UNREFINED MINIMAL K-TYPES FOR p-ADIC GROUPS

Allen Moy and Gopal Prasad

Dedicated to Armand Borel

1. Introduction

Let G be a reductive group over a p-adic field k. The study of admissible representations of G(k) via their restriction to compact open subgroups was begun by Mautner, Shalika and Tanaka for groups of type A_{1}. In contrast to real reductive groups where the representation theory of a maximal compact subgroup K is given in terms of slight modifications to Cartan’s theory of the highest weight, the representation theory of a (compact) parahoric subgroup 𝒫 is quite complicated. There is still no comprehensive theory for classifying the irreducible representations. In the case of GL_{n}(k), Howe, in [8], defined the notion of an essential character of a filtration subgroup K_{i} of GL_{n}(𝔬) based on realizing characters of K_{i}/K_{j} (2i ≧ j) as cosets in M_{n}(k). In particular, Howe proved that any admissible representation of GL_{n}(k) has only finitely many essential characters.

In [12], based on work of Howe and the first author in regard to Hecke algebra isomorphisms, the first author gave a more precise formulation of Howe’s ideas and defined the notion of an unrefined minimal K-type as certain representations of parahoric filtrations subgroups 𝒫_{i} in terms of semisimple and nilpotent elements in M_{n}(k). It was conjectured in [12] that every irreducible admissible representation of GL_{n}(k) contained an unrefined minimal K-type. This conjecture was proved by Howe and the first author via a combinatorial argument and it was also shown that any two unrefined minimal K-types contained in an irreducible representation must be closely related, namely they must be associates of one another.

The term “minimal K-type” was used by the first author in analogy with a similar notion in the case of real groups. In both cases there is a measure of the depth of a representation. In the real case Vogan [16] defines the depth of an irreducible representation of K as the length of the highest weight with a rho shift. A minimal K-type of an admissible irreducible representation π is then defined as a K-type whose depth is minimal among all K-types occuring in π. In particular, in the real case, a representation of K can be a minimal K-type in one representation and not in another. In the p-adic case an unrefined minimal K-type is intrinsically defined (see 5.1), i.e. its definition depends only on the

Typeset by AMS-TEX

representation and not on how it sits in the restriction of an admissible representation relative to other representations of filtration subgroups. Because the definition is intrinsic, one must now prove that there is an unrefined minimal K-type in any irreducible admissible representation π. In both the real and p-adic setting a minimal or unrefined minimal Ktype is a very important constituent of a representation. In the real case Vogan has proved that a minimal K-type occurs with multiplicity one and used minimal K-types as an anchor in classifying the irreducible admissible representations. Unrefined minimal K-types and their refinements to “refined” minimal K-types should hopefully play a similar role in the classification of irreducible admissible representations of a p-adic group.

Based on insight obtained from the combinatorial proof with Howe, the first author discovered that the existence of unrefined minimal K-types would follow quite easily if certain cosets, which realize characters of filtrations subgroups, satisfy a descent property whenever a coset contains a nilpotent element. The descent property for M_{n}(k) was established in [5] and [9] and for Lie algebras of certain classical groups in [11]. The formulation of the descent property in terms of the Lie algebra leads to certain unnecessary restrictions. In the case of GL_{n}(k), the vector space M_{n}(k) is both naturally the Lie algebra and the dual of the Lie algebra. This double role of M_{n}(k) obscures the distinction between the Lie algebra and its dual.

In this paper, the existence of unrefined minimal K-types is established for all reductive p-adic groups (Theorem 5.2). We also prove that any two unrefined minimal K-types occuring in an irreducible admissible representation are associates of each other. Depth is defined in terms of the congruence level of a filtration subgroup and it is proved that an unrefined minimal K-type minimizes depth. To each point x in the Bruhat–Tits building of G/k there is a naturally attached filtration {𝒫_{x,r_{i}}} of the parahoric subgroup 𝒫_{x} which is the isotropy at x. These filtrations include as special cases the filtrations introduced in [14]. The point x also defines filtrations {𝔤_{x,r_{i}}} (resp. {𝔤_{x,−r_{i}}^{∗}}) of the Lie algebra 𝔤 (resp. its dual 𝔤^{∗}). In particular, cosets in the dual 𝔤^{∗} naturally parametrize characters of the abelian group 𝒫_{x,r_{i}}/𝒫_{x,r_{i+1}}, r_{i} > 0. The descent property can be formulated in the proper setting of the nilpotent, i.e. unstable, elements in the dual, and its truth implies the existence of unrefined minimal K-types.

The proof of existence of unrefined minimal K-types presented in this paper is conceptually different from the approach followed by earlier authors to prove the existence in particular cases. Our proofs are uniform and do not require explicit realizations of the reductive groups over local fields.

The existence and associativity properties of unrefined minimal K-type allows one to attach a nonnegative rational number ϱ(π) – the depth of an unrefined minimal K-type contained in π – to any irreducible admissible representation π. This number should be important for certain aspects of the representation π. Thus, it is quite natural to conjecture that Harish–Chandra’s local character expansion of the character Θ_{π} of π is valid for all regular g ∈ 𝒫_{x,ϱ(π)^{+}} for any point x in the Bruhat–Tits building of G/k.

Acknowledgements. The authors were supported in part by NSF grants DMS–9203933 and DMS–9204296.

2. Parahoric subgroups and their natural filtrations

The goal of this section is to introduce natural filtrations of any parahoric subgroup in terms of the (absolute) affine root system.

2.1. Let k be a nonarchimedean local field and K be a fixed maximal unramified extension of k. Let 𝔬 (resp. 𝔒) be the ring of integers in k (resp. K) and 𝔣 (resp. 𝔉) be the residue field of k (resp. K). Note that 𝔉 is an algebraic closure of 𝔣. Let Γ = Gal(K/k) be the Galois group of K/k; Γ has a natural identification with the Galois group of 𝔉/𝔣. We fix a uniformizing element ϖ in 𝔬. Let ω be the discrete valuation of K such that ω(K^{×}) = Z.

Let G be an absolutely quasi-simple, simply connected algebraic group defined over k. Let 𝔤 be the Lie algebra of G/k and 𝔤 = 𝔤 ⊗_{k} K. Recall that the Bruhat–Tits building of G/K is a contractible simplicial complex on which G(K) and the Galois group Γ operate by simplicial automorphisms. The Bruhat–Tits building of G/k is the set of points in the building (of G/K) fixed under Γ. For a point x of the Bruhat–Tits building of G/K, P_{x} will denote the isotropy subgroup at x in the natural action of G(K) on the building; P_{x} is a parahoric subgroup of G(K) and all the parahoric subgroups of G(K) arise this way. If x is fixed under Γ, then P_{x} is defined over k (i.e. it is Γ-stable) and we shall denote P_{x} ∩G(k) by 𝒫_{x}.

In the sequel when we say that a point lies in a particular simplex of the building, we shall mean that it lies in the interior of the simplex.

2.2. Let S be a maximal k-split torus of G and let T be a maximal K-split torus of G defined over k and containing S. According to the Bruhat–Tits theory such a torus T exists ([4: 5.1.12]). Since the residue field of K is algebraically closed, according to a well known result of Steinberg, G is quasi-split over K i.e., it contains a Borel subgroup defined over K. Equivalently, the centralizer Z of T in G is a (maximal) torus; it is defined over k since T is. Let N be the normalizer of T in G.

Let X^{∗}(T) be the group of characters of T, and X_{∗}(T) be the group of 1-parameter subgroups of T (recall that a 1-parameter subgroup of T is a rational homomorphism λ : GL_{1} → T). There is a nondegenerate pairing

⟨ , ⟩ : X^{∗}(T) × X_{∗}(T) → Z,

defined as follows: For χ ∈ X^{∗}(T) and λ ∈ X_{∗}(T), χ ∘ λ is an endomorphism of GL_{1}. Now End(GL_{1}) = Z and ⟨χ, λ⟩ is set to be equal to the integer corresponding to χ ∘ λ. Let V = X_{∗}(T) ⊗_{Z} R, and V^{∗} = X^{∗}(T) ⊗_{Z} R. Then V^{∗} is the vector space dual of V.

The apartment A associated with the torus T in the Bruhat–Tits building of G/K is an affine space under V. As T is defined over k, A is stable under the action of the Galois group Γ on the building of G/K. There is a natural transitive action of V on A by translations; for x ∈ A and v ∈ V, we shall denote the translate of x by v by x + v. In particular, if λ is a 1-parameter subgroup of T and r is a real number, then the translate of x ∈ A under rλ(∈ V) is denoted x + rλ. For a given x ∈ A, the subset {x + rλ | r ∈ R and λ ∈ X_{∗}(T)} is clearly dense in A.

Let F be the space of R-valued affine-linear functions on A. The function which takes the value 1 at all the points of A will be denoted by δ in the sequel.

2.3. Let Φ(⊂ X^{∗}(T)) be the set of roots of G with respect to T and Π be the basis of the root system determined by a Borel subgroup containing T and defined over the splitting field of T. For a root b ∈ Φ, let U_{b} be the corresponding root subgroup. It is a connected unipotent subgroup of G defined over K, normalized by Z, and of dimension ≦ 3. The Lie algebra of U_{b} consists of root spaces corresponding to the roots which are positive integral multiples of b. For b ∈ Φ, let G_{b} be the subgroup generated by U_{b} and U_{−b}; G_{b} is a simply connected semi-simple subgroup of G defined over K. Set Z_{b} = G_{b} ∩Z. Then Z_{b} is a maximal torus of G_{b} and Z is a direct product of the subtori Z_{a}, a ∈ Π. Let Z_{0}^{b} be the maximal bounded subgroup of Z_{b}(K) and for any positive integer s, let Z_{sδ}^{b} be the congruence subgroup described in [14: 2.6]. Let Z_{0} be the maximal bounded subgroup of Z(K); Z_{0} is a direct product of the subgroups Z_{0}^{a}, a ∈ Π.

2.4. The subgroup U_{ψ}. To every affine function ψ ∈ F with gradient b belonging to Φ one associates a subgroup U_{ψ} of U_{b}(K). (This subgroup has been denoted by X_{ψ} in [15].) As we shall need to make extensive use of this subgroup, for the reader’s convenience we reproduce its description from [15: 1.4] below. An equivalent description of the subgroup, which uses a “Chevalley basis”over the splitting field of the torus Z, is given in [14: §2].

The group X_{K}^{∗}(Z) of K-rational characters of Z can be identified with a subgroup of finite index of the character group X^{∗}(T). Let ν : Z(K) → V be the homomorphism defined by

χ(ν(z)) = −ω(χ(z)),

for z ∈ Z(K) and χ ∈ X_{K}^{∗}(Z). Then the maximal bounded subgroup Z_{0} of Z(K) is precisely the kernel of ν, and Λ := Z(K)/Z_{0} is a free abelian group whose rank is the K-rank of G (which is equal to dimV). The affine Weyl group N(K)/Z_{0} is an extension of the K-Weyl group N(K)/Z(K) by Λ. There is an extension of ν to a homomorphism of N(K) in the group of affine transformations of A. This extension will be denoted again by ν.

Let b ∈ Φ, and u ∈ U_{b}(K) − {1}. It is known ([2: §5]) that the intersection U_{−b}uU_{−b} ∩ N(K) consists of a single element m(u) whose image in the K-Weyl group is the reflection r_{b} associated with b. Hence r(u) := ν(m(u)) is an affine reflection whose gradient is r_{b}.

Let α(b, u) denote the affine function on A whose gradient is b and which vanishes on the hyperplane of points fixed by r(u). Now for any affine function ψ ∈ F with gradient b ∈ Φ, let

U_{ψ} = {u ∈ U_{b}(K) | u = 1 or α(b, u) ≧ ψ}.

Then U_{ψ} is a subgroup and it is normalized by Z_{0}. If b is a multipliable root (i.e. 2b is also a root), then U_{2ψ} ⊂ U_{ψ}.

2.5. An affine function ψ with gradient b ∈ Φ is called an affine root of G (relative to T and K) if U_{ψ} is not contained in U_{ψ+ϵ} · U_{2b}(K) (= U_{ψ+ϵ} if 2b is not a root) for any ϵ > 0. Let Ψ(⊂ F) be the set of affine-roots of G relative to T and K. As T is defined over k, there is a natural action of the Galois group Γ on Ψ. Now let x be a point of the apartment A, then the parahoric subgroup P_{x} is generated by Z_{0} and the subgroups U_{ψ}, ψ ∈ Ψ such that ψ(x) ≧ 0.

2.6. Filtration of parahoric subgroups. Let x ∈ A. There is a natural filtration of P_{x} defined as follows: For any nonnegative real number r, let P_{x,r} be the subgroup of P_{x} generated by Z_{nδ}^{a}, a ∈ Π and n ∈ Z, n ≧ r, and the U_{ψ}, for ψ ∈ Ψ such that ψ(x) ≧ r. It is obvious that P_{x,0} = P_{x}, and if s ≧ r, then P_{x,s} ⊂ P_{x,r}. For r ≧ 0, let P_{x,r^{+}} = ∪_{s>r}P_{x,s}. It follows easily from Lemmas 2.4 and 2.7 of [14] that for all r ≧ 0, P_{x,r} is a normal subgroup of P_{x}. In fact, for any nonnegative real numbers r and s, the commutator subgroup [P_{x,r}, P_{x,s}] is contained in P_{x,r+s}. This implies in particular that for all r > 0, P_{x,r}/P_{x,r^{+}} is abelian; moreover as [P_{x,0^{+}}, P_{x,r}] ⊂ P_{x,r^{+}}, the conjugation action of P_{x} = P_{x,0} on P_{x,r} induces an action of the group P_{x,0}/P_{x,0^{+}} on P_{x,r}/P_{x,r^{+}}. If x is fixed under Γ, i.e. if it is a point of the Bruhat–Tits building of G/k, then for all r ≧ 0, P_{x,r} is Γ-stable.

The apartment A′ corresponding to a maximal K-split torus T′ of G contains x if, and only if, T′(K) ∩ P_{x} is the maximal bounded subgroup of T′(K) (see [15: 3.6.1]), and if this is the case, then T′ (resp. A′) is conjugate to T (resp. A) under an element of P_{x}. This at once implies that the filtration introduced above is independent of the choice of the apartment containing x.

Now let y = g · x, g ∈ G(K), be a point of the Bruhat–Tits building of G/K. Then P_{y} = gP_{x}g^{−1}. For r ≧ 0, we set P_{y,r} = gP_{x,r}g^{−1}. Then P_{y,r} depends only on r and y and not on the choice of the conjugating element g. Note that if x and y are points contained in the same simplex, then P_{x} = P_{y} and P_{x,0^{+}} = P_{y,0^{+}}; however, for r > 0, P_{x,r} may not in general be equal to P_{y,r}.

For a parahoric subgroup P = P_{x} defined over k, the group P_{x}/P_{x,0^{+}} has a natural identification with the group of 𝔉-rational points of a connected reductive 𝔣-group (see 3.2 below). The pro-nil radical R_{n}(P) of P is by definition the subgroup P_{x,0^{+}}. The pro-nil radical R_{n}(𝒫) of the parahoric subgroup 𝒫 = P ∩ G(k) of G(k) is R_{n}(P) ∩ G(k).

2.7. Standard parahoric subgroups. We fix a chamber (i.e., a simplex of maximal dimension) C lying in A, which is stable under Γ. Then given a point y of the Bruhat–Tits building of G/K, there is a unique point x in the closure \overline{C} of C (\overline{C} consists of points lying in C and all its faces) and an element g ∈ G(K) such that y = g · x. If y is fixed under Γ, then so is x and the element g can be chosen to be k-rational. The isotropy I of C in G(K) is an Iwahori subgroup defined over k. I determines a basis Δ of the affine-root system Ψ. An affine root ψ ∈ Ψ is said to be positive (ψ > 0) if it is a nonnegative integral linear combination of roots in Δ. Equivalently, ψ > 0 if, and only if, U_{ψ} ⊂ I. An affine root ψ is said to be negative (ψ < 0) if −ψ is positive. As I is defined over k, Δ is stable under the action of the Galois group Γ on the affine root system Ψ. For a subset Θ of Δ, let P_{Θ} be the subgroup of G(K) generated by I and the U_{−θ}, θ ∈ Θ. Then P_{Θ} is a parahoric subgroup of G(K) and any parahoric subgroup (of G(K)) containing I equals P_{Θ} for a unique Θ ⊂ Δ. The parahoric subgroup P_{Θ} is defined over k if, and only if, Θ is stable under Γ. Any parahoric subgroup P of G(K) is conjugate (in G(K)) to a unique P_{Θ}, moreover if P is defined over k, then the unique P_{Θ} to which it can be conjugated is also defined over k, and then P and P_{Θ} are in fact conjugate to each other under an element of G(k). The parahoric subgroups P_{Θ}, for Θ ⊂ Δ, will be referred to as the standard parahoric subgroups. These are the isotropy subgroups of points lying in \overline{C}.

Let x be a point of \overline{C} and let Θ = \{θ ∈ Δ | θ(x) = 0\}, then P_{x} = P_{Θ}. Since x ∈ \overline{C}, if for an affine root ψ, ψ(x) > 0, then ψ is positive. This implies that P_{x,r} is contained in the Iwahori subgroup I for all r > 0.

It is obvious that for a suitable point x in the simplex fixed by a parahoric subgroup P, the filtration subgroups P_{x,r} of P = P_{x}, defined above, coincide with the filtration subgroups introduced in [14: 2.14].

3. The associated filtrations of the Lie algebra 𝔤 and its dual 𝔤^{∗}

Every point of the Bruhat–Tits building of G/k determines a natural filtration of the Lie algebra 𝔤 and its dual 𝔤^{∗}. The purpose of this section is to describe these filtrations.

3.1. If G/K is not a triality form of type ^{6}D_{4}, let L be the smallest Galois extension of K over which G splits. If G/K is a triality form of type ^{6}D_{4}, let L be a fixed extension of K of degree 3 contained in the Galois extension of K, of degree 6, over which G splits. There are three such extensions, all isomorphic to each other over K. We fix a uniformizing element ϖ_{L} of L. Let ℓ = [L : K]. Then ϖ_{L}^{ℓ}/ϖ is a unit.

For a root b ∈ Φ, let L_{b} equal K or L according as b is a long or a short root. (If all the roots of the root system Φ are of equal length, then by convention all roots are long; if Φ is nonreduced, then all divisible roots are long and all nondivisible roots are short.) Let ϖ_{b} = ϖ if L_{b} = K and ϖ_{b} = ϖ_{L} if L_{b} = L. We shall denote the ring of integers of L_{b} by 𝔬(L_{b}) and let ℓ_{b} = [L : L_{b}]. If ψ is an affine root whose gradient is b, set L_{ψ} = L_{b}, ϖ_{ψ} = ϖ_{b} and ℓ_{ψ} = ℓ_{b}; then for an integer n, ψ + nδ is an affine root if, and only if, n is a multiple of ℓ_{ψ} (cf. [14: §2]). We shall denote the affine root ψ + ℓ_{ψ}δ by ψ^{+}.

If b ∈ Φ is a nonmultipliable root, then U_{b}(K) is isomorphic to L_{b}. Also if b is a multipliable root, then U_{b}(K)/U_{2b}(K) is isomorphic to L_{b} as well. For any affine root ψ, U_{ψ}/U_{ψ^{+}} is isomorphic to 𝔬(L_{ψ})/ϖ_{ψ}𝔬(L_{ψ}); see [14: §2].

3.2. Let x be a Γ-invariant point of \overline{C}. We shall now describe the filtration of the Lie algebra 𝔤 associated with x.

The Bruhat–Tits theory associates a smooth affine 𝔬-group scheme ℐ (resp. 𝒢_{x}) to I (resp. P_{x}) whose generic fiber ℐ ⊗_{𝔬} k (resp. 𝒢_{x} ⊗_{𝔬} k) is G and whose group of 𝔒-rational points is P_{x} (resp. I). Let L(ℐ) and L(𝒢_{x}) be the Lie algebras of ℐ and 𝒢_{x} respectively. Since the generic fibers of both ℐ and 𝒢_{x} are equal to G, we have L(ℐ) ⊗_{𝔬} k = 𝔤 = L(𝒢_{x}) ⊗_{𝔬} k. In particular, both L(ℐ) and L(𝒢_{x}) are lattices in 𝔤. We denote them by 𝔦 and 𝔤_{x} respectively. Set

\begin{aligned}𝔦 & = 𝔦 ⊗_{𝔬} 𝔒 \\ 𝔤_{x} & = 𝔤_{x} ⊗_{𝔬} 𝔒. \end{aligned}

The inclusion of I into P_{x} induces an 𝔬-group scheme homomorphism of ℐ into 𝒢_{x}; we use it to identify 𝔦 with a 𝔬-Lie subalgebra of 𝔤_{x} and 𝔦 with a 𝔒-Lie subalgebra of 𝔤_{x}. Note that 𝔤_{x} is itself a lattice in the Lie algebra 𝔤 = 𝔤 ⊗_{k} K.

The special fiber 𝒢_{x}⊗_{𝔬}𝔣 of 𝒢_{x} is a connected algebraic group defined over the residue field 𝔣; it admits a Levi decomposition over 𝔣. Let M_{x} be the quotient of 𝒢_{x} ⊗_{𝔬} 𝔣 by its unipotent radical. Then P_{x,0}/P_{x,0^{+}} has a natural identification with the group of 𝔉-rational points of the reductive group M_{x}.

Let 𝔷 ⊂ 𝔤 be the Lie algebra of Z, and let

𝔷 = 𝔷 ⊗_{k} K.

For any b ∈ Φ, let 𝔷_{b} ⊂ 𝔷 be the subalgebra corresponding to the subgroup Z_{b} and let 𝔤_{b} be the root subspace in 𝔤 corresponding to b.

It is known that for each b ∈ Φ, both

𝔦^{b} = 𝔦 ∩ 𝔤_{b} and 𝔷_{0}^{b} = 𝔦 ∩ 𝔷_{b}

are “canonically” isomorphic to 𝔬(L_{b}) as 𝔒-modules. For any nonnegative integral multiple n of ℓ_{b}, denote the 𝔒-submodule of 𝔦^{b} (resp. 𝔷_{0}^{b}) corresponding to the submodule ϖ_{b}^{n/ℓ_{b}}𝔬(L_{b}) of 𝔬(L_{b}) by 𝔦_{n}^{b} (resp. 𝔷_{n}^{b}). Let ψ be an affine root, b be its gradient, and let ψ_{b} be the smallest positive affine root with gradient b. Then ψ − ψ_{b} = nδ; where n is a multiple of ℓ_{b}. Define the 𝔒-submodule

𝔲_{ψ} = \{ \begin{matrix}𝔦_{n}^{b} if n ≧ 0 \\ ϖ^{n} 𝔦_{−n(ℓ−1)}^{b} if n < 0. \end{matrix}

If n is a negative integral multiple of ℓ_{b}, set 𝔷_{n}^{b} = ϖ^{n} 𝔷_{−n(ℓ−1)}^{b}. It is clear that the 𝔲_{ψ}’s and the 𝔷_{n}^{b}’s do not depend on the choice of the uniformizing elements ϖ of K and ϖ_{L} of L.

Now for any real number r, let 𝔷_{r} be the 𝔒-submodule of 𝔷 spanned by the 𝔷_{n}^{a}’s, a ∈ Π, and n ∈ ℓ_{a}Z with n ≧ r. Let 𝔤_{x,r} be the 𝔒-submodule of 𝔤 spanned by 𝔷_{r} and the 𝔲_{ψ}’s for affine roots ψ such that ψ(x) ≧ r. It is obvious that 𝔤_{x,0} = 𝔤_{x}, and for r ≧ 0, 𝔤_{x,r} is an ideal in 𝔤_{x}. For any r, 𝔤_{x,r} is stable under the action of the Galois group Γ and the adjoint action of P_{x} on 𝔤.

For r ≦ s, we have 𝔤_{x,r} ⊃ 𝔤_{x,s}. Set 𝔤_{x,r^{+}} = ∪_{s>r}𝔤_{x,s}. The induced action of P_{x,0^{+}} on

𝔤_{x,r}/𝔤_{x,r^{+}} is trivial, so there is a natural action of M_{x}(𝔉) = P_{x,0}/P_{x,0^{+}} on it. Note that for all r, ϖ𝔤_{x,r} = 𝔤_{x,r+ℓ} and so 𝔤_{x,r^{+}} ⊃ ϖ𝔤_{x,r}.

For all r > 0, there is a natural isomorphism of P_{x,r}/P_{x,r^{+}} with 𝔤_{x,r}/𝔤_{x,r^{+}} which is Γ × M_{x}(𝔉) equivariant.

The 𝔬-submodule of 𝔤_{x,r}(resp. 𝔤_{x,r^{+}}) consisting of the elements fixed under Γ will be denoted by 𝔤_{x,r}(resp. 𝔤_{x,r^{+}}) in the sequel.

3.3. Now let y be a point of the Bruhat–Tits building of G/k which is conjugate to x and choose g ∈ G(k) such that y = g · x. We define

𝔤_{y,r} = Adg(𝔤_{x,r}) and 𝔤_{y,r} = Adg(𝔤_{x,r}).

Then since P_{x} keeps 𝔤_{x,r} stable, it is clear that 𝔤_{y,r} and 𝔤_{y,r} depend only on y and r and not on the choice of g(∈ G(k)). If y ∈ A, it can be conjugated in \overline{C} by an element of N(k), and hence 𝔤_{y,r} is the 𝔒-submodule of 𝔤 spanned by 𝔷_{r} and the 𝔲_{ψ}’s for affine roots ψ such that ψ(y) ≧ r.

3.4. To every point x of the Bruhat–Tits building of G/K, we associate a monotone increasing sequence of nonnegative numbers as follows: If x is in \overline{C}, let \{r_{i}|i ∈ N ∪ \{0\}\} be the set of nonnegative values assumed by the affine functions in Ψ ∪ {nδ|n ∈ N} at x arranged in a monotone increasing sequence. Now to an arbitrary point x of the building of G/K, we associate the sequence associated to the (unique) point of \overline{C} conjugate to x. Then for r_{i−1} < s ≦ r_{i}, we have P_{x,s} = P_{x,r_{i}} and 𝔤_{x,s} = 𝔤_{x,r_{i}}. In particular, P_{x,r_{i}^{+}} = P_{x,r_{i+1}} and 𝔤_{x,r_{i}^{+}} = 𝔤_{x,r_{i+1}}.

3.5. Filtration of the dual. Let 𝔤^{∗} = Hom_{k}(𝔤, k) and 𝔤^{∗} = Hom_{K}(𝔤, K). There is a natural action of the Galois group Γ on 𝔤^{∗} by semi-linear automorphisms, and 𝔤^{∗} is the k-subspace of 𝔤^{∗} of elements fixed under Γ. In this paper we shall always view 𝔤^{∗} as a G-module under the coadjoint action of G. An element X of 𝔤^{∗} will be called nilpotent (resp. semi-simple), if in the coadjoint action, the G-orbit through X contains zero in its closure (resp. is closed) in the Zariski-topology. As confusion is unlikely, we shall denote the coadjoint action also by Ad. Let

𝔤^{∗} = 𝔤_{0}^{∗} ⊕ ⨁_{b∈Φ}𝔤_{b}^{∗}

be the weight space decomposition of 𝔤^{∗} with respect to T. Here,

𝔤_{b}^{∗} = {X ∈ 𝔤^{∗} | Ad(t)X = b(t)X for all t ∈ T }.

Note that for X ∈ 𝔤_{b}^{∗} and Y ∈ 𝔤_{c}, if X(Y) ≠ 0, then b = −c. Thus, we can identify 𝔤_{0}^{∗} with the dual 𝔷^{∗} = Hom(𝔷, K) of 𝔷, and for every b ∈ Φ, 𝔤_{b}^{∗} with the dual of 𝔤_{−b}. For an affine root ψ, with gradient b, let

𝔲_{ψ}^{∗} = { X ∈ 𝔤_{b}^{∗} | X(Y) ∈ 𝔒 for all Y ∈ 𝔲_{−ψ}}.

Let x be a Γ-invariant point of the apartment A. For r ∈ R, let

𝔤_{x,r^{+}}^{•} = {X ∈ 𝔤^{∗} | X(Y) ∈ 𝔒 for all Y ∈ 𝔤_{x,r^{+}}}.

For t ∈ R, let

𝔷_{t}^{∗} = {X ∈ 𝔷^{∗} | X(Y) ∈ 𝔒 for all Y ∈ 𝔷_{s} and s > −t − ℓ},

and let 𝔤_{x,t}^{∗} be the 𝔒-submodule of 𝔤^{∗} spanned by 𝔷_{t}^{∗} and the 𝔲_{ψ}^{∗}’s for affine roots ψ such that ψ(x) ≧ t + (ℓ − ℓ_{ψ}). Then

𝔤_{x,r^{+}}^{•} = 𝔤_{x,−r−ℓ}^{∗} ,

and

(∗) 𝔤_{x,−r}^{∗} = ϖ𝔤_{x,−r−ℓ}^{∗} = {X ∈ 𝔤^{∗} | X(Y) ∈ ϖ𝔒 for all Y ∈ 𝔤_{x,r^{+}}}.

From the above description it is clear that for all r, 𝔤_{x,−r}^{∗} is stable under the action of the Galois group Γ and also under the coadjoint action of the parahoric subgroup P_{x} on

𝔤^{∗}. For r ≧ s, 𝔤_{x,−r}^{∗} ⊃ 𝔤_{x,−s}^{∗}. Set 𝔤_{x,−r^{+}}^{∗} = ∪_{s<r}𝔤_{x,−s}^{∗}. Then for all r, 𝔤_{x,−r^{+}}^{∗} ⊃ ϖ𝔤_{x,−r}^{∗}.

The induced action of P_{x,0^{+}} on 𝔤_{x,−r}^{∗}/𝔤_{x,−r^{+}}^{∗} is trivial and hence there is a natural action of M_{x}(𝔉) = P_{x,0}/P_{x,0^{+}} on it.

The 𝔬-submodule of 𝔤_{x,−r}^{∗} (resp. 𝔤_{x,−r^{+}}^{∗}) consisting of the elements fixed under Γ will be denoted by 𝔤_{x,−r}^{∗} (resp. 𝔤_{x,−r^{+}}^{∗}).

3.6. If y is a point of the Bruhat–Tits building of G/k which is conjugate to x under an element g of G(k), we set

𝔤_{y,−r}^{∗} = Adg(𝔤_{x,−r}^{∗}) and 𝔤_{y,−r}^{∗} = Adg(𝔤_{x,−r}^{∗}).

Then 𝔤_{y,−r}^{∗} and 𝔤_{y,−r}^{∗} are well defined i.e., they depend only on y and r and not on the choice of the conjugating element g.

3.7. Now let x be a point of the Bruhat–Tits building of G/k and let {r_{i}} be the sequence associated to it in 3.4. Then 𝔤_{x,r_{j}^{+}} = 𝔤_{x,r_{j+1}} for all j. For i ≧ 1, the 𝔣-bilinear map

\begin{matrix}𝔤_{x,−r_{i}}^{∗}/𝔤_{x,−r_{i−1}}^{∗} × 𝔤_{x,r_{i}}/𝔤_{x,r_{i+1}} & ⟶ 𝔣 \\ (X, Y) & ⟼ X(Y) mod ϖ𝔬 \end{matrix}

is a nondegenerate M_{x}(𝔣)-invariant pairing. This nondegenerate pairing composed with a fixed nontrivial character 𝔣 ⟶ C^{×} provides an M_{x}(𝔣)-equivariant isomorphism of the Pontrjagin dual of 𝔤_{x,r_{i}}/𝔤_{x,r_{i+1}} with 𝔤_{x,−r_{i}}^{∗}/𝔤_{x,−r_{i−1}}^{∗}.

3.8. Let 𝒫_{x} = P_{x} ∩ G(k) and for r ≧ 0, 𝒫_{x,r} = P_{x,r} ∩ G(k), 𝒫_{x,r^{+}} = P_{x,r^{+}} ∩ G(k). Then for r > 0, the natural isomorphism of P_{x,r}/P_{x,r^{+}} with 𝔤_{x,r}/𝔤_{x,r^{+}} gives an isomorphism of 𝒫_{x,r}/𝒫_{x,r^{+}} onto 𝔤_{x,r}/𝔤_{x,r^{+}} which is M_{x}(𝔣)-equivariant (cf. [14: 2.24]). Thus the above also provides an identification of the Pontrjagin dual of 𝒫_{x,r_{i}}/𝒫_{x,r_{i+1}} with 𝔤_{x,−r_{i}}^{∗}/𝔤_{x,−r_{i−1}}^{∗}. The character of 𝒫_{x,r_{i}}/𝒫_{x,r_{i+1}} corresponding to the coset 𝔛 = X + 𝔤_{x,−r_{i−1}}^{∗} will be denoted by χ_{𝔛}. We shall say that this character is nondegenerate if the coset X + 𝔤_{x,−r_{i−1}}^{∗} does not contain any nilpotent elements.

4. Unstable elements in integral representations

4.1. Let P be a parahoric subgroup of G(K) and T be a maximal K-split torus such that the apartment corresonding to T in the Bruhat–Tits building of G/K contains the simplex whose isotropy is P, or, equivalently, P contains the maximal bounded subgroup T_{0} of T (K). We assume that both P and T are defined over k and further that T contains a maximal k-split torus S of G.

Let 𝒯 be the smooth affine 𝔬-group scheme with generic fiber T and which is diagonalizable over 𝔒. (The ring of regular function of 𝒯 is (𝔒[X^{∗}(T)])^{Γ}; where 𝔒[X^{∗}(T)] is the group ring of the character group X^{∗}(T) with coefficients in 𝔒.) The group of 𝔒-rational points of 𝒯 is T_{0}. Let 𝒮 be the closed 𝔬-subgroup scheme of 𝒯 corresponding to the subtorus S of T. Let S = 𝒮 ⊗_{𝔬} 𝔣 and T = 𝒯 ⊗_{𝔬} 𝔣. Then S is the maximal 𝔣-split subtorus of T. There is a canonical identification of the character group X^{∗}(S) (resp. X^{∗}(T)) of S (resp. T) with the character group X^{∗}(S) (resp. X^{∗}(T)) of S (resp. T), and of the group X_{∗}(S) (resp. X_{∗}(T)) of 1-parameter subgroups of S (resp. T) with the group X_{∗}(S) (resp. X_{∗}(T)) of 1-parameter subgroups of S (resp. T).

Let V be a free 𝔬-module of finite rank and ρ : 𝒯 → GL(V) be a rational representation of 𝒯 defined over 𝔬. Then ρ is completely reducible over 𝔒 i.e., V ⊗_{𝔬} 𝔒 is a direct sum of the weight submodules (see [6: Proposition on p.177]). ρ induces a rational representation ρ_{k} of T = 𝒯 ⊗_{𝔬} k on V ⊗_{𝔬} k and a representation \overline{ρ} of T = 𝒯 ⊗_{𝔬} 𝔣 on V ⊗_{𝔬} 𝔣. The weights of ρ_{k} and \overline{ρ} are the same if we use the canonical identification of X^{∗}(T) with X^{∗}(T).

Now let 𝒢 be the smooth affine 𝔬-group scheme associated to P by the Bruhat–Tits theory. The generic fiber 𝒢 ⊗_{𝔬} k of 𝒢 is G/k, the group of its 𝔒-rational points is P and the special fiber \overline{𝒢} = 𝒢 ⊗_{𝔬} 𝔣 is a connected algebraic 𝔣-group which admits a Levi decomposition defined over 𝔣. Let M be the quotient of \overline{𝒢} by its unipotent radical. Then M is a reductive 𝔣-group. There is a natural embedding of T in M, defined over 𝔣, which corresponds to the inclusion of T_{0} in P. In this embedding, T is a maximal torus of M and S(⊂ T) is a maximal 𝔣-split torus.

4.2. In the rest of this section we shall use the notation introduced in §§2, 3. Thus x is a Γ-invariant element of \overline{C}, P_{x} = P_{Θ} is the associated parahoric subgroup, and 𝒢_{x} is the 𝔬-group scheme associated with the parahoric subgroup P_{x}. Denote by M_{x} the quotient of the special fiber 𝒢_{x} ⊗_{𝔬} 𝔣 (of 𝒢_{x}) by its unipotent radical; M_{x} is a connected reductive group defined over 𝔣. The filtration lattices 𝔤_{x,r} (resp. 𝔤_{x,s}^{∗}) are free 𝔬-modules. The adjoint (resp. coadjoint) action of 𝒢_{x} on 𝔤_{x,r} (resp. 𝔤_{x,s}^{∗}) can be placed in the context of a rational representation ρ : 𝒢_{x} ⟶ GL(V) of 𝒢_{x} on a finite rank free 𝔬-module V. We also denote by ρ, the extended representation of G = 𝒢_{x} ⊗_{𝔬} k on V ⊗_{𝔬} k. In what follows, we shorten ρ(g)X to either g · X or gX. As usual, we shall say that an element X ∈ V is unstable if the Zariski closure of the orbit G · X contains zero. If k is of characteristic zero and X ∈ V is an unstable element then, according to a result of Kempf [10], there exists a 1-parameter subgroup λ : GL_{1} → G, defined over k, such that Lim_{t→0} λ(t)X = 0.

Nilpotent elements contained in 𝔤_{x,r} (resp. 𝔤_{x,s}^{∗}) are unstable for the adjoint (resp. coadjoint) action of 𝒢_{x}.

4.3. Proposition. Suppose that ρ : 𝒢_{x} ⟶ GL(V) is a rational representation of 𝒢_{x} on a free finite rank 𝔬-module V and W is a 𝒢_{x}(𝔬)-submodule containing ϖV. Assume that the induced action of 𝒢_{x} ⊗_{𝔬} 𝔣 on V/W has the unipotent radical acting trivially so that ρ gives rise to a representation of M_{x} on V/W. Let X ∈ V be such that there is a 1-parameter subgroup λ : GL_{1} ⟶ G, defined over K, so that

Lim_{t→0} λ(t)X = 0.

Then the M_{x}-orbit through the image \overline{X} of X in V/W contains zero in its closure.

Proof. Let Q be a special maximal parahoric subgroup of G(K) containing the Iwahori subgroup I. Let

𝔓_{λ} = { g ∈ G | Lim_{t→0} λ(t)gλ(t)^{−1} exists in G }.

Then 𝔓_{λ} is a parabolic subgroup of G defined over K. Since by Iwasawa decomposition ([15: 3.3.2]), G(K) = Q · 𝔓_{λ}(K), and for any p ∈ 𝔓_{λ}(K), Lim_{t→0} pλ(t)p^{−1}X = 0, after

replacing the 1-parameter subgroup λ by a conjugate under an element of 𝔓_{λ}(K), we may, and we shall, assume that it is contained in the maximal K-split torus T′ := qT q^{−1}; where q is an element of Q. Then T′ ⊂ 𝔓_{λ}.

Let 𝒬 be the 𝔒-group scheme associated to the parahoric subgroup Q by the Bruhat– Tits theory. The generic fiber 𝒬 ⊗_{𝔒} K of 𝒬 is G/K and the special fiber Q := 𝒬 ⊗_{𝔒} 𝔉 is a connected algebraic 𝔉-group. The group of 𝔒-rational points of 𝒬 is Q and the reduction “mod 𝔭”map Q → Q(𝔉) is known to be surjective. Let R_{u}(Q) be the unipotent radical of Q and M_{Q} = Q/R_{u}(Q). Now observe that the image of I ⊂ Q under the composite of the reduction map Q → Q(𝔉) and the natural projection Q(𝔉) → M_{Q}(𝔉) is the group of 𝔉-rational points of a Borel subgroup B of M_{Q}; the image of 𝔓_{λ}(K) ∩ Q is the group of 𝔉-rational points of a parabolic subgroup P (of M_{Q}) and the image of T′(K) ∩ Q is the group of 𝔉-rational points of a maximal torus T′ contained in P (recall that T′ ⊂ 𝔓_{λ} and as T′ = qT q^{−1} with q in Q, T′(K) ∩ Q is the maximal bounded subgroup of T′(K)). By the Bruhat decomposition, the intersection B ∩ P contains a maximal torus T′′ ([1: 14.13]). Then T′′ is conjugate to T′ in P. Fix a p ∈ 𝔓_{λ}(K) ∩ Q such that the image of p in M_{Q}(𝔉) conjugates T′ to T′′ and let T′′ = pT′p^{−1}. Then as T′′ ⊂ B, the maximal bounded subgroup of T′′(K) is contained in I which in turn is contained in P_{x}. This implies that T′′ embedds in a natural way in M_{x} as a maximal 𝔉-torus. Let λ′′ be the conjugate of the 1-parameter subgroup λ under p. Then as p is in 𝔓_{λ}(K), Lim_{t→0} λ′′(t)X = 0. On reduction

λ′′ gives a 1-parameter subgroup \overline{λ′′} of T′′ such that Lim_{t→0} \overline{λ′′}(t)\overline{X} = 0. Hence T′′ · \overline{X}, and

so a fortiori M_{x} · \overline{X}, contains zero in its closure.

Notation. For an affine root φ, we let φ̇ denote its gradient, then φ̇ ∈ Φ(⊂ X^{∗}(T)). For λ ∈ X_{∗}(T), we set ⟨φ, λ⟩ = ⟨φ̇, λ⟩.

4.4. Proposition. Let V, W, ρ and X ∈ V be as in the preceding proposition. Then there is a p ∈ 𝒫_{x} = P_{x} ∩ G(k) and a 1-parameter subgroup μ : GL_{1} ⟶ S(⊂ G), defined over k, such that

(1) Lim_{t→0} μ(t)(pX) ≡ 0 (mod W),

(2) for all θ ∈ Θ, ⟨θ, μ⟩ ≧ 0.

Proof. Let \overline{X} denote the projection of X on V/W. Then according to the preceding proposition, the closure of M_{x} · \overline{X} contains zero. Now in view of a result of Kempf [10], there is a 1-parameter subgroup λ : GL_{1} → M_{x} defined over the finite field 𝔣 such that Lim_{t→0} λ(t)X = 0. Let T be the maximal 𝔣-torus of M_{x} corresponding to T; it contains a maximal 𝔣-split torus S of M_{x}. Now we recall that the set consisting of restrictions to S of the gradients of the affine roots belonging to Θ is a basis of the root system of M_{x} with respect to S (X^{∗}(S) identified with X^{∗}(S) here), cf. [15: 3.5.1]. By conjugacy of maximal 𝔣-split tori under M_{x}(𝔣) and the fact that the 𝔣-Weyl group of M_{x} acts transitively on the set of Weyl chambers in X_{∗}(S), we conclude now that there is a conjugate λ′ of λ, under an element of M_{x}(𝔣), such that λ′ is contained in X_{∗}(S)(⊂ X_{∗}(M_{x})) and ⟨θ, λ′⟩ ≧ 0 for all θ ∈ Θ. Let μ be the lift of λ′ to S. Then μ has the desired properties. To determine p ∈ 𝒫_{x}, use the fact that the natural map 𝒫_{x} → M_{x}(𝔣) is surjective.

5. Unrefined minimal K-types and the main theorem

Given a point x of the Bruhat–Tits building of G/k, let P_{x} be the associated parahoric subgroup. Let 𝒫_{x} = P_{x} ∩ G(k) and for r ≧ 0, let 𝒫_{x,r} = P_{x,r} ∩ G(k). Let the sequence {r_{i}} be as in 3.4. For i ≧ 1, realize the characters of 𝒫_{x,r_{i}}/𝒫_{x,r_{i+1}} as in 3.8.

5.1. Definition. An unrefined minimal K-type is a pair (𝒫_{x,r}, χ), where x is a point of the Bruhat–Tits building of G/k, r is a nonnegative real number, χ is a representation of 𝒫_{x,r} trivial on 𝒫_{x,r^{+}} and

i) if r = 0, then χ is a cuspidal representation of 𝒫_{x,0}/𝒫_{x,0^{+}} inflated to 𝒫_{x}(= 𝒫_{x,0}) ii) if r > 0, then 𝒫_{x,r} ≠ 𝒫_{x,r^{+}} and χ is a nondegenerate character of 𝒫_{x,r}/𝒫_{x,r^{+}}.

In the remainder of the paper, we shall drop the adjective “unrefined”. The nonnegative number r is called the depth of the minimal K-type. Let y be another point of the Bruhat-Tits building of G/k, and let {s_{j}} be the monotone increasing sequence of nonnegative real numbers associated to it as in 3.4. Then two minimal K-types (𝒫_{x,r_{i}} , χ) and (𝒫_{y,s_{j}} , ξ), are said to be associates if they have the same depth (i.e. r_{i} = s_{j}), and

i) in the case of zero depth, there is an element g ∈ G(k) so that 𝒫_{x} ∩ 𝒫_{gy} surjects onto both M_{x}(𝔣) and M_{gy}(𝔣) and χ is isomorphic to Ad(g)ξ.

ii) in the case of positive depth, the G(k) orbit of the coset X + 𝔤_{x,−r_{i−1}}^{∗}, which realizes

χ, intersects the coset Y + 𝔤_{y,−s_{j−1}}^{∗}, which realizes ξ. We can now formulate the main theorem.

5.2. Theorem. Assume that k is of characteristic zero. Given an irreducible admissible complex representation (π, V_{π}) of G(k), there is a nonnegative rational number ϱ(π) with the following properties.

(1) For some x in the Bruhat–Tits building of G/k, the space V_{π}^{𝒫_{x,ϱ(π)^{+}}} of 𝒫_{x,ϱ(π)^{+}}- fixed vectors is nonzero and ϱ(π) is the smallest number with this property.

(2) For any y in the Bruhat–Tits building, if W = V_{π}^{𝒫_{y,ϱ(π)^{+}}} ≠ {0}, then

i) if ϱ(π) = 0, any irreducible 𝒫_{y,ϱ(π)}-submodule of W contains a minimal Ktype of depth zero of a parahoric Q ⊂ 𝒫;

ii) if ϱ(π) > 0, any irreducible 𝒫_{y,ϱ(π)}-submodule of W is a minimal K-type.

Moreover, any two minimal K-types contained in π are associates of each other.

6. Three key propositions

6.1. Optimal points. Let

Σ = {ψ ∈ Ψ | ψ > 0 and ψ − ℓδ < 0}.

Then Σ is a finite set. For any nonempty subset 𝔖 of Σ, let f_{𝔖} be the real valued function on \overline{C} defined as follows:

f_{𝔖}(x) = min\{ψ(x) − (ℓ − ℓ_{ψ}) | ψ ∈ 𝔖\} for x ∈ \overline{C}.

Now for each nonempty Γ-stable subset 𝔖 of Σ, we fix an element x_{𝔖} of \overline{C} such that

i) the functionf_{𝔖} takes its maximum value at x_{𝔖}, ii) ψ(x_{𝔖}) is a rational number for all affine roots ψ, iii) x_{𝔖} is fixed under Γ.

Note that the set of points where f_{𝔖} takes its maximum value is the intersection of \overline{C} with a Γ-stable hyperplane of A which is defined over Q. Hence there exit elements in \overline{C} satisfying the above conditions. We shall call x_{𝔖} an optimal point for the subset 𝔖 of Σ. Let 𝒪 be the (finite) set consisting of the optimal points x_{𝔖}’s, for all Γ-stable nonempty subsets 𝔖 of Σ.

Finding optimal points explicitly is a problem of linear programming. For the basic results and techniques of this theory see [13]. That there exists a point x_{𝔖} such that the function f_{𝔖} takes its maximum value at x_{𝔖}, and for all affine roots ψ, ψ(x_{𝔖}) is rational, also follows from an observation on page 33 of [13].

Remark. It can be shown that if G = SL_{n}, then given any nonempty subset 𝔖 of Σ, the barycenter of a suitable face of C is an optimal point for 𝔖.

6.2. We say that a subset Ξ of the affine root system Ψ is bounded below if there exists an integer n such that for all ψ ∈ Ξ, ψ +ℓnδ > 0. Now let Ξ be a nonempty Γ-stable subset of Ψ which is bounded below and let n be the smallest integer such that every root in the set Ξ^{+} := {ψ + ℓnδ | ψ ∈ Ξ} is positive. To Ξ we associate the subset 𝔖(Ξ) := Σ ∩ Ξ^{+} of Σ, and set x_{Ξ} = x_{𝔖(Ξ)}; where Σ, and for a Γ-stable nonempty subset 𝔖 of Σ, the optimal point x_{𝔖}(∈ \overline{C}), are as in 6.1.

6.3. Proposition. Assume that k is of characteristic zero. Let x be a Γ-invariant point of \overline{C}. Let r be a real number such that 𝔤_{x,r}^{∗} properly contains 𝔤_{x,r^{+}}^{∗} and let X ∈ 𝔤_{x,r}^{∗} be a nilpotent element. Then there is a p ∈ 𝒫_{x} = P_{x} ∩ G(k) and a y ∈ 𝒪 such that for some s > r

pX + 𝔤_{x,r^{+}}^{∗} ⊂ 𝔤_{y,s}^{∗} ,

or, equivalently, X + 𝔤_{x,r^{+}}^{∗} ⊂ 𝔤_{p^{−1}y,s}^{∗}.

Proof. From Proposition 4.4 applied to the case where V = 𝔤_{x,r}^{∗} and W = 𝔤_{x,r^{+}}^{∗}, we conclude that there is a p ∈ 𝒫_{x} and a 1-parameter subgroup μ : GL_{1} → S defined over k such that Lim_{t→0} μ(t)pX ≡ 0 (mod 𝔤_{x,r^{+}}^{∗}) and ⟨θ, μ⟩ ≧ 0 for all θ ∈ Θ. Denote pX by Y.

Then Y ≡Y_{0} + ΣY_{ψ} (mod 𝔤_{x,r^{+}}^{∗}); where Y_{0} ∈ 𝔷_{r}^{∗}, Y_{ψ} ∈ 𝔲_{ψ}^{∗}, and the summation is over the set of affine roots ψ such that ψ(x) = r + (ℓ − ℓ_{ψ}). Now let 𝔖 be the set of affine roots ψ such that ψ(x) = r + (ℓ − ℓ_{ψ}) and Y_{ψ}≢0 (mod 𝔤_{x,r^{+}}^{∗}). As X is k-rational, 𝔖 is Γ-stable.

Since μ is a 1-parameter subgroup contained in S, the coadjoint action of μ on 𝔷^{∗} is trivial. From this and the fact that Lim_{t→0} μ(t)Y ≡ 0 (mod 𝔤_{x,r^{+}}^{∗}), we conclude that

(1) Y_{0} = 0

(2) ⟨ψ, μ⟩ > 0 for ψ ∈ 𝔖. Now, for ϵ ≧ 0, consider the element x + ϵμ ∈ A. As x is Γ-invariant and μ is defined over k, the point x + ϵμ is Γ-invariant for all ϵ. Recall that for any affine root θ, θ(x + ϵμ) = θ(x) + ϵ⟨θ, μ⟩. Now,

(1) if θ ∈ Θ, then θ(x) = 0 and ⟨θ, μ⟩ ≧ 0 so that θ(x + ϵμ) ≧ 0 for all ϵ ≧ 0,

(2) if θ ∈ Δ − Θ, then θ(x) > 0 and hence for all sufficiently small ϵ ≧ 0, we have

θ(x + ϵμ) ≧ 0. Therefore, for all sufficiently small ϵ ≧ 0, x + ϵμ ∈ \overline{C}.

For ψ ∈ 𝔖 and ϵ > 0 we have ψ(x + ϵμ) = ψ(x) + ϵ⟨ψ, μ⟩ > ψ(x) (= r + (ℓ − ℓ_{ψ})) since ⟨ψ, μ⟩ > 0. Also, for the affine roots ψ such that ψ(x) > r + (ℓ − ℓ_{ψ}), it is clear that ψ(x + ϵμ) > r + (ℓ − ℓ_{ψ}) for all sufficiently small positive ϵ. Hence, if z = x + ϵμ, where ϵ is a sufficiently small positive number, we can find a real number u > r such that 𝔤_{z,u}^{∗} ⊃ Y + 𝔤_{x,r^{+}}^{∗}.

Let Ξ = 𝔖 ∪ {ψ | 𝔲_{ψ}^{∗} ⊂ 𝔤_{x,r^{+}}^{∗}}. Then Ξ is clearly Γ-stable and bounded below. Take y = x_{Ξ}; where x_{Ξ} is as in 6.2. Then as y is an optimal point for the subset Ξ, there exists a s, s ≧ u > r such that 𝔤_{y,s}^{∗} ⊃ Y + 𝔤_{x,r^{+}}^{∗}. This proves the proposition.

The following is a converse to the above proposition.

6.4. Proposition. Let x be an element of the Bruhat–Tits building of G/k, and r be a real number such that 𝔤_{x,r}^{∗} contains 𝔤_{x,r^{+}}^{∗} properly. Suppose X ∈ 𝔤_{x,r}^{∗} is such that the coset X + 𝔤_{x,r^{+}}^{∗} does not contain any nilpotent elements. Then for all y in the building, 𝔤_{y,s}^{∗} ∩ (X + 𝔤_{x,r^{+}}^{∗}) = ∅ for s > r.

Proof. We argue by contradiction. Assume there is a y in the building and a real number s > r such that 𝔤_{y,s}^{∗} ∩ (X + 𝔤_{x,r^{+}}^{∗}) ≠ ∅. After replacing x, y and X by their conjugates under an element of G(k), we assume that both x and y lie on A. Replacing X by X + Z for a suitable Z ∈ 𝔤_{x,r^{+}}^{∗}, we further assume that X ∈ 𝔤_{y,s}^{∗}. Let X = X_{0} + ΣX_{ψ}; where X_{0} ∈ 𝔷_{s}^{∗}, X_{ψ} ∈ 𝔲_{ψ}^{∗} and the summation is over a (finite) set of affine roots ψ with distinct gradients and such that ψ(x) ≧ r + (ℓ − ℓ_{ψ}). Let 𝔖 be the set of affine roots ψ such that ψ(x) = r + (ℓ − ℓ_{ψ}) and X_{ψ}≢0 (mod 𝔤_{x,r^{+}}^{∗}). Then, as 𝔤_{y,s}^{∗} ∩ 𝔷^{∗} = 𝔷_{s}^{∗} = 𝔤_{x,s}^{∗} ∩ 𝔷^{∗}, and s > r, we conclude that X_{0} ∈ 𝔤_{x,r^{+}}^{∗}. Hence X ≡ Σ_{ψ∈𝔖}X_{ψ} (mod 𝔤_{x,r^{+}}^{∗}). Since X, and so also X_{ψ}, is in 𝔤_{y,s}^{∗}, ψ(y) ≧ s+(ℓ−ℓ_{ψ}) > r+(ℓ−ℓ_{ψ}) for every affine root ψ such that X_{ψ} ≠ 0. Now as the set of elements in A of the form x+uλ, for u ∈ R and λ ∈ X_{∗}(T), is dense, we can find an u > 0 and a 1-parameter subgroup λ defined over k such that ψ(x + uλ) > r + (ℓ − ℓ_{ψ}) for all ψ ∈ 𝔖. But as ψ(x + uλ) = ψ(x) + u⟨ψ, λ⟩ = r + (ℓ − ℓ_{ψ}) + u⟨ψ, λ⟩, it follows that ⟨ψ, λ⟩ > 0 for all ψ ∈ 𝔖. This implies that Lim_{t→0} λ(t)Σ_{ψ∈𝔖}X_{ψ} = 0. Thus Σ_{ψ∈𝔖}X_{ψ} is nilpotent and we have shown that X is congruent to a nilpotent element modulo 𝔤_{x,r^{+}}^{∗}. This contradiction completes the proof of the proposition.

6.5. Proposition. Let x and y be two points of the Bruhat–Tits building of G/k and let r and s be real numbers so that 𝔤_{x,r} ⊃ 𝔤_{y,s}. Then P_{x,r} ⊃ P_{y,s} and hence, 𝒫_{x,r} ⊃ 𝒫_{y,s}.

Proof. After replacing x and y by their conjugates under a suitable element of G(k), we assume that both x and y lie in the apartment A. Recall that P_{x,r} (resp. P_{y,s}) is generated by the subgroups U_{ψ}, ψ ∈ Ψ such that ψ(x) ≧ r, and the subgroups Z_{nδ}^{a}, a ∈ Π, n ≧ r (resp. by the subgroups U_{ψ}, ψ ∈ Ψ such that ψ(y) ≧ s, and the subgroups Z_{nδ}^{a}, a ∈ Π, n ≧ s). As 𝔤_{x,r} contains 𝔤_{y,s}, 𝔤_{x,r} contains 𝔤_{y,s}. Now since 𝔲_{ψ} ⊂ 𝔤_{x,r} (resp. 𝔲_{ψ} ⊂ 𝔤_{y,s}) if, and only if, ψ(x) ≧ r (resp. ψ(x) ≧ s) and 𝔷_{n}^{a} ⊂ 𝔤_{x,r} (resp. 𝔷_{n}^{a} ⊂ 𝔤_{y,s}) if, and only if, n ≧ r (resp. n ≧ s), the assertion of the proposition is obvious.

7. Proof of Theorem 5.2

In this section the local field k is assumed to be of characteristic zero.

Suppose (π, V_{π}) is an irreducible admissible complex representation of G(k) and 𝒪 be as in 6.1.

Given a point x of the Bruhat–Tits building of G/k, we shall let {r_{i}}, with r_{0} = 0, denote the monotone increasing sequence associated to it in 3.4. If y is another point of the building, we shall denote the monotone increasing sequence associated to it by {s_{j}} below.

7.1. Existence of minimal K-types. We claim (π, V_{π}) contains a minimal K-type. Let r be the smallest nonnegative rational number such that there is a point x ∈ 𝒪 so that the subspace W of elements of V_{π} fixed under 𝒫_{x,r^{+}} is nontrivial. (The existence of r is assured since 𝒪 is finite.) Then r = r_{i} for some i. As 𝒫_{x,r^{+}} is a normal subgroup of 𝒫_{x,r}, there is an induced representation of 𝒫_{x,r}/𝒫_{x,r^{+}} on W. If r = 0, then 𝒫_{x,r}/𝒫_{x,r^{+}} = M_{x}(𝔣). By Harish–Chandra’s theory of Eisenstein series for reductive groups over finite fields, [7: Vol IV], there is a parahoric subgroup Q ⊂ 𝒫_{x} and a cuspidal representation χ of Q/R_{n}(Q) whose inflation to Q is contained in W_{|Q}. In particular, π contains a minimal K-type of depth zero in this case. Therefore, we can assume that i > 0 and thus the group 𝒫_{x,r_{i}}/𝒫_{x,r_{i+1}} is abelian. Realize its characters as the cosets of 𝔤_{x,−r_{i−1}}^{∗} in 𝔤_{x,−r_{i}}^{∗} (see 3.8). Let χ = χ_{X+𝔤{∗ \atop x,−r_{i−1}}} be any character of 𝒫_{x,r_{i}} which occurs in the decomposition of

W into irreducible 𝒫_{x,r_{i}}-submodules. We claim that X + 𝔤_{x,−r_{i−1}}^{∗} contains no nilpotent elements and therefore χ is a minimal K-type. To prove the claim suppose X + 𝔤_{x,−r_{i−1}}^{∗} contains a nilpotent element. Then we may assume that X itself is nilpotent. According to Proposition 6.3, there is an optimal point y ∈ 𝒪, p ∈ 𝒫_{x}, and a j ≧ 0 such that −s_{j} > −r_{i}, i.e. s_{j} < r_{i}, and X +𝔤_{x,−r_{i−1}}^{∗} ⊂ 𝔤_{p^{−1}y,−s_{j}}^{∗}. This implies that if z = p^{−1}y, 𝔤_{x,−r_{i−1}}^{∗} ⊂ 𝔤_{z,−s_{j}}^{∗}, from which we conclude by taking duals (see 3.5(∗)) that 𝔤_{x,r_{i}} ⊃ 𝔤_{z,s_{j+1}}. From Proposition 6.5 we infer now that

𝒫_{x,r_{i}} ⊃ 𝒫_{z,s_{j+1}}.

For Y ∈ 𝔤_{z,s_{j+1}} we have X(Y) ∈ ϖ𝔬 since X ∈ 𝔤_{z,−s_{j}}^{∗} (3.5(∗)). So the restriction of χ

to 𝒫_{z,s_{j+1}} is trivial. Hence V_{π}^{𝒫_{z,s_{j+1}}} , and therefore V_{π}^{𝒫_{y,s_{j+1}}} also, is nontrivial. Now note that 𝒫_{y,s_{j+1}} = 𝒫_{y,s_{j}^{+}} and as s_{j} < r_{i} = r, this contradicts the minimality of r. Hence any irreducible 𝒫_{x,r}-submodule in W is a nondegenerate representation. This completes the proof of existence of a minimal K-type.

7.2. Associativity of minimal K-types. Suppose x and y are two points of the Bruhat– Tits building of G/k. Let χ be the representation of the group 𝒫_{x,r_{i}} on an irreducible 𝒫_{x,r_{i}}-submodule V_{χ} of V_{π}. We assume that (𝒫_{x,r_{i}}, χ) is a minimal K-type. Let ξ be the representation of 𝒫_{y,s_{j}} on an irreducible 𝒫_{y,s_{j}}-submodule V_{ξ} of V_{π} which is fixed pointwise by 𝒫_{y,s_{j+1}}. Let E_{ξ} be an 𝒫_{y,s_{j}}-equivarient projection of V_{π} onto V_{ξ}. Since V_{π} is irreducible, there is a g ∈ G(k) so that

φ = E_{ξ}π(g^{−1}) : V_{χ} ⟶ V_{ξ}

is nonzero. For h ∈ 𝒫_{x,r_{i}} ∩ g𝒫_{y,s_{j}}g^{−1} we have

(∗) φ ∘ χ(h) = ξ(g^{−1}hg) ∘ φ.

We shall consider now three cases according to whether r_{i} and s_{j} are both greater than zero, r_{i} > 0 and s_{j} = 0, or r_{i} = s_{j} = 0.

Case 1. Both r_{i}, s_{j} > 0. Let X + 𝔤_{x,−r_{i−1}}^{∗} and Y + 𝔤_{y,−s_{j−1}}^{∗} be the cosets which give the characters χ and ξ respectively. As χ is nondegenerate, X + 𝔤_{x,−r_{i−1}}^{∗} does not contain any nilpotent elements. By the intertwining principle (∗), the two characters h ↦ χ(h) and h ↦ ξ(g^{−1}hg) agree on 𝒫_{x,r_{i}} ∩ g𝒫_{y,s_{j}}g^{−1}. This implies (X − Ad(g)Y)(Z) ∈ ϖ𝔬 for all Z ∈ 𝔤_{x,r_{i}} ∩ Ad(g)𝔤_{y,s_{j}}, and so X − Ad(g)Y lies in 𝔤_{x,−r_{i−1}}^{∗} + Ad(g)𝔤_{y,−s_{j−1}}^{∗}. Hence, X+𝔤_{x,−r_{i−1}}^{∗} and Ad(g)(Y +𝔤_{y,−s_{j−1}}^{∗})(⊂ Ad(g)(𝔤_{y,−s_{j}}^{∗}) = 𝔤_{gy,−s_{j}}^{∗}) intersect. By Proposition 6.4, then −s_{j} ≦ −r_{i}, i.e. s_{j} ≧ r_{i}. In particular, if (𝒫_{y,s_{j}}, ξ) is another minimal K-type in π, it follows that χ and ξ are associates.

Case 2. r_{i} > 0, s_{j} = 0. The representation ξ of 𝒫_{y} is the inflation of a representation of 𝒫_{y,0}/𝒫_{y,0^{+}}. In particular, this means the trivial representation of 𝒫_{y,0^{+}}, which corresponds to the coset 𝔤_{y,0}^{∗}, occurs in π. By the same reasoning as in case 1, the two cosets 𝔤_{gy,0}^{∗} and X + 𝔤_{x,−r_{i−1}}^{∗} must intersect. However, X + 𝔤_{x,−r_{i−1}}^{∗} contains no nilpotent elements. This contradicts Proposition 6.4. Hence this case can not occur.

In the above argument, interchanging the roles of x and y, we conclude that if ξ is a minimal K-type and r_{i} = 0, then s_{j} = 0.

Case 3. r_{i} = 0, s_{j} = 0. In this case the only assertion which requires a proof is that when ξ is also a minimal K-type, then it is an associate of χ. So we assume ξ is also a minimal K-type. Then, χ (resp. ξ) is the inflation to 𝒫_{x,0} (resp. 𝒫_{y,0}) of a cuspidal representation of M_{x}(𝔣) = 𝒫_{x,0}/𝒫_{x,0^{+}} (resp. M_{y}(𝔣) = 𝒫_{y,0}/𝒫_{y,0^{+}}). Observe that the image of 𝒫_{x} ∩ 𝒫_{gy} in M_{x}(𝔣) (resp. in M_{gy}(𝔣)) is the group of 𝔣-rational points of a parabolic 𝔣-subgroup P_{x} (resp. P_{gy}) of M_{x} (resp. M_{gy}). If P_{x} = M_{x} and P_{gy} = M_{gy}, then φ is an isomorphism of χ to Ad(g)ξ. Thus, χ and ξ are associates. Suppose P_{x} ⫋ M_{x}. Using the fact that there is an apartment of the Bruhat–Tits building of G/K containing both x and gy and the description of parahoric subgroups given in 2.5, it is easy to check that the image of 𝒫_{x} ∩ R_{n}(𝒫_{gy}) in M_{x}(𝔣) contains the unipotent radical of P_{x}(𝔣). Let 𝒰 be the inverse image in 𝒫_{x} ∩ R_{n}(𝒫_{gy}) of the unipotent radical of P_{x}(𝔣). As 𝒰 is contained in R_{n}(𝒫_{gy}), the restriction of Ad(g)ξ to 𝒰 is trivial. On the other hand, since χ is assumed to be cuspidal, the restriction of χ to 𝒰 does not contain the trivial representation of 𝒰. This is a contradiction; therefore P_{x} = M_{x}. A similar argument shows that P_{gy} = M_{gy}.

7.3. Minimality of ϱ(π). With the notation as in 7.1, we claim ϱ(π) := r is the smallest nonnegative number such that there is a point y in the building with V_{π}^{𝒫_{y,ϱ(π)^{+}}} ≠ {0}. By 7.1, V_{π}^{𝒫_{x,ϱ(π)^{+}}} ≠ {0}. Suppose by way of contradiction that there is a nonnegative real number s < r and a point y in the building so that V_{π}^{𝒫_{y,s^{+}}} ≠ {0}. By case 2 in 7.2, s > 0. Let ξ be any character of 𝒫_{y,s}/𝒫_{y,s^{+}} occurring in V_{π}^{𝒫_{y,s^{+}}} . By case 1 in 7.2, s ≧ r, a contradiction.

7.4. It remains to show that if ϱ(π) > 0, and y is a point in the building of G/k with V_{π}^{𝒫_{y,ϱ(π)^{+}}} ≠ {0}, then any irreducible 𝒫_{y,ϱ(π)}-constituent of V_{π}^{𝒫_{y,ϱ(π)^{+}}} is a minimal K-type. Let j be such that s_{j} = ϱ(π) and let Y + 𝔤_{y,−s_{j−1}}^{∗} be the coset which represents such a constituent. If Y + 𝔤_{y,−s_{j−1}}^{∗} contains a nilpotent element, the argument of 7.1 shows that ϱ(π) is not minimal, a contradiction. Thus the constituent must be a minimal K-type. This completes the proof of Theorem 5.2.

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Department of Mathematics, University of Michigan, Ann Arbor, MI 48109