w = 1. Again we may write D(I)_{r} = ⨁ δ_{u}U(𝔤, P_{0})_{r} for a finite number of u ∈ U_{P,0}^{−}, so that D(I)_{r} ⊗_{U(𝔤,P_{0})_{r}} M_{r} = ⨁_{u} δ_{u} ⊗ M_{r}. We will show that if u ∉ U_{P,0}^{−} ∩ U(𝔤, P_{0})_{r}, then H^{0}(Ad(u^{−1})𝔲, M_{r}) = 0. Here we will use Step 2 in the proof of [OS2, Theorem 5.7]. Let M̂ be the formal completion of M, i.e. M̂ = ∏_{μ} M_{μ} which is a 𝔤-module. The action of 𝔲_{P}^{−} can be extended to an action of U_{P}^{−} as explained in loc.cit. If 𝔵 ∈ 𝔤 and u ∈ U_{P}^{−}, the action of ad(u)𝔵 on M_{r} is the restriction of the composite u ∘ 𝔵 ∘ u^{−1} on M̂. As a consequence, we get

H^{0}(ad(u^{−1})𝔲, M_{r}) = M_{r} ∩ u^{−1} · H^{0}(𝔲, M̂).

But

H^{0}(𝔲, M̂) = H^{0}(𝔲, M) = Kv^{+} = K_{λ}

where v^{+} is a highest weight vector of M. So if H^{0}(ad(u^{−1})𝔲, M_{r}) ≠ 0, we have u^{−1}v^{+} ∈ M_{r}. By the proof of [OS2, Theorem 5.7], this gives a contradiction if u ∉ U_{P}^{−} ∩ U_{r}(𝔤, P_{0}). Hence by using identity (3.1), we obtain finally an isomorphism of Fréchet spaces

H^{0}(𝔲, D(I) ⊗_{U(𝔤,P_{0})} M) ≃ H^{0}(𝔲, M).

By combining the result above together with Lemma 3.4, we get thus an isomorphism

H^{0}(𝔲, F_{P}^{G}(M, V′)) ≃ H^{0}(𝔲, M) ⊗_{K} V′ ≃ K_{λ} ⊗_{K} V′

Now H^{0}(U, F_{P}^{G}(M, V)′) is a subspace of H^{0}(𝔲, F_{P}^{G}(M, V)′) and this latter one is stable by the action of U. Thus we deduce that

H^{0}(U, F_{P}^{G}(M, V)′) = H^{0}(U, H^{0}(𝔲, F_{P}^{G}(M, V)′))
= H^{0}(U, K_{λ} ⊗_{K} V′)
= K_{λ} ⊗_{K} J_{U∩L_{P}} (V)′.
□

Next we can formulate our result about intertwining between subquotients of the principal series. For this recall that by Lemma X.4.6 of [BoWa] the smooth induction i_{B}^{G} has Jordan-Hölder factors v_{P_{I}}^{G} indexed by subsets I ⊂ Δ which appear all with multiplicity one. Moreover, by X.3.2, (1) to (5), of [BoWa], if Z is an irreducible subquotient of i_{B}^{G}, then there is a smooth character δ of T such that Z is the unique non-zero irreducible subrepresentation of i_{B}^{G}(δ) and δ contributes to J_{U}(Z).

Corollary 3.6. Let L(λ_{1}) and L(λ_{2}) be two simple objects in the category O_{alg}. For i = 1, 2, let P_{i} be a std psgp maximal for L(λ_{i}) and let V_{i} be a simple subquotient of the smooth parabolic induction i_{B}^{P_{i}}. Then the two irreducible representations F_{P_{1}}^{G}(L(λ_{1}), V_{1}) and F_{P_{2}}^{G}(L(λ_{2}), V_{2}) are isomorphic if and only if P_{1} = P_{2}, V_{1} = V_{2} and λ_{1} = λ_{2}.

Proof. Suppose that there is a non-trivial isomorphism between the irreducible representations F_{P_{1}}^{G}(L(λ_{1}), V_{1}) and F_{P_{2}}^{G}(L(λ_{2}), V_{2}). Let δ_{2} be a smooth character of T such that V_{2} ↪ i_{B}^{P_{2}}(δ_{2}). By the sequence of embeddings

F_{P_{2}}^{G}(L(λ_{2}), V_{2}) ⊂ F_{P_{2}}^{G}(L(λ_{2}), i_{B}^{P_{2}}(δ_{2})) ≃ F_{B}^{G}(L(λ_{2}), δ_{2}) ⊂ Ind_{B}^{G}(λ′_{2} ⊗_{K} δ_{2}),

we obtain a non-trivial map F_{P_{1}}^{G}(L(λ_{1}), V_{1}) → Ind_{B}^{G}(λ′_{2} ⊗ δ_{2}) and by Frobenius reciprocity a non-trivial T -equivariant map

\overline{H}_{0}(U, F_{P_{1}}^{G}(L(λ_{1}), V_{1})) = λ′_{1} ⊗_{K} J_{U∩L_{P_{1}}} (V_{1}) → λ′_{2} ⊗ δ_{2}.

It follows that λ_{1} = λ_{2} and P_{1} = P_{2}. By [BoWa, X.3.2.(1)], we know that J_{U∩L_{P_{1}}} (V_{1}) is a direct sum of smooth characters of T. By Frobenius reciprocity, these are exactly the smooth characters δ such that V_{1} is an irreducible subobject of i_{B}^{P_{2}}(δ). As δ_{2} is one of them, we can conclude by [BoWa, X.3.2.(4)] that V_{1} = V_{2}. □

4. The locally analytic Steinberg representation

The section deals with the proof of our main theorem. Here we start with the proof of the acyclicity of the evaluated locally analytic Tits complex. As before let P = P_{I} be a std psgp attached to a subset I ⊂ Δ. Let

I_{P}^{G} = I_{P}^{G}(1) = Ind_{P}^{G}(1)

be the locally analytic G-representation induced from the trivial representations 1. If Q ⊃ P is another parabolic subgroup, we get an injection I_{Q}^{G} ↪ I_{P}^{G}. We denote by

V_{P}^{G} = I_{P}^{G}/ ∑_{Q⊋P} I_{Q}^{G}

the generalized locally analytic Steinberg representation associated to P. For P = G, we have V_{G}^{G} = I_{G}^{G} = 1. The next result is the locally analytic analogue of the classical situations, cf. [BoWa, Ch. X,4], [Leh], [DOR].

Theorem 4.1. Let I ⊂ Δ. Then the following complex is an acyclic resolution of V_{P_{I}}^{G} by locally analytic G-representations,

0 → I_{G}^{G} → ⨁_{{I⊂K⊂Δ \atop |Δ\K|=1}} I_{P_{K}}^{G} → ⨁_{{I⊂K⊂Δ \atop |Δ\K|=2}} I_{P_{K}}^{G} → ⋯ → ⨁_{{I⊂K⊂Δ \atop |K\I|=1}} I_{P_{K}}^{G} → I_{P_{I}}^{G} → V_{P_{I}}^{G} → 0.

Here the differentials d_{K′,K} : I_{P_{K′}}^{G} → I_{P_{K}}^{G} are defined as follows. We fix a total ordering on Δ. For two subsets K ⊂ K′ of Δ, we let

p_{K′,K} : I_{P_{K′}}^{G} ⟶ I_{P_{K}}^{G}

be the natural homomorphism induced by the surjection G/P_{K} → G/P_{K′}. For arbitrary subsets K, K′ ⊂ Δ, with |K′| − |K| = 1 and K′ = {k_{1} < … < k_{r}}, we put

d_{K′,K} = \begin{cases}(−1)^{i}p_{K′,K} & K′ = K ∪ \{k_{i}\} \\ 0 & K ⊄ K′ \end{cases} .

It is easy to check that p_{K′,K} is nothing else but the composite

I_{P_{K′}}^{G} = F_{P_{K′}}^{G} (M_{K′}(1), 1) ⟶^{F_{P_{K′}}^{G} (q_{K,K′} ,incl.)} F_{P_{K′}}^{G} (M_{K}(1), i_{P_{K}}^{P_{K′}}) ≃

≃ F_{P_{K}}^{G} (M_{K}(1), 1) ≃ I_{P_{K}}^{G} .

More generally, we will prove a variant of the above theorem. For this, let λ ∈ X_{+} be a dominant weight. For a std psgp P = P_{I} ⊂ G, we consider the finite-dimensional algebraic P -representation V_{I}(λ) = V_{P} (λ) with highest weight λ. We set

I_{P}^{G}(λ) := Ind_{P}^{G}(V_{P} (λ)′).

In particular, we get I_{G}^{G}(λ) = V (λ)′. If Q ⊂ G is another parabolic subgroup with P ⊂ Q, then there is a map

I_{Q}^{G}(λ) → I_{P}^{G}(λ)

similarly as above for V (λ) = 1. More precisely, by the transitivity of parabolic induction we deduce that I_{P}^{G}(λ) ≃ Ind_{Q}^{G}(Ind_{P}^{Q}(V_{P} (λ)′)). As V_{Q}(λ)′ is the space of algebraic vectors in Ind_{P}^{Q}(V_{P} (λ)′), we see that the above map is injective and has closed image.

We define analogously as above the twisted generalized locally analytic Steinberg representation by

V_{P}^{G}(λ) := I_{P}^{G}(λ)/ ∑_{Q⊋P} I_{Q}^{G}(λ).

Theorem 4.2. Let λ ∈ X_{+} and let I ⊂ Δ. Then the following complex is an acyclic resolution of V_{P_{I}}^{G}(λ) by locally analytic G-representations,

(4.1) 0 → I_{G}^{G}(λ) → ⨁_{{I⊂K⊂Δ \atop |Δ\K|=1}} I_{P_{K}}^{G} (λ) → ⨁_{{I⊂K⊂Δ \atop |Δ\K|=2}} I_{P_{K}}^{G} (λ) → …

⋯ → ⨁_{{I⊂K⊂Δ \atop |K\I|=1}} I_{P_{K}}^{G} (λ) → I_{P_{I}}^{G} (λ) → V_{P_{I}}^{G}(λ) → 0.

Proof. The proof is by induction on the semi-simple rank rk_{ss}(G) = |Δ| of G. We start with the case |Δ| = 1. Then the complex above coincides with

0 → I_{G}^{G}(λ) → I_{B}^{G}(λ) → V_{B}^{G}(λ) → 0

and the claim is trivial. Now, let |Δ| > 1. We consider for any subset K ⊂ Δ, the resolution (2.5) :

0 ← Ind_{P_{K}}^{G} (V_{K}(w · λ)′) ← ⨁_{{w∈^{K} W \atop ℓ(w)=ℓ(w^{K})−1}} Ind_{P_{K}}^{G} (V_{K}(w · λ)′) ← …

⋯ ← ⨁_{{w∈^{K} W \atop ℓ(w)=1}} Ind_{P_{K}}^{G} (V_{K}(w · λ)′) ← I_{P_{K}}^{G} (λ) ← i_{P_{K}}^{G} (λ) ← 0.

Here we set i_{P_{K}}^{G} (λ) := i_{P_{K}}^{G} ⊗_{K} V (λ)′. We abbreviate for any w ∈ ^{K}W and any integer i ≥ 0,

I_{P_{K}}^{G} (w) := Ind_{P_{K}}^{G} (V_{K}(w · λ)′),

I_{P_{K}}^{G} [i] := ⨁_{{w∈^{K} W \atop ℓ(w)=i}} I_{P_{K}}^{G} (w).

and

V_{P_{K}}^{G} (w) = I_{P_{K}}^{G} (w)/ ∑_{{K′⊋K \atop w∈^{K′} W}} I_{P_{K′}}^{G} (w).

The complexes above induce hence a double complex

0→ i_{G}^{G}(λ) → ⨁_{{I⊂K⊂Δ \atop |Δ\K|=1}} i_{P_{K}}^{G} (λ) →…→  ⨁_{{I⊂K⊂Δ \atop |K\I|=1}} i_{P_{K}}^{G} (λ)  →  i_{P_{I}}^{G} (λ)
∥  ↓  ↓  ↓
0→ I_{G}^{G}(λ) → ⨁_{{I⊂K⊂Δ \atop |Δ\K|=1}} I_{P_{K}}^{G} (λ) →…→  ⨁_{{I⊂K⊂Δ \atop |K\I|=1}} I_{P_{K}}^{G} (λ)  →  I_{P_{I}}^{G} (λ)
↓  ↓  ↓  ↓
0→ 0 → ⨁_{{I⊂K⊂Δ \atop |Δ\K|=1}} I_{P_{K}}^{G} [1] →…→  ⨁_{{I⊂K⊂Δ \atop |K\I|=1}} I_{P_{K}}^{G} [1]  →  I_{P_{I}}^{G} [1]
↓  ↓  ↓  ↓
0→ 0 → ⨁_{{I⊂K⊂Δ \atop |Δ\K|=1}} I_{P_{K}}^{G} [2] →…→  ⨁_{{I⊂K⊂Δ \atop |K\I|=1}} I_{P_{K}}^{G} [2]  →  I_{P_{I}}^{G} [2]
↓  ↓  ↓  ↓
⋮  ⋮  ⋮  ⋮
↓  ↓  ↓  ↓
0→ 0 →  0  →…→ ⨁_{{I⊂K⊂Δ \atop |K\I|=1}} I_{P_{K}}^{G} [ℓ(^{I}w) − 1] → I_{P_{I}}^{G} [ℓ(^{I}w) − 1]
↓  ↓  ↓  ↓
0→ 0 →  0  →…→  0  → I_{P_{I}}^{G} (^{I}w).

To prove the commutativity of this diagram, it suffices to prove the following fact. Let K′ ⊂ K ⊂ Δ with |K′| = |K| − 1. If C(K) and C(K′) are the BGG resolutions (2.5) obtained with P = P_{K} and P = P_{K′}, then the maps p_{K,K′} induce a morphism of complexes C(K) → C(K′). Let w ∈ ^{K}W. Choose w′ ≥ w with ℓ(w′) = ℓ(w) + 1 such that w′ ∈ ^{K′}W. Clearly, we have w ∈ ^{K′}W, so that we have to consider two cases, depending on whether w′ ∈ ^{K}W or not. If w′ ∈ ^{K}W, it is tautological from the definitions of i_{w′,w} and q_{K,K′} that the following diagram is commutative

\xymatrix{M_{K′}(w′ · λ) \ar[r]^{q_{K′,K}} \ar[d]^{i_{w′,w}} & M_{K}(w′ · λ) \ar[d]^{i_{w′,w}} \\ M_{K′}(w · λ) \ar[r]^{q_{K′,K}} & M_{K}(w · λ).}

If w′ ∉ ^{K}W, we claim that w′ = s_{α}w with K = K′ ∪ {α}. Namely, we must have ℓ(s_{α}w′) < ℓ(w′) and by the exchange condition a reduced expression of w′ is of the form w′ = s_{α}s_{2} ⋯ s_{r}. By [Hum3, Thm. 5.10], w is obtained as a a subexpression of this reduced expression. We have w ∈ ^{K}W so that a reduced expression of w cannot begin by s_{α}, so that w′ = s_{α}w. We conclude from this fact that M(w′ · λ) is included in the kernel of M(w · λ) → M_{K}(w · λ) and finally that the composite

M_{K′}(w′ · λ) ⟶^{i_{w′,w}} M_{K′}(w · λ) ⟶^{q_{w′,w}} M_{K}(w · λ)

is zero.

Applying the functor F_{P_{K′}}^{G}, we obtain the commutativity of the following square

\xymatrix{I_{P_{K}}^{G} (w · λ) \ar@{^{(}->}[r] \ar[d]^{F(i_{w′,w})} & I_{P_{K′}}^{G} (w · λ) \ar[d]^{F(i_{w′,w})} \\ I_{P_{K}}^{G} (w′ · λ) \ar@{^{(}->}[r] & I_{P_{K′}}^{G} (w′ · λ)}

where we have replaced I_{P_{K}}^{G} (w′ · λ) by 0 if w′ ∉ ^{K}W. This is enough to prove our assertion on BGG resolutions. For proving our theorem, it suffices now to show that each row of the big double complex, except from the second one, is exact apart from the very right hand side. Indeed, let

C^{p,q} = ⨁_{{I⊂K⊂Δ \atop |K\I|=−p}} I_{P_{K}}^{G} [q − 1]

(with convention that I_{P_{K}}^{G} [−1] = i_{P_{K}}^{G} (λ)). The double complex above is a second quadrant double complex. There are two spectral sequences converging towards the cohomology of the total complex associated to C^{·,·}. The first is E_{2,I}^{p,q} = H_{h}^{p}H_{v}^{q}(C^{·,·}) = 0. The second is E_{2,II}^{p,q} = H_{v}^{p}H_{h}^{q}(C^{·,·}). By our hypothesis, we have E_{2,II}^{p,q} = 0 except in the cases p = 0 or q = 1. Moreover, the term E_{2,II}^{0,0} is the kernel of the natural map v_{P_{I}}^{G} (λ) → V_{P_{I}}^{G}(λ) which is 0, this last fact can be shown using description of Jordan-Hölder factors of the I_{P_{K}}^{G} (λ) together with Corollary 3.6. Finally this second spectral sequence degenerates in E_{2} and we deduce that E_{1,II}^{p,1} = 0 for p ≤ −1, this shows that all the lines are exact apart form the very right hand side. Now let’s prove that each row, except from the second one, is exact apart from the very right hand side. The upper line satisfies this property, since it is tensor product of the (generalized) smooth Tits complex by the algebraic representation V (λ)′. For w ∈ ^{I}W, let I(w) = {α ∈ Δ | ℓ(s_{α}w) > ℓ(w)} ⊂ Δ be the unique maximal subset such that w ∈ ^{I(w)}W and set P_{w} = P_{I(w)}. Hence it suffices to show the exactness of the evaluated sequence

0 → I_{P_{w}}^{G}(w) → ⋯ → ⨁_{{I⊂K⊂I(w) \atop |K\I|=1}} I_{P_{K}}^{G} (w) → I_{P_{I}}^{G} (w) → V_{P_{I}}^{G}(w) → 0

for each w ∈ ^{I}W. This complex can be rewritten as follows. We have I_{P_{K}}^{G} (w) = Ind_{P_{w}}^{G}(I_{P_{K}}^{P_{w}}(w)). Since the induction functor Ind_{P_{w}}^{G} is exact, it suffices to show the exactness of the sequence

0 → I_{P_{w}}^{P_{w}}(w) → ⋯ → ⨁_{{I⊂K⊂I(w) \atop |K\I|=1}} I_{P_{K}}^{P_{w}}(w) → I_{P_{I}}^{P_{w}}(w) → V_{P_{I}}^{P_{w}}(w) → 0

where V_{P_{I}}^{P_{w}}(w) = I_{P_{I}}^{P_{w}}(w)/ ∑_{K⊋I,w∈^{K}W} I_{P_{K}}^{P_{w}}(w). But we have I_{P_{K}}^{P_{w}}(w) = I_{L_{w}∩P_{K}}^{L_{w}} (w) and rk_{ss}(L_{w}) < rk_{ss}(G). Thus the complex above is by induction hypothesis exact. Hence the claim of our theorem follows. □

As a byproduct of the proof we get the following result (Note that V_{P_{I}}^{G}(^{I}w) = I_{P_{I}}^{G} (^{I}w).):

Corollary 4.3. For any dominant weight λ ∈ Λ_{+} and any subset I ⊂ Δ, there is an acyclic complex

0 → v_{P_{I}}^{G} (λ) → V_{P_{I}}^{G}(λ) → ⨁_{{w∈^{I} W \atop ℓ(w)=1}} V_{P_{I}}^{G}(w) → ⋯ → ⨁_{{w∈^{I} W \atop ℓ(w)=ℓ(^{I} w)−1}} V_{P_{I}}^{G}(w) → V_{P_{I}}^{G}(^{I}w) → 0.

□

Example 4.4. Let G = GL_{2} and λ = 0. Then W = {1, s} and the above complex is given by

0 → v_{B}^{G} → V_{B}^{G} → I_{B}^{G}(s) → 0.

This complex was already considered by Morita [Mo] resp. Schneider and Teitelbaum [ST5] and coincides with the dual of the K-valued “de Rham” complex

0 → O(X)/K → Ω^{1}(X) → H_{dR}^{1}(X) → 0

of the Drinfeld half space X = ℙ^{1} \ ℙ^{1}(L).

Lemma 4.5. The representation V_{P}^{G}(λ) has a Jordan-Hölder series of finite length.

Proof. The representation V_{P}^{G}(λ) is a quotient of I_{P}^{G}(λ) which coincides by definition of the functor F_{P}^{G} with F_{P}^{G}(M_{P} (λ)). As explained in Section 1, the latter object has a finite Jordan-Hölder series, hence the same holds true for V_{P}^{G}(λ). □

¿From the previous lemma we see that a Jordan-Hölder series of V_{B}^{G}(λ) is induced by such a series of the induced representation I_{P}^{G}(λ). Now we are able to give a recipe for the determination of the composition factors with respect to V_{B}^{G}(λ). For two reflections w, w′ in W, let m(w′, w) ∈ ℤ_{≥0} be the multiplicity of the simple U(𝔤)-module L(w · 0) in the Verma module M(w′ · 0). It is known that m(w′, w) ≥ 1 if and only if w′ ≤ w for the Bruhat order ≤ on W. These multiplicities can be computed using Kazhdan-Lusztig polynomials, as it was conjectured by Kazhdan and Lusztig in [KaLu, Conj. 1.5] and proved independently by Beilinson-Bernstein [BeBe] and Brylinski-Kashiwara [BrKa]. Let’s recall that if w ∈ W is an element of the Weyl group, the support supp(w) of w is the set of simple reflections appearing in one (and so in any) reduced expression of w. In the following we identify the set of simple reflections with Δ. Then w ∈ W_{I} if and only if supp(w) ⊂ I. Recall from the proof of Theorem 4.2, that we defined for every w ∈ W a subset I(w) ⊂ Δ with w ∈ ^{I(w)}W and that the subset I(w) is maximal with this property. On the other hand, by [Hum2, p. 186 − 187], the simple module L(w · λ) lies in O^{𝔭_{I}} if and only if w ∈ ^{I}W. It follows that for w ∈ W, the parabolic Lie Algebra 𝔭_{I(w)} is maximal for L(w · λ).

Theorem 4.6. Fix w ∈ W and let I = I(w) ⊂ Δ be as above. For a subset J ⊂ Δ with J ⊂ I, let v_{P_{J}}^{P_{I}} be the generalized smooth Steinberg representation of L_{P_{I}}. Then the multiplicity of the irreducible G-representation F_{P_{I}}^{G} (L(w·λ), v_{P_{J}}^{P_{I}}) in V_{B}^{G}(λ) is

(4.2)  ∑_{{w′∈W \atop supp(w′)=J}} (−1)^{ℓ(w′)+|J|}m(w′, w),

and we obtain in this way all the Jordan-Hölder factors of V_{B}^{G}(λ). Moreover, this multiplicity is non-zero if and only if J ⊂ supp(w). In particular, the locally algebraic representation v_{B}^{G}(λ) is the only locally algebraic subquotient of V_{B}^{G}(λ). Proof. By Theorem 4.2, we get the following formula for the multiplicity of the simple object F_{P_{I}}^{G} (L(w · λ), v_{P_{J}}^{P_{I}}) in V_{B}^{G}(λ):

[V_{B}^{G}(λ) : F_{P_{I}}^{G} (L(w · λ), v_{P_{J}}^{P_{I}})] = ∑_{K⊂Δ}(−1)^{|K|}[I_{P_{K}}^{G} (λ) : F_{P_{I}}^{G} (L(w · λ), v_{P_{J}}^{P_{I}})].

By definition we have I_{P_{K}}^{G} (λ) = F_{P_{K}}^{G} (M_{K}(λ)). Since the Jordan-Hölder series of M_{K}(λ) lies in O_{alg}^{𝔭_{K}}, as well, we deduce by by Corollary 3.6 and the discussion in Section 2 that K ⊂ I is a necessary condition for the non-vanishing of the expression [I_{P_{K}}^{G} (λ) : F_{P_{I}}^{G} (L(w · λ), v_{P_{J}}^{P_{I}})]. On the other hand, v_{P_{J}}^{P_{I}} is a subquotient of i_{P_{K}}^{P_{I}} if and only if K ⊂ J. Again by Corollary 3.6 and Section 2, the term F_{P_{I}}^{G} (L(w · λ), v_{P_{J}}^{P_{I}}) appears in I_{P_{K}}^{G} (λ) only if K ⊂ J. So, let K ⊂ J and consider the JH-component Q = L(w · λ) of M_{K}(λ) in O_{alg}. Then F_{P_{K}}^{G} (Q, 1) = F_{P_{I}}^{G} (Q, i_{P_{K}}^{P_{I}}) by the PQ-formula. In a Jordan-Hölder series of F_{P_{I}}^{G} (Q, i_{P_{K}}^{P_{I}}) the term F_{P_{I}}^{G} (Q, v_{P_{J}}^{P_{I}}) appears with multiplicity one by Corollary 3.6, so that

[I_{P_{K}}^{G} (λ) : F_{P_{I}}^{G} (L(w · λ), v_{P_{J}}^{P_{I}})] = [M_{K}(λ) : L(w · λ)].

As a consequence, we have

[V_{B}^{G}(λ) : F_{P_{I}}^{G} (L(w · λ), v_{P_{J}}^{P_{I}})] = ∑_{K⊂J}(−1)^{|K|}[M_{K}(λ) : L(w · λ)].

Now we make use of the character formula

chM_{K}(λ) = ∑_{w′∈W_{K}}(−1)^{ℓ(w′)}chM(w′ · λ),

cf. [Hum2, Proposition 9.6]. We obtain

[V_{B}^{G}(λ) : F_{P_{I}}^{G} (L(w · λ), v_{P_{J}}^{P_{I}})] =

= ∑_{K⊂J}(−1)^{|K|} ∑_{w′∈W_{K}}(−1)^{ℓ(w′)}[M(w′ · λ) : L(w · λ)]

= ∑_{w′∈W}(−1)^{ℓ(w′)}[M(w′ · λ) : L(w · λ)] ∑_{supp(w′)⊂K⊂J}(−1)^{|K|}.

But the sum

∑_{supp(w′)⊂K⊂J}(−1)^{|K|} = (1 − 1)^{|J\supp(w′)|}

is non-zero if and only if supp(w′) = J, so we deduce the formula. The nonvanishing criterion follows from Corollary 5.3 in the appendix. □

Example 4.7. a) Let G = GL_{3}, Δ = {α_{1}, α_{2}}. Let s_{i} be the element of W corresponding to α_{i} and abbreviate P_{i} = P_{{α_{i}}}. In this case, m(w′, w) = 1 if and only if w′ ≤ w for the Bruhat order (cf. [Hum2, 8.3.(c)]). As a consequence, we obtain the following Jordan-Hölder factors all with multiplicity one :

v_{B}^{G}
F_{P_{1}}^{G}(L(s_{2} · 0), v_{B}^{P_{1}}), F_{P_{2}}^{G}(L(s_{1} · 0), v_{B}^{P_{2}})
F_{P_{1}}^{G}(L(s_{2}s_{1} · 0), 1), F_{P_{1}}^{G}(L(s_{2}s_{1} · 0), v_{B}^{P_{1}})
F_{P_{2}}^{G}(L(s_{1}s_{2} · 0), 1), F_{P_{2}}^{G}(L(s_{1}s_{2} · 0), v_{B}^{P_{2}})
F_{B}^{G}(L(s_{1}s_{2}s_{1} · 0), 1).

This particular result was already shown in [Schr].

b) Let G = GL_{4}. Here there are exactly two subquotients of V_{B}^{G} with multiplicity greater than one. We fix notation as follows. Let Δ = {α_{1}, α_{2}, α_{3}} such that s_{1} and s_{3} commute. We use the more compact notation P_{i} = P_{{α_{i}}}, P_{i,j} = P_{{α_{i},α_{j}}}, W^{i,j} = W^{{α_{i},α_{j}}} etc. The two subquotients with multiplicity greater than one are then F_{P_{1,3}}^{G}(L(s_{2}s_{3}s_{1}s_{2} · 0), v_{B}^{P_{1,3}}) and F_{P_{2}}^{G}(L(s_{1}s_{3}s_{2}s_{3}s_{1} · 0), v_{B}^{P_{2}}) and they have both multiplicity 2. The length of V_{B}^{G} is 50 and it contains 48 non-isomorphic Jordan-Hölder components. Let us remark that this example also shows that Kazhdan-Lusztig multiplicities are the reason for unexpected components in V_{B}^{G}. For instance, the component F_{P_{2}}^{G}(L(s_{1}s_{3}s_{2}s_{1}s_{3} · 0, 1)) appears in I_{P_{2}}^{G}, but passing to V_{B}^{G} kills only one of its two occurrences in I_{B}^{G}. Here is the idea to carry out this computation. The number of components of the form F_{P_{I}}^{G} (L(w · λ, V) for a fixed w ∈ W^{I} with I = I(w) maximal and V a smooth irreducible subquotient of i_{L_{I}∩B}^{L_{I}}, is exactly the number of subsets of I ∩ supp(w), that is 2^{|I∩supp(w)|}. Let W_{p}^{I} be the subset of W^{I} consisting of elements for which I is maximal, an easy computation shows that

(4.3)  |W_{p}^{I}| = ∑_{J⊂Δ\I}(−1)^{|J|}|W^{I∪J}| = ∑_{J⊂Δ\I}(−1)^{|J|}|W |/|W_{I∪J}|.

If I = ∅ or I = Δ, then |W_{p}^{I}| = 1 and |I ∩ supp(w)| = 0. Furthermore, we compute easily that

\begin{matrix}|W_{p}^{1,2}| = |W_{p}^{2,3}| = |W_{p}^{1}| = |W_{p}^{3}| = 3 \\ |W_{p}^{1,3}| = |W_{p}^{2}| = 5. \end{matrix}

We see for example that W_{p}^{1,2} = {s_{3}, s_{3}s_{2}, s_{3}s_{2}s_{1}} so that |{s_{1}, s_{2}} ∩ supp(w)| takes the values 0, 1 and 2 when w is varying in W_{p}^{1,2}. This procedure gives us the cardinality of the composition factors appearing in V_{B}^{G}. As for the multiplicities, we can use a software package like CHEVIE ([GHLMP]) to see that the only simple modules L(w · 0) having multiplicities greater than one in M(0) are given by w = s_{2}s_{1}s_{3}s_{2} and w = s_{1}s_{3}s_{2}s_{1}s_{3}.

5. Appendix

In this appendix, we prove the last assertion of Theorem 4.6 by interpreting the alternate sum (4.2) as the multiplicity of L(w · λ) in the H_{0} of a subcomplex of the BGG complex. Then we show that the submodule of M(λ) generated by the M(w · λ) with w a Coxeter element in W_{J} is a quotient of this H_{0}. We fix the same notation as before. Let λ ∈ X_{+} and recall from Section 1 that the BGG resolution of L(λ) = V (λ) takes the form

⋯ → C_{i}(𝔤) ⟶^{ϵ_{i}} C_{i−1}(𝔤) ⟶^{ϵ_{i−1}} ⋯ ⟶^{ϵ_{1}} C_{0}(𝔤) → L(λ) → 0

with

C_{i}(𝔤) = ⨁_{ℓ(w)=i} M(w · λ).

Let I be a subset of Δ. For i ≥ 0, let C_{i}(𝔤, I) ⊂ C_{i}(𝔤) be defined by

C_{i}(𝔤, I) := ⨁_{w∈W_{I},ℓ(w)=i} M(w · λ).

As ϵ_{i}(C_{i}(𝔤, I)) ⊂ C_{i−1}(𝔤, I), we obtain a subcomplex C_{•}(𝔤, I) of C_{•}(𝔤). Now we abbreviate W_{I}^{(i)} for the subset of W_{I} consisting of elements of length i.

Lemma 5.1. The complex C_{•}(𝔤, I) is isomorphic to the complex U(𝔤) ⊗_{U(𝔭_{I})} C_{•}(𝔩_{I}), hence it is exact in positive degrees.

Proof. We can write ϵ_{i}|_{C_{i}(𝔤,I)} = ∑_{w_{1},w_{2}} a_{w_{1},w_{2}}^{i}i_{w_{1},w_{2}} where the sum is over all

w_{1} ∈ W_{I}^{(i)} and w_{2} ∈ W_{I}^{(i−1)} such that w_{2} ≤ w_{1}. By [Ro, Lemma 10.2 ] we have a_{w_{1},w_{2}}^{i} ≠ 0 for each such pair (w_{1}, w_{2}). We fix injections U(𝔩_{I})⊗_{U(𝔟∩𝔩_{I})} K_{w·λ} ↪ U(𝔩_{I}) ⊗_{U(𝔟∩𝔩_{I})} K_{λ}, w ∈ W_{I}, giving rise to well defined maps

i_{w,w′}^{I} : U(𝔩_{I}) ⊗_{U(𝔟∩𝔩_{I})} K_{w·λ} ↪ U(𝔩_{I}) ⊗_{U(𝔟∩𝔩_{I})} K_{w′·λ}

such that after applying U(𝔤) ⊗_{U(𝔭_{I})} −, we have the equality id ⊗ i_{w,w′}^{I} = i_{w,w′}. Now consider the complex C̃_{•}(𝔩_{I}) where C̃_{•}(𝔩_{I}) = C_{•}(𝔩_{I}) and where the differential maps are given by ϵ̃_{i} = ∑_{w_{1}≤w_{2}∈W_{I}} a_{w_{1},w_{2}}^{i}i_{w_{1},w_{2}}^{I}, so that C_{•}(𝔤, I) = U(𝔤) ⊗_{U(𝔭_{I})} C̃_{•}(𝔩_{I}). As the Bruhat order on W_{I} is induced by the Bruhat order of W, the squares (see [Ro, Definition 10.3]) of W with elements in W_{I} are exactly the squares of W_{I}, so that C̃_{•}(𝔩_{I}) is a complex. By Corollary 10.7 or Lemma 10.5 of [Ro] the complex C̃_{•}(𝔩_{I}) is exact in positive degrees and even isomorphic to the BGG complex C_{•}(𝔩_{I}). □

For every integer i ≥ 0, let C_{•}^{≤i}(𝔤, I) be the subcomplex of C_{•}(𝔤, I) defined by

C_{j}^{≤i}(𝔤, I) = ⨁_{{∃J⊂I,|J|≤i \atop w∈W{(j) \atop J}}} M(w · λ)

and set \overline{C}_{•}(𝔤, I) = C_{•}(𝔤, I)/C_{•}^{≤|I|−1}(𝔤, I).

Proposition 5.2. The complex C_{•}^{≤i}(𝔤, I) is exact in degrees ≥ i + 1 and the complex \overline{C}_{•}(𝔤, I) in degree ≥ |I| + 1.

Proof. We will prove this proposition by induction on i. For i = 0, this is clear. Now we can apply induction hypothesis to the long exact sequence coming from the exact sequence of complexes

(5.1)  0 → C_{•}^{≤i−1}(𝔤, I) → C_{•}^{≤i}(𝔤, I) → ⨁_{J⊂I,|J|=i} \overline{C}_{•}(𝔤, J) → 0

(Here we use the identity W_{J_{1}} ∩W_{J_{2}} = W_{J_{1}∩J_{2}} for the appearance of the direct sum.). In the special case, where |I| = i, we can deduce from Lemma 5.1 that C_{•}(𝔤, I) is acyclic in strictly positive degrees, hence by induction on i that H_{j}(\overline{C}_{•}(𝔤, I)) = 0 for j > i. In general, when |I| ≠ i, either |I| < i and there is nothing to prove, or |I| > i. But for J ⊂ I with |J| = i, we already know that H_{j}(\overline{C}_{•}(𝔤, J)) = 0 for j > i by what precedes. Hence, using (5.1) and our induction hypothesis on i, we can conclude that C_{•}^{≤i}(𝔤, I) is acyclic in degree > i. The claim follows. □

Corollary 5.3. For I ⊂ Δ, let M_{I} be the submodule of M(λ) generated by the Verma modules M(w ·λ) where w is a Coxeter element in W_{I}. For w ∈ W, we have

(5.2)  [M_{I} : L(w · λ)] ≤ ∑_{supp(w′)=I}(−1)^{|I|+ℓ(w′)}[M(w′ · λ) : L(w · λ)].

Hence this sum is non-zero if and only if I ⊂ supp(w).

Proof. As the complex \overline{C}_{•}(𝔤, I) is exact in degree > |I| and \overline{C}_{j}(𝔤, I) = 0 for j < |I|, it is sufficient to prove that M_{I} is a quotient of H_{|I|}(\overline{C}_{•}(𝔤, I)). Remark that M′ = ϵ_{|I|}(⨁_{supp(w)=I,ℓ(w)=|I|} M(w)) is a quotient of H_{|I|}(\overline{C}_{•}(𝔤, I)). Then the image of M′ by ∑_{supp(w)=I,ℓ(w)=|I|−1} i_{w,1} is exactly M_{I}, and we deduce the result. It is not difficult to see that there exists a Coxeter element w′ of W_{I} such that w′ ≤ w if and only if I ⊂ supp(w). □

References

[BeBe] A. Beilinson and J. Bernstein, Localisation de 𝔤-modules, C. R. Acad. Sci. Paris 292, No. 1, 15–18 (1981).

[BGG] J. Bernstein, I. M. Gelfand, S. I. Gelfand, Differential operators on the base affine space and a study of 𝔤-modules, in ”Lie Groups ant Their Representations, Halsted Press, New York (1975).

[Bou] N. Bourbaki, Topologie générale, chapitre 9, Hermann, Paris (1974).

[BoWa] A. Borel, N. R. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Annals of Mathematics Studies, 94, Princeton University Press (1980).

[Br] C. Breuil, Invariant L et série spéciale p-adique, Ann. Sci. École Norm. Sup. (4) 37, no. 4, 559–610 (2004).

[BrKa] J. L. Brylinski and M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math. 64, No. 3, 387–410 (1981).

[Car] R.W. Carter, Finite groups of Lie type. Conjugacy classes and complex characters, Reprint of the 1985 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Ltd., Chichester (1993).

[Ca1] W. Casselman, On a p-adic vanishing theorem of Garland, Bull. Amer. Math. Soc. 80, 1001–1004 (1974).

[Ca2] W. Casselman, A new nonunitarity argument for p-adic representations, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28, no. 3, 907–928 (1982).

[DOR] J.-F. Dat, S. Orlik, M. Rapoport, Period Domains over Finite and padic Fields, Cambridge Tracts in Mathematics (No. 183), Cambridge University Press, Cambridge (2010).

[DM] F. Digne, J. Michel, Representations of finite groups of Lie type, London Mathematical Society Student Texts, 21, Cambridge University Press, Cambridge (1991).

[Em] M. Emerton, Locally analytic vectors in representations of locally p-adic analytic groups, To appear in Memoirs of the AMS.

[Fe] C.T. Féaux de Lacroix, Einige Resultate über die topologischen Darstellungen p-adischer Liegruppen auf unendlich dimensionalen Vektorräumen über einem p-adischen Körper, Schriftenreihe des Mathematischen Instituts der Universität Münster, 3. Serie, 23, Univ. Münster (1999).

[GHLMP] M. Geck, G. Hiss, F. Lübeck, G. Malle, and G. Pfeiffer. CHEVIE – A system for computing and processing generic character tables for finite groups of Lie type, Weyl groups and Hecke algebras. Appl. Algebra Engrg. Comm. Comput., 7:175–210 (1996).

[Hum1] J. E. Humphreys, The Steinberg representation, Bull. Amer. Math. Soc. (N.S.) 16, no. 2, 247–263 (1987).

[Hum2] J. E. Humphreys, Representations of semisimple Lie algebras in the BGG category O, Graduate Studies in Mathematics, American Mathematical Society (2008).

[Hum3] J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, Cambridge University Press (1990).

[KaLu] D. Kazhdan and G. Luszig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53, No.2, 165–184 (1979).

[K] J. Kohlhaase, Invariant distributions on p-adic analytic groups, Duke Math. J. 137, no. 1, p. 19–62 (2007).

[Ku] S. Kumar, Kac-Moody groups, their flag varieties and representation theory, Progress in Mathematics, 204, Birkhäuser Boston, Inc., Boston, MA (2002).

[Ja] J.C. Jantzen, Representations of algebraic groups, Pure and Applied Mathematics, Vol. 131, Academic Press (1987).

[Le] J. Lepowsky, Generalized Verma modules, the Cartan-Helgason theorem, and the Harish-Chandra homomorphism, J. Algebra 49, 470-495 (1977).

[Leh] G.I. Lehrer, The spherical building and regular semisimple elements, Bull. Austral. Math. Soc. 27, no. 3, 361–379 (1983).

[Mo] Y. Morita, Analytic representations of SL_{2} over a 𝔭-adic number field II, Automorphic forms of several variables, Progr. Math. 46, 282–297 (1984).

[O] S. Orlik, Equivariant vector bundles on Drinfeld’s upper half space, Invent. Math. 172, No.3, 585–656 (2008).

[OS1] S. Orlik, M. Strauch, On the irreducibility of locally analytic principal series representations, Representation Theory 14, 713-746, (2010).

[OS2] S. Orlik, M. Strauch, On the Jordan-Hölder series of locally analytic principal series representations, to appear in Journal of the AMS.

[Ro] A. Rocha-Caridi, Splitting criteria for 𝔤-modules induced from a parabolic and the Berňsteĭn-Gelfand-Gelfand resolution of a finitedimensional, irreducible 𝔤-module, Trans. Amer. Math. Soc. 262, No.2, 335–366 (1980).

[S] P. Schneider, Nonarchimedean functional analysis, Springer Monographs in Mathematics, Springer Verlag (2002).

[ST1] P. Schneider, J. Teitelbaum, An integral transform for p-adic symmetric spaces, Duke Math. J. 86, no. 3, 391–433 (1997).

[ST2] P. Schneider, J. Teitelbaum, Algebras of p-adic distributions and admissible representations, Invent. Math. 153, No.1, 145-196 (2003).

[ST3] P. Schneider, J. Teitelbaum, Locally analytic distributions and p-adic representation theory, with applications to GL_{2}, J. Amer. Math. Soc. 15, no. 2, 443–468 (2002).

[ST4] P. Schneider, J. Teitelbaum, U(𝔤)-finite locally analytic representations, Represent. Theory 5, p. 111–128 (2001).

[ST5] P. Schneider, J. Teitelbaum, p-adic boundary values, Cohomologies padiques et applications arithmétiques, I. Astérisque No. 278, 51–125 (2002).

[SICM] P. Schneider, Continuous representation theory of p-adic Lie groups, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 1261–1282 (2006).

[Schr] B. Schraen, Représentations localement analytiques de GL_{3}(ℚ_{p}), Annales Scientifiques de l’ É.N.S., vol. 44-1, pp 43 - 145, (2011).

[Sp] T. A. Springer, Reductive Groups, Proc. Symp. Pure Math. Vol 33, I, AMS, 3–28 (1979).

[T] J. Teitelbaum, Admissible analytic representations. An introduction to three questions, Talk at Harvard Eigensemester 2006.

Sascha Orlik Fachgruppe Mathematik und Informatik Bergische Universität Wuppertal Gaußstrasse 20 42119 Wuppertal Germany orlik@math.uni-wuppertal.de Benjamin Schraen Laboratoire de Mathématiques de Versailles UMR CNRS 8100 45 avenue des États Unis

- Bâtiment Fermat F–78035 Versailles Cedex France benjamin.schraen@math.uvsq.fr 672

Documenta Mathematica 19 (2014)