FUNDAMENTAL HERMITE CONSTANTS

OF LINEAR ALGEBRAIC GROUPS

Takao Watanabe

Abstract. Let G be a connected reductive algebraic group defined over a global field k and Q a maximal k-parabolic subgroup of G. The constant γ(G, Q, k) attached to (G, Q) is defined as an analogue of Hermite’s constant. This constant depends only on G, Q and k in contrast to the previous definition of generalized Hermite constants ([W1]). Some functorial properties of γ(G, Q, k) are proved. In the case that k is a function field of one variable over a finite field, γ(GL_{n}, Q, k) is computed.

Let k be an algebraic number field of finite degree over ℚ and let G be a connected reductive algebraic group defined over k. In [W1], we introduced a constant γ_{π}^{G} attached to an absolutely irreducible strongly k-rational representation π: G ⟶ GL(V_{π}) of G. More precisely, if G(𝔸) denotes the adele group of G and G(𝔸)^{1} the unimodular part of G(𝔸), it is defined by

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

where x_{π} is a non-zero k-rational point of the highest weight line in the representation space V_{π} and || · || is a height function on the space GL(V_{π}(𝔸))V_{π}(k). This constant is called a generalized Hermite constant by the reason that, in the case when k = ℚ, G = GL_{n} and π = π_{d} is the d-th exterior representation of GL_{n}, γ_{π_{d}}^{GL_{n}} is none other than the Hermite - Rankin constant ([R]):

γ_{n,d} = max_{g∈GL_{n}(ℝ)} min_{\begin{matrix}x_{1},··· ,x_{d}∈ℤ^{n} \\ x_{1}∧···∧x_{d}≠0 \end{matrix}} \frac{det(^{t}x_{i}^{t}ggx_{j})_{1≤i,j≤d}}{| det g|^{2d/n}} .

This research was partly supported by Grant-in-Aid for Scientific Research (No. 12640023), Ministry of Education, Culture, Sports, Science and Technology, Japan

2000 Mathematics Subject Classification. Primary 11R56; Secondary 11G35, 14G25

Key words and phrases. Hermite constant, Tamagawa number, linear algebraic group.

When GL_{n} is defined over a general k, then γ_{π_{d}}^{GL_{n}} coincides with the following generalization of γ_{n,d} due to Thunder ([T2]):

γ_{n,d}(k) = max_{g∈GL_{n}(𝔸)} min_{X∈Gr_{d}(k^{n})} \frac{H_{g}(X)^{2}}{| det g|_{𝔸}^{2d/(n[k:ℚ])}} ,

where Gr_{d}(k^{n}) is the Grassmannian variety of d-dimensional subspaces in k^{n} and H_{g} a twisted height on Gr_{d}(k^{n}). In a general G, γ_{π}^{G} has a geometrical representation similarly to γ_{n,d}(k). In order to describe this, we change our primary object from a representation π to a parabolic subgroup of G. Thus, we first fix a k-parabolic subgroup Q of G, and then take a representation π such that the stabilizer Q_{π} of the highest weight line of π in G is equal to Q. The mapping g ↦ π(g^{−1})x_{π} gives rise to a k-rational embedding of the generalized flag variety Q\G into the projective space ℙV_{π}. Taking a k-basis of V_{π}(k), we get a height H_{π} on ℙV_{π}(k), and on Q(k)\G(k) by restriction. In this notation, γ_{π}^{G} is represented as

γ_{π}^{G} = max_{g∈G(𝔸)^{1}} min_{x∈Q(k)\G(k)} H_{π}(xg)^{2}.

In this paper, we investigate γ_{π}^{G} more closely when Q is a maximal k-parabolic subgroup of G. Especially, we shall show that π and H_{π} are not essentials of the constant γ_{π}^{G}, to be exact, there exists a constant γ(G, Q, k) depending only on G, Q and k such that the equality γ_{π}^{G} = γ(G, Q, k)^{c_{π}} holds for any π with Q_{π} = Q, where c_{π} is a positive constant depending only on π. This γ(G, Q, k) is called the fundamental Hermite constant of (G, Q) over k. We emphasize that there is a similarity between the definition of γ(G, Q, k) and a representation of the original Hermite’s constant γ_{n,1} as the maximum of some lattice constants. Remember that γ_{n,1} is represented as

γ_{n,1}^{1/2} = max_{\begin{matrix}g∈GL_{n}(ℝ) \\ | det g|=1 \end{matrix}} min\{T > 0: B_{T}^{n} ∩ gℤ^{n} ≠ \{0\}\} ,

where B_{T}^{n} stands for the ball of radius T with center 0 in ℝ^{n}. Corresponding to ℝ^{n}, we consider the adelic homogeneous space Y_{Q} = Q(𝔸)^{1}\G(𝔸)^{1} as a base space. The set X_{Q} of k-rational points of Q\G plays a role of the standard lattice ℤ^{n}. In addition, there is a notion of ”the ball” B_{T} of radius T in Y_{Q}, whose precise definition will be given in Section 2. Then γ(G, Q, k) is defined by

γ(G, Q, k) = max_{g∈G(𝔸)^{1}} min{T > 0: B_{T} ∩ X_{Q}g ≠ ∅}.

Independency of γ(G, Q, k) on π and H_{π} allows us to study some functorial properties of fundamental Hermite constants. For instance, the following theorems will be verified in Section 4.

Theorem. If β: G ⟶ G′ is a surjective k-rational homomorphism of connected reductive groups defined over k such that its kernel is a central k-split torus in G, then γ(G, Q, k) = γ(G′, β(Q), k).

Theorem. If R_{k/ℓ} denotes the functor of restriction of scalars for a subfield ℓ ⊂ k, then γ(R_{k/ℓ}(G), R_{k/ℓ}(Q), ℓ) = γ(G, Q, k).

Theorem. If both Q and R are standard maximal k-parabolic subgroups of G and M_{R} is a standard Levi subgroup of R, then one has an inequality of the form

γ(G, Q, k) ≤ γ(M_{R}, M_{R} ∩ Q, k)^{ω_{1}}γ(G, R, k)^{ω_{2}} ,

where ω_{1} and ω_{2} are rational numbers explicitly determined from Q and R.

These theorems are including the duality theorem: γ_{n,j}(k) = γ_{n,n−j}(k) for 1 ≤ j ≤ n − 1 and Rankin’s inequality ([R], [T2]): γ_{n,i}(k) ≤ γ_{j,i}(k)γ_{n,j}(k)^{i/j} for 1 ≤ i < j ≤ n − 1 as a particular case.

Since no any serious problem arises from replacing k with a function field of one variable over a finite field, we shall develop a theory of fundamental Hermite constants for any global field. In the case of number fields, the main theorem of [W1] gives a lower bound of γ(G, Q, k). An analogous result will be proved for the case of function fields in the last half of this paper. The case of G = GL_{n} is especially studied in detail because this case gives an analogue of the classical Hermite – Rankin constants. When k is a function field, it is almost trivial from definition that γ(G, Q, k) is a power of the cardinal number q of the constant field of k. Thus, the possible values of γ(G, Q, k) are very restricted if both lower and upper bounds are given. This is a striking difference between the number fields and the function fields. For example, it will be proved that γ(GL_{n}, Q, k) = 1 for all maximal Q and all n ≥ 2 provided that the genus of k is zero, i.e., k is a rational function field over a finite field.

The paper is organized as follows. In Section 1, we recall the Tamagawa measures of algebraic groups and homogenous spaces. In Sections 2 and 3, the constant γ(G, Q, k) is defined, and then a relation between γ(G, Q, k) and γ_{π}^{G} is explained. The functorial properties of γ(G, Q, k) is proved in Section 4. In Section 5, we will give a lower bound of γ(G, Q, k) when k is a function field, and compute γ(GL_{n}, Q, k) in Section 6.

Notation. 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 a global field, i.e., an algebraic number field of finite degree over ℚ or an algebraic function field of one variable over a finite field. In the latter case, we identify the constant field of k with the finite field 𝔽_{q} with q elements. Let 𝔙 be 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. If v is finite, 𝔒_{v} denotes the ring of integers in k_{v}, 𝔭_{v} the maximal ideal of 𝔒_{v}, 𝔣_{v} the residual field 𝔒_{v}/𝔭_{v} and q_{v} the order of 𝔣_{v}. We fix, once and for all, a Haar measure μ_{v} on k_{v} normalized so that μ_{v}(𝔒_{v}) = 1 if v ∈ 𝔙_{f}, μ_{v}([0, 1]) = 1 if v is a real place and μ_{v}(\{a ∈ k_{v}: a\overline{a} ≤ 1\}) = 2π if v is an imaginary place. Then the absolute value | · |_{v} on k_{v} is defined as |a|_{v} = μ_{v}(aC)/μ_{v}(C), where C is an arbitrary compact subset of k_{v} with nonzero measure.

Let 𝔸 be the adele ring of k, | · |_{𝔸} = ∏_{v∈𝔙} | · |_{v} the idele norm on the idele group 𝔸^{×} and μ_{𝔸} = ∏_{v∈𝔙} μ_{v} an invariant measure on 𝔸. The measure μ_{𝔸} is characterized by

μ_{𝔸}(𝔸/k) = \{ \begin{matrix}|D_{k}|^{1/2} & (if k is an algebraic number field of discriminant D_{k}). \\ q^{g(k)−1} & (if k is a function field of genus g(k)). \end{matrix}

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_{B}\μ_{A} (resp. μ_{A}/μ_{B}) denotes a unique right (resp. left) A-invariant measure on the homogeneous space B\A (resp. A/B) matching with μ_{A} and μ_{B}.

1. Tamagawa measures.

Let G be a connected affine algebraic group defined over k. For any k-algebra A, G(A) stands for the set of A-rational points of G. Let X^{∗}(G) and X_{k}^{∗}(G) be the free ℤ-modules consisting of all rational characters and all k-rational characters of G, respectively. The absolute Galois group Gal(\overline{k}/k) acts on X^{∗}(G). The representation of Gal(\overline{k}/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 = rank X_{k}^{∗}(G). Let ω^{G} be a nonzero right invariant gauge form on G defined over k. From ω^{G} and the fixed Haar measure μ_{v} on k_{v}, one can construct a right invariant Haar measure ω_{v}^{G} on G(k_{v}). Then, the Tamagawa measure on G(𝔸) is well defined by

ω_{𝔸}^{G} = μ_{𝔸}(𝔸/k)^{− dim G}ω_{∞}^{G}ω_{f}^{G} ,

where

ω_{∞}^{G} = ∏_{v∈𝔙_{∞}} ω_{v}^{G} and ω_{f}^{G} = σ_{k}(G)^{−1} ∏_{v∈𝔙_{f}} L_{v}(1, σ_{G})ω_{v}^{G} .

For each g ∈ G(𝔸), we define the homomorphism ϑ_{G}(g): X_{k}^{∗}(G) ⟶ ℝ_{+}^{×} by ϑ_{G}(g)(χ) = |χ(g)|_{𝔸} for χ ∈ X_{k}^{∗}(G). Then ϑ_{G} is a homomorphism from G(𝔸) into Hom_{ℤ}(X_{k}^{∗}(G), ℝ_{+}^{×}).

We write G(𝔸)^{1} for the kernel of ϑ_{G}. The Tamagawa measure ω_{G(𝔸)^{1}} on G(𝔸)^{1} is defined as follows:

• The case of ch(k) = 0. If a ℤ-basis χ_{1}, ⋯, χ_{n} of X_{k}^{∗}(G) is fixed, then Hom_{ℤ}(X_{k}^{∗}(G), ℝ_{+}^{×}) is identified with (ℝ_{+}^{×})^{n} and ϑ_{G} gives rise to an isomorphism from G(𝔸)^{1}\G(𝔸) onto (ℝ_{+}^{×})^{n}. Put the Lebesgue measure dt on ℝ and the invariant measure dt/t on ℝ_{+}^{×}. Then ω_{G(𝔸)^{1}} is the measure on G(𝔸)^{1} such that the quotient measure ω_{G(𝔸)^{1}}\ω_{𝔸}^{G} is the pullback of the measure ∏_{i=1}^{n} dt_{i}/t_{i} on (ℝ_{+}^{×})^{n} by ϑ_{G}. The measure ω_{G(𝔸)^{1}} is independent of the choice of a ℤ-basis of X_{k}^{∗}(G).

• The case of ch(k) > 0. The value group of the idele norm | · |_{𝔸} is the cyclic group q^{ℤ} generated by q (cf. [We2]). Thus the image Imϑ_{G} of ϑ_{G} is contained in Hom_{ℤ}(X_{k}^{∗}(G), q^{ℤ}) and G(𝔸)^{1} is an open normal subgroup of G(𝔸). Since the index of Imϑ_{G} in Hom_{ℤ}(X_{k}^{∗}(G), q^{ℤ}) is finite ([Oe, I, Proposition 5.6]),

(1.1)  d_{G}^{∗} = (log q)^{rankX_{k}^{∗}(G)}[Hom_{ℤ}(X_{k}^{∗}(G), q^{ℤ}): Imϑ_{G}]

is well defined. The measure ω_{G(𝔸)^{1}} is defined to be the restriction of the measure (d_{G}^{∗})^{−1}ω_{𝔸}^{G} to G(𝔸)^{1}.

In both cases, we put the counting measure ω_{G(k)} on G(k). The volume of G(k)\G(𝔸)^{1} with respect to the measure ω_{G} = ω_{G(k)}\ω_{G(𝔸)^{1}} is called the Tamagawa number of G and denoted by τ(G).

In the following, let G be a connected reductive group defined over k. We fix a maximally k-split torus S of G, a maximal k-torus S_{1} of G containing S, a minimal k-parabolic subgroup P of G containing S and a Borel subgroup B of P containing S_{1}. Denote by Φ_{k} and Δ_{k} the relative root system of G with respect to S and the set of simple roots of Φ_{k} corresponding to P, respectively. 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 standard kparabolic subgroup R of G, R has a unique Levi subgroup M_{R} containing M. We denote by U_{R} the unipotent radical of R. Throughout this paper, we fix a maximal compact subgroup K of G(𝔸) satisfying the following property; For every standard k-parabolic subgroup R of G, K ∩ M_{R}(𝔸) is a maximal compact subgroup of M_{R}(𝔸) and M_{R}(𝔸) possesses an Iwasawa decomposition (M_{R}(𝔸) ∩ U(𝔸))M(𝔸)(K ∩ M_{R}(𝔸)). We set K^{M_{R}} = K ∩ M_{R}(𝔸), P^{R} = M_{R} ∩ P and U^{R} = M_{R} ∩ U.

Let R be a standard k-parabolic subgroup of G and Z_{R} be the greatest central k-split torus in M_{R}. The restriction map X_{k}^{∗}(M_{R}) ⟶ X_{k}^{∗}(Z_{R}) is injective. Since X_{k}^{∗}(M_{R}) has the same rank as X_{k}^{∗}(Z_{R}), both indexes

d_{R} = [X_{k}^{∗}(Z_{R}) : X_{k}^{∗}(M_{R})] and d̂_{R} = [X_{k}^{∗}(Z_{R}/Z_{G}) : X_{k}^{∗}(M_{R}/Z_{G})]

are finite. We define another Haar measure ν_{M_{R}(𝔸)} of M_{R}(𝔸) as follows. Let ω_{𝔸}^{M} and ω_{𝔸}^{U^{R}} be the Tamagawa measures of M(𝔸) and U^{R}(𝔸), respectively. The modular character δ_{P^{R}}^{−1} of P^{R}(𝔸) is a function on M(𝔸) which satisfies the integration formula

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

Let ν_{K^{M_{R}}} be the Haar measure on K^{M_{R}} normalized so that the total volume equals one. Then the mapping

f ↦ ∫_{U^{R}(𝔸)×M(𝔸)×K^{M_{R}}} f(nmh)δ_{P^{R}}(m)^{−1}dω_{𝔸}^{U^{R}}(u)dω_{𝔸}^{M}(m)dν_{K^{M_{R}}} (h) , (f ∈ C_{0}(M_{R}(𝔸)))

defines an invariant measure on M_{R}(𝔸) and is denoted by ν_{M_{R}(𝔸)}. There exists a positive constant C_{R} such that

ω_{𝔸}^{M_{R}} = C_{R}ν_{M_{R}(𝔸)} .

We have the following compatibility formula:

(1.2) ∫_{G(𝔸)} f(g)dω_{𝔸}^{G}(g) = \frac{C_{G}}{C_{R}} ∫_{U_{R}(𝔸)×M_{R}(𝔸)×K} f(umh)δ_{R}(m)^{−1}dω_{𝔸}^{U_{R}}dω_{𝔸}^{M_{R}}(m)dν_{K}(h)

for f ∈ C_{0}(G(𝔸)), where δ_{R}^{−1} is the modular character of R(𝔸).

On the homogeneous space Y_{R} = R(𝔸)^{1}\G(𝔸)^{1}, we define the right G(𝔸)^{1}-invariant measure ω_{Y_{R}} by ω_{R(𝔸)^{1}}\ω_{G(𝔸)^{1}}. We note that both G(𝔸)^{1} and R(𝔸)^{1} are unimodular.

2. Definition of fundamental Hermite constants.

Throughout this paper, Q denotes a standard maximal k-parabolic subgroup of G. There is an only one simple root α ∈ Δ_{k} such that the restriction of α to Z_{Q} is non-trivial. Let n_{Q} be the positive integer such that n_{Q}^{−1}α|_{Z_{Q}} is a ℤ-basis of X_{k}^{∗}(Z_{Q}/Z_{G}). We write α_{Q} and α̂_{Q} for n_{Q}^{−1}α|_{Z_{Q}} and d̂_{Q}n_{Q}^{−1}α|_{Z_{Q}}, respectively. Then α̂_{Q} is a ℤ-basis of the submodule X_{k}^{∗}(M_{Q}/Z_{G}) of X_{k}^{∗}(Z_{Q}/Z_{G}). If we set e_{Q} = n_{Q} dim U_{Q} and ê_{Q} = n_{Q} dim U_{Q}/d̂_{Q}, then

δ_{Q}(z) = |α_{Q}(z)|_{𝔸}^{e_{Q}} and δ_{Q}(m) = |α̂_{Q}(m)|ê_{Q} ; 𝔸

holds for z ∈ Z_{Q}(𝔸) and m ∈ M_{Q}(𝔸).

Define a map z_{Q}: G(𝔸) ⟶ Z_{G}(𝔸)M_{Q}(𝔸)^{1}\M_{Q}(𝔸) by z_{Q}(g) = Z_{G}(𝔸)M_{Q}(𝔸)^{1}m if g = umh, u ∈ U_{Q}(𝔸), m ∈ M_{Q}(𝔸) and h ∈ K. This is well defined and a left Z_{G}(𝔸)Q(𝔸)^{1}- invariant. Since Z_{G}(𝔸)^{1} = Z_{G}(𝔸) ∩ G(𝔸)^{1} ⊂ M_{Q}(𝔸)^{1}, z_{Q} gives rise to a map from Y_{Q} = Q(𝔸)^{1}\G(𝔸)^{1} to M_{Q}(𝔸)^{1}\(M_{Q}(𝔸) ∩ G(𝔸)^{1}). Namely, we have the following commutative diagram:

Y_{Q}  ⟶^{z_{Q}} M_{Q}(𝔸)^{1}\(M_{Q}(𝔸) ∩ G(𝔸)^{1})
↓  ↓
Z_{G}(𝔸)Q(𝔸)^{1}\G(𝔸) ⟶^{z_{Q}} Z_{G}(𝔸)M_{Q}(𝔸)^{1}\M_{Q}(𝔸)

In this diagram, the vertical arrows are injective, and in particular, these are bijective if ch(k) = 0. We further define a function H_{Q}: G(𝔸) ⟶ ℝ_{+}^{×} by H_{Q}(g) = |α̂_{Q}(z_{Q}(g))|_{𝔸}^{−1} for g ∈ G(𝔸). This has the following property:

• The case of ch(k) = 0. Let Z_{G}^{+} and Z_{Q}^{+} be the subgroups of Z_{G}(𝔸) and Z_{Q}(𝔸), respectively, defined as in [W1]. Then H_{Q} gives a bijection from Z_{G}^{+}\Z_{Q}^{+} onto ℝ_{+}^{×}. If (H_{Q}|_{Z_{G}^{+}\Z_{Q}^{+}})^{−1} denotes the inverse map of this bijection, then the map

i_{Q}: ℝ_{+}^{×} × K ⟶ Y_{Q}: (t, h) ↦ Q(𝔸)^{1}(H_{Q}|_{Z_{G}^{+}\Z_{Q}^{+}})^{−1}(t)h

is surjective.

• The case of ch(k) > 0. The value group |α̂_{Q}(M_{Q}(𝔸) ∩ G(𝔸)^{1})|_{𝔸} is a subgroup of q^{ℤ}. Let q_{0} = q_{0}(Q) be the generator of |α̂_{Q}(M_{Q}(𝔸) ∩ G(𝔸)^{1})|_{𝔸} that is greater than one. Then H_{Q} gives a surjection from Y_{Q} onto the cyclic group q_{0}^{ℤ}.

We set X_{Q} = Q(k)\G(k), which is regarded as a subset of Y_{Q}. Let B_{T} = {y ∈ Y_{Q}: H_{Q}(y) ≤ T } for T > 0. The volume of B_{T} is given by

ω_{Y_{Q}}(B_{T}) = \{ \begin{matrix}\frac{C_{G}d_{Q}}{C_{Q}d_{G}e_{Q}} T ê_{Q} & (ch(k) = 0) \\ \frac{C_{G}d_{Q}^{∗}}{C_{Q}d_{G}^{∗}} \frac{q_{0}^{[log_{q_{0}} T]ê_{Q}}}{1 − q_{0}^{−ê_{Q}}} & (ch(k) > 0) \end{matrix}

where [log_{q_{0}} T] is the largest integer which is not exceeding log_{q_{0}} T (cf. [W1, Lemma1] and Lemma 1 in §5).

Proposition 1. For T > 0 and any g ∈ G(𝔸)^{1}, B_{T} ∩ X_{Q}g is a finite set. Hence, one can define the function

Γ_{Q}(g) = min{T > 0: B_{T} ∩ X_{Q}g ≠ ∅} = min_{y∈X_{Q}g} H_{Q}(y)

on G(𝔸)^{1}. Then the maximum

γ(G, Q, k) = max_{g∈G(𝔸)^{1}} Γ_{Q}(g)

exists.

Proposition 1 will be proved in the next section.

Definition. The constant γ(G, Q, k) is called the fundamental Hermite constant of (G, Q) over k.

We often write γ_{Q} for γ(G, Q, k) if k and G are clear from the context. The constant γ_{Q} is characterized as the greatest positive number T_{0} such that B_{T} ∩X_{Q}g_{T} = ∅ for any T < T_{0} and some g_{T} ∈ G(𝔸)^{1} . It is obvious by definition that γ_{Q} ∈ q_{0}^{ℤ} if ch(k) > 0.

Remark. Let Ỹ_{Q} = Z_{G}(𝔸)Q(𝔸)^{1}\G(𝔸). Then, for any g ∈ G(𝔸), X_{Q}g is regarded as a subset of Ỹ_{Q}. In some cases, it is more convenient to consider the constant

γ̃(G, Q, k) = max_{g∈G(𝔸)} min_{y∈X_{Q}g} H_{Q}(y) .

In general, γ(G, Q, k) ≤ γ̃(G, Q, k) holds. If ch(k) = 0 or G is semisimple, then γ(G, Q, k) = γ̃(G, Q, k) because of Ỹ_{Q} = Y_{Q}.

Remark. If ch(k) = 0, one can consider the more general Hermite constant defined by

γ(G, Q, D, k) = max_{g∈G(𝔸)^{1}} min{T > 0: i_{Q}((0, T] × D) ∩ X_{Q}g ≠ ∅}

for an open and closed subset D of K.

3. A relation between γ_{Q} and a generalized Hermite constant.

We recall the definition of generalized Hermite constants ([W1, §2]). Let V_{π} be a finite dimensional \overline{k}-vector space defined over k and π: G ⟶ GL(V_{π}) be an absolutely irreducible k-rational representation. The highest weight space in V_{π} with respect to B is denoted by x_{π}. Let Q_{π} be the stabilizer of x_{π} in G and λ_{π} the rational character of Q_{π} by which Q_{π} acts on x_{π}. In the following, we assume Q = Q_{π} and π is strongly k-rational, i.e., x_{π} is defined over k. Then λ_{π} is a k-rational character of Q_{π}. It is known that such π always exists (cf. [Ti1], [W1]). We use a right action of G on V_{π} defined by a · g = π(g^{−1})a for g ∈ G and a ∈ V_{π}. Then the mapping g ↦ x_{π} · g gives rise to a k-rational embedding of Q\G into the projective space ℙV_{π}. We fix a k-basis e_{1}, ⋯, e_{n} of the k-vector space V_{π}(k) and define a local height H_{v} on V_{π}(k_{v}) for each v ∈ 𝔙 as follows:

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

The global height H_{π} on V_{π}(k) is defined to be a product of all H_{v}, that is, H_{π}(a) = ∏_{v∈𝔙} H_{v}(a). By the product formula, H_{π} is invariant by scalar multiplications. Thus, H_{π} defines a height on ℙV_{π}(k), and on X_{Q} by restriction. The height H_{π} is extended to GL(V_{π}(𝔸))ℙV_{π}(k) by

H_{π}(ξ\overline{a}) = ∏_{v∈𝔙} H_{v}(ξ_{v}a)

for ξ = (ξ_{v}) ∈ GL(V_{π}(𝔸)) and \overline{a} = ka ∈ ℙV_{π}(k), a ∈ V_{π}(k) − \{0\}. Put

Φ_{π,ξ}(g) = H_{π}(ξ(x_{π} · g))/H_{π}(ξx_{π}), (g ∈ G(𝔸)).

Since this satisfies

Φ_{π,ξ}(gg′) = |λ_{π}(g)^{−1}|_{𝔸}Φ_{π,ξ}(g′), (g ∈ Q(𝔸), g′ ∈ G(𝔸)),

Φ_{π,ξ} defines a function on Y_{Q}. We can and do choose a ξ ∈ GL(V_{π}(𝔸)) so that Φ_{π,ξ} is right K-invariant. Then, in the case of ch(k) = 0, the generalized Hermite constant attached to π is defined by

(3.1)  γ_{π} = max_{g∈G(𝔸)^{1}} min_{x∈X_{Q}} Φ_{π,ξ}(xg)^{2/[k:ℚ]} .

Let us prove Proposition 1. We take positive rational numbers e_{π} and ê_{π} such that

|λ_{π}(z)|_{𝔸} = |α_{Q}(z)|_{𝔸}^{e_{π}} and |λ_{π}(m)|_{𝔸} = |α̂_{Q}(m)|ê_{π} ; 𝔸

for z ∈ Z_{Q}(𝔸) ∩ G(𝔸)^{1} and m ∈ M_{Q}(𝔸) ∩ G(𝔸)^{1}. Then, by definition,

Φ_{π,ξ}(y) = H_{Q}(y)ê_{π} , (y ∈ Y_{Q}).

Therefore, one has

B_{T} ∩ X_{Q} = {x ∈ X_{Q}: H_{π}(ξx) ≤ H_{π}(ξx_{π})T ê_{π}}.

Since ♯{x ∈ ℙV_{π}(k): H_{π}(ξx) ≤ c} is finite for a fixed constant c (cf. [S]), B_{T} ∩ X_{Q} is a finite set. If g ∈ G(𝔸)^{1} is given, then there is a T_{g} > 0 depending on g such that B_{T} g^{−1} ⊂ B_{T_{g}}. This implies that ♯(B_{T} ∩ X_{Q}g) = ♯(B_{T} g^{−1} ∩ X_{Q}) is also finite. Furthermore, we obtain

Γ_{Q}(g) = min_{x∈X_{Q}} Φ_{π,ξ}(xg)^{1/ê_{π}} .

In [W1, Proposition 2], we proved in the case of ch(k) = 0 that the function in g ∈ G(𝔸)^{1} defined by the right hand side attains its maximum. The same proof works well for the case of ch(k) > 0 by using the reduction theory due to Harder ([H]). We mention its proof for the sake of completeness. If necessary, by replacing G with G/(Kerπ)^{0}, we may assume Kerπ is finite. Let

S(𝔸)_{c} = {z ∈ S(𝔸): |β(z)|_{𝔸}^{−1} ≤ c for all β ∈ Δ_{k}}

and

S(𝔸)′_{c} = {z ∈ S(𝔸): c^{−1} ≤ |β(z)|_{𝔸}^{−1} ≤ c for all β ∈ Δ_{k}}

for a sufficiently large constant c > 1. By reduction theory, there are compact subsets Ω_{1} ∈ P (𝔸) and Ω_{2} ∈ G(𝔸) such that K ⊂ Ω_{2} and G(𝔸) = G(k)Ω_{1}S(𝔸)_{c}Ω_{2}. Set 𝔖(c) = Ω_{1}S(𝔸)_{c}Ω_{2} ∩ G(𝔸)^{1} and 𝔖(c)′ = Ω_{1}S(𝔸)′_{c}Ω_{2} ∩ G(𝔸)^{1}. There is a constant c′ such that

min_{x∈X_{Q}} Φ_{π,ξ}(xω_{1}zω_{2}) ≤ Φ_{π,ξ}(ω_{1}zω_{2}) ≤ c′|λ_{π}(z)|_{𝔸}^{−1}

holds for all ω_{1} ∈ Ω_{1}, z ∈ S(𝔸)_{c} and ω_{2} ∈ Ω_{2}. The highest weight λ_{π} can be written as a ℚ-linear combination of simple roots modulo X_{k}^{∗}(Z_{G}) ⊗_{ℤ} ℚ, i.e.,

λ_{π}|_{S} ≡ ∑_{β∈Δ_{k}} c_{β}β mod X_{k}^{∗}(Z_{G}) ⊗_{ℤ} ℚ .

A crucial fact is c_{β} > 0 for all β ∈ Δ_{k} (cf. [W1, Proof of Proposition 2]). From this and the above inequality, it follows

sup_{g∈𝔖(c)} min_{x∈X_{Q}} Φ_{π,ξ}(xg) = sup_{g∈𝔖(c)′} min_{x∈X_{Q}} Φ_{π,ξ}(xg) .

This implies that the function g ↦ min_{x∈X_{Q}} Φ_{π,ξ}(xg) attains its maximum since 𝔖(c)′ is relatively compact in G(𝔸)^{1} modulo G(k). Therefore, the maximum

(3.2)  γ_{Q} = max_{g∈G(𝔸)^{1}} min_{x∈X_{Q}} Φ_{π,ξ}(xg)^{1/ê_{π}}

exists. This completes the proof of Proposition 1.

Next theorem is obvious by (3.1), (3.2), e_{π} = d̂_{Q}ê_{π}, e_{Q} = d̂_{Q}ê_{Q} and [W1, Theorem 1].

Theorem 1. If ch(k) = 0, then the Hermite constant attached to a strongly k-rational representation π is given by

γ_{π} = γ_{Q}^{2ê_{π}/[k:ℚ]} .

One has an estimate of the form

(3.3)  (\frac{C_{Q}d_{G}e_{Q}τ(G)}{C_{G}d_{Q}τ(Q)})^{1/ê_{Q}} ≤ γ_{Q} .

Example 1. Let V be an n dimensional vector space defined over an algebraic number field k and e_{1}, ⋯, e_{n} a k-basis of V (k). We identify the group of linear automorphisms of V with GL_{n}. For 1 ≤ j ≤ n − 1, Q_{j} denotes the stabilizer of the subspace spanned by e_{1}, ⋯, e_{j} in GL_{n} and π_{j}: GL_{n} ⟶ GL(⋀^{j} V) the j-th exterior representation. A k-basis of V_{π_{j}}(k) = ⋀^{j} V (k) is formed by the elements e_{I} = e_{i_{1}} ∧⋯∧e_{i_{j}} with I = {1 ≤ i_{1} < i_{2} < ⋯ < i_{j} ≤ n}. The global height H_{π_{j}} is defined similarly as above with respect to the basis {e_{I}}_{I}. By definition and H_{π_{j}}(e_{1} ∧ ⋯ ∧ e_{j}) = 1, we have

γ_{n,j}(k) = γ_{π_{j}} = max_{g∈GL_{n}(𝔸)^{1}} min_{x∈Q_{j}(k)\GL_{n}(k)} H_{π_{j}}(x · g)^{2/[k:ℚ]}

= max_{g∈GL_{n}(𝔸)} min_{\begin{matrix}x_{1},··· ,x_{j}∈V (k) \\ x_{1}∧···∧x_{j}≠0 \end{matrix}} \frac{H_{π_{j}}(gx_{1} ∧ ⋯ ∧ gx_{j})^{2/[k:ℚ]}}{| deg g|_{𝔸}^{2j/(n[k:ℚ])}} .

Let gcd(j, n − j) be the greatest common divisor of j and n − j. It is easy to see that

(3.4)  d̂_{Q_{j}} = \frac{j(n − j)}{gcd(j, n − j)},  ê_{Q_{j}} = gcd(j, n − j),  ê_{π_{j}} = \frac{gcd(j, n − j)}{n}.

Therefore,

γ(GL_{n}, Q_{j}, k) = γ_{n,j}(k)^{n[k:ℚ]/(2 gcd(j,n−j))},

and in particular, γ(GL_{n}, Q_{1}, ℚ)^{2/n} is none other than the classical Hermite’s constant γ_{n,1}. By [T2] and [W1, Example 2], we have

(\frac{|D_{k}|^{j(n−j)/2}n}{Res_{s=1}ζ_{k}(s)} \frac{∏_{i=n−j+1}^{n} Z_{k}(i)}{∏_{j=2}^{j} Z_{k}(j)})^{1/ gcd(j,n−j)} ≤ γ(GL_{n}, Q_{j}, k) ,

γ(GL_{n}, Q_{j}, k) ≤ (\frac{2^{r_{1}+r_{2}}|D_{k}|^{1/2}}{π^{r/2}} Γ(1 + \frac{n}{2})^{r_{1}/n}Γ(1 + n)^{r_{2}/n})^{jn/ gcd(j,n−j)} ,

where ζ_{k}(s) denotes the Dedekind zeta function of k, Γ(s) the gamma function, Z_{k}(s) = (π^{−s/2}Γ(s/2))^{r_{1}}((2π)^{1−s}Γ(s))^{r_{2}}ζ_{k}(s), r_{1} and r_{2} the numbers of real and imaginary places of k, respectively. When j = 1, the next inequality was proved in [O-W]:

γ(GL_{n}, Q_{1}, k) ≤ |D_{k}|^{1/[k:ℚ]} \frac{γ(GL_{n[k:ℚ]}, Q_{1}, ℚ)}{[k : ℚ]} .

4. Some properties of fundamental Hermite constants.

First, we consider the exact sequence

1 ⟶ Z ⟶ G ⟶^{β} G′ ⟶ 1

of connected reductive groups defined over a global field k. We assume the following two conditions for Z:

(4.1) Z is central in G.

(4.2) Z is isomorphic to a product of tori of the form R_{k′/k}(GL_{1}), where each k′/k is a finite separable extension and R_{k′/k} denotes the functor of restriction of scalars from k′ to k.

By [B, Theorem 22.6], the assumption (4.1) implies that P′ = β(P), S′ = β(S) and Q′ = β(Q) give a minimal k-parabolic subgroup, a maximal k-split torus and a maximal standard k-parabolic subgroup of G′, respectively, and furthermore, the homomorphism (β|_{S})^{∗}: X_{k}^{∗}(S′) ⟶ X_{k}^{∗}(S) induced from β maps bijectively the relative root system Φ′_{k} of (G′, S′) onto Φ_{k}. From the assumption (4.2), it follows that β gives rise to the isomorphisms G(k)/Z(k) ≅ G′(k), G(𝔸)/Z(𝔸) ≅ G′(𝔸) and X_{Q} ≅ X_{Q′} (cf. [Oe, III 2.2]). By the commutative diagram

Z(𝔸)^{1}  ⟶  G(𝔸)^{1}  ⟶^{β}  G′(𝔸)^{1}
↓  ↓  ↓
Z(𝔸)  ⟶  G(𝔸)  ⟶^{β}  G′(𝔸)
ϑ_{Z} ↓  ^{ϑ_{G}}↓  ^{ϑ_{G′}} ↓
Hom_{ℤ}(X_{k}^{∗}(Z), ℝ_{+}^{×}) ⟶ Hom_{ℤ}(X_{k}^{∗}(G), ℝ_{+}^{×}) ⟶^{(β^{∗})^{∗}} Hom_{ℤ}(X_{k}^{∗}(G′), ℝ_{+}^{×})

we obtain the isomorphisms G(𝔸)^{1}/Z(𝔸)^{1} ≅ G′(𝔸)^{1}, Q(𝔸)^{1}/Z(𝔸)^{1} ≅ Q′(𝔸)^{1} and Y_{Q} ≅ Y_{Q′}. Since Z ∩ Z_{G} is the greatest k-split subtorus of Z, the character group X_{k}^{∗}(Z/Z ∩ Z_{G}) is trivial. Therefore, β induces an isomorphism X_{k}^{∗}(M_{Q′}/Z_{G′}) ⟶ X_{k}^{∗}(M_{Q}/Z_{G}) and maps α_{Q′} to α̂_{Q}. The next proposition is now obvious.

Theorem 2. If the exact sequence

1 ⟶ Z ⟶ G ⟶^{β} G′ ⟶ 1

of connected reductive groups defined over k satisfies the conditions (4.1) and (4.2), then γ(G, Q, k) equals γ(G′, β(Q), k).

Example 2. If β: GL_{n} ⟶ P GL_{n} denotes a natural quotient morphism, then γ(GL_{n}, Q, k) = γ(P GL_{n}, β(Q), k).

Example 3. Let D be a division algebra of finite dimension m^{2} over k and D^{∘} the opposition algebra of D. There are inner k-forms G and G′ of GL_{mn} such that G(k) = GL_{n}(D) and G′(k) = GL_{n}(D^{∘}). We put

w_{0} = ( \begin{matrix}0 & 0 & ⋯ & 0 & 1 \\ 0 & 0 & ⋯ & 1 & 0 \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 1 & ⋯ & 0 & 0 \\ 1 & 0 & ⋯ & 0 & 0 \end{matrix} ) ∈ GL_{n}(D^{∘}).

Then the morphism β: G ⟶ G′ defined by β(g) = w_{0}(^{t}g^{−1})w_{0}^{−1} yields a k-isomorphism. If we take a maximal k-parabolic subgroup Q_{j} of G as

Q_{j}(k) = \{( \begin{matrix}a & ∗ \\ 0 & b \end{matrix} ) : a ∈ GL_{j}(D), b ∈ GL_{n−j}(D)\}

for 1 ≤ j ≤ n − 1, then β(Q_{j}(k)) equals

Q′_{n−j}(k) = \{( \begin{matrix}a′ & ∗ \\ 0 & b′ \end{matrix} ) : a′ ∈ GL_{n−j}(D^{∘}), b′ ∈ GL_{j}(D^{∘})\} .

Therefore,

γ(G, Q_{j}, k) = γ(G′, Q′_{n−j}, k).

This relation was first proved in [W3]. Particularly, if m = 1, this is none other than the duality relation

γ(GL_{n}, Q_{j}, k) = γ(GL_{n}, Q_{n−j}, k) .

Remark. When ch(k) = 0, for a given connected reductive k-group G, there exists a group extension

1 ⟶ Z ⟶ G̃ ⟶ G ⟶ 1

defined over k such that Z satisfies (4.1) and (4.2), and in addition , the derived group of G̃ is simply connected. Such an extension of G is called z-extension(cf. [K, §1]).

Second, we consider a restriction of scalars. Take a subfield ℓ of k such that k/ℓ is a finite separable extension and put G′ = R_{k/ℓ}(G), P′ = R_{k/ℓ}(P) and Q′ = R_{k/ℓ}(Q). The adele ring of ℓ is denoted by 𝔸_{ℓ}. Since the functor R_{k/ℓ} yields a bijection from the set of k-parabolic subgroups of G to the set of ℓ-parabolic subgroups of G′ ([B-Ti, Corollaire 6.19]), P′ and Q′ give a minimal ℓ-parabolic subgroup and a maximal standard ℓ-parabolic subgroup of G′, respectively. Although the torus R_{k/ℓ}(S) does not necessarily split over ℓ, the greatest ℓ-split subtorus S′ of R_{k/ℓ}(S) gives a maximal ℓ-split torus of G′. For an arbitrary connected k-subgroup R of G and R′ = R_{k/ℓ}(R), we introduce a canonical homomorphism β^{∗}: X_{k}^{∗}(R) ⟶ X_{ℓ}^{∗}(R′). If A is an ℓ-algebra, there is a canonical identification R′(A) with R(A ⊗_{ℓ} k). Then, for χ ∈ X_{k}^{∗}(R), β^{∗}(χ) is defined to be

β^{∗}(χ)(a) = N_{A⊗k/A}(χ(a)) , (a ∈ R′(A) = R(A ⊗_{ℓ} k))

for any ℓ-algebra A, where N_{A⊗k/A}: (A⊗_{ℓ}k)^{×} ⟶ A^{×} denotes the norm. This β^{∗} is bijective ([Oe, II Theorem 2.4]), and if R = S, then β^{∗} maps Φ_{k} to the relative root system Φ′_{ℓ} of (G′, S′) ([B-Ti, 6.21]). From the commutative diagram

R(𝔸)  R′(𝔸_{ℓ})
ϑ_{R}↓  ^{ϑ_{R′}} ↓
Hom_{ℤ}(X_{k}^{∗}(R), ℝ_{+}^{×}) ⟶^{(β^{∗})^{∗}} Hom_{ℤ}(X_{k}^{∗}(R′), ℝ_{+}^{×})

it follows R(𝔸)^{1} = R′(𝔸_{ℓ})^{1}. Accordingly, Q(𝔸)^{1}\G(𝔸)^{1} = Q′(𝔸_{ℓ})^{1}\G′(𝔸_{ℓ})^{1}. Since Z_{G′} is the greatest ℓ-split torus in R_{k/ℓ}(Z_{G}), the natural quotient morphism M_{Q′}/Z_{G′} ⟶ M_{Q′}/R_{k/ℓ}(Z_{G}) induces an isomorphism X_{ℓ}^{∗}(M_{Q′}/R_{k/ℓ}(Z_{G})) ≅ X_{ℓ}^{∗}(M_{Q′}/Z_{G′}). The composition of this and β^{∗} yields an isomorphism between X_{k}^{∗}(M_{Q}/Z_{G}) and X_{ℓ}^{∗}(M_{Q′}/Z_{G′}). This maps α̂_{Q} to α_{Q′}. Then, by definition of β^{∗},

|α_{Q′}(m)|_{𝔸_{ℓ}} = |N_{𝔸/𝔸_{ℓ}}(α̂_{Q}(m))|_{𝔸_{ℓ}} = |α̂_{Q}(m)|_{𝔸}

for m ∈ M_{Q′}(𝔸_{ℓ}) ∩ G′(𝔸_{ℓ})^{1} = M_{Q}(𝔸) ∩ G(𝔸)^{1}, In other words, H_{Q′} is equal to H_{Q} on Y_{Q′} = Y_{Q}. As a consequence, we proved the following

Theorem 3. If k/ℓ is a finite separable extension, then γ(R_{k/ℓ}(G), R_{k/ℓ}(Q), ℓ) is equal to γ(G, Q, k).

Finally, we show a generalization of Rankin’s inequality. Let R and Q be two different maximal standard k-parabolic subgroups of G. We set Q^{R} = M_{R} ∩ Q, M_{Q}^{R} = M_{R} ∩ M_{Q}, U_{Q}^{R} = M_{R} ∩ U_{Q} and X_{Q}^{R} = Q^{R}(k)\M_{R}(k). Then Q^{R} is a maximal standard parabolic subgroup of M_{R} with a Levi decomposition U_{Q}^{R}M_{Q}^{R}. We write α_{Q}^{R} for the ℤ-basis α_{Q^{R}} of X_{k}^{∗}(M_{Q}^{R}/Z_{R}), z_{Q}^{R} for the map z_{Q^{R}}: M_{R}(𝔸) ⟶ Z_{R}(𝔸)M_{Q}^{R}(𝔸)^{1}\M_{Q}^{R}(𝔸) and H_{Q}^{R} for the function H_{Q^{R}}: M_{R}(𝔸) ⟶ ℝ_{+}^{×} defined by m ↦ |α_{Q}^{R}(z_{Q}^{R}(m))|_{𝔸}^{−1}. The fundamental Hermite constants of (M_{R}, Q^{R}) are given by

γ(M_{R}, Q^{R}, k) = max_{m∈M_{R}(𝔸)^{1}} min_{y∈X_{Q}^{R}m} H_{Q}^{R}(y) and γ̃(M_{R}, Q^{R}, k) = max_{m∈M_{R}(𝔸)} min_{y∈X_{Q}^{R}m} H_{Q}^{R}(y).

The exact sequence

1 ⟶ Z_{R}/Z_{G} ⟶ M_{Q}^{R}/Z_{G} ⟶ M_{Q}^{R}/Z_{R} ⟶ 1

induces the exact sequence

1 ⟶ X_{k}^{∗}(M_{Q}^{R}/Z_{R}) ⟶ X_{k}^{∗}(M_{Q}^{R}/Z_{G}) ⟶ X_{k}^{∗}(Z_{R}/Z_{G}).

From α̂_{R}|_{Z_{R}} = d̂_{R}α_{R} ≠ 0, it follows that the ℚ-vector space X_{k}^{∗}(M_{Q}^{R}/Z_{G}) ⊗_{ℤ} ℚ is spanned by α_{Q}^{R} and α̂_{R}|_{M_{Q}^{R}}, and hence there are ω_{1}, ω_{2} ∈ ℚ such that

(4.3)  α̂_{Q}|_{M_{Q}^{R}} = ω_{1}α_{Q}^{R} + ω_{2}α̂_{R}|_{M_{Q}^{R}} .

Theorem 4. Being notations as above, one has the inequality γ(G, Q, k) ≤ γ̃(M_{R}, Q^{R}, k)^{ω_{1}}γ(G, R, k)^{ω_{2}}.

Proof. Since X_{Q}^{R} is naturally regarded as a subset of X_{Q}, the inequality

Γ_{Q}(g) = min_{x∈X_{Q}} H_{Q}(xg) ≤ min_{x∈X_{Q}^{R}} H_{Q}(xg)

holds for g ∈ G(𝔸)^{1}. By the Iwasawa decomposition, we write g = umh, where u ∈ U_{R}(𝔸), m ∈ M_{R}(𝔸) ∩ G(𝔸)^{1} and h ∈ K. Then, for x ∈ M_{R}(k), xux^{−1} ∈ U_{R}(𝔸) ⊂ Q(𝔸)^{1}, and

H_{Q}(xg) = H_{Q}((xux^{−1})xmh) = H_{Q}(xm) = |α̂_{Q}(z_{Q}(xm))|_{𝔸}^{−1} .

If we write xm = u_{1}m_{1}h_{1}, u_{1} ∈ U_{Q}^{R}(𝔸), m_{1} ∈ M_{Q}^{R}(𝔸) and h_{1} ∈ K^{M_{R}} by the Iwasawa decomposition M_{R}(𝔸) = U_{Q}^{R}(𝔸)M_{Q}^{R}(𝔸)K^{M_{R}}, then

H_{Q}(xm) = |α̂_{Q}(m_{1})|_{𝔸}^{−1} = |α_{Q}^{R}(m_{1})|_{𝔸}^{−ω_{1}}|α̂_{R}(m_{1})|_{𝔸}^{−ω_{2}}
= |α_{Q}^{R}(z_{Q}^{R}(xm))|_{𝔸}^{−ω_{1}}|α̂_{R}(xm)|_{𝔸}^{−ω_{2}} = H_{Q}^{R}(xm)^{ω_{1}}|α̂_{R}(m)|_{𝔸}^{−ω_{2}}
= H_{Q}^{R}(xm)^{ω_{1}}H_{R}(g)^{ω_{2}} .

Therefore,

Γ_{Q}(g) ≤ ( min_{x∈X_{Q}^{R}} H_{Q}^{R}(xm))^{ω_{1}} H_{R}(g)^{ω_{2}} ≤ γ̃(M_{R}, Q^{R}, k)^{ω_{1}}H_{R}(g)^{ω_{2}} .

As Γ_{Q} is left G(k)-invariant, the inequality

Γ_{Q}(g) ≤ γ̃(M_{R}, Q^{R}, k)^{ω_{1}}H_{R}(xg)^{ω_{2}}

holds for all x ∈ G(k). Taking the minimum, we get

Γ_{Q}(g) ≤ γ̃(M_{R}, Q^{R}, k)^{ω_{1}}Γ_{R}(g)^{ω_{2}} .

The assertion follows from this. □

Notice that γ̃(M_{R}, Q^{R}, k) = γ(M_{R}, Q^{R}, k) in the case of number fields.

16 FUNDAMENTAL HERMITE CONSTANTS

Corollary. If ch(k) = 0, then γ(G, Q, k) ≤ γ(M_{R}, Q^{R}, k)^{ω_{1}}γ(G, R, k)^{ω_{2}}. Example 4. We use the same notations as in Example 1. For i, j ∈ ℤ with 1 ≤ i < j ≤ n−1, both R = Q_{j} and Q = Q_{i} are maximal standard k-parabolic subgroups of GL_{n}. Then, M_{R} = GL_{j} × GL_{n−j}, M_{Q} = GL_{i} × GL_{n−i} and M_{Q}^{R} = GL_{i} × GL_{j−i} × GL_{n−j}. We denote an element of M_{Q}^{R} by

diag(a, b, c) = ( \begin{matrix}a & & 0 \\ & b \\ 0 & & c \end{matrix} ) , (a ∈ GL_{i}, b ∈ GL_{j−i}, c ∈ GL_{n−j}).

It is easy to see

α_{Q}^{R}(diag(a, b, c)) = (det a)^{(j−i)/ gcd(i,j−i)}(det b)^{−i/ gcd(i,j−i)}
α̂_{R}|_{M_{Q}^{R}}(diag(a, b, c)) = (det a)^{(n−j)/ gcd(j,n−j)}(det b)^{(n−j)/ gcd(j,n−j)}(det c)^{−j/ gcd(j,n−j)}
α̂_{Q}|_{M_{Q}^{R}}(diag(a, b, c)) = (det a)^{(n−i)/ gcd(i,n−i)}(det b)^{−i/ gcd(i,n−i)}(det c)^{−i/ gcd(i,n−i)} .

Thus,

ω_{1} = \frac{n}{j} \frac{gcd(i, j − i)}{gcd(i, n − i)} , ω_{2} = \frac{i}{j} \frac{gcd(j, n − j)}{gcd(i, n − i)} .

Theorem 4 deduces

γ(GL_{n}, Q_{i}, k) ≤ γ̃(M_{Q_{j}}, Q_{i}^{Q_{j}}, k)^{\frac{n}{j} \frac{gcd(i,j−i)}{gcd(i,n−i)}} γ(GL_{n}, Q_{j}, k)^{\frac{i}{j} \frac{gcd(j,n−j)}{gcd(i,n−i)}} .

If ch(k) = 0, then, by Example 1, this reduces to Rankin’s inequality γ_{n,i}(k) ≤ γ_{j,i}(k)γ_{n,j}(k)^{i/j} .

5. A lower bound of γ_{Q}.

We prove an analogous inequality to (3.3) when ch(k) > 0. Thus we assume ch(k) > 0 throughout this section. Lemma 1. If f is a right K-invariant measurable function on Y_{Q},

∫_{Y_{Q}} f(y)dω_{Y_{Q}}(y) = \frac{C_{G}d_{Q}^{∗}}{C_{Q}d_{G}^{∗}} ∑_{M_{Q}(𝔸)^{1}ξ∈M_{Q}(𝔸)^{1}\(M_{Q}(𝔸)∩G(𝔸)^{1})} δ_{Q}(ξ)^{−1}f(ξ).

Proof. Let ϕ ∈ C_{0}(G(𝔸)^{1}) be a right K-invariant function. By the definition of invariant measures, we have

∫_{G(𝔸)^{1}} ϕ(g)dω_{G(𝔸)^{1}}(g) = (d_{G}^{∗})^{−1} ∫_{G(𝔸)^{1}} ϕ(g)dω_{𝔸}^{G}(g)

= \frac{C_{G}}{C_{Q}d_{G}^{∗}} ∫_{U_{Q}(𝔸)×(M_{Q}(𝔸)∩G(𝔸)^{1})} ϕ(um)δ_{Q}(m)^{−1}dω_{𝔸}^{U_{Q}}(u)dω_{𝔸}^{M_{Q}}(m)

= \frac{C_{G}d_{Q}^{∗}}{C_{Q}d_{G}^{∗}} ∑_{M_{Q}(𝔸)^{1}ξ∈M_{Q}(𝔸)^{1}\(M_{Q}(𝔸)∩G(𝔸)^{1})} δ_{Q}(ξ)^{−1}f(ξ) ,

where

f(ξ) = ∫_{U_{Q}(𝔸)×M_{Q}(𝔸)^{1}} ϕ(umξ)dω_{𝔸}^{U_{Q}}(u)dω_{M_{Q}(𝔸)^{1}}(m) = ∫_{Q(𝔸)^{1}} ϕ(gξ)dω_{Q(𝔸)^{1}}(g).

On the other hand,

∫_{G(𝔸)^{1}} ϕ(g)dω_{G(𝔸)^{1}}(g) = ∫_{Y_{Q}} ∫_{Q(𝔸)^{1}} ϕ(gy)dω_{Q(𝔸)^{1}}(g)dω_{Y_{Q}}(y)
= ∫_{Y_{Q}} f(y)dω_{Y_{Q}}(y). □

Theorem 5. If ch(k) > 0, one has

(\frac{C_{Q}d_{G}^{∗}τ(G)}{C_{G}d_{Q}^{∗}τ(Q)}(1 − q_{0}^{−ê_{Q}}))^{1/ê_{Q}} < q_{0}^{j_{0}+1} ≤ γ_{Q} ,

where the integer j_{0} is given by

j_{0} = max\{j ∈ ℤ: q_{0}^{jê_{Q}} ≤ \frac{C_{Q}d_{G}^{∗}τ(G)}{C_{G}d_{Q}^{∗}τ(Q)}(1 − q_{0}^{−ê_{Q}})\}

and q_{0} = q_{0}(Q) is the generator of the value group |α̂_{Q}(M_{Q}(𝔸) ∩ G(𝔸)^{1})|_{𝔸} which is greater than one.

Proof. For j ∈ ℤ, we define the function ψ_{j}: q_{0}^{ℤ} ⟶ ℝ by

ψ_{j}(q_{0}^{i}) = \{ \begin{matrix}1 & (i ≤ j). \\ 0 & (i > j). \end{matrix}

18 FUNDAMENTAL HERMITE CONSTANTS

Then, by Lemma 1,

I_{j} = ∫_{Y_{Q}} ψ_{j}(H_{Q}(y))dω_{Y_{Q}}(y)

= \frac{C_{G}d_{Q}^{∗}}{C_{Q}d_{G}^{∗}} ∑_{M_{Q}(𝔸)^{1}ξ∈M_{Q}(𝔸)^{1}\(M_{Q}(𝔸)∩G(𝔸)^{1})} δ_{Q}(ξ)^{−1}ψ_{j}(H_{Q}(ξ)).

Since H_{Q} is bijective from M_{Q}(𝔸)^{1}\(M_{Q}(𝔸) ∩ G(𝔸)^{1}) to q_{0}^{ℤ} and δ_{Q}(m)^{−1} = H_{Q}(m)ê_{Q} for m ∈ M_{Q}(𝔸), we have

I_{j} = \frac{C_{G}d_{Q}^{∗}}{C_{Q}d_{G}^{∗}} ∑_{i=−∞}^{j} q_{0}^{iê_{Q}} = \frac{C_{G}d_{Q}^{∗}}{C_{Q}d_{G}^{∗}} \frac{q_{0}^{jê_{Q}}}{1 − q_{0}^{−ê_{Q}}} .

If j satisfies I_{j} < τ(G)/τ(Q), then

I_{j} = \frac{1}{τ(Q)} ∫_{G(k)\G(𝔸)^{1}} ∑_{x∈X_{Q}} ψ_{j}(H_{Q}(xg))dω_{G}(g) < \frac{τ(G)}{τ(Q)} .

Therefore, at least one g_{0} ∈ G(𝔸)^{1},

∑_{x∈X_{Q}} ψ_{j}(H_{Q}(xg_{0})) < 1

holds, and hence ψ_{j}(H_{Q}(xg_{0})) = 0 for all x ∈ X_{Q}. This implies

min_{x∈X_{Q}} H_{Q}(xg_{0}) ≥ q_{0}^{j+1} ,

and

γ_{Q} ≥ q_{0} sup\{q_{0}^{j}: \frac{C_{G}d_{Q}^{∗}}{C_{Q}d_{G}^{∗}} \frac{q_{0}^{jê_{Q}}}{1 − q_{0}^{−ê_{Q}}} < \frac{τ(G)}{τ(Q)}\} = q_{0}^{1+j_{0}}

> (\frac{C_{Q}d_{G}^{∗}τ(G)}{C_{G}d_{Q}^{∗}τ(Q)}(1 − q_{0}^{−ê_{Q}}))^{1/ê_{Q}} . □

Remark. In §6, Example 5, we will see an example of γ_{Q} satisfying

(\frac{C_{Q}d_{G}^{∗}τ(G)}{C_{G}d_{Q}^{∗}τ(Q)}(1 − q_{0}^{−ê_{Q}}))^{1/ê_{Q}} < γ_{Q} < q_{0} (\frac{C_{Q}d_{G}^{∗}τ(G)}{C_{G}d_{Q}^{∗}τ(Q)}(1 − q_{0}^{−ê_{Q}}))^{1/ê_{Q}} .

If G splits over k, this lower bound is described more precisely. For v ∈ 𝔙_{f}, we choose each v component K_{v} of K as follows:

(5.1) K_{v} is a hyperspecial maximal compact subgroup G_{v}(𝔒_{v}) of G(k_{v}), and

(5.2) K_{v} ∩ M_{Q}(k_{v}) is a hyperspecial maximal compact subgroup M_{Q,v}(𝔒_{v}) of M_{Q}(k_{v}), where G_{v} and M_{Q,v} stand for the smooth affine group schemes defined over 𝔒_{v} with generic fiber G and M_{Q}, respectively (cf. [Ti2]).

Then it is known by [Oe, I Proposition 2.5] that

ω_{𝔸}^{G}(K) = μ_{𝔸}(𝔸/k)^{− dim G}σ_{k}(G)^{−1} ∏_{v∈𝔙_{f}} L_{v}(1, σ_{G})q_{v}^{− dim G}|G_{v}(𝔣_{v})|
ω_{𝔸}^{M_{Q}}(K^{M_{Q}}) = μ_{𝔸}(𝔸/k)^{− dim M_{Q}}σ_{k}(M_{Q})^{−1} ∏_{v∈𝔙_{f}} L_{v}(1, σ_{M_{Q}})q_{v}^{− dim M_{Q}}|M_{Q,v}(𝔣_{v})|
ω_{𝔸}^{U_{Q}}(K ∩ U_{Q}(𝔸)) = μ_{𝔸}(𝔸/k)^{− dim U_{Q}} .

In the integral formula (1.2), if we put the characteristic function of K as f, then

\frac{C_{G}}{C_{Q}} = \frac{ω_{𝔸}^{G}(K)}{ω_{𝔸}^{U_{Q}}(K ∩ U_{Q}(𝔸))ω_{𝔸}^{M_{Q}}(K^{M_{Q}})} .

Since G splits over k, σ_{G} is the trivial representation of Gal(\overline{k}/k) of dimension rankX^{∗}(G) = dim Z_{G}. As Q is a maximal parabolic subgroup, we have

\frac{σ_{k}(G)}{σ_{k}(M_{Q})} = \frac{(Res_{s=1}ζ_{k}(s))^{dim Z_{G}}}{(Res_{s=1}ζ_{k}(s))^{dim Z_{Q}}} = \frac{1}{Res_{s=1}ζ_{k}(s)} = \frac{q^{g(k)−1}(q − 1) log q}{h_{k}} ,

where ζ_{k}(s) denotes the congruence zeta function of k and h_{k} the divisor class number of k. Summing up, we obtain

Theorem 6. If ch(k) > 0 and G splits over k, then

(\frac{(1 − q_{0}^{−ê_{Q}})q^{(g(k)−1) dim G/Q}}{Res_{s=1}ζ_{k}(s)} \frac{d_{G}^{∗}τ(G)}{d_{Q}^{∗}τ(Q)} ∏_{v∈𝔙}(1 − q_{v}^{−1})q_{v}^{dim G/M_{Q}} \frac{|M_{Q,v}(𝔣_{v})|}{|G_{v}(𝔣_{v})|})^{1/ê_{Q}} < γ_{Q} .

6. Computations of γ(GL_{n}, Q, k) when ch(k) > 0.

In this section, we assume ch(k) > 0. We concentrate our attention on G = GL_{n} because this case gives an analogue of classical Hermite’s constant. We use the same notations as in Example 1 of §3. Namely, V denotes an n dimensional vector space defined over k, e_{1}, ⋯, e_{n} a k-basis of V (k), Q_{j} the stabilizer of the subspace spanned by e_{1}, ⋯, e_{j} in GL_{n} and π_{j}: GL_{n} ⟶ GL(V_{π_{j}}) the j-th exterior representation of GL_{n} for 1 ≤ j ≤ n − 1. We take K as ∏_{v∈𝔙} GL_{n}(𝔒_{v}). The global height H_{j} = H_{π_{j}} on V_{π_{j}}(k) is defined to be

H_{j}(∑_{I} a_{I}e_{I}) = ∏_{v∈𝔙} sup_{I}(|a_{I}|_{v}) .

As an analogue of the number fields case, we can define the constant

γ_{n,j}(k) = max_{g∈GL_{n}(𝔸)} min_{\begin{matrix}x_{1},··· ,x_{j}∈V (k) \\ x_{1}∧···∧x_{j}≠0 \end{matrix}} \frac{H_{j}(gx_{1} ∧ ⋯ ∧ gx_{j})}{| det g|_{𝔸}^{j/n}}.

It is immediate to see that

\frac{H_{j}(g^{−1}e_{1} ∧ ⋯ ∧ g^{−1}e_{j})}{| det g^{−1}|_{𝔸}^{j/n}} = H_{Q_{j}}(g)^{gcd(j,n−j)/n}

for g ∈ GL_{n}(𝔸), and hence

γ_{n,j}(k) = γ̃(GL_{n}, Q_{j}, k)^{gcd(j,n−j)/n} .

In general, Z_{GL_{n}}(𝔸)GL_{n}(𝔸)^{1} is not equal to GL_{n}(𝔸) in contrast to the number fields case. It is obvious that Z_{GL_{n}}(𝔸)GL_{n}(𝔸)^{1} is an index finite normal subgroup of GL_{n}(𝔸). Let Ξ = {ξ} be a complete set of representatives for the cosets of Z_{GL_{n}}(𝔸)GL_{n}(𝔸)^{1}\GL_{n}(𝔸). If we put

γ_{n,j}(k)_{ξ} = max_{g∈Z_{GLn} (𝔸)GL_{n}(𝔸)^{1}ξ} min_{\begin{matrix}x_{1},··· ,x_{j}∈V (k) \\ x_{1}∧···∧x_{j}≠0 \end{matrix}} \frac{H_{j}(gx_{1} ∧ ⋯ ∧ gx_{j})}{| det g|_{𝔸}^{j/n}}

= \frac{1}{| det ξ|_{𝔸}^{j/n}} max_{g∈GL_{n}(𝔸)^{1}ξ} min_{\begin{matrix}x_{1},··· ,x_{j}∈V (k) \\ x_{1}∧···∧x_{j}≠0 \end{matrix}} H_{j}(gx_{1} ∧ ⋯ ∧ gx_{j})

for ξ ∈ Ξ, then

γ_{n,j}(k) = max_{ξ∈Ξ} γ_{n,j}(k)_{ξ},

and in particular, for the unit element ξ = 1,

γ_{n,j}(k)_{1} = γ(GL_{n}, Q_{j}, k)^{gcd(j,n−j)/n}.

Since 1 ≤ γ_{n,j}(k)_{1} by the definition of H_{j}, we obtain

(6.1)  1 ≤ γ(GL_{n}, Q_{j}, k) ≤ γ_{n,j}(k)^{n/ gcd(j,n−j)} .

Lemma 2. γ_{n,j}(k) ≤ q^{jg(k)}.

Proof. By [T1, §5, Corollary 1], for a given g ∈ GL_{n}(𝔸), there are linearly independent vectors x_{1}, ⋯, x_{n} of V (k) with

H_{1}(gx_{1}) ⋯ H_{1}(gx_{n}) ≤ q^{ng(k)}| det g|_{𝔸} .

We may assume H_{1}(gx_{1}) ≤ H_{1}(gx_{2}) ≤ ⋯ ≤ H_{1}(gx_{n}). Then,

H_{j}(gx_{1} ∧ ⋯ ∧ gx_{j}) ≤ H_{1}(gx_{1}) ⋯ H_{1}(gx_{j})
≤ (H_{1}(gx_{1}) ⋯ H_{1}(gx_{n}))^{j/n}
≤ q^{jg(k)}| det g|_{𝔸}^{j/n} .

This implies the assertion. We note that our definition of the global height H_{j} is slightly different from [T1]. □

Theorem 7. We have the following estimate.

(\frac{q^{(g(k)−1)(j(n−j)+1)}(q − 1)(1 − q^{−n})}{h_{k}} \frac{∏_{i=n−j+1}^{n} ζ_{k}(i)}{∏_{i=2}^{j} ζ_{k}(i)})^{1/ gcd(j,n−j)}

< γ(GL_{n}, Q_{j}, k) ≤ γ̃(GL_{n}, Q_{j}, k) ≤ q^{njg(k)/ gcd(j,n−j)} = q_{0}(Q_{j})^{jg(k)} .

Proof. Recall that q_{0}(Q_{j}) is the generator of the value group |α̂_{Q_{j}}(M_{Q_{j}}(𝔸) ∩ GL_{n}(𝔸)^{1})|_{𝔸} which is greater than one. Since

M_{Q_{j}} = \{diag(a, b) = ( \begin{matrix}a & 0 \\ 0 & b \end{matrix} ) : a ∈ GL_{j}, b ∈ GL_{n−j}\} ,

any diag(a, b) ∈ M_{Q_{j}}(𝔸) ∩ GL_{n}(𝔸)^{1} satisfies

| det a|_{𝔸} = | det b|_{𝔸}^{−1}.

The ℤ-basis α̂_{Q_{j}} of X^{∗}(M_{Q_{j}}/Z_{GL_{n}}) is given by

α̂_{Q_{j}}(diag(a, b)) = (det a)^{(n−j)/ gcd(j,n−j)}(det b)^{−j/ gcd(j,n−j)} .

Hence, |α̂_{Q_{j}}(diag(a, b))|_{𝔸} = | det a|^{n/ gcd(j,n−j)} holds for diag(a, b) ∈ M_{Q_{j}}(𝔸) ∩ GL_{n}(𝔸)^{1}. This and {| det a|_{𝔸}: a ∈ GL_{j}(𝔸)} = q^{ℤ} conclude q_{0}(Q_{j}) = q^{n/ gcd(j,n−j)}. The upper estimate is obvious from Lemma 2 and (6.1). Since the order of the finite group GL_{n}(𝔣_{v}) is equal to (q_{v}^{n} − 1)(q_{v}^{n} − q_{v}) ⋯ (q_{v}^{n} − q_{v}^{n−1}), one has

∏_{v∈𝔙}(1 − q_{v}^{−1})q_{v}^{dim GL_{n}/M_{Q_{j}}} \frac{|GL_{j}(𝔣_{v}) × GL_{n−j}(𝔣_{v})|}{|GL_{n}(𝔣_{v})|} = \frac{∏_{i=n−j+1}^{n} ζ_{k}(i)}{∏_{i=2}^{j} ζ_{k}(i)} .

It is known that τ(GL_{n}) = τ(GL_{j} × GL_{n−j}) = 1 (cf. [We1, Theorem 3.2.1] and [Oe, III Theorem 5.2]). From the surjectivity of ϑ_{GL_{n}}, it follows d_{GL_{n}}^{∗} = log q, d_{Q_{j}}^{∗} = d_{GL_{j}×GL_{n−j}}^{∗} = (log q)^{2} and

\frac{1}{Res_{s=1}ζ_{k}(s)} \frac{d_{GL_{n}}^{∗}τ(GL_{n})}{d_{Q_{j}}^{∗}τ(Q_{j})} = \frac{q^{g(k)−1}(q − 1)}{h_{k}} .

Then, the lower bound is a result of Theorem 6 and ê_{Q_{j}} = gcd(j, n − j). □

Corollary 1. If g(k) = 0, i.e., k is a rational function field over 𝔽_{q}, then γ(GL_{n}, Q_{j}, k) = γ̃(GL_{n}, Q_{j}, k) = 1 for all n and j.

It is known that the zeta function ζ_{k}(s) is of the form

ζ_{k}(s) = \frac{L_{k}(q^{−s})}{(1 − q^{−s})(1 − q^{1−s})},

where L_{k}(t) is a polynomial of degree 2g(k) with integer coefficients. If we write L_{k}(t) as

L_{k}(t) = a_{0} + a_{1}t + ⋯ + a_{2g(k)}t^{2g(k)} ,

then a_{i}’s have the following properties:

1) a_{0} = 1, a_{2g(k)} = q^{g(k)} and a_{2g(k)−i} = q^{g(k)−i}a_{i} for 1 ≤ i ≤ g(k).

2) a_{1} = N(k) − (q + 1), where N(k) = ♯{v ∈ 𝔙: [𝔣_{v} : 𝔽_{q}] = 1}.

3) L_{k}(1) = h_{k}.

In this notation, Theorem 4 deduces the following inequality.

Corollary 2. If j = 1, then

\frac{q^{g(k)n}(q − 1)L_{k}(q^{−n})}{h_{k}(q^{n} − q)} < γ(GL_{n}, Q_{1}, k) ≤ γ̃(GL_{n}, Q_{1}, k) ≤ q^{g(k)n} = q_{0}(Q_{1})^{g(k)} .

Example 5. If g(k) = 0, then L_{k}(t) = 1 and h_{k} = 1. So that we have

\frac{q − 1}{q^{n} − q} < γ(GL_{n}, Q_{1}, k) = 1 < q^{n} \frac{q − 1}{q^{n} − q} = q_{0}(Q_{1})\frac{q − 1}{q^{n} − q} .

Put

ϵ_{n}(k) = \frac{q^{n}(q − 1)L_{k}(q^{−n})}{h_{k}(q^{n} − q)} .

By Corollary 2, if 1 ≤ ϵ_{n}(k) holds for k, then both γ(GL_{n}, Q_{1}, k) and γ̃(GL_{n}, Q_{1}, k) must be equal to q^{g(k)n}.

Example 6. If g(k) = 1, then

ϵ_{n}(k) = \frac{(q − 1)(q^{2n} + a_{1}q^{n} + q)}{(q + a_{1} + 1)(q^{2n} − qq^{n})} .

We have the inequality:

1 ≤ \frac{q^{2n} + a_{1}q^{n} + q}{q^{2n} − qq^{n}} .

This is obvious by the Hasse – Weil bound |a_{1}| ≤ 2\sqrt{q}. Hence, if a_{1} ≤ −2, i.e., h_{k} ≤ q − 1, then γ(GL_{n}, Q_{1}, k) = γ̃(GL_{n}, Q_{1}, k) = q^{n} for all n ≥ 2.

Remark. In the case of number fields, the explicit values of γ(GL_{n}, Q_{1}, k) are very little known. One knows only γ(GL_{n}, Q_{1}, ℚ) for 2 ≤ n ≤ 8 and γ(GL_{2}, Q_{1}, k) for a few quadratic number fields k (cf. [BCIO], [O-W]).

References

[BCIO] R. Baeza, R. Coulangeon, M. I. Icaza and M. O’Ryan, Hermite’s constant for quadratic number fields, Experimental Math. 10 (2001), 543 - 551.

[B] A. Borel, Linear Algebraic Groups, 2nd ed., Springer-Verlag, 1991. [B-Ti] A. Borel and J. Tits, Groupes réductifs, Publ. Math. I.H.E.S. 27 (1965), 55 - 150.

[G] R. Godement, Domaines fondamentaux des groupes arithmétiques, Exp. 257, Séminaire Bourbaki 15 (1962/1963).

[G-L] P. M. Gruber and C. G. Lekkerkerker, Geometry of Numbers, 2nd ed., North-Holland, 1987.

[H] G. Harder, Minkowskische Reductionstheorie über Functionenkörpern, Invent. Math. 7 (1969), 33 - 54.

[K] R. Kottwitz, Rational conjugacy classes in reductive groups, Duke Math. J. 49 (1982), 785 - 806.

[R] R. A. Rankin, On positive definite quadratic forms, J. London Math. Soc. 28 (1953), 309 - 319.

[Oe] J. Oesterlé, Nombres de Tamagawa et groupes unipotents en caractéristique p, Invent. Math. 78 (1984), 13 - 88.

[O-W] S. Ohno and T. Watanabe, Estimates of Hermite constants for algebraic number fields, Comm.

Math. Uni. Sancti Pauli 50 (2001), 53 - 63.

[S] J. H. Silverman, The theory of height functions, Arithmetic Geometry, Papers from the conference held at the University of Connecticut. Edited by G. Cornell and J. Silverman, Springer Verlag, 1986, pp. 151 – 166.

[T1] J. L. Thunder, An adelic Minkowski-Hlawka theorem and an application to Siegel’s lemma, J. reine angew. Math. 475 (1996), 167 - 185.

[T2] , Higher dimensional analogues of Hermite’s constant, Michigan Math. J. 45 (1998), 301 - 314.

[Ti1] J. Tits, Représentations linéaires irréductibles d’un groupe réductif sur un corps quelconque, J. f. reine. u. angew. Math. 247 (1971), 196 - 220.

[Ti2] , Reductive groups over local fields, Proc. Symp. Pure Math., Amer. Math. Soc. 33 (1979), 29 - 69.

[W1] T. Watanabe, On an analog of Hermite’s constant, J. Lie Theory 10 (2000), 33 - 52.

[W2] , Upper bounds of Hermite constants for orthogonal groups, Comm. Math. Uni. Sancti Pauli 48 (1999), 25 - 33.

[W3] , Hermite constants of division algebras, Monatshefte für Math. 135 (2002), 157 - 166.

[W4] , The Hardy–Littlewood property of flag varieties, preprint.

[We1] A. Weil, Adeles and Algebraic Groups, Birkhäuser, 1982.

[We2] , Basic Number Theory, Springer Verlag, 1974.

Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043 Japan

  E-mail address: watanabe@math.wani.osaka-u.ac.jp