Journal of Lie Theory Volume 10 (2000) 33–52 C ◯2000 Heldermann Verlag

On an analog of Hermite’s constant

Takao Watanabe^{∗}

Communicated by G. A. Margulis

Abstract. To any strongly k -rational representation π of a connected reductive algebraic group G defined over a number field k, we attach a constant γ_{π} as an analog of Hermite’s constant, and we give a lower estimate of γ_{π} in the case that the stabilier of the highest weight space of π in G is a maximal parabolic subgroup.

Introduction

In the theory of quadratic forms, the number

γ_{n} = max_{L} min_{x∈L\{0}}^{t}xx , (L runs over unimodular lattices in ℝ^{n})

is called Hermite’s constant and its exact value is known only for n ≤ 8 (cf.[10]). An estimate of γ_{n} is one of interesting problems in the geometry of numbers. Minkowski’s first convex-bodies theorem and Minkowski-Hlawka theorem show that

(0.1)  (\frac{2ζ(n)}{V (n)})^{2/n} ≤ γ_{n} ≤ 4 (\frac{1}{V (n)})^{2/n} ,

where V (n) denotes the volume of the unit ball in ℝ^{n} and ζ(·) the Riemann zeta function.

As a generalization of Hermite’s constant, Rankin [16] defined the constant γ_{n,d}, (1 ≤ d ≤ n − 1), by

γ_{n,d} = max_{L} min_{{x_{1},···,x_{d}∈L \atop x_{1}∧···∧x_{d}≠0}} det(^{t}x_{i}x_{j})_{i,j} , (L runs over unimodular lattices in ℝ^{n})

and proved the inequality

γ_{n,d} ≤ γ_{m,d}(γ_{n,m})^{d/m}

ISSN 0949–5932 / $2.50 ^{_{C} ◯} Heldermann Verlag for 1 ≤ d < m ≤ n − 1. On the other hand, by extending a base field, Icaza [9] defined Hermite-Humbert’s constant γ_{n}(k) as an analog of γ_{n} for any number field k and gave an upper bound analogous to (0.1). These two directions of genralizations of Hermite’s constant are unified by Thunder [17]. Namely, Thunder defined the constant γ_{n,d}(k) for any number field k by using the notion of height on flag varieties. He also extend the inequality (0.1) and Rankin’s inequality to the case of γ_{n,d}(k) by the adelic variant of Minkowski’s first convex-bodies theorem and Minkowski-Hlawka theorem. Thunder’s lower bound is given as follows;

∗ Partly supported by the Grants-in-Aid for Encouragement of Young Scientists, The Ministry of Education, Science and Culture, Japan

(0.2)  γ_{n,d}(k)^{n[k:ℚ]/2} ≥ \frac{w_{k}n}{h_{k}R_{k}} \begin{matrix}∏_{j=n−d+1}^{n} & \frac{ζ_{k}(j)|D_{k}|^{j/2}}{j^{r_{1}+r_{2}}2^{jr_{2}}V (j)^{r_{1}}V (2j)^{r_{2}}} \\ ∏_{ℓ=2}^{d} & \frac{ζ_{k}(ℓ)|D_{k}|^{ℓ/2}}{ℓ^{r_{1}+r_{2}}2^{ℓr_{2}}V (ℓ)^{r_{1}}V (2ℓ)^{r_{2}}} \end{matrix} ,

where ζ_{k} is the Dedekind zeta function, h_{k} the class number, R_{k} the regulator, D_{k} the discriminant, r_{1} and r_{2} the number of real and imaginary places of k, respectively.

In this paper, we generalize the constnat γ_{n,d}(k) and its estimate (0.2) within the framework of Borel’s reduction theory of reductive algebraic groups ([1],[8]). Observe that Rankin’s constant γ_{n,d} is written as

γ_{n,d} = max_{g∈SL_{n}(ℝ)/SL_{n}(ℤ)} min_{{x_{1},···x_{d}∈ℤ^{n} \atop x_{1}∧···∧x_{d}≠0}} ||π_{d}(g)(x_{1} ∧ ⋯ ∧ x_{d})||_{∞}^{2} ,

where (π_{d}, V_{d}) denotes the d-th exterior representation of GL_{n} and || · ||_{∞} the usual Euclidean metric on V_{d}(ℝ). If we choose an appropriate height function || · ||_{𝔸_{ℚ}} on GL(V_{d}, 𝔸_{ℚ})V_{d}(ℚ) and a highest weight vector v_{d} ∈ V_{d}(ℚ), then γ_{n,d} is adelically written as

γ_{n,d} = max_{g∈SL_{n}(𝔸_{ℚ})/SL_{n}(ℚ)} min_{γ∈SL_{n}(ℚ)} ||π_{d}(gγ)v_{d}||_{𝔸_{ℚ}}^{2} .

In this form, we replace SL_{n}, (π_{d}, V_{d}) and v_{d} by an arbitrary connected reductive algebraic group G over k, its strongly k -rational representation (π, V_{π}) and a highest weight vector x_{0} in V_{π}(k), respectively. Then it is known by Borel’s reduction theory that the constant

γ_{π} = max_{g∈G(𝔸)^{1}/G(k)} min_{γ∈G(k)} ||π(gγ)x_{0}||_{𝔸}^{2/[k:ℚ]}

exists (Proposition 2). If G = GL_{n} and π = π_{d}, then γ_{π} coincides with γ_{n,d}(k). In order to estimate γ_{π}, we can use Thunder’s technique based on the mean value argument. Let Q be the stabilizer of the highest weight space of π in G, which is a k -parabolic subgroup of G. If Q is maximal, we shall prove the estimate of the form

(0.3)  γ_{π} ≥ (\frac{C_{Q}d_{G}e_{Q}τ(G)}{C_{G}d_{Q}τ(Q)})^{\frac{2e_{ξ}}{[k:ℚ]e_{Q}}} .

Here τ(G) and τ(Q) denote the Tamagawa numbers of G and Q, respectively, and other constants C_{Q}, d_{Q}, e_{Q}, e_{ξ} and similarly C_{G}, d_{G} are defined in (1.1), (1.2), (3.1) and (3.2) below. We note that this lower bound depends only on the equivalence class ξ of π. If G is split over k, then the constants C_{G}, C_{Q}, ⋯ are explicitly described in terms of the Dedekind zeta function and the Gamma function (Theorem 2). As an example, we shall compute the lower bound in the case that G is a classical split group and observe that (0.3) coincides with (0.2) if G = GL_{n} and π = π_{d}. Although Thunder applied Weil’s method to compute the lower bound of (0.2), we make use Langlands’ method to compute the lower bound of (0.3). At present, we have no any result analogous to Minkowsi’s first convex-bodies theorem except for the case of G = GL_{n}. This is a reason of the lack of an upper bound of γ_{π}. In a subsequent paper, we will study this problem when G is an orthogonal group.

Notations. As usual, ℤ, ℚ, ℝ and ℂ denote the ring of integers, the field of rational, real and complex numbers, respectively. The group of positive real numbers is denoted by ℝ_{+}^{×}.

Let k be an algebraic number field of finite degree over ℚ, 𝔒 the ring of integers in k and 𝔙 the set of all places of k. We write 𝔙_{∞} and 𝔙_{f} for the sets of all infinite places and all finite places of k, respectively. For v ∈ 𝔙, k_{v} denotes the completion of k at v with the normalized multiplicative valuation | · |_{v}. If v is finite, 𝔒_{v} denotes the ring of integers in k_{v}. We set k_{∞} = ∏_{v∈𝔙_{∞}} k_{v} and denote by 𝔸_{f} the ring of finite adeles of k. Thus the adele ring of k is 𝔸 = k_{∞} × 𝔸_{f}. The idele norm on 𝔸^{×} is denoted by | · |_{𝔸}. We fix an algebraic closure \overline{k} of k and write Γ_{k} for the absolute Galois group.

Let G be a connected affine algebraic group defined over k. For any k -algebra R, G(R) stands for the set of R-rational points of G. We denote by X^{∗}(G) and by X_{k}^{∗}(G) the free ℤ-modules consisting of all rational characters and all k -rational characters of G, respectively. The absolute Galois group Γ_{k} acts on X^{∗}(G). The representation of Γ_{k} in the space X^{∗}(G) ⊗_{ℤ} ℚ is denoted by σ_{G} and the corresponding Artin L-function is denoted by L(s, σ_{G}) = ∏_{v∈𝔙_{f}} L_{v}(s, σ_{G}). We set σ_{k}(G) = lim_{s→1}(s − 1)^{n}L(s, σ_{G}), where n is the rank of X_{k}^{∗}(G). For a left invariant gauge form ω^{G} on G defined over k, we associate a left invariant Haar measure ω_{v}^{G} on G(k_{v}). Then, the Tamagawa measure on G(𝔸) is well defined by ω_{𝔸}^{G} = |D_{k}|^{− dim G/2}ω_{∞}^{G}ω_{f}^{G}, where ω_{∞}^{G} = ∏_{v∈𝔙_{∞}} ω_{v}^{G}, ω_{f}^{G} = σ_{k}(G)^{−1} ∏_{v∈𝔙_{f}} L_{v}(1, σ_{G})ω_{v}^{G} and |D_{k}| is the absolute value of the discriminant of k. For χ ∈ X_{k}^{∗}(G), let |χ|_{𝔸} be the continuous homomorphism G(𝔸) → ℝ_{+}^{×} defined by |χ|_{𝔸}(g) = |χ(g)|_{𝔸}. We write G(𝔸)^{1} for the intersection of kernels of all such |χ|_{𝔸} ’s. If χ_{1}, ⋯, χ_{n} is a ℤ-basis of X_{k}^{∗}(G), then the mapping

g ↦ (|χ_{1}(g)|_{𝔸}, ⋯, |χ_{n}(g)|_{𝔸})

yields an isomorphism from the quotient group G(𝔸)/G(𝔸)^{1} to (ℝ_{+}^{×})^{n}. We put the Lebesgue measure dx on ℝ and the invariant measure dx/x on ℝ_{+}^{×}. Then there exists uniquely a Haar measure ω_{G(𝔸)^{1}} of G(𝔸)^{1} such that the Haar measure on G(𝔸)/G(𝔸)^{1} matching with ω_{𝔸}^{G} and ω_{G(𝔸)^{1}} is equal to the pullback of the measure ∏_{i=1}^{n} dx_{i}/x_{i} on (ℝ_{+}^{×})^{n} by the above isomorphism. The measure ω_{G(𝔸)^{1}} is independent of the choice of a ℤ-basis of X_{k}^{∗}(G). Since G(k) is a discrete subgroup of G(𝔸)^{1} , we put the counting measure ω_{G(k)} on G(k). Then the Tamagawa number τ(G) is defined to be the volume of the quotient space G(𝔸)^{1}/G(k) with respect to ω_{G(𝔸)^{1}}/ω_{G(k)} . Here, in general, if μ_{A} and μ_{B} denote Haar measures on a locally compact unimodular group A and its closed unimodular subgroup B, respectively, then μ_{A}/μ_{B} denotes a unique left A-invariant measure on the homogeneous space A/B matching with μ_{A} and μ_{B} in the sense that μ_{A} = μ_{A}/μ_{B} · μ_{B} holds.

If X is a locally compact topological space, we denote by C_{0}(X) the space of all compactly supported continuous functions on X.

1. Strongly k -rational representations

In this section, let G be a connected reductive group defined over k. We recall the notion of strongly k -rational representatons of G (cf.[3],[18]).

We fix a maximally k -split torus S of G and a maximal k -torus T of G containing S. We also fix a minimal k -parabolic subgroup P of G containing S and a Borel subgroup B of P containing T. Then, we denote by Φ (resp. Φ_{k}) the absolute (resp. relative) root system of G with respect to T (resp. S) and by Δ (resp. Δ_{k}) the set of simple roots of Φ (resp. Φ_{k}) corresponding to B (resp. P). The restriction homomorphism ρ_{k} : X^{∗}(T) → X^{∗}(S) maps Δ into Δ_{k} ∪ {0}. We fix an admissible and compatible inner product (,) on X^{∗}(T) ⊗_{ℤ} ℝ and X^{∗}(S) ⊗_{ℤ} ℝ ([3, (6.10)]). We define the ∗-action of Γ_{k} on X^{∗}(T). For given σ ∈ Γ_{k}, there is an element w in the absolute Weyl group such that w(σ(Δ)) = Δ. Then we set σ^{∗} = w ∘ σ, which acts on X^{∗}(T) by

σ^{∗}χ(t) = σ(χ(σ^{−1}(w^{−1}(t)))) , (χ ∈ X^{∗}(T), t ∈ T)

This action is independent of the choice of w.

Let 𝔏 be the weight lattice of T and 𝔏_{+} the set of dominant weights in 𝔏 with respect to B, i.e.,

𝔏 = {λ ∈ X^{∗}(T) : (λ, a^{∨}) ∈ ℤ for all a ∈ Δ} ,
𝔏_{+} = {λ ∈ 𝔏 : (λ, a^{∨}) ≥ 0 for all a ∈ Δ} ,

where a^{∨} = 2(a, a)^{−1}a is the coroot of a. The set 𝔏_{+} is stable under the ∗- action of Γ_{k}, so 𝔏_{+}^{Γ_{k}} stands for the subset consisting of Γ_{k} -invariant elements in 𝔏_{+} . For a ∈ Δ, the fundamental weight ℓ_{a} ∈ X^{∗}(T) ⊗_{ℤ} ℚ is defined by the conditions (ℓ_{a}, b^{∨}) = δ_{a,b}, (b ∈ Δ). Any dominant weight λ is written as a non-negative integral linear combination of the ℓ_{a} ’s. If G is semisimple and simplyconnected, then the ℓ_{a} are ℤ-basis of X^{∗}(T). For k -root α ∈ Δ_{k}, we define the fundamental k -weight m_{α} ∈ X^{∗}(S) ⊗_{ℤ} ℚ by

m_{α} = ∑_{a∈ρ_{k}^{−1}(α)} ρ_{k}(ℓ_{a}) .

Let ℜ(G) be the set of equivalent classes of all irreducible \overline{k} -rational representations of G. If λ_{ξ} denotes the highest weight of ξ ∈ ℜ(G), then the correspondence ξ → λ_{ξ} yields a bijection from ℜ(G) to 𝔏_{+} (cf.[18]). For ξ ∈ ℜ(G), let π be a representation contained in ξ. We write V_{π} for the representation space of π. The highest weight space in V_{π} is denoted by D_{π}. Let Q_{π} be the stabilizer of D_{π} in G and let λ_{π} be the \overline{k} -rational character of Q_{π} by which Q_{π} acts on D_{π}. Then Q_{π} is a standard \overline{k} -parabolic subgroup of G and the restriction of λ_{π} to T is equal to λ_{ξ}. An equivalent class ξ ∈ ℜ(G) is said to be k -rational if ξ contains a k -rational representation π : G → GL(V_{π}), where V_{π} is defined over k. Then, such a k -rational representation π is unique up to k -equivalence ([3, (12.3)]). Moreover, ξ ∈ ℜ(G) is said to be strongly k -rational if ξ is k -rational and, for a k -rational representation π ∈ ξ, the highest weight space D_{π} is defined over k. Then Q_{π} is defined over k and λ_{π} ∈ X_{k}^{∗}(Q_{π}). We denote by ℜ_{k}^{s}(G) the set of all strongly k -rational ξ ∈ ℜ(G). The next proposition is due to Borel and Tits ([3, §12], [18]).

Proposition 1. (1) If ξ ∈ ℜ(G) is k -rational, then λ_{ξ} ∈ 𝔏_{+}^{Γ_{k}}. (Note that the converse statement is not true in general.)

(2) ξ ∈ ℜ(G) is strongly k -rational if and only if λ_{ξ} ∈ 𝔏_{+}^{Γ_{k}} and (λ_{ξ}, a^{∨}) = 0 for all a ∈ ρ_{k}^{−1}(0) ∩ Δ. Then λ_{ξ} is completely determined by ρ_{k}(λ_{ξ}) and ρ_{k}(λ_{ξ}) is a non-negative integral linear combination of the fundamental k -weights.

(3) For each α ∈ Δ_{k}, there exists a positive integer d_{α} such that nd_{α}m_{α} ∈ {ρ_{k}(λ_{ξ}) : ξ ∈ ℜ_{k}^{s}(G)} for all positive integers n.

By the statment (2), ξ ↦ ρ_{k}(λ_{ξ}) yields a bijection from ℜ_{k}^{s}(G) to ρ_{k}(𝔏_{+}^{Γ_{k}}). If G is semisimple and simplyconnected, then one has

ρ_{k}(𝔏_{+}^{Γ_{k}}) = { ∑_{α∈Δ_{k}} c_{α}m_{α} : 0 ≤ c_{α} ∈ ℤ for all α ∈ Δ_{k} } .

We say that ξ ∈ ℜ_{k}^{s}(G) is maximal if ρ_{k}(λ_{ξ}) is a positive integer multiple of some one fundamental k -weight m_{α}. Then, for a k -rational representation π ∈ ξ, Q_{π} is the standard maximal k -parabolic subgroup associated to Δ_{k}\{α}, i.e., the centralizer of the torus (∩_{β∈Δ_{k}\{α}}Ker β)^{0} in G is a Levi subgroup of Q_{π}.

In the rest of this section, we define some notations. Let M be the centralizer of S in G. Then P has a Levi decomposition P = MU, where U is the unipotent radical of P. For every stanadard k -parabolic subgroup Q of G, Q has a unique Levi subgroup M_{Q} containing M. We denote by U_{Q} the unipotent radical of Q. Throughout this paper, we fix a maximal compact subgroup K^{G} = ∏_{v∈𝔙} K_{v}^{G} of G(𝔸) satisfying the following property; For every standard k -parabolic subgroup Q of G, K^{G} ∩ Q(𝔸) = (K^{G} ∩ M_{Q}(𝔸))(K^{G} ∩ U_{Q}(𝔸)), K^{G} ∩ M_{Q}(𝔸) is a maximal compact subgroup of M_{Q}(𝔸) and M_{Q}(𝔸) posseses an Iwasawa decomposition (K^{G}∩M_{Q}(𝔸))M(𝔸)(M_{Q}(𝔸)∩U(𝔸)). It is known that such maximal compact subgroup of G(𝔸) exists. We set K^{M_{Q}} = K^{G} ∩ M_{Q}(𝔸), P^{Q} = M_{Q} ∩ P and U^{Q} = M_{Q} ∩ U.

Let Q be a standard k -parabolic subgroup of G. We include the case Q = G. Let Z_{Q} be the greatest central k -split torus in M_{Q}. The restriction map X_{k}^{∗}(M_{Q}) → X^{∗}(Z_{Q}) is injective. Since X_{k}^{∗}(M_{Q}) has the same rank as X^{∗}(Z_{Q}), the index

(1.1)  d_{Q} = [X^{∗}(Z_{Q}) : X_{k}^{∗}(M_{Q})]

is finite. If χ_{1}, ⋯, χ_{r} is a ℤ-basis of X^{∗}(Z_{Q}), the map z ↦ (χ_{1}(z), ⋯, χ_{r}(z)) defines an isomorphism from Z_{Q}(𝔸) to (𝔸^{×})^{r}. We regard ℝ_{+}^{×} as a subgroup of 𝔸^{×} by identifying x ∈ ℝ_{+}^{×} with the idele x_{𝔸} = (x_{v}) such that x_{v} = x if v ∈ 𝔙_{∞} and x_{v} = 1 if v ∈ 𝔙_{f}. Let Z_{Q}^{+} denote the inverse image of (ℝ_{+}^{×})^{r} by the isomorpism Z_{Q}(𝔸) → (𝔸^{×})^{r}. Then M_{Q}(𝔸) has the direct product decomposition M_{Q}(𝔸) = Z_{Q}^{+}M_{Q}(𝔸)^{1} . We define the Haar measure μ_{Z_{Q}^{+}} on Z_{Q}^{+}

by the pull-back of the invariant measure ∏_{i=1}^{r} dx_{i}/x_{i} on (ℝ_{+}^{×})^{r} with respect to the isomorphsim z ↦ (|χ_{1}(z)|_{𝔸}, ⋯, |χ_{r}(z)|_{𝔸}) from Z_{Q}^{+} onto (ℝ_{+}^{×})^{r}. It follows from definition of ω_{M(𝔸)^{1}} that the Tamagawa measure ω_{𝔸}^{M_{Q}} is decomposed to d_{Q}μ_{Z_{Q}^{+}} · ω_{M_{Q}(𝔸)^{1}} . We note that Z_{Q}^{+} and μ_{Z_{Q}^{+}} are independent of the choice of a

basis of X^{∗}(Z_{Q}). In particular, Z_{G}^{+} is a subgroup of Z_{Q}^{+} for the surjectivity of the restriction map X^{∗}(Z_{Q}) → X^{∗}(Z_{G}).

We define another Haar measure ν_{M_{Q}(𝔸)} of M_{Q}(𝔸) as follows. Let ω_{𝔸}^{M} and ω_{𝔸}^{U^{Q}} be the Tamagawa measures of M(𝔸) and U^{Q}(𝔸), respectively. There is the function δ_{P^{Q}} on M(𝔸) such that the integration formula

∫_{U^{Q}(𝔸)} f(mum^{−1})dω_{𝔸}^{U^{Q}}(u) = δ_{P^{Q}}(m)^{−1} ∫_{U^{Q}(𝔸)} f(u)dω_{𝔸}^{U^{Q}}(u)

holds for any function f ∈ C_{0}(U^{Q}(𝔸)). In other words, δ_{P^{Q}}^{−1} is the modular character of P^{Q}(𝔸). Let ν_{K^{M_{Q}}} be the Haar measure on K^{M_{Q}} normalized so that the total volume equals one. Then the mapping

f ↦ ∫_{K^{M_{Q}}} ∫_{M(𝔸)} ∫_{U^{Q}(𝔸)} f(hmu)δ_{P^{Q}}(m)dν_{K^{M_{Q}}} (h)dω_{𝔸}^{M}(m)dω_{𝔸}^{U^{Q}}(u)

for f ∈ C_{c}(M_{Q}(𝔸)) defines a Haar measure on M_{Q}(𝔸) and is denoted by ν_{M_{Q}(𝔸)} . We also define the Haar measure ν_{M_{Q}(𝔸)^{1}} on M_{Q}(𝔸)^{1} by ν_{M_{Q}(𝔸)^{1}} = ν_{M_{Q}(𝔸)}/d_{Q}μ_{Z_{Q}^{+}} . There exists a positive constant C_{Q} such that

(1.2)  ω_{𝔸}^{M_{Q}} = C_{Q}ν_{M_{Q}(𝔸)} .

We define left G(𝔸)^{1} -invariant measures on the homogeneous space Y_{Q} = G(𝔸)^{1}/Q(𝔸)^{1} . Since Q(𝔸)^{1} = M(𝔸)^{1}U_{Q}(𝔸), ν_{M(𝔸)^{1}}ω_{𝔸}^{U_{Q}} defines a Haar measure, say ν_{Q(𝔸)^{1}} , of Q(𝔸)^{1} . We note that both G(𝔸)^{1} and Q(𝔸)^{1} are unimodular. Therefore, both ω_{Y_{Q}} = ω_{G(𝔸)^{1}}/ω_{Q(𝔸)^{1}} and ν_{Y_{Q}} = ν_{G(𝔸)^{1}}/ν_{Q(𝔸)^{1}} yield G(𝔸)^{1} -invariant measures on Y_{Q}. We identify Y_{Q} with G(𝔸)/Z_{G}^{+}Q(𝔸)^{1} and set K^{Q} = K^{G} ∩ Q(𝔸). Then the mapping

ι_{Q} : K^{G}/K^{Q} × Z_{Q}^{+}/Z_{G}^{+} → Y_{Q} : (hK^{Q}, zZ_{G}^{+}) ↦ hzZ_{G}^{+}Q(𝔸)^{1}

is an isomorphism. Set μ_{Z_{Q}^{+}/Z_{G}^{+}} = μ_{Z_{Q}^{+}}/μ_{Z_{G}^{+}} .

Lemma 1. For any left K^{G} -invariant function f ∈ C_{0}(Y_{Q}), one has

∫_{Y_{Q}} f(y)dν_{Y_{Q}}(y) = \frac{d_{Q}}{d_{G}} ∫_{Z_{Q}^{+}/Z_{G}^{+}} f(ι_{Q}(K^{Q}, zZ_{G}^{+}))δ_{Q}(z)dμ_{Z_{Q}^{+}/Z_{G}^{+}}(zZ_{G}^{+}) ,

Watanabe 39

where δ_{Q}^{−1} is the modular character of Q(𝔸). Proof. Let ι_{Q}^{∗}ν_{Y_{Q}} be the measure on K^{G}/K^{Q} × Z_{Q}^{+}/Z_{G}^{+} obtained by the pull-back of ν_{Y_{Q}}. We define the measure μ_{Z_{Q}^{+}/Z_{G}^{+}} on Z_{Q}^{+}/Z_{G}^{+} by the mapping

φ ↦ ∫_{K^{G}/K^{Q}×Z_{Q}^{+}/Z_{G}^{+}} φ(zZ_{G}^{+})dι_{Q}^{∗}ν_{Y_{Q}}(hK^{Q}, zZ_{G}^{+}) , (φ ∈ C_{0}(Z_{Q}^{+}/Z_{G}^{+})) Then we have

∫_{Y_{Q}} f(y)dν_{Y_{Q}}(y) = ∫_{Z_{Q}^{+}/Z_{G}^{+}} f(ι_{Q}(K^{Q}, zZ_{G}^{+}))dμ_{Z_{Q}^{+}/Z_{G}^{+}}(zZ_{G}^{+})

for any left K^{G} -invariant function f ∈ C_{0}(Y_{Q}). We determine μ_{Z_{Q}^{+}/Z_{G}^{+}} . Let ψ ∈ C_{0}(G(𝔸)^{1}) be a left K^{G} -invariant function on G(𝔸)^{1} . On the one hand, we have ∫_{G(𝔸)^{1}} ψ(g)dν_{G(𝔸)^{1}}(g) = ∫_{K^{G}} ∫_{M(𝔸)/Z_{G}^{+}} ∫_{U(𝔸)} ψ(hmu)δ_{P} (m)dν_{K^{G}}(h)d(ω_{𝔸}^{M}/d_{G}μ_{Z_{G}^{+}})(m)dω_{𝔸}^{U}(u) = ∫_{Z_{P}^{+}/Z_{G}^{+}} ∫_{M(𝔸)^{1}} ∫_{U(𝔸)} ψ(zm_{1}u)δ_{P} (z)d(d_{P} μ_{Z_{P}^{+}}/d_{G}μ_{Z_{G}^{+}})(z)dω_{M(𝔸)^{1}}(m_{1})dω_{𝔸}^{U}(u)

= \frac{d_{Q}}{d_{G}} ∫_{Z_{P}^{+}/Z_{Q}^{+}} ∫_{Z_{Q}^{+}/Z_{G}^{+}} ∫_{M(𝔸)^{1}} ∫_{U(𝔸)} ψ(z_{1}z_{2}m_{1}u)δ_{P^{Q}}(z_{1})δ_{Q}(z_{2})

× d(d_{P} μ_{Z_{P}^{+}}/d_{Q}μ_{Z_{Q}^{+}})(z_{1})dμ_{Z_{Q}^{+}/Z_{G}^{+}}(z_{2})dω_{M(𝔸)^{1}}(m_{1})dω_{𝔸}^{U}(u). On the other hand, ∫_{G(𝔸)^{1}} ψ(g)dν_{G(𝔸)^{1}}(g)

= ∫_{Y_{Q}} {∫_{Q(𝔸)^{1}} ψ(gq)dν_{Q(𝔸)^{1}}(q)} dν_{Y_{Q}}(gQ(𝔸)^{1}) = ∫_{K^{G}/K^{Q}×Z_{Q}^{+}/Z_{G}^{+}} { ∫_{K^{M_{Q}}} ∫_{M(𝔸)/Z_{Q}^{+}} ∫_{U^{Q}(𝔸)} ∫_{U_{Q}(𝔸)} ψ(ι_{Q}(h_{1}K^{Q}, zZ_{G}^{+})h_{2}mu_{1}u_{2})

× δ_{P^{Q}}(m)dν_{K^{M_{Q}}} (h_{2})d(ω_{𝔸}^{M}/d_{Q}μ_{Z_{Q}^{+}})(m)dω_{𝔸}^{U^{Q}}(u_{1})dω_{𝔸}^{U_{Q}}(u_{2})}

× dι_{Q}^{∗}ν_{Y_{Q}}(h_{1}K_{Q}, zZ_{G}^{+})

= ∫_{Z_{Q}^{+}/Z_{G}^{+}} { ∫_{Z_{P}^{+}/Z_{Q}^{+}} ∫_{M(𝔸)^{1}} ∫_{U(𝔸)} ψ(z_{2}z_{1}m_{1}u)δ_{P^{Q}}(z_{1})d(d_{P} μ_{Z_{P}^{+}}/d_{Q}μ_{Z_{Q}^{+}})(z_{1})

× dω_{M(𝔸)^{1}}(m_{1})dω_{𝔸}^{U}(u)}dμ_{Z_{Q}^{+}/Z_{G}^{+}}(z_{2}) = ∫_{Z_{P}^{+}/Z_{Q}^{+}} ∫_{Z_{Q}^{+}/Z_{G}^{+}} ∫_{M(𝔸)^{1}} ∫_{U(𝔸)} ψ(z_{1}z_{2}m_{1}u)δ_{P^{Q}}(z_{1})d(d_{P} μ_{Z_{P}^{+}}/d_{Q}μ_{Z_{Q}^{+}})(z_{1})

× dμ_{Z_{Q}^{+}/Z_{G}^{+}}(z_{2})dω_{M(𝔸)^{1}}(m_{1})dω_{𝔸}^{U}(u) . Therefore, we obtain μ_{Z_{Q}^{+}/Z_{G}^{+}} = d_{Q}d_{G}^{−1}δ_{Q}(z_{2})μ_{Z_{Q}^{+}/Z_{G}^{+}} .

Corollary . For any left K^{G} -invariant function f ∈ C_{0}(Y_{Q}), one has

∫_{Y_{Q}} f(y)dω_{Y_{Q}}(y) = \frac{C_{G}d_{Q}}{C_{Q}d_{G}} ∫_{Z_{Q}^{+}/Z_{G}^{+}} f(ι_{Q}(K^{Q}, zZ_{G}^{+}))δ_{Q}(z)dμ_{Z_{Q}^{+}/Z_{G}^{+}}(zZ_{G}^{+}) .

2. An analog of Hermite’s constant

We fix a strongly k -rational ξ ∈ ℜ_{k}^{s}(G) and a k -rational representation π ∈ ξ. We also fix a free 𝔒-lattice L in the k -vector space V_{π}(k) and a 𝔒-basis e_{1}, ⋯, e_{n} of L, where n = dim π. For every v ∈ 𝔙, we define the norm || · ||_{L,v} of the k_{v} -vector space V_{π}(k_{v}) as follows;

(2.1) ||x_{1}e_{1} + ⋯ + x_{n}e_{n}||_{L,v} = \{ \begin{matrix}sup(|x_{1}|_{v}, ⋯, |x_{n}|_{v}) & (v ∈ 𝔙_{f}) \\ (|x_{1}|_{v}^{2} + ⋯ + |x_{n}|_{v}^{2})^{1/2} & (v is real) \\ |x_{1}|_{v} + ⋯ + |x_{n}|_{v} & (v is imaginary). \end{matrix}

We define height functions on GL_{n}(𝔸)V_{π}(k). Let || · ||_{v}, v ∈ 𝔙 be a norm on V_{π}(k_{v}) compatible with | · |_{v}. Assume that || · ||_{v} = || · ||_{L,v} holds for almost all v ∈ 𝔙. Then we set

||x||_{𝔸} = ∏_{v∈𝔙} ||x_{v}||_{v} , (x ∈ GL_{n}(𝔸)V_{π}(k)).

Such || · ||_{𝔸} is called a height function on GL_{n}(𝔸)V_{π}(k) (cf.[8]). The following property is obvious by its definition.

(2.2)  ||ax||_{𝔸} = |a|_{𝔸}||x||_{𝔸} for any a ∈ 𝔸^{×} and x ∈ GL_{n}(𝔸)V_{π}(k).

A height function || · ||_{𝔸} is said to be K^{G} -invariant if ||π(h)x||_{𝔸} = ||x||_{𝔸} holds for all h ∈ K^{G} and x ∈ GL_{n}(𝔸)V_{π}(k). Furthermore, || · ||_{𝔸} is said to be normalized if ||x_{0}||_{𝔸} = 1 for a highest weight vector x_{0} ∈ D_{π}(k)\{0}. Then one has ||x||_{𝔸} = 1 for any highest weight vector x ∈ D_{π}(k)\{0} because of the one-dimensionality of D_{π}(k) and (2.2). It is obvious that K^{G} -invariant and normalized height functions on GL_{n}(𝔸)V_{π}(k) exist.

In the following, we fix a K^{G} -invariant normalized height function || · ||_{𝔸} on GL_{n}(𝔸)V_{π}(k). Let x_{0} ∈ D_{π}(k)\{0} be a highest weight vector of π. For each g ∈ G(𝔸)^{1} , we define the function Φ_{g} : G(k) → ℝ_{+}^{×} by Φ_{g}(γ) = ||π(gγ)x_{0}||_{𝔸}. From π(γ)x_{0} = λ_{π}(γ)x_{0} , (γ ∈ Q_{π}(k)) and (2.2), it follows that Φ_{g} is right Q_{π}(k)-invariant and is independent of the choice of x_{0} . By the reduction theory (cf.[1, §16],[8]), it is known that Φ_{g} attains its minimum value at a point in the intersection of G(k) and a Siegel set of G(𝔸). We set Ψ(g) = min_{γ∈G(k)/Q_{π}(k)} Φ_{g}(γ). Then it is easy to see that Ψ is a continuous function on K^{G}\G(𝔸)^{1}/G(k) and is bounded from above (cf.[1, 16.10]).

Proposition 2. Ψ has the maximum value.

Proof. Set H = (Ker π)^{0} , P_{H} = P ∩H and S_{H} = S∩H. Let Δ_{k}^{H} be the set of simple k -roots of H with respect to S_{H} and P_{H}. We also set Δ_{k}^{G/H} = Δ_{k}^{G}\Δ_{k}^{H} and S_{G/H} = (∩_{α∈Δ_{k}^{H}} Ker α)^{0} . Then Z_{P}^{+} = S_{G/H}^{+}S_{H}^{+} . For κ > 0 and every standard k -parabolic subgroup Q, define a subset Z_{Q}^{+}(κ) of Z_{Q}^{+} by

Z_{Q}^{+}(κ) = {z ∈ Z_{Q}^{+} : |α(z)|_{𝔸} < κ for all α ∈ Δ_{k} } .

By the reduction theory, we can choose κ > 1 and a compact subset Ω in P (𝔸) so that G(𝔸) = K^{G}ΩZ_{P}^{+}(κ)G(k). Since π is trivial on H(𝔸), it is enough to prove that Ψ attains its maximum value at a point in ΩZ_{P}^{+}(κ)S_{H}^{+}/Z_{G}^{+}S_{H}^{+} = ΩS_{G/H}^{+}(κ)/Z_{G}^{+} . By compactness of Ω, there is a constant C such that

(2.3)  Ψ(ωz) ≤ ||π(ωz)x_{0}||_{𝔸} ≤ C||π(z)x_{0}||_{𝔸} = C|λ_{π}(z)|_{𝔸}

holds for all ω ∈ Ω and z ∈ Z_{P}^{+} . Since ρ_{k}(λ_{π}) ∈ X^{∗}(S/S_{H}) ⊂ X^{∗}(S_{G/H}) ⊗_{ℤ} ℚ, ρ_{k}(λ_{π}) is represented by a ℚ-linear combination of α ∈ Δ_{k}^{G/H} modulo X^{∗}(Z_{G}) ⊗_{ℤ} ℚ, i.e., we have

(2.4)  ρ_{k}(λ_{π}) ≡ ∑_{α∈Δ_{k}^{G/H}} c_{α}α mod X^{∗}(Z_{G}) ⊗_{ℤ} ℚ .

For the moment, we assume that the following condition is satisfied;

(2.5)  c_{α} > 0 for all α ∈ Δ_{k}^{G/H}.

Let Q_{β} be the standard maximal k -parabolic subgroup of G correspoinding to Δ_{k}\{β} for β ∈ Δ_{k}^{G/H}. By (2.3) and (2.4), one has for ω ∈ Ω, z ∈ Z_{Q_{β}}^{+}(κ_{1}) and κ_{1} > 0

Ψ(ωz) ≤ C|λ_{π}(z)|_{𝔸} ≤ Cκ_{1}^{c_{β}} .

Under the condition (2.5), this concludes that if κ_{1} is sufficiently small, then Ψ(ωz) is also small for all ω ∈ Ω and z ∈ S_{Q_{β}}^{+}(κ_{1}). In other words, if we define a subset S_{G/H}^{+}(κ, κ_{1}) of S_{G/H}^{+}(κ) by

S_{G/H}^{+}(κ, κ_{1}) = S_{G/H}^{+}(κ)\ ⋃_{β∈Δ_{k}^{G/H}} Z_{Q_{β}}^{+}(κ_{1}) ,

then we have

sup_{g∈K^{G}\G(𝔸)^{1}/G(k)} Ψ(g) = sup_{g∈ΩS_{G/H}^{+}(κ,κ_{1})/Z_{G}^{+}} Ψ(g) .

Therefore, Ψ(g) has the maximum value since ΩS_{G/H}^{+}(κ, κ_{1})/Z_{G}^{+} is relatively compact in G(𝔸)^{1} . To complete the proof, we have to show (2.5). Since it is an assetion concerning root systems, we may assume that H = 1. Furthermore, it is reduced to the cases that G is almost k -simple. Thus ρ_{k}(λ_{π}) is non-zero and is represented as a non-negative integral linear combination of the fundamental k -weights m_{α}, α ∈ Δ_{k}. Let

m_{α} = ∑_{β∈Δ_{k}} d_{α,β}β , (d_{α,β} ∈ ℚ) .

It is sufficient to show

d_{α,β} > 0 for all α, β ∈ Δ_{k}.

For an absolute fundamental weight ℓ_{a}, let

ℓ_{a} = ∑_{b∈Δ} d′_{a,b}b , (d′_{a,b} ∈ ℚ).

Then we have

d_{α,β} = ∑_{a∈ρ_{k}^{−1}(α)} ∑_{b∈ρ_{k}^{−1}(β)} d′_{a,b} .

Therefore, we need to show that d′_{a,b} ≥ 0 for all a, b and there are a ∈ ρ_{k}^{−1}(α) and b ∈ ρ_{k}^{−1}(β) such that d′_{a,b} > 0. Since G is almost k -simple, there are a finite field extention L/k and an absolutely almost simple group G_{1} defined over L so that G is k -isomorphic to the scalar restriction R_{L/k}(G_{1}) of G_{1} . Let Φ^{G_{1}} be the absolute root system of G_{1} and Δ^{G_{1}} a set of simple roots in Φ^{G_{1}} . Then Φ is an orthogonal direct sum of [L : k] copies of Φ^{G_{1}} , and hence Δ is of the form

Δ = ⨆_{i=1}^{[L:k]} Δ^{i} , Δ^{i} = Δ^{G_{1}} .

We have d′_{a,b} = 0 if a ∈ Δ^{i}, b ∈ Δ^{j} and i ≠ j. Since ρ_{k}(Δ^{i}) = Δ_{k} ∪{0} for each i, it is enough to prove d′_{a,b} > 0 for a, b ∈ Δ^{i}. This follows from a classification table of the simple root systems ([4]).

Let γ′_{π} = max_{g∈K^{G}\G(𝔸)^{1}/G(k)} Ψ(g). Then we call γ_{π} = (γ′_{π})^{2/[k:ℚ]} the Hermite constant attached to π and ||·||_{𝔸}. We write γ_{π}(||·||_{𝔸}) for γ_{π} if we need to emphasize the dependence of || · ||_{𝔸}. In an example below, we shall clarify a relation of γ_{π} and the original Hermite’s constant.

Example 1. Let V be an n-dimensional vector space defined over k. We fix a free 𝔒-lattice L in V and its 𝔒-basis e_{1}, ⋯, e_{n}. We identify G = GL(V) with GL_{n} with respect to the basis e_{1}, ⋯, e_{n}. Let || · ||_{L,𝔸} = ∏_{v∈𝔙} || · ||_{L,v} be the height function on GL_{n}(𝔸)V (k), where || · ||_{L,v} is defined similary as (2.1). Let S be the subgroup of diagonal matirces in G and P the subgroup of upper triangular matrices in G. For g ∈ S, ϵ_{i}(g) denotes the i-th diagonal element of g for 1 ≤ i ≤ n. Thus ϵ_{i} is a rational character of S. As usual, the root system Φ = Φ_{k} is given by {ϵ_{i} − ϵ_{j} : 1 ≤ i, j ≤ n, i ≠ j}. We have Δ = Δ_{k} = {α_{i} = ϵ_{i} − ϵ_{i+1} : 1 ≤ i ≤ n − 1}. The fundamental weights ℓ_{d} are given by

ℓ_{d} = ϵ_{1} + ⋯ + ϵ_{d} .

Then

𝔏_{+} = 𝔏_{+}^{Γ_{k}} = {∑_{d=1}^{n} c_{d}ℓ_{d} : 0 ≤ c_{d} ∈ ℤ for all d} .

We take the maximal compact subgroup K^{G} as follows;

K^{G} = ∏_{v∈𝔙} K_{v}^{G} , K_{v}^{G} = {g ∈ G(k_{v}) : ||gx||_{L,v} = ||x||_{L,v} for all x ∈ V (k_{v})}.

We consider the classes ξ_{d} ∈ ℜ(G) corresponding to the fundamental weights ℓ_{d}. Let π_{d} ∈ ξ_{d} be the d-th exterior representation of G, i.e. V_{π_{d}} = ⋀^{d} V . Then v_{d} = e_{1} ∧ ⋯ ∧ e_{d} is a highest weight vector of π_{d}. We denote by Q_{d} the stabilizer of the highest weight space D_{π_{d}} of π_{d} in G. The Levi-subgroup M_{Q_{d}} is isomorphic to GL_{d} × GL_{n−d}. Let L_{d} = ⋀^{d} L be a lattice in V_{π_{d}}. We define the height function || · ||_{L_{d},𝔸} on GL(V_{π_{d}}, 𝔸)V_{π_{d}}(k) similary as above. By definition,

γ_{π_{d}}(|| · ||_{L_{d},𝔸}) = ( max_{g∈G(𝔸)^{1}/G(k)} min_{γ∈G(k)/Q_{d}(k)} ||π_{d}(gγ)v_{d}||_{L_{d},𝔸})^{2/[k:ℚ]} .

This γ_{π_{d}} is equal to the constant γ_{n,d}(k) defined by Thunder, and further, if k = ℚ, this is equal to the constnat defined by Rankin.

3. A lower bound of γ_{π}

We fix ξ ∈ ℜ_{k}^{s}(G), a k -rational representation π ∈ ξ and a K^{G} -invariant normalized height function || · ||_{𝔸} on GL_{n}(𝔸)V_{π}(k), where n = dim π. Let x_{0} ∈ D_{π}(k). We define the function f_{π} : G(𝔸) → ℝ_{+}^{×} by f_{π}(g) = ||π(g)x_{0}||_{𝔸}. We have

f_{π}(gh) = |λ_{π}(h)|_{𝔸}f_{π}(g) for all h ∈ Q_{π}(𝔸) and g ∈ G(𝔸).

Therefore, f_{π} is regarded as a function on Y_{Q_{π}} = G(𝔸)^{1}/Q_{π}(𝔸)^{1} . In the following, we assume that ξ is maximal, i.e., ρ_{k}(λ_{ξ}) is a positive integer multiple of one fundamental k -weight m_{α}. Then Q_{π} is the standard maximal k -parabolic subgroup associated to Δ_{k}\{α}. For simplicity, we omit the subscript π and write Q for Q_{π}. Let n_{Q} be the positive integer such that n_{Q}^{−1}α|_{Z_{Q}} is a ℤbase of X^{∗}(Z_{Q}/Z_{G}). We set α_{Q} = n_{Q}^{−1}α|_{Z_{Q}}. Then the Haar measuer μ_{Z_{Q}^{+}/Z_{G}^{+}}

equals the pull-back of the measure dx/x by the isomorphism |α_{Q}|_{𝔸} : Z_{Q}^{+}/Z_{G}^{+} → ℝ_{+}^{×}. Note that dim U_{Q}α|_{Z_{Q}} is the restriction of the k -rational character m ↦ det(Ad(m)|_{Lie(U_{Q})}) of M_{Q} to Z_{Q}, where by Ad(m)|_{Lie(U_{Q})} we mean the adjoing action of m on the Lie algebra of U_{Q}. Since δ_{Q}(m) = | det(Ad(m)|_{Lie(U_{Q})})|_{𝔸}, we have

(3.1)  δ_{Q}(z) = |α_{Q}(z)|_{𝔸}^{e_{Q}} ,  (z ∈ Z_{Q}(𝔸))

where e_{Q} = n_{Q} dim U_{Q}. The quotient morphism Z_{Q} → Z_{Q}/Z_{G} induces an isomorphism X^{∗}(Z_{Q}/Z_{G}) ⊗_{ℤ} ℚ → X^{∗}(Z_{Q} ∩ G^{ss}) ⊗_{ℤ} ℚ, where G^{ss} denotes the derived group of G. Under the identification X^{∗}(Z_{Q} ∩ G^{ss}) ⊗_{ℤ} ℚ ≅ X^{∗}(Z_{Q}/Z_{G}) ⊗_{ℤ} ℚ, there exists the positive rational number e_{ξ} such that

(3.2)  ρ_{k}(λ_{ξ})|_{Z_{Q}∩G^{ss}} = e_{ξ}α_{Q} .

Then f_{π}(ι_{Q}(K^{Q}, zZ_{G}^{+})) = |α_{Q}(z)|_{𝔸}^{e_{ξ}} holds for any zZ_{G}^{+} ∈ Z_{Q}^{+}/Z_{G}^{+} .

44 Watanabe Lemma 2. For any measurable function φ on ℝ, one has

∫_{Y_{Q}} φ(f_{π}(g))dω_{Y_{Q}}(g) = \frac{C_{G}d_{Q}}{C_{Q}d_{G}e_{ξ}} ∫_{ℝ_{+}^{×}} φ(x)x^{\frac{e_{Q}}{e_{ξ}}} \frac{dx}{x} .

Proof. By Corollary to Lemma 1, ∫_{Y_{Q}} φ(f_{π}(g))dω_{Y_{Q}}(g)

= \frac{C_{G}d_{Q}}{C_{Q}d_{G}} ∫_{Z_{Q}^{+}/Z_{G}^{+}} φ(f_{π}(ι_{Q}(K^{Q}, zZ_{G}^{+})))δ_{Q}(z)dμ_{Z_{Q}^{+}/Z_{G}^{+}}(zZ_{G}^{+})

= \frac{C_{G}d_{Q}}{C_{Q}d_{G}} ∫_{Z_{Q}^{+}/Z_{G}^{+}} φ(|α_{Q}(z)|_{𝔸}^{e_{ξ}})|α_{Q}(z)|_{𝔸}^{e_{Q}}dμ_{Z_{Q}^{+}/Z_{G}^{+}}(zZ_{G}^{+})

= \frac{C_{G}d_{Q}}{C_{Q}d_{G}} ∫_{ℝ_{+}^{×}} φ(x^{e_{ξ}})x^{e_{Q}} \frac{dx}{x} .

By change of variables, we obtain the assertion. Theorem 1. One has

γ_{π}(|| · ||_{𝔸}) ≥ (\frac{C_{Q}d_{G}e_{Q}τ(G)}{C_{G}d_{Q}τ(Q)})^{\frac{2e_{ξ}}{[k:ℚ]e_{Q}}} .

As is known from the proof, this estimate is independent of the choice of a K^{G} - invariant normalized height function || · ||_{𝔸}. Proof. Fix a positive real number t such that

\frac{C_{G}d_{Q}}{C_{Q}d_{G}e_{Q}} t^{e_{Q}/e_{ξ}} < \frac{τ(G)}{τ(Q)} .

Let ε be a sufficiently small positive real number satisfying

\frac{C_{G}d_{Q}}{C_{Q}d_{G}e_{Q}} (t^{e_{Q}/e_{ξ}} + ε) < \frac{τ(G)}{τ(Q)} .

We define the function φ(x) by

φ(x) = \{ \begin{matrix}1 & (0 ≤ x ≤ t) \\ (\frac{x}{t})^{\frac{e_{Q}}{e_{ξ}} (−1−ε^{−1}t^{e_{Q}/e_{ξ}})} & (t < x). \end{matrix}

Then, by Lemma 2,

\begin{aligned}∫_{Y_{Q}} φ(f_{π}(g))dω_{Y_{Q}}(g) & = \frac{C_{G}d_{Q}}{C_{Q}d_{G}e_{ξ}} ∫_{ℝ_{+}^{×}} φ(x)x^{e_{Q}/e_{ξ}} \frac{dx}{x} \\ & = \frac{C_{G}d_{Q}}{C_{Q}d_{G}e_{Q}} (t^{e_{Q}/e_{ξ}} + ε) < \frac{τ(G)}{τ(Q)} . \end{aligned}

Thus we obtain

\begin{matrix}∫_{Y_{Q}} φ(f_{π}(g))dω_{Y_{Q}}(g) \\ = \frac{1}{τ(Q)} ∫_{G(𝔸)^{1}/G(k)} ∑_{γ∈G(k)/Q(k)} φ(f_{π}(gγ))d(ω_{G(𝔸)^{1}}/ω_{G(k)})(g) < \frac{τ(G)}{τ(Q)} . \end{matrix}

Therefore there exists g ∈ G(𝔸)^{1} such that ∑_{γ∈G(k)/Q(k)} φ(f_{π}(gγ)) < 1. This implies φ(f_{π}(gγ)) < 1 for all γ ∈ G(k)/Q(k). By definition of φ, we have f_{π}(gγ) > t for all γ ∈ G(k)/Q(k). Namely

min_{γ∈G(k)/Q(k)} f_{π}(gγ) > t .

Hence

γ′_{π} = max_{g∈G(𝔸)^{1}/G(k)} min_{γ∈G(k)/Q(k)} f_{π}(gγ) ≥ sup\{t > 0 : \frac{C_{G}d_{Q}}{C_{Q}d_{G}e_{Q}} t^{e_{Q}/e_{ξ}} < \frac{τ(G)}{τ(Q)}\}

= (\frac{C_{Q}d_{G}e_{Q}τ(G)}{C_{G}d_{Q}τ(Q)})^{\frac{e_{ξ}}{e_{Q}}} .

4. Examples

First, we compute C_{G}. We use the same notations as in Sections 1 and 3. We state an integration formula. Let w_{0} be an element in the normalizer of S in G which represents the longest element in the relative Weyl group of G with respect to S. We set U^{−} = w_{0}^{−1}Uw_{0} . Then the homogeneous space U^{−}\G has the right G-invariant gauge form ω^{U^{−}}\ω^{G} defined over k matching with ω^{U^{−}} and ω^{G}. By the Bruhat decomposition, P is regarded as a Zariski open subset of U^{−}\G. Then (ω^{U^{−}}\ω^{G})|_{P} yields a right invariant gauge form on P defined over k. Since a right invariant gauge form on P defined over k is unique up to scalar multiplications, we can choose the gauge form ω^{M} so that (ω^{U^{−}}\ω^{G})|_{P} = ω^{U}ω^{M}. Then, for each v ∈ 𝔙 and φ ∈ C_{0}(G(k_{v})), we have

(4.1) \begin{matrix}∫_{G(k_{v})} φ(g)dω_{v}^{G} \\ = ∫_{U^{−}(k_{v})} ∫_{U(k_{v})} ∫_{M(k_{v})} φ(u^{−}um)dω_{v}^{U^{−}}(u^{−})dω_{v}^{U}(u)dω_{v}^{M}(m) \\ = ∫_{U^{−}(k_{v})} ∫_{M(k_{v})} ∫_{U(k_{v})} φ(u^{−}mu)δ_{P,v}(m)dω_{v}^{U^{−}}(u^{−})dω_{v}^{M}(m)dω_{v}^{U}(u) \end{matrix}

where δ_{P,v}^{−1} denotes the modular character of P (k_{v}). We define some notations. We fix v ∈ 𝔙 for the moment. By the Iwasawa decompositon, g ∈ U^{−}(k_{v}) is decomposed to h_{v}(g)m_{v}(g)u_{v}(g) with h_{v}(g) ∈ K_{v}^{G}, m_{v}(g) ∈ M(k_{v}), u_{v}(g) ∈ U(k_{v}). Then the function η_{v} of U^{−}(k_{v}) defined by η_{v}(g) = δ_{P,v}(m_{v}(g))^{−1} is well-defined. It is known that the integral

I_{v}(G, P) = ∫_{U^{−}(k_{v})} η_{v}(g)dω_{v}^{U^{−}}(g)

converges. We set I_{∞}(G, P) = ∏_{v∈𝔙_{∞}} I_{v}(G, P). Let K_{∞}^{G} and K_{f}^{G} denote the infinite and the finite part of K^{G}, respectively. Similarly, K_{∞}^{P} and K_{f}^{P} are defined.

Lemma 3. One has

C_{G} = |D_{k}|^{(dim P −dim G)/2} \frac{ω_{f}^{G}(K_{f}^{G})}{ω_{f}^{P} (K_{f}^{P})}I_{∞}(G, P) .

Proof. The proof is essentially same as [12]. See also [11, Proposition 5.2]. For a given φ ∈ C_{0}(K_{∞}^{P}\P (k_{∞})), let Φ be a left K^{G} -invariant function on G(𝔸) such that the support of Φ is contained in G(k_{∞})K_{f}^{G} and Φ(h_{∞}p_{∞}h_{f}) = φ(p_{∞}) for h_{∞} ∈ K_{∞}^{G}, p_{∞} ∈ P (k_{∞}), h_{f} ∈ K_{f}^{G}. On the one hand, we have

∫_{G(𝔸)} Φ(g)dω_{𝔸}^{G}(g)
= C_{G} ∫_{G(𝔸)} Φ(g)dν_{G(𝔸)}(g)
= C_{G} ∫_{M(𝔸)} ∫_{U(𝔸)} Φ(mu)δ_{P} (m)dω_{𝔸}^{M}(m)dω_{𝔸}^{U}(u)
= C_{G}|D_{k}|^{− dim P/2}ω_{f}^{P} (K_{f}^{P})
× ∫_{M(k_{∞})} ∫_{U(k_{∞})} φ(m_{∞}u_{∞})δ_{P,∞}(m_{∞})dω_{∞}^{M}(m_{∞})dω_{∞}^{U}(u_{∞}) ,

where δ_{P,∞} = ∏_{v∈𝔙_{∞}} δ_{P,v}. On the other hand, by (4.1), we have

∫_{G(𝔸)} Φ(g)dω_{𝔸}^{G}(g)
= |D_{k}|^{− dim G/2}ω_{f}^{G}(K_{f}^{G}) ∫_{G(k_{∞})} Φ(g)dω_{∞}^{G}(g)
= |D_{k}|^{− dim G/2}ω_{f}^{G}(K_{f}^{G}) ∫_{U^{−}(k_{∞})} ∫_{M(k_{∞})} ∫_{U(k_{∞})} Φ(u^{−}mu)δ_{P,∞}(m)
× dω_{∞}^{U^{−}}(u^{−})dω_{∞}^{M}(m)dω_{∞}^{U}(u)
= |D_{k}|^{− dim G/2}ω_{f}^{G}(K_{f}^{G}) ∫_{U^{−}(k_{∞})} ∫_{M(k_{∞})} ∫_{U(k_{∞})} φ(mu)η_{∞}(u^{−})δ_{P,∞}(m)
× dω_{∞}^{U^{−}}(u^{−})dω_{∞}^{M}(m)dω_{∞}^{U}(u)
= |D_{k}|^{− dim G/2}ω_{f}^{G}(K_{f}^{G})I_{∞}(G, P)
× ∫_{M(k_{∞})} ∫_{U(k_{∞})} φ(mu)δ_{P,∞}(m)dω_{∞}^{M}(m)dω_{∞}^{U}(u) .

These imply the assertion.

Corollary . If Q is a standard k -parabolic subgroup of G, then

\frac{C_{Q}}{C_{G}} = |D_{k}|^{dim U_{Q}/2} \frac{ω_{f}^{Q}(K_{f}^{Q})I_{∞}(M_{Q}, P^{Q})}{ω_{f}^{G}(K_{f}^{G})I_{∞}(G, P)} .

Proof. It follows from the following two equations;

\begin{aligned}\frac{ω_{f}^{P} (K_{f}^{P})}{ω_{f}^{G}(K_{f}^{G})} · \frac{ω_{f}^{M_{Q}}(K_{f}^{M_{Q}})}{ω_{f}^{P^{Q}}(K_{f}^{G} ∩ P^{Q}(𝔸_{f}))} & = \frac{ω_{f}^{U}(K_{f}^{G} ∩ U(𝔸_{f}))ω_{f}^{M_{Q}}(K_{f}^{M_{Q}})}{ω_{f}^{G}(K_{f}^{G})ω_{f}^{U^{Q}}(K_{f}^{G} ∩ U^{Q}(𝔸_{f}))} \\ & = \frac{ω_{f}^{U_{Q}}(K_{f}^{G} ∩ U_{Q}(𝔸_{f}))ω_{f}^{M_{Q}}(K_{f}^{M_{Q}})}{ω_{f}^{G}(K_{f}^{G})} . \end{aligned}

\frac{|D_{k}|^{(dim P^{Q}−dim M_{Q})/2}}{|D_{k}|^{(dim P −dim G)/2}} = \frac{|D_{k}|^{− dim U^{Q}/2}}{|D_{k}|^{− dim U/2}} = |D_{k}|^{dim U_{Q}/2} .

From now on, we assume that G is split over k. Then G has a model over 𝔒, i.e., there is a smooth affine group scheme 𝔊 of finite type over 𝔒 such that the generic fiber 𝔊×_{𝔒} k is isomorphic to G and the special fiber 𝔊×_{𝔒} 𝔒_{𝔭}/𝔭 is a split reductive algebraic group over 𝔒_{𝔭}/𝔭 for all finite primes 𝔭 ∈ 𝔙_{f}. In this case, we choose K_{𝔭}^{G} as 𝔊(𝔒_{𝔭}) for 𝔭 ∈ 𝔙_{f}. Let r_{k}(G) be the k -rank of the derived group G^{ss} of G and let a_{1}^{G}, ⋯, a_{r_{k}(G)}^{G} be positive integers such that the polynomial in X

∏_{i=1}^{r_{k}(G)}(X^{2a_{i}^{G}−1} + 1)

is the Poincaré polynomial of the compact form of G^{ss}(ℂ). Then we have

ω_{f}^{G}(K_{f}^{G}) = σ_{k}(G)^{−1} ∏_{i=1}^{r_{k}(G)} ζ_{k}(a_{i}^{G})^{−1} ,

where ζ_{k}(·) denotes the Dedekind zeta function of k (cf.[15]). In a similar fashion, one has

ω_{f}^{Q}(K_{f}^{Q}) = ω_{f}^{M_{Q}}(K_{f}^{M_{Q}}) = σ_{k}(M_{Q})^{−1} ∏_{i=1}^{r_{k}(M_{Q})} ζ_{k}(a_{i}^{M_{Q}})^{−1}.

Since the representations σ_{G} and σ_{M_{Q}} are trivial, σ_{k}(G)/σ_{k}(M_{Q}) is equal to

(Res_{s=1}ζ_{k}(s))^{dim Z_{G}−dim Z_{Q}} .

Furthermore, it is known that I_{∞}(G, P) is described as

(4.2) I_{∞}(G, P) = \{ ∏_{{α∈Φ_{k} \atop α>0}} \sqrt{π} \frac{Γ(c_{α}/2)}{Γ((c_{α} + 1)/2)}\}^{r_{1}} \{ ∏_{{α∈Φ_{k} \atop α>0}} 2π \frac{Γ(c_{α})}{Γ(c_{α} + 1)}\}^{r_{2}} ,

where r_{1} and r_{2} denote the numbers of real and imaginary places of k, respectively, Γ(·) is the Gamma function and the constant c_{α} is defined by

c_{α} = (\frac{1}{2} ∑_{{β∈Φ_{k} \atop β>0}} β, α^{∨})

for each positive root α ∈ Φ_{k} (cf.[6],[12]). Summing up, we obtain the following estimate of γ_{π}; Theorem 2. Let G and K_{f}^{G} be as above, ξ ∈ ℜ_{k}^{s}(G) = ℜ(G) be a maximal class, π ∈ ξ be a k -rational representation of G and Q = Q_{π} be the stabilizer of the highest weight space D_{π} in G. Then, for any K^{G} -invariant normalized height || · ||_{𝔸} on GL(V_{π}, 𝔸)V_{π}(k), one has

γ_{π}(|| · ||_{𝔸}) ≥ \{\frac{|D_{k}|^{dim U_{Q}/2}}{Res_{s=1}ζ_{k}(s)} \frac{∏_{i=1}^{r_{k}(G)} ζ_{k}(a_{i}^{G})}{∏_{j=1}^{r_{k}(M_{Q})} ζ_{k}(a_{j}^{M_{Q}})} \frac{I_{∞}(M_{Q}, P^{Q})}{I_{∞}(G, P)} \frac{d_{G}e_{Q}τ(G)}{d_{Q}τ(Q)}\}^{\frac{2e_{ξ}}{[k:ℚ]e_{Q}}} .

The factors I_{∞}(G, P) and I_{∞}(M_{Q}, P^{Q}) are calculated by the formula (4.2).

We shall apply this theorem to classical split groups.

Example 2. The case of G = GL(n). We use the same notations as in Example 1. We fix 1 ≤ d ≤ n−1 and set Q = Q_{d}. For a positive integer q, let ξ_{d}^{q} ∈ ℜ(G) be the class whose highest weight is equal to qℓ_{d} and let π_{d}^{q} ∈ ξ_{d}^{q} be a k -rational representation. It follows from easy calculations that

d_{G} = n, d_{Q} = d(n − d), e_{Q} = d(n − d), e_{ξ_{d}^{q}} = q \frac{d(n − d)}{n}

\frac{ω_{f}^{G}(K_{f}^{G})}{ω_{f}^{M_{Q}}(K_{f}^{M_{Q}})} = \frac{1}{Res_{s=1}ζ_{k}(s)} \frac{∏_{i=n−d+1}^{n} ζ_{k}(i)}{∏_{j=2}^{d} ζ_{k}(j)}

\frac{I_{∞}(M_{Q}, P^{Q})}{I_{∞}(G, P)} = (\frac{∏_{i=n−d+1}^{n} π^{−i/2}Γ(i/2)}{∏_{j=2}^{d} π^{−j/2}Γ(j/2)})^{r_{1}} (\frac{∏_{i=n−d+1}^{n}(2π)^{1−i}Γ(i)}{∏_{j=2}^{n}(2π)^{1−j}Γ(j)})^{r_{2}}

We note that

τ(G) = τ(Q) = 1 and Res_{s=1}ζ_{k}(s) = |D_{k}|^{−1/2} \frac{2^{r_{1}}(2π)^{r_{2}}h_{k}R_{k}}{w_{k}} ,

where h_{k} is the class number of k, R_{k} the regulator of k and w_{k} the number of roots of 1 in k. If we put Z_{k}(s) = (π^{−s/2}Γ(s/2))^{r_{1}}((2π)^{1−s}Γ(s))^{r_{2}}ζ_{k}(s), we obtain

γ_{π_{d}^{q}} ≥ (\frac{|D_{k}|^{d(n−d)/2}n}{Res_{s=1}ζ_{k}(s)} \frac{∏_{i=n−d+1}^{n} Z_{k}(i)}{∏_{j=2}^{d} Z_{k}(j)})^{2q/([k:ℚ]n)} .

When q = 1, this estimate is the same as Thunder’s one (0.2).

Example 3. The case of G = SO(n, n + 1), Sp(2n) or SO(n, n). Let V be an m-dimensional vector space over k equipped with a non-degenerate ε-symmetric bilinear form Ψ with Witt index n. We consider the following three cases;

(B_{n}) m = 2n + 1 and ε = 1.
(C_{n}) m = 2n and ε = −1.
(D_{n}) m = 2n and ε = 1.

We choose a k -basis e_{1}, ⋯, e_{m} of V (k) by which Ψ is represented as follows;

Ψ = \{ \begin{matrix}( \begin{matrix}0 & 0 & J_{n} \\ 0 & 2 & 0 \\ J_{n} & 0 & 0 \end{matrix} ) & (B_{n}) \\ ( \begin{matrix}0 & J_{n} \\ −J_{n} & 0 \end{matrix} ) & (C_{n}) \\ ( \begin{matrix}0 & J_{n} \\ J_{n} & 0 \end{matrix} ) & (D_{n}) \end{matrix} where J_{n} = ( \begin{matrix}0 & 0 & ⋯ & 0 & 1 \\ 0 & 0 & ⋯ & 1 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 1 & ⋯ & 0 & 0 \\ 1 & 0 & ⋯ & 0 & 0 \end{matrix} ) .

Let L be the free 𝔒-lattice generated by e_{1}, ⋯, e_{m}. We set

G = {g ∈ SL(V) : ^{t}gΨg = Ψ} ,
K_{v}^{G} = {g ∈ G(k_{v}) : ||gx||_{L,v} = ||x||_{L,v} for all x ∈ V (k_{v})} , (v ∈ 𝔙)

Let S be the group of diagonal matrices in G and P the group of upper triangular matrices in G. For g ∈ S, ϵ_{i}(g) denotes the i-th diagonal element of g. Then ϵ_{1}, ⋯, ϵ_{n} is a ℤ-basis of X^{∗}(S). The simple roots α_{i} and the fundamental weights ℓ_{i} are given as follows;

α_{i} = ϵ_{i} − ϵ_{i+1}, (1 ≤ i ≤ n − 1), α_{n} = \{ \begin{matrix}ϵ_{n} & (B_{n}) \\ 2ϵ_{n} & (C_{n}) \\ ϵ_{n−1} + ϵ_{n} & (D_{n}) \end{matrix}

ℓ_{i} = \{ \begin{matrix}\frac{1}{2}(ϵ_{1} + ⋯ + ϵ_{n−2} + ϵ_{n−1} − ϵ_{n}) & (D_{n} and i = n − 1) \\ \frac{1}{2}(ϵ_{1} + ⋯ + ϵ_{n}) & (B_{n} and i = n or D_{n} and i = n) \\ ϵ_{1} + ⋯ + ϵ_{i} & (otherwise) \end{matrix}

We fix 1 ≤ d ≤ n and a positive integer q so that qℓ_{d} ∈ X^{∗}(S). Let ξ_{d}^{q} ∈ ℜ(G) be the class whose highest weight is equal to qℓ_{d} and let π_{d}^{q} ∈ ξ_{d}^{q} be a k - rational representation of G. Then Q_{d} = Q_{π_{d}^{q}} be the standard maximal parabolic subgroup of G associated with Δ\{α_{d}}. We denote by U_{d} the unipotent radical of Q_{d} and by M_{d} the Levi subgroup of Q_{d} containing S. For convenience, we define the constant δ(D_{n}) by

δ(D_{n}) = \{ \begin{matrix}1 & (D_{n} and d = n − 1) \\ 0 & (otherwise) \end{matrix}

50 Watanabe It is easy to check the following

dim U_{d} = \{ \begin{matrix}\frac{1}{2}d(d + 1) + 2d(n − d) & (B_{n} or C_{n}) \\ \frac{1}{2}(d + δ(D_{n}))(d + δ(D_{n}) − 1) + 2d(n − d − δ(D_{n})) & (D_{n}) \end{matrix}

and

(4.3)

d_{G} = 1 , d_{Q_{d}} = d + δ(D_{n}) ,
e_{Q_{d}} = \{ \begin{matrix}2 dim U_{d} & (C_{n} and d = n or D_{n} and d ≥ n − 1) \\ dim U_{d} & (otherwise) \end{matrix}
\frac{τ(G)}{τ(Q_{d})} = \{ \begin{matrix}2 & (B_{n} and d = n or D_{n} and d ≥ n − 1) \\ 1 & (otherwise) \end{matrix}
e_{ξ_{d}^{q}} = \{ \begin{matrix}qn/2 & (B_{n} and d = n or D_{n} and d ≥ n − 1) \\ qd & (otherwise) \end{matrix}

Furthermore, we have

(4.4)

∏_{i=1}^{r_{k}(G)} ζ_{k}(a_{i}^{G}) ∏_{j=1}^{r_{k}(M_{d})} ζ_{k}(a_{j}^{M_{d}})^{−1}

= \frac{1}{∏_{j=2}^{d+δ(D_{n})} ζ_{k}(j)} × \{ \begin{matrix}∏_{i=n−d+1}^{n} ζ_{k}(2i) & (B_{n} or C_{n}) \\ \frac{ζ_{k}(n)}{ζ_{k}(n − d)} ∏_{i=n−d}^{n−1} ζ_{k}(2i) & (D_{n} and d ≤ n − 2) \\ ζ_{k}(n) ∏_{i=1}^{n−1} ζ_{k}(2i) & (D_{n} and d ≥ n − 1) \end{matrix}

We define the function Λ_{k}(s) by

Λ_{k}(s) = {π^{−s/2}Γ(s/2)}^{r_{1}} {(2π)^{1−s}Γ(s)}^{r_{2}} .

For x ∈ ℝ, we write [x] for the largest integer not exceeding x, and define c_{n,d} by

(B_{n}) ∏_{h=n−d+1}^{n} \frac{Λ_{k}(2h)}{Λ_{k}(2h − 1)} ∏_{i=1}^{[(n−d)/2]} \frac{Λ_{k}(2i)}{Λ_{k}(2n − 2d − 2i + 1)} ∏_{j=1}^{[n/2]} \frac{Λ_{k}(2n − 2j + 1)}{Λ_{k}(2j)},
(C_{n}) \frac{Λ_{k}(n + 1)}{Λ_{k}(n − d + 1)} ∏_{i=1}^{[(n−d)/2]} \frac{Λ_{k}(2i + 1)}{Λ_{k}(2n − 2d − 2i + 2)} ∏_{j=1}^{[n/2]} \frac{Λ_{k}(2n − 2j + 2)}{Λ_{k}(2j + 1)},
(D_{n}) ∏_{i=1}^{[(n−d)/2]} \frac{Λ_{k}(2i − 1)}{Λ_{k}(2n − 2d − 2i)} ∏_{j=1}^{[n/2]} \frac{Λ_{k}(2n − 2j)}{Λ_{k}(2j − 1)}.

Then we have

(4.5)  \frac{I_{∞}(M_{d}, P^{Q_{d}})}{I_{∞}(G, P)} = c_{n,d} ∏_{i=1}^{d+δ(D_{n})} \frac{Λ_{k}(n − d + i)}{Λ_{k}(i)} .

(4.3)–(4.5) yield an explicit description of a lower bound of γ_{π_{d}^{q}}. For instance, when d = q = 1, then π_{1} : G → GL(V) is the natural representation and

γ_{π_{1}} ≥ \{ \begin{matrix}\{\frac{|D_{k}|^{n−1/2}(2n − 1)}{Res_{s=1}ζ_{k}(s)}Z_{k}(2n)\}^{2/((2n−1)[k:ℚ])} & (B_{n} or C_{n}) \\ \{\frac{|D_{k}|^{n−1}(2n − 2)}{Res_{s=1}ζ_{k}(s)} \frac{Z_{k}(2(n − 1))Z_{k}(n)}{Z_{k}(n − 1)}\}^{1/((n−1)[k:ℚ])} & (D_{n}) \end{matrix}

Acknowledgement. The author would like to thank Professor Morishita for useful discussions.

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Department of Mathematics Graduate School of Science Osaka University Toyonaka, Osaka, 560-0043 Japan watanabe@math.wani.osaka-u.ac.jp

Received January 6, 1998 and in final form July 5, 1998