The normalization constant of a certain invariant measure on GL_{n}(D_{A})

Yoshihide Nakamura and Takao Watanabe ^{∗}

June 16, 2004

Abstract

The ratio of the Tamagawa measure and a certain invariant measure on the group GL_{n}(D_{A}) is computed, where D_{A} is the adèle of a division algebra D over a global field. An explicit formula of the ratio is described in terms of the special values of the zeta function of D. This formula yields (i) an explicit lower bound of the Hermite–Rankin constant γ_{n,m}(D) of D and (ii) an explicit asymptotic behavior of the distribution of rational points on Brauer–Severi variety.

Introduction

Let G be a connected reductive algebraic group defined over a global field k and G(A) the adèle group of G. Since G(A) is a locally compact unimodular group, it has a non-trivial invariant measure. The invariant measure ω_{A}^{G} on G(A) induced from the invariant gauge form ω^{G} on G defined over k is called the Tamagawa measure, which is a canonical invariant measure on G(A) in a sense. There is another useful invariant measure on G(A) defined as follows: We fix a parabolic subgroup R of G defined over k and a maximal compact subgroup K of G(A) which possesses an Iwasawa decomposition G(A) = R(A)K. Let ω_{A}^{R} denote the Tamagawa measure of R(A) and ω_{K} the invariant measure on K normalized so that ω_{K}(K) = 1. Then the product ω_{A}^{R} ·ω_{K} defines an invariant measure, say ω_{(G(A),R(A))}, on G(A). Since an invariant measure is unique up to constant, there is the positive constant C_{G,R,K} such that ω_{A}^{G} = C_{G,R,K} · ω_{(G(A),R(A))}. We call C_{G,R,K} the normalization constant of ω_{(G(A),R(A))}.

In general, the constant C_{G,R,K} has a description by an Euler product such as

C_{G,R,K} = ∏_{v} ϵ_{v}J_{v} ,

where v runs over all places of k and ϵ_{v} are elementary constants determined by G and R. Every J_{v} is an integral of the form

J_{v} = ∫_{U_{R}^{−}(k_{v})} η_{v}(u_{v})dω_{v}^{U_{R}^{−}}(u_{v}) ,

∗The second author was partly supported by Grant-in-Aid for Scientific Research, Ministry of Education, Culture, Science and Technology, Japan.

02000 Mathematics Subject Classification. Primary 11R52; Secondary 11H50

where U_{R}^{−} denotes the unipotent radical of the opposite k-parabolic subgroup of R and η_{v} the function on G(k_{v}) induced by the modular character of R(k_{v}). In §2.1, we will show this formula in detail. In principle, the constant C_{G,R,K} can be explicitly computed by using this formula and the reduction of J_{v} to the cases of semisimple rank one groups due to Gindikin–Karpelevič formula (see §2.2 and §2.3). Indeed, an explicit formula of C_{G,R,K} is known in the case where G is a k-quasisplit group ([L]), an orthogonal group ([Ik]) and a unitary group ([Ic]). However, except for the case that G is a k-quasisplit group, its actual computation is not easy.

In this paper, we give an explicit formula of C_{G,R,K} in the case that G is an inner k-from of general linear groups, i.e., G is the algebraic group determined by G(k) = M_{n}(D)^{×} = GL_{n}(D), where D is a division k-algebra. We fix a minimal k-parabolic subgroup P of G and a certain maximal compact subgroup K of G(A) such that G(A) = P (A)K. Since C_{G,R,K} = C_{G,P,K}/C_{M_{R},M_{R}∩P,M_{R}(A)∩K} holds for any standard k-parabolic subgroup R of G with a Levi subgroup M_{R}, it is sufficient to compute C_{G,P,K}. Then the integral J_{v} occurring in the Euler product of C_{G,P,K} is decomposed into a product of integrals over a division k_{v}-algebra D(v) which is equivalent to D ⊗_{k} k_{v} in the Brauer group of k_{v}. By computing the integrals over D(v), we obtain the value of J_{v}, and as a consequence, the explicit formula of C_{G,P,K} is described in terms of special values of the zeta function Z_{D}(s) of D (see §3.6).

Our motivation of computing C_{G,R,K} is the following. In [Wa], the second author introduced the fundamental Hermite constant γ(G, Q; k) of the pair of a connected reductive k-group G and a maximal k-parabolic subgroup Q of G. Then the constant C_{G,Q,K} appeared in the Minkowski–Hlawka type lower bound of γ(G, Q; k). Thus an explicit formula of C_{G,R,K} yields an explicit lower bound of γ(G, Q, k). In the case of G(k) = GL_{n}(D), we will take up this application in §4.2. Moreover, we will apply the formula of C_{G,R,K} to give an explicit asymptotic behavior of the distribution of rational points on Brauer–Severi variety in §4.3.

Notations

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_{∞}, 𝔙_{R}, 𝔙_{C} and 𝔙_{f} for the sets of all infinite places, all real places, all imaginary places and all finite places of k, respectively. For v ∈ 𝔙, k_{v} denotes the completion of k at v. If v ∈ 𝔙_{f}, 𝔬_{v} denotes the maximal compact subring of k_{v} and q_{v} the cardinality of the residual field of k_{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 ∈ 𝔙_{R} and μ_{v}(\{a ∈ k_{v} : a\overline{a} ≤ 1\}) = 2π if v ∈ 𝔙_{C}. 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 adèle 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}

The zeta function ζ_{k}(s) of k is defined to be

ζ_{k}(s) = ∏_{v∈𝔙_{f}}(1 − q_{v}^{−s})^{−1} .

The residue of ζ_{k}(s) at s = 1 is denoted by ρ_{k}.

Let k_{1} be an arbitrary field. If 𝔄_{1} is a central simple k_{1}-algebra, then Nr_{𝔄_{1}/k_{1}} and τ_{𝔄_{1}/k_{1}} stand for the reduced norm and the reduced trace of 𝔄_{1}, respectively. The unit group of 𝔄_{1} is denoted by 𝔄_{1}^{×}.

1 Normalization constant of an invariant measure

1.1 Tamagawa measure

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 the Z-basis χ_{1}, ⋯, χ_{n}.

• The case of ch(k) > 0. The value group of the idele norm | · |_{A} is the cyclic group q^{Z} generated by q. 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 ([O, I, Proposition 5.6]),

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(k)}\ω_{G(A)^{1}} is called the Tamagawa number of G and denoted by τ(G).

1.2 Another Haar measure on G(A) and its normalization constant

In the following, let G be a connected reductive group defined over k. We fix a maximal k-split torus S in G and a minimal k-parabolic subgroup P of G which contains S. The centralizer of S in G gives a Levi subgroup M_{P} of P. Thus P has a Levi decomposition: P = M_{P} U_{P}, where U_{P} denotes the unipotent radical of P. Let R be a k-parabolic subgroup of G such that P ⊂ R. Such R is called a standard k-parabolic subgroup. There exists a unique Levi subgroup M_{R} of R such that M_{P} ⊂ M_{R}. The unipotent radical of R is denoted by U_{R}. 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 furthermore M_{R}(A) possesses an Iwasawa decomposition (M_{R}(A) ∩ U_{P} (A))M_{P} (A)(K ∩ M_{R}(A)).

If a standard k-parabolic subgroup R of G is given, then one can define another Haar measure ω_{(G(A),R(A))} of G(A) as follows. Let ω_{A}^{M_{R}} and ω_{A}^{U_{R}} be the Tamagawa measures of M_{R}(A) and U_{R}(A), respectively. The modular character δ_{R}^{−1} of R(A) is a function on M_{R}(A) which satisfies the integration formula

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

Let ω_{K} be the Haar measure on K normalized so that the total volume equals one. Then the mapping

f ↦ ∫_{U_{R}(A)×M_{R}(A)×K} f(umh)δ_{R}(m)^{−1}dω_{A}^{U_{R}}(u)dω_{A}^{M_{R}}(m)dω_{K}(h) , (f ∈ C_{0}(G(A)))

defines an invariant measure on G(A) and is denoted by ω_{(G(A),R(A))}.

Since a non-trivial invariant measure on G(A) is unique up to constant, there exists a positive constant C_{G,R,K} such that

ω_{A}^{G} = C_{G,R,K} · ω_{(G(A),R(A))} .

We call C_{G,R,K} the normalization constant of ω_{(G(A),R(A))}. For simplicity, we often write C_{G,R} for C_{G,R,K}. It is easy to show the following compatibility of three constants C_{G,R,K}, C_{G,P,K} and C_{M_{R},M_{R}∩P,M_{R}(A)∩K}:

C_{G,R,K} = \frac{C_{G,P,K}}{C_{M_{R},M_{R}∩P,M_{R}(A)∩K}} .

2 A formula of C_{G,R}

2.1 An expression of C_{G,R} by a product of integrals

Let G, R and K be the same as in §1.2. We consider the right G-homogeneous space 𝔛_{R} = U_{R}\G. Since U_{R} is a split unipotent subgroup, one has 𝔛_{R}(A) = U_{R}(A)\G(A).

Since both U_{R} and G are unimodular, ω^{U_{R}}\ω^{G} gives a unique (up to constant) G-invariant gauge form on 𝔛_{R} defined over k. The G(A)-invariant measure on 𝔛_{R}(A) defined from ω^{U_{R}}\ω^{G} is equal to

ω_{A}^{U_{R}}\ω_{A}^{G} = C_{G,R}δ_{R}^{−1}ω_{A}^{M_{R}}ω_{K} .  (1)

We take the opposite parabolic subgroup R^{−} of R. We denote by U_{R}^{−} the unipotent radical of R^{−}, i.e., U_{R}^{−} = U_{R^{−}}. Then one has the Levi decomposition R^{−} = U_{R}^{−}M_{R} and R ∩ R^{−} = M_{R}. By [B-T, Proposition 4.10 d)], the product morphism U_{R} × R^{−} ⟶ G is injective and gives an isomorphism of variety from U_{R} × R^{−} onto a Zariski open set in G. Thus R^{−} is regarded as a Zariski open subset of 𝔛_{R}. Since (ω^{U_{R}}\ω^{G})|_{R^{−}} yields a right invariant gauge form on R^{−} defined over k, there exists a constant λ ∈ k^{×} such that

(ω^{U_{R}}\ω^{G})|_{R^{−}} = λω^{U_{R}^{−}}ω^{M_{R}} .  (2)

For each v ∈ 𝔙, define the function η_{v} : G(k_{v}) ⟶ R_{+} by η_{v}(umh) = δ_{R}(m) for u ∈ U_{R}(k_{v}), m ∈ M_{R}(k_{v}) and h ∈ K_{v}. We take a right K-invariant Φ ∈ C_{0}(𝔛_{R}(A)) of the form Φ = ∏_{v∈𝔙} Φ_{v}, Φ_{v} ∈ C_{0}(𝔛_{R}(k_{v})). On the one hand, by (1), we have

∫_{𝔛_{R}(A)} Φ(x)d(ω_{A}^{U_{R}}\ω_{A}^{G})(x) = C_{G,R} ∫_{M_{R}(A)×K} Φ(mh)δ_{R}(m)^{−1}dω_{A}^{M_{R}}(m)dω_{K}(h)
= C_{G,R} ∫_{M_{R}(A)} Φ(m)δ_{R}(m)^{−1}dω_{A}^{M_{R}}(m).  (3)

On the other hand, by (2),

∫_{𝔛_{R}(A)} Φ(x)d(ω_{A}^{U_{R}}\ω_{A}^{G})(x)
= \frac{μ_{A}(A/k)^{dim U_{R}−dim G}}{σ_{k}(G)} ∏_{v∈𝔙_{∞}} ∫_{𝔛_{R}(k_{v})} Φ_{v}(x_{v})d(ω_{v}^{U_{R}}\ω_{v}^{G})(x_{v})

× ∏_{v∈𝔙_{f}} L_{v}(1, σ_{G}) ∫_{𝔛_{R}(k_{v})} Φ_{v}(x_{v})d(ω_{v}^{U_{R}}\ω_{v}^{G})(x)

= \frac{μ_{A}(A/k)^{dim U_{R}−dim G}}{σ_{k}(G)} ∏_{v∈𝔙_{∞}} ∫_{M_{R}(k_{v})×U_{R}^{−}(k_{v})} Φ_{v}(m_{v}u_{v})δ_{R}(m_{v})^{−1}|λ|_{v}dω_{v}^{M_{R}}(m_{v})dω_{v}^{U_{R}^{−}}(u_{v})

× ∏_{v∈𝔙_{f}} L_{v}(1, σ_{G}) ∫_{M_{R}(k_{v})×U_{R}^{−}(k_{v})} Φ_{v}(m_{v}u_{v})δ_{R}(m_{v})^{−1}|λ|_{v}dω_{v}^{M_{R}}(m_{v})dω_{v}^{U_{R}^{−}}(u_{v}).

= \frac{μ_{A}(A/k)^{dim U_{R}−dim G}}{σ_{k}(G)} ∏_{v∈𝔙_{∞}} ∫_{M_{R}(k_{v})×U_{R}^{−}(k_{v})} Φ_{v}(m_{v}u_{v})δ_{R}(m_{v})^{−1}dω_{v}^{M_{R}}(m_{v})dω_{v}^{U_{R}^{−}}(u_{v})

× ∏_{v∈𝔙_{f}} L_{v}(1, σ_{G}) ∫_{M_{R}(k_{v})×U_{R}^{−}(k_{v})} Φ_{v}(m_{v}u_{v})δ_{R}(m_{v})^{−1}dω_{v}^{M_{R}}(m_{v})dω_{v}^{U_{R}^{−}}(u_{v})

since |λ|_{A} = 1. We decompose u_{v} ∈ U_{R}^{−}(k_{v}) into u′_{v}m′_{v}h′_{v}, u′_{v} ∈ U_{R}(k_{v}), m′_{v} ∈ M_{R}(k_{v}) and h′_{v} ∈ K_{v}. Then one has

∫_{M_{R}(k_{v})×U_{R}^{−}(k_{v})} Φ_{v}(m_{v}u_{v})δ_{R}(m_{v})^{−1}dω_{v}^{M_{R}}(m_{v})dω_{v}^{U_{R}^{−}}(u_{v})
= ∫_{M_{R}(k_{v})×U_{R}^{−}(k_{v})} Φ_{v}((m_{v}u′_{v}m_{v}^{−1})(m_{v}m′_{v})h′_{v})δ_{R}(m_{v})^{−1}dω_{v}^{M_{R}}(m_{v})dω_{v}^{U_{R}^{−}}(u_{v})
= ∫_{M_{R}(k_{v})×U_{R}^{−}(k_{v})} Φ_{v}(m_{v}m′_{v})δ_{R}(m_{v})^{−1}dω_{v}^{M_{R}}(m_{v})dω_{v}^{U_{R}^{−}}(u_{v})
= ∫_{M_{R}(k_{v})×U_{R}^{−}(k_{v})} Φ_{v}(m_{v})δ_{R}(m_{v}(m′_{v})^{−1})^{−1}dω_{v}^{M_{R}}(m_{v})dω_{v}^{U_{R}^{−}}(u_{v})

= (∫_{M_{R}(k_{v})} Φ_{v}(m_{v})δ_{R}(m_{v})^{−1}dω_{v}^{M_{R}}(m_{v})) (∫_{U_{R}^{−}(k_{v})} η_{v}(u_{v})dω_{v}^{U_{R}^{−}}(u_{v})) .

By the definition of Tamagawa measures,

∫_{M_{R}(A)} Φ(m)δ_{R}(m)^{−1}dω_{A}^{M_{R}}(m) = \frac{μ_{A}(A/k)^{− dim M_{R}}}{σ_{k}(M_{R})} ∏_{v∈𝔙_{∞}} ∫_{M_{R}(k_{v})} Φ_{v}(m_{v})δ_{R}(m_{v})^{−1}dω_{v}^{M_{R}}(m_{v})

× ∏_{v∈𝔙_{f}} L_{v}(1, σ_{M_{R}}) ∫_{M_{R}(k_{v})} Φ_{v}(m_{v})δ_{R}(m_{v})^{−1}dω_{v}^{M_{R}}(m_{v}).

Therefore,

∫_{𝔛_{R}(A)} Φ(x)d(ω_{A}^{U_{R}}\ω_{A}^{G})(x) = \frac{μ_{A}(A/k)^{dim R−dim G}σ_{k}(M_{R})}{σ_{k}(G)} ∫_{M_{R}(A)} Φ(m)δ_{R}(m)^{−1}dω_{A}^{M_{R}}(m)
× ∏_{v∈𝔙_{∞}} J_{v} ∏_{v∈𝔙_{f}} \frac{L_{v}(1, σ_{G})}{L_{v}(1, σ_{M_{R}})}J_{v} ,  (4)

where

J_{v} = ∫_{U_{R}^{−}(k_{v})} η_{v}(u_{v})dω_{v}^{U_{R}^{−}}(u_{v}) .

From (3), (4) and dim R − dim G = − dim U_{R}, we obtain the following.

Theorem 1 Notations being as above, we have

C_{G,R} = \frac{μ_{A}(A/k)^{− dim U_{R}}σ_{k}(M_{R})}{σ_{k}(G)} ∏_{v∈𝔙_{∞}} J_{v} ∏_{v∈𝔙_{f}} \frac{L_{v}(1, σ_{G})}{L_{v}(1, σ_{M_{R}})}J_{v} .

2.2 Reduction of J_{v} to the case of minimal k_{v}-parabolic subgroups

We explain how to compute the local integral J_{v}. Let P^{(v)} be a minimal parabolic subgroup of G defined over k_{v} such that P^{(v)}(k_{v}) ⊂ R(k_{v}). Then P^{(v)} has a Levi subgroup M^{(v)} such that M^{(v)}(k_{v}) ⊂ M_{R}(k_{v}). Let U^{(v)} be the unipotent radical of P^{(v)} and U^{(v)−} be the unipotent radical of the opposite parabolic subgroup of P^{(v)}. We set P_{M_{R}}^{(v)} =

P^{(v)} ∩ M_{R}, U_{M_{R}}^{(v)} = U^{(v)} ∩ M_{R} and U_{M_{R}}^{(v)−} = U^{(v)−} ∩ M_{R}. Then P_{M_{R}}^{(v)} is a minimal parabolic subgroup of M_{R} defined over k_{v} with the unipotent radical U_{M_{R}}^{(v)} and a Levi subgroup M^{(v)}. The unipotent group U_{R}(k_{v}) is a normal subgroup of U^{(v)}(k_{v}), and U^{(v)}(k_{v}) has a semidirect product decomposition U_{R}(k_{v})U_{M_{R}}^{(v)}(k_{v}). Let δ_{P^{(v)}}^{−1} : M^{(v)}(k_{v}) ⟶ R_{+} and δ_{P_{M_{R}}^{(v)}}^{−1} : M^{(v)}(k_{v}) ⟶ R_{+} be the modular characters of P^{(v)}(k_{v}) and P_{M_{R}}^{(v)}(k_{v}), respectively. One has a relation

δ_{R}^{−1}|_{M^{(v)}(k_{v})} = δ_{P^{(v)}}^{−1} · δ_{P_{M_{R}}^{(v)}} .  (5)

Define the function η_{P^{(v)}}^{G} : G(k_{v}) ⟶ R_{+} by η_{P^{(v)}}^{G}(umh) = δ_{P^{(v)}}(m) for u ∈ U^{(v)}(k_{v}), m ∈ M^{(v)}(k_{v}) and h ∈ K_{v}. In a similar fashion, the function η_{P_{M_{R}}^{(v)}}^{M_{R}} : M_{R}(k_{v}) ⟶ R_{+} is

defined by η_{P_{M_{R}}^{(v)}}^{M_{R}} (umh) = δ_{P_{M_{R}}^{(v)}} (m) for u ∈ U_{M_{R}}^{(v)}(k_{v}), m ∈ M^{(v)}(k_{v}) and h ∈ K_{v} ∩ M_{R}(k_{v}). We set

J_{v}^{G} = ∫_{U^{(v)−}(k_{v})} η_{P^{(v)}}^{G}(u)dω_{U^{(v)−}(k_{v})}(u), J_{v}^{M_{R}} = ∫_{U_{M_{R}}^{(v)−}(k_{v})} η_{P_{M_{R}}^{(v)}}^{M_{R}} (u)dω_{U_{M_{R}}^{(v)−}(k_{v})}(u) .

Here, we fix invariant measures ω_{U^{(v)−}(k_{v})} on U^{(v)−}(k_{v}) and ω_{U_{M_{R}}^{(v)−}(k_{v})} on U_{M_{R}}^{(v)−}(k_{v}) such that

ω_{U^{(v)−}(k_{v})} = ω_{v}^{U_{R}^{−}} · ω_{U_{M_{R}}^{(v)−}(k_{v})} .

Let us compute J_{v}^{G} following the decomposition U^{(v)−} = U_{R}^{−}(k_{v})U_{M_{R}}^{(v)−}(k_{v}):

J_{v}^{G} = ∫_{U^{(v)−}(k_{v})} η_{P^{(v)}}^{G}(u)dω_{U^{(v)−}(k_{v})}(u) = ∫_{U_{M_{R}}^{(v)−}(k_{v})} dω_{U_{M_{R}}^{(v)−}(k_{v})}(u_{2}) ∫_{U_{R}^{−}(k_{v})} η_{P^{(v)}}^{G}(u_{1}u_{2})dω_{v}^{U_{R}^{−}}(u_{1}) .

Let u_{i} = α_{i}β_{i}γ_{i}, α_{i} ∈ U^{(v)}(k_{v}), β_{i} ∈ M^{(v)}(k_{v}) and γ_{i} ∈ K_{v} for i = 1, 2. Since

η_{P^{(v)}}^{G}(u_{1}u_{2}) = η_{P^{(v)}}^{G}(α_{2}β_{2}(α_{2}β_{2})^{−1}u_{1}(α_{2}β_{2})) = δ_{P^{(v)}}(β_{2})η_{P^{(v)}}^{G}((α_{2}β_{2})^{−1}u_{1}(α_{2}β_{2})),

one has

J_{v}^{G} = ∫_{U_{M_{R}}^{(v)−}(k_{v})} dω_{U_{M_{R}}^{(v)−}(k_{v})}(u_{2}) ∫_{U_{R}^{−}(k_{v})} δ_{P^{(v)}}(β_{2})δ_{R^{−}}(β_{2})η_{P^{(v)}}^{G}(u_{1})dω_{v}^{U_{R}^{−}}(u_{1})
= ∫_{U_{M_{R}}^{(v)−}(k_{v})} δ_{P^{(v)}}(β_{2})δ_{R^{−}}(β_{2})dω_{U_{M_{R}}^{(v)−}(k_{v})}(u_{2}) ∫_{U_{R}^{−}(k_{v})} δ_{P^{(v)}}(β_{1})dω_{v}^{U_{R}^{−}}(u_{1}) .

By δ_{R^{−}}(β_{2}) = δ_{R}(β_{2})^{−1}, δ_{R}(β_{1}) = δ_{P^{(v)}}(β_{1}) and (5), we obtain

J_{v}^{G} = ∫_{U_{M_{R}}^{(v)−}(k_{v})} δ_{P_{M_{R}}^{(v)}} (β_{2})dω_{U_{M_{R}}^{(v)−}(k_{v})}(u_{2}) · ∫_{U_{R}^{−}(k_{v})} δ_{R}(β_{1})dω_{v}^{U_{R}^{−}}(u_{1})
= J_{v}^{M_{R}} · J_{v} .

Therefore, one has

J_{v} = \frac{J_{v}^{G}}{J_{v}^{M_{R}}} .  (6)

2.4 Gindikin-Karpelevič formula of J_{v}^{G}

We set

J_{v}^{G}(s) = ∫_{U^{(v)−}(k_{v})} η_{P^{(v)}}^{G}(u)^{s+1/2}dω_{U^{(v)−}(k_{v})}(u),

where s is a complex number with ℜ(s) > 0. We recall the Gindikin–Karpelevič formula of J_{v}^{G}(s) (cf. [K, Chap.VII, §5, Corollary 7.5]). Let S^{(v)} be a maximal k_{v}-split torus of M^{(v)}, Σ_{v}(G) the relative root system of G with respect to S^{(v)} and Σ_{v}^{+}(G) the set of positive roots of Σ_{v}(G) corresponding to the minimal k_{v}-parabolic subgroup P^{(v)}. We set

𝔞_{v} = X_{k_{v}}^{∗}(S^{(v)}/Z_{G}^{(v)}) ⊗_{Z} R ,

where Z_{G}^{(v)} denotes the maximal central k_{v}-split torus of G. Note that the real vector space 𝔞_{v} is identified with X_{k_{v}}^{∗}(M^{(v)}/Z_{G}^{(v)}) ⊗_{Z} R since M^{(v)}/S^{(v)} is anisotropic over k_{v}. The set of simple roots of Σ_{v}^{+}(G) gives a basis of 𝔞_{v}, and hence Σ_{v}(G) is regarded as a subset of 𝔞_{v}. Thus, for each β ∈ Σ_{v}(G), the function ξ_{β}^{G} : G(A) ⟶ R_{+} is well defined by ξ_{β}^{G}(umh) = |β(m)|_{v} for u ∈ U^{(v)}(k_{v}), m ∈ M^{(v)}(k_{v}) and h ∈ K_{v}. We fix an admissible inner product (·, ·) on 𝔞_{v} and define the coroot β^{∨} of β ∈ Σ_{v}(G) by

β^{∨} = \frac{2}{(β, β)}β .

For β ∈ Σ_{v}^{+}(G), the connected component (Kerβ)^{0} of the kernel of β is a subtorus of S^{(v)}. We denote by G_{(β)} the centralizer of (Kerβ)^{0} in G. Then G_{(β)} is a reductive k_{v}-subgroup of G with semisimple k_{v}-rank one. We set P_{(β)} = G_{(β)} ∩ P^{(v)}, M_{(β)} = G_{(β)} ∩ M^{(v)}, U_{(β)} = G_{(β)} ∩ U^{(v)}, U_{(β)}^{−} = G_{(β)} ∩ U^{(v)−} and K_{(β)} = G_{(β)}(k_{v}) ∩ K_{v}. We assume that G_{(β)}(k_{v}) = P_{(β)}(k_{v})K_{(β)} holds for all β ∈ Σ_{v}^{+}(G). Then we define the function η_{β} : G_{(β)}(k_{v}) ⟶ R_{+} by η_{β}(umh) = δ_{P_{(β)}}(m) for u ∈ U_{(β)}(k_{v}), m ∈ M_{(β)}(k_{v}) and h ∈ K_{(β)}, where δ_{P_{(β)}}^{−1} : M_{(β)}(k_{v}) ⟶ R_{+} denote the modular character of P_{(β)}(k_{v}). Moreover, we write ρ_{v}^{G} for the half-sum of positive roots and ξ_{ρ_{v}^{G}} : G(A) ⟶ R_{+} for the function corresponding to ρ_{v}^{G}, i.e.,

ρ_{v}^{G} = \frac{1}{2} ∑_{β∈Σ_{v}^{+}}(dim U_{(β)})β , ξ_{ρ_{v}^{G}} = ∏_{β∈Σ_{v}^{+}}(ξ_{β}^{G})^{dim U_{(β)}/2} .

There is a relation ξ_{ρ_{v}^{G}}^{2} = η_{P^{(v)}}^{G}. With these notations, the Gindikin–Karpelevič formula of J_{v}^{G}(s) is stated as follows:

J_{v}^{G}(s) = ∏_{\begin{matrix}β ∈ Σ_{v}^{+}(G) \\ β/2 ∉ Σ_{v}^{+}(G) \end{matrix}} ∫_{U_{(β)}^{−}(k_{v})} ξ_{β}^{G}(u)^{(ρ_{v}^{G},β^{∨})s}η_{β}(u)^{1/2}dω_{U_{(β)}^{−}(k_{v})}(u) .  (7)

Here, we fix a family of invariant measures ω_{U_{(β)}^{−}(k_{v})}, β ∈ Σ_{v}^{+}(G) such that

ω_{U^{(v)−}(k_{v})} = ∏_{\begin{matrix}β ∈ Σ_{v}^{+} \\ β/2 ∉ Σ_{v}^{+} \end{matrix}} ω_{U_{(β)}^{−}(k_{v})}

holds. In principle, C_{G,R} can be computed by Theorem 1 and formulas (6), (7).

3 An explicit formula of C_{G,P} in the case of G(k) = GL_{n}(D)

3.1 Central simple algebras

Let D be a central division k-algebra of degree d^{2}. Let D_{v} = D ⊗_{k} k_{v} for v ∈ 𝔙 and D_{A} = D ⊗_{k} A. Since D_{v} is a central simple k_{v}-algebra, it is isomorphic with an algebra M_{d/d_{v}}(D(v)), where D(v) is a division k_{v}-algebra of degree d_{v}^{2}. The set 𝔙 is divided into two subsets 𝔙_{1} = {v ∈ 𝔙 : d_{v} = 1} and 𝔙_{2} = {v ∈ 𝔙 : d_{v} > 1}. We write 𝔙_{R,1}, 𝔙_{R,2}, 𝔙_{f,1} and 𝔙_{f,2} for 𝔙_{R} ∩𝔙_{1}, 𝔙_{R} ∩𝔙_{2}, 𝔙_{f} ∩𝔙_{1} and 𝔙_{f} ∩𝔙_{2}, respectively. We fix a maximal order 𝔒_{D} of D. For v ∈ 𝔙_{f}, the completion of 𝔒_{D} in D_{v} is denoted by 𝔒_{D_{v}}, which is a maximal order of D_{v}. Since any maximal order of D_{v} is conjugate to 𝔒_{D_{v}}, there is an isomorphism from D_{v} onto M_{d/d_{v}}(D(v)) such that the image of 𝔒_{D_{v}} equals M_{d/d_{v}}(𝔒_{D(v)}), where 𝔒_{D(v)} denotes a unique maximal order of D(v).

For every v ∈ 𝔙_{f}, we denote by 𝔡_{v} the different of 𝔒_{D_{v}}/𝔬_{v}, i.e.,

𝔡_{v}^{−1} = {a ∈ D_{v} : τ_{D_{v}/k_{v}}(a𝔒_{D_{v}}) ⊂ 𝔬_{v}}.

Then the different 𝔡_{𝔒_{D}} of 𝔒_{D} is given by ∏_{v∈𝔙_{f}} 𝔡_{v}. The absolute norm N𝔡_{D/k} of 𝔡_{𝔒_{D}} is defined to be

N𝔡_{D/k} = ∏_{v∈𝔙_{f}} |𝔒_{D_{v}}/𝔡_{v}| ,

which is independent of the choice of the maximal order 𝔒_{D} (cf. [R, Theorems (25.3) and (25.7)])

Now we consider the central simple k-algebra 𝔄 = M_{n}(D) and its maximal order 𝔒_{𝔄} = M_{n}(𝔒_{D}). We identify 𝔄_{v} = 𝔄 ⊗_{k} k_{v} with M_{n}(D_{v}) for v ∈ 𝔙 and 𝔄_{A} = 𝔄 ⊗_{k} A with M_{n}(D_{A}). For v ∈ 𝔙_{f}, set 𝔒_{𝔄_{v}} = M_{n}(𝔒_{D_{v}}), which is a maximal order of 𝔄_{v}. Hereafter, G denotes an affine algebraic k-group defined by G(k) = 𝔄^{×} = GL_{n}(D). The adèle group G(A) of G is the unit group of 𝔄_{A}. If v ∈ 𝔙_{∞}, we define an involution a ↦ a^{∗} of 𝔄_{v} as follows. We fix an algebra isomorphism 𝔄_{v} ≅ M_{nd/d_{v}}(D(v)). Then, for a = (a_{ij}) ∈ 𝔄_{v} (a_{ij} ∈ D(v)), the involution a^{∗} is defined to be a^{∗} = (\overline{a}_{ij})^{t}, where the superscript t means the transpose of a matrix and a_{ij} ↦ \overline{a}_{ij} denotes the canonical involution of the division algebra D(v), i.e., it is the identity map, the complex conjugate or the quaternion conjugate according as v ∈ 𝔙_{R,1}, v ∈ 𝔙_{C} or v ∈ 𝔙_{R,2}. By using this involution, we define the subgroup K_{v} of G(k_{v}) = 𝔄_{v}^{×} by K_{v} = {a ∈ 𝔄_{v}^{×} : a^{−1} = a^{∗}}. If v ∈ 𝔙_{f}, set K_{v} = 𝔒_{𝔄_{v}}^{×}. Then K = ∏_{v∈𝔙} K_{v} gives a maximal compact subgroup of G(A). Let P be the minimal k-parabolic subgroup of G which consists of upper triangular matrices in G. We will compute the constant C_{G,P} = C_{G,P,K}.

3.2 Self-dual measures

It is convenient to use a self-dual measure on D_{A} in order to compute C_{G,P}. We recall its construction. We fix a non-trivial character ψ : A/k ⟶ C^{1} as follows. If ch(k) > 0, we arbitrarily choose a non-trivial ψ. If ch(k) = 0, we define the character ψ_{0} on the adèle group A_{Q} of Q by

ψ_{0}(x) = e^{−2π\sqrt{−1}x_{∞}} ∏_{p: prime} e^{2π\sqrt{−1}(x_{p} mod Z_{p})}

for x = (x_{∞}, x_{2}, x_{3}, ⋯) ∈ A_{Q}, and then set ψ = ψ_{0} ∘ Tr_{k/Q}. For every v ∈ 𝔙, ψ induces a character ψ_{v} : k_{v} ⟶ C^{1}. Let ℭ be an arbitrary central simple k-algebra and ℭ_{v} = ℭ⊗_{k} k_{v} for v ∈ 𝔙 and ℭ_{A} = ℭ ⊗_{k} A. An invariant measure ν_{ℭ_{v}} on the locally compact additive group ℭ_{v} is called the self-dual measure with respect to ψ_{v} if

Φ(x) = ∫_{ℭ_{v}} {∫_{ℭ_{v}} Φ(z)ψ_{v}(τ_{ℭ_{v}/k_{v}}(yz))dν_{ℭ_{v}}(z)} ψ_{v}(−τ_{ℭ_{v}/k_{v}}(xy))dν_{ℭ_{v}}(y)

holds for any Schwartz–Bruhat function Φ on ℭ_{v}. The product measure ν_{ℭ_{A}} = ∏_{v∈𝔙} ν_{ℭ_{v}} on ℭ_{A} satisfies

Φ(x) = ∫_{ℭ_{A}} {∫_{ℭ_{A}} Φ(z)ψ_{v}(τ_{ℭ/k}(yz))dν_{ℭ_{A}}(z)} ψ_{v}(−τ_{ℭ/k}(xy))dν_{ℭ_{A}}(y)

for any Schwartz–Bruhat function Φ on ℭ_{A}. The invariant measure ν_{ℭ_{A}} is called the self-dual measure of ℭ_{A} with respect to ψ.

For v ∈ 𝔙, let ν_{D(v)} be the self-dual measure on D(v) with respect to ψ_{v}. It is known

by [T, Propositions 5, 6, 7 and 8] that the product measure ν_{D(v)}^{d^{2}/d_{v}^{2}} coincides with the self-dual measure on M_{d/d_{v}}(D(v)) with respect to ψ_{v}. Hence one can identify ν_{D_{v}} with

ν_{D(v)}^{d^{2}/d_{v}^{2}}. Note that this identification is independent of the choice of the algebra isomorphism D_{v} ≅ M_{d/d_{v}}(D(v)) because of Skolem–Noether theorem. Therefore, we have

ν_{D_{A}} = ∏_{v∈𝔙}ν_{D_{v}} = ∏_{v∈𝔙}ν_{D(v)}^{d^{2}/d_{v}^{2}} .

As was shown in the proof of [T, Theorem 2], ν_{D_{A}} is the Tamagawa measure of D_{A}, namely ν_{D_{A}}(D_{A}/D) = 1.

We define another invariant measure μ_{D_{A}} on D_{A}. If v ∈ 𝔙_{1}, i.e., D(v) = k_{v}, then we put μ_{D(v)} = μ_{v}, where μ_{v} is the measure on k_{v} introduced in Notations. For v ∈ 𝔙_{2}, μ_{D(v)} is defined to be the invariant measure on D(v) normalized so that μ_{D(v)}(𝔒_{D(v)}) = 1 if v ∈ 𝔙_{f,2} and μ_{D(v)}({x ∈ D(v) : Nr_{D(v)/k_{v}}(x) ≤ 1}) = 4π^{2} if v ∈ 𝔙_{R,2}. For every

v ∈ 𝔙, we set μ_{D_{v}} = μ_{D(v)}^{d^{2}/d_{v}^{2}}, which gives an invariant measure on D_{v} ≅ M_{d/d_{v}}(D(v)). By Skolem–Noether Theorem, μ_{D_{v}} is independent of the choice of the algebra isomorphism D_{v} ≅ M_{d/d_{v}}(D(v)). In particular, one has μ_{D_{v}}(𝔒_{D_{v}}) = 1 for v ∈ 𝔙_{f}. The product measure μ_{D_{A}} = ∏_{v∈𝔙} μ_{D_{v}} is an invariant measure on D_{A}. For every v ∈ 𝔙, there is the positive constant κ_{v} such that μ_{D(v)} = κ_{v}ν_{D(v)}. One has μ_{D_{v}} = κ_{v}^{d^{2}/d_{v}^{2}}ν_{D_{v}}.

Lemma 1 μ_{D_{A}}(D_{A}/D) = ∏_{v∈𝔙} κ_{v}^{d^{2}/d_{v}^{2}} = μ_{A}(A/k)^{d^{2}}N𝔡_{D/k}^{1/2}.

Proof. We define the Schwartz–Bruhat function Φ_{A} = ∏_{v∈𝔙} Φ_{v} on D_{A} as follows: If v ∈ 𝔙_{f}, let Φ_{v} be the characteristic function of 𝔒_{D_{v}}. If v ∈ 𝔙_{∞}, we set Φ_{v}(x) = e^{−[k_{v}:R]d_{v}πTr(x^{∗}x)}, where Tr(x^{∗}x) denotes the trace of the Hermitian matrix x^{∗}x. One hand, we have

∫_{D_{A}} Φ_{A}(x)dμ_{D_{A}}(x) = 1.

On the other hand, by [T, §II, Propositions 1 and 2],

∫_{D_{A}} Φ_{A}(x)dν_{D_{A}}(x) = μ_{A}(A/k)^{−d^{2}}N𝔡_{D/k}^{−1/2} ,

which proves the lemma.

3.3 A formula of C_{G,P}

Let M_{P} be the Levi subgroup of P consisting of diagonal matrices in G and S be the maximal k-split torus of M_{P}, i.e.,

M_{P} (k) = \{diag(a_{1}, ⋯, a_{n}) = ( \begin{matrix}a_{1} & & & & 0 \\ & . & . & . \\ 0 & & & & a_{n} \end{matrix} ) : a_{1}, ⋯, a_{n} ∈ D^{×}\}

S(k) = {diag(a_{1}, ⋯, a_{n}) : a_{1}, ⋯, a_{n} ∈ k^{×}}.

Let Σ(G) be the relative root system of G with respect to S and Σ^{+}(G) be the set of positive roots of Σ(G) corresponding to P. For each α ∈ Σ(G), U_{α} denotes the root subgroup of G. We fix an isomorphism U_{α}(k) ≅ D and define the invariant measures ν_{U_{α}(k_{v})} on U_{α}(k_{v}) for v ∈ 𝔙 and ν_{U_{α}(A)} on U_{α}(A) as

ν_{U_{α}(k_{v})} = ν_{D_{v}}, ν_{U_{α}(A)} = ∏_{v∈𝔙}ν_{U_{α}(k_{v})} = ν_{D_{A}}.

We set

ν_{U_{P}^{−}(k_{v})} = ∏_{α∈Σ^{+}(G)} ν_{U_{−α}(k_{v})}, ν_{U_{P}^{−}(A)} = ∏_{α∈Σ^{+}(G)} ν_{U_{−α}(A)} = ∏_{v∈𝔙}ν_{U_{P}^{−}(k_{v})}.

Since ν_{D_{A}} is the Tamagawa measure on D_{A}, ν_{U_{P}^{−}(A)} coincides with the Tamagawa measure

on the unipotent group U_{P}^{−}(A), i.e., ω_{A}^{U_{P}^{−}} = ν_{U_{P}^{−}(A)}.

For v ∈ 𝔙, we define the local integral I_{v} by

I_{v} = ∫_{U_{P}^{−}(k_{v})} η_{v}(u_{v})dν_{U_{P}^{−}(k_{v})}(u_{v}) ,

where the function η_{v} : G(k_{v}) ⟶ R_{+} is defined by

η_{v}(u · diag(a_{1}, ⋯, a_{n}) · h) = ∏_{i=1}^{n} |Nr_{D_{v}/k_{v}}(a_{i})|_{v}^{d(n−2i+1)}

for u ∈ U_{P} (k_{v}), a_{1}, ⋯, a_{n} ∈ D_{v}^{×} and h ∈ K_{v}. Since

\frac{σ_{k}(M_{P})}{σ_{k}(G)} = ρ_{k}^{n−1}, \frac{L_{v}(1, σ_{G})}{L_{v}(1, σ_{M_{P}})} = (1 − q_{v}^{−1})^{n−1}

and

ω_{A}^{U_{P}^{−}} = μ_{A}(A/k)^{− dim U_{P}} ∏_{v∈𝔙}ω_{v}^{U_{P}^{−}} = ∏_{v∈𝔙}ν_{U_{P}^{−}(k_{v})},

Theorem 1 leads us to

C_{G,P} = ρ_{k}^{n−1} ∏_{v∈𝔙_{∞}} I_{v} ∏_{v∈𝔙_{f}}(1 − q_{v}^{−1})^{n−1}I_{v} .  (8)

3.4 Reduction of I_{v} to the case of GL_{2}(D(v))

We fix a place v ∈ 𝔙. Let S^{(v)} be the maximal k_{v}-split torus in M_{P} and P^{(v)} be a minimal k_{v}-parabolic subgroup of G such that S^{(v)} ⊂ P^{(v)} ⊂ P. The unipotent radical of P^{(v)} is denoted by U^{(v)}. The centralizer M^{(v)} of S^{(v)} in G is a Levi subgroup of P^{(v)}. As in §2.3, we set P_{M_{P}}^{(v)} = P^{(v)} ∩ M_{P}, U_{M_{P}}^{(v)} = U^{(v)} ∩ M_{P} and U_{M_{P}}^{(v)−} = U^{(v)−} ∩ M_{P}. Let Σ_{v}(G) be the relative root system of G with respect to S^{(v)} and Σ_{v}^{+}(G) be the set of positive roots of Σ_{v}(G) corresponding to P^{(v)}. For every β ∈ Σ_{v}(G), U_{(β)} stands for the root subgroup of G. We fix an isomorphism U_{(β)}(k_{v}) ≅ D(v) and define the invariant measures ν_{U_{(β)}(k_{v})} on

U_{(β)}(k_{v}), ν_{U^{(v)−}(k_{v})} on U^{(v)−}(k_{v}) and ν_{U_{M_{P}}^{(v)−}(k_{v})} on U_{M_{P}}^{(v)−}(k_{v}) as

ν_{U_{(β)}(k_{v})} = ν_{D(v)},  ν_{U^{(v)−}(k_{v})} = ∏_{β∈Σ_{v}^{+}(G)} ν_{U_{(−β)}(k_{v})},  ν_{U_{M_{P}}^{(v)−}(k_{v})} = ∏_{{β∈Σ_{v}^{+}(G) \atop β|_{S}=0}} ν_{U_{(−β)}(k_{v})} .

For a k-root α ∈ Σ(G), one has

U_{α}(k_{v}) = ∏_{{β∈Σ_{v}(G) \atop β|_{S}=α}} U_{(β)}(k_{v}) .

From ν_{D_{v}} = ν_{D(v)}^{d^{2}/d_{v}^{2}}, it follows

ν_{U_{α}(k_{v})} = ∏_{{β∈Σ_{v}(G) \atop β|_{S}=α}} ν_{U_{(β)}(k_{v})} .

This implies the relation ν_{U^{(v)−}(k_{v})} = ν_{U_{P}^{−}(k_{v})} · ν_{U_{M_{P}}^{(v)−}(k_{v})}. Therefore, if we set

I_{v}^{G}(s) = ∫_{U^{(v)−}(k_{v})} η_{P^{(v)}}^{G}(u)^{s+1/2}dν_{U^{(v)−}(k_{v})}(u) ,
I_{v}^{M_{P}} (s) = ∫_{U_{M_{P}}^{(v)−}(k_{v})} η_{P_{M_{P}}^{(v)}}^{M_{P}} (u)^{s+1/2}dν_{U_{M_{P}}^{(v)−}(k_{v})}(u)

for ℜ(s) > 0 with the notations in §2.3, then I_{v} · I_{v}^{M_{P}} (1/2) = I_{v}^{G}(1/2) holds similarly as (6).

Let K_{v}^{GL_{2}} be a maximal compact subgroup of GL_{2}(D(v)) defined by the same way as K_{v}. We define the function η_{v}^{GL_{2}} : GL_{2}(D(v)) ⟶ R_{+} as follows:

η_{v}^{GL_{2}} (( \begin{matrix}1 & b \\ 0 & 1 \end{matrix} ) ( \begin{matrix}a_{1} & 0 \\ 0 & a_{2} \end{matrix} ) h) = |Nr_{D(v)/k_{v}}(a_{1})|_{v}^{d_{v}}|Nr_{D(v)/k_{v}}(a_{2})|_{v}^{−d_{v}}

for b ∈ D(v), a_{1}, a_{2} ∈ D(v)^{×} and h ∈ K_{v}^{GL_{2}}. We set

I_{v}^{GL_{2}}(s) = ∫_{D(v)} η_{v}^{GL_{2}} (( \begin{matrix}1 & 0 \\ b & 0 \end{matrix} ))^{s+1/2} dν_{D(v)}(b)

for ℜ(s) > 0. Then, by the Gindikin–Karpelevič formula,

I_{v}^{G}(s) = ∏_{β∈Σ_{v}^{+}(G)} ∫_{U_{(−β)}(k_{v})} ξ_{β}^{G}(u)^{(ρ_{v}^{G},β^{∨})s}η_{β}(u)^{1/2}dν_{U_{(−β)}(k_{v})}(u)
= ∏_{β∈Σ_{v}^{+}(G)} I_{v}^{GL_{2}}((ρ_{v}^{G}, β^{∨})s/d_{v}^{2})
= ∏_{1≤i<j≤nd/d_{v}} I_{v}^{GL_{2}}((j − i)s),

and, in a similar fashion,

I_{v}^{M_{P}} (s) = ( ∏_{1≤i<j≤d/d_{v}} I_{v}^{GL_{2}}((j − i)s))^{n} .

Therefore,

I_{v} = ( ∏_{1≤i<j≤d/d_{v}} I_{v}^{GL_{2}}((j − i)/2))^{−n} ∏_{1≤i<j≤nd/d_{v}} I_{v}^{GL_{2}}((j − i)/2) .  (9)

3.5 Computations of I_{v}^{GL_{2}}(s) An Iwasawa decomposition of the unipotent matrix ( \begin{matrix}1 & 0 \\ x & 1 \end{matrix} ) ∈ GL_{2}(D(v)) is given as follows:

• If v ∈ 𝔙_{f},

( \begin{matrix}1 & 0 \\ x & 1 \end{matrix} ) = \{ \begin{matrix}( \begin{matrix}1 & 0 \\ x & 1 \end{matrix} ) & (x ∈ 𝔒_{D(v)}). \\ ( \begin{matrix}1 & x^{−1} \\ 0 & 1 \end{matrix} ) ( \begin{matrix}x^{−1} & 0 \\ 0 & x \end{matrix} ) ( \begin{matrix}0 & −1 \\ 1 & x^{−1} \end{matrix} ) & (x ∉ 𝔒_{D(v)}). \end{matrix}

• If v ∈ 𝔙_{R,1},

( \begin{matrix}1 & 0 \\ x & 1 \end{matrix} ) = ( \begin{matrix}1 & \frac{x}{1+x^{2}} \\ 0 & 1 \end{matrix} ) ( \begin{matrix}\frac{1}{\sqrt{1+x^{2}}} & 0 \\ 0 & \sqrt{1 + x^{2}} \end{matrix} ) ( \frac{1}{\sqrt{1+x^{2}}} −\frac{x}{\sqrt{1+x^{2}}}_{\frac{x}{\sqrt{1+x^{2}}} \frac{1}{\sqrt{1+x^{2}}}} ) .

• If v ∈ 𝔙_{C},

( \begin{matrix}1 & 0 \\ x & 1 \end{matrix} ) = ( \begin{matrix}1 & \frac{x̄}{1+|x|_{v}} \\ 0 & 1 \end{matrix} ) ( \begin{matrix}\frac{1}{\sqrt{1+|x|_{v}}} & 0 \\ 0 & \sqrt{1 + |x|_{v}} \end{matrix} ) ( \frac{1}{\sqrt{1+|x|_{v}}} −\frac{x̄}{\sqrt{1+|x|_{v}}}_{\frac{x}{\sqrt{1+|x|_{v}}} \frac{1}{\sqrt{1+|x|_{v}}}} ) .

• If v ∈ 𝔙_{R,2},

( \begin{matrix}1 & 0 \\ x & 1 \end{matrix} ) = ( \begin{matrix}1 & \frac{x̄}{1+|x|^{2}} \\ 0 & 1 \end{matrix} ) ( \begin{matrix}\frac{1}{\sqrt{1+|x|^{2}}} & 0 \\ 0 & \sqrt{1 + |x|^{2}} \end{matrix} ) ( \frac{1}{\sqrt{1+|x|^{2}}} −\frac{x̄}{\sqrt{1+|x|^{2}}}_{\frac{x}{\sqrt{1+|x|^{2}}} \frac{1}{\sqrt{1+|x|^{2}}}} ) ,

where |x| = Nr_{D(v)/k_{v}}(x)^{1/2} for x ∈ D(v).

Lemma 2

I_{v}^{GL_{2}}(s) = κ_{v}^{−1} × \{ \begin{matrix}\frac{1 − q_{v}^{−2d_{v}s−d_{v}}}{1 − q_{v}^{−2d_{v}s}} & (v ∈ 𝔙_{f}). \\ π^{1/2} \frac{Γ(s)}{Γ(s + 1/2)} & (v ∈ 𝔙_{R,1}). \\ π/s & (v ∈ 𝔙_{C}). \\ \frac{π^{2}}{s(4s + 1)} & (v ∈ 𝔙_{R,2}). \end{matrix}

Proof. Let v ∈ 𝔙_{f} and π_{D(v)} be a prime element of D(v). Since κ_{v}ν_{D(v)} = μ_{D(v)}, one has

κ_{v}I_{v}^{GL_{2}}(s) = 1 + ∑_{t=1}^{∞} ∫_{π_{D(v)}^{−t}𝔒_{D(v)}^{×}} |N_{D(v)/k_{v}}(x)|_{v}^{−2d_{v}s−d_{v}} dμ_{D(v)}(x)

= 1 + ∑_{t=1}^{∞} q_{v}^{−(2s+1)td_{v}} ∫_{π_{D(v)}^{−t}𝔒_{D(v)}^{×}} dμ_{D(v)}(x)

= 1 + ∑_{t=1}^{∞} q_{v}^{−2td_{v}s}(1 − q_{v}^{−d_{v}})

= 1 + (1 − q_{v}^{−d_{v}})\frac{q_{v}^{−2d_{v}s}}{1 − q_{v}^{−2d_{v}s}}
= \frac{1 − q_{v}^{−2d_{v}s−d_{v}}}{1 − q_{v}^{−2d_{v}s}} .

If v ∈ 𝔙_{R,2},

κ_{v}I_{v}^{GL_{2}}(s) = ∫_{D(v)}(1 + |x|^{2})^{−4s−2}dμ_{D(v)}(x)
= 4 ∫_{0}^{∞}(1 + r^{2})^{−4s−2}r^{3}dr ∫_{0}^{2π} dθ ∫_{−π/2}^{π/2} cos φdφ ∫_{−π/2}^{π/2}(cos ψ)^{2}dψ
= \frac{π^{2}}{s(4s + 1)}.

The other cases are also easy.

3.6 An explicit formula of C_{G,P}

To describe I_{v}, we define functions F_{1}(s), F_{2}(s), F_{3}(s) in s ∈ C as

F_{1}(s) = π^{−s/2}Γ(s/2),  F_{2}(s) = (2π)^{1−s}Γ(s),  F_{3}(s) = (2π)^{2−s}Γ(s) .

By the formula (9) and Lemma 2, we have the following conclusion.

Lemma 3 Notations being as above, we have

I_{v} = κ_{v}^{− \frac{1}{2} \frac{d^{2}}{d^{2}_{v}} n(n−1)} × \{ \begin{matrix}∏_{{1≤i≤d \atop i≡0 (d_{v})}}(1 − q_{v}^{−i})^{−(n−1)} ∏_{{d+1≤i≤nd \atop i≡0 (d_{v})}}(1 − q_{v}^{−i})^{−1} & (v ∈ 𝔙_{f}) \\ ∏_{1≤i≤d} F_{1}(i)^{n−1} ∏_{d+1≤i≤nd} F_{1}(i)^{−1} & (v ∈ 𝔙_{R,1}) \\ ∏_{1≤i≤d} F_{2}(i)^{n−1} ∏_{d+1≤i≤nd} F_{2}(i)^{−1} & (v ∈ 𝔙_{C}) \\ ∏_{{1≤i≤d \atop i≡0 (2)}} F_{3}(i)^{n−1} ∏_{{d+1≤i≤nd \atop i≡0 (2)}} F_{3}(i)^{−1} & (v ∈ 𝔙_{R,2}). \end{matrix}

It is convenient to introduce a zeta function of D in order to formulate an explicit formula of C_{G,P}. We first define the constant C_{D} as follows:

• If ch(k) = 0,

C_{D} = ρ_{k}μ_{D_{A}}(D_{A}/D) ∏_{2≤i≤d} ζ_{k}(i)F_{1}(i)^{r_{1}+r_{3}}F_{2}(i)^{r_{2}}

× ∏_{v∈𝔙_{f,2}} ( ∏_{{1≤i≤d−1 \atop i≢0 (d_{v})}} 1 − q_{v}^{−i}) · ∏_{{1≤i≤d−1 \atop i≢0 (2)}} i^{r_{3}} ,

where r_{1}, r_{2} and r_{3} denote the cardinality of 𝔙_{R,1}, 𝔙_{C} and 𝔙_{R,2}, respectively.

• If ch(k) > 0,

C_{D} = (log q)ρ_{k}μ_{D_{A}}(D_{A}/D) ∏_{2≤i≤d} ζ_{k}(i) · ∏_{v∈𝔙_{f,2}} ∏_{{1≤i≤d−1 \atop i≢0 (d_{v})}}(1 − q_{v}^{−i}).

Then the zeta function of D is defined by

Z_{D}(s) = C_{D}^{−1} ∏_{0≤i≤d−1} ζ_{k}(s − i)F_{1}(s − i)^{r_{1}+r_{3}}F_{2}(s − i)^{r_{2}}

× ∏_{v∈𝔙_{f,2}} ( ∏_{{1≤i≤d−1 \atop i≢0 (d_{v})}}(1 − q_{v}^{−(s−i)})) · ∏_{{1≤i≤d−1 \atop i≢0(2)}}(s − i)^{r_{3}} .

By [T, Propositions 7 and 8], Z_{D}(s) has a simple pole at s = d with the residue

ρ_{D} = \{ \begin{matrix}μ_{D_{A}}(D_{A}/D)^{−1} & (ch(k) = 0) \\ (log q)^{−1}μ_{D_{A}}(D_{A}/D)^{−1} & (ch(k) > 0) \end{matrix}

By the formula (8) and Lemmas 1 and 3, the constant C_{G,P} is expressed in terms of Z_{D}(s). Theorem 2 If G(k) = GL_{n}(D) and P a minimal k-parabolic subgroup of G, then

C_{G,P} = μ_{D_{A}}(D_{A}/D)^{−n(n−1)/2}ρ_{D}^{n−1} ∏_{2≤i≤n} Z_{D}(id)^{−1}.

We take positive integers n_{1}, ⋯, n_{t} such that n = n_{1} + ⋯ + n_{t}. For such n_{1}, ⋯, n_{t}, R_{(n_{1},··· ,n_{t})} denotes the standard k-parabolic subgroup of G whose Levi subgroup M_{R_{(n_{1},··· ,n_{t})}}(k) is isomorphic with GL_{n_{1}}(D) × ⋯ × GL_{n_{t}}(D).

Corollary 1 Let R = R_{(n_{1},··· ,n_{t})} be a standard k-parabolic subgroup of G. Then we have

C_{G,R} = μ_{D_{A}}(D_{A}/D)^{− \frac{1}{2} (n^{2}−∑_{1≤j≤t} n_{j}^{2})}ρ_{D}^{t−1} \frac{∏_{1≤j≤t} ∏_{2≤i≤n_{j}} Z_{D}(id)}{∏_{2≤i≤n} Z_{D}(id)}.

This is a consequence of Theorem 2 and the relation C_{G,R} = C_{G,P} /C_{M_{R},M_{R}∩P}.

4 Applications

4.1 Fundamental Hermite constants of GL_{n}(D)

We use the same notations as in §3. For 1 ≤ m ≤ n − 1, Q_{m} denotes the standard maximal k-parabolic subgroup R_{(m,n−m)} of G. We recall the fundamental Hermite constants γ(G, Q_{m}, k) introduced in [Wa].

In the following, we fix m and write Q for Q_{m}. The Levi subgroup M_{Q} is given by

M_{Q}(k) = \{diag(a, b) = ( \begin{matrix}a & 0 \\ 0 & b \end{matrix} ) : a ∈ GL_{m}(D), b ∈ GL_{n−m}(D)\} .

Denote by Z_{G} and Z_{Q} the central maximal k-split tori of G and M_{Q}, respectively, i.e.,

Z_{G}(k) = {λI_{n} : λ ∈ k^{×}} and Z_{Q}(k) = {diag(λI_{m}, μI_{n−m}) : λ, μ ∈ k^{×}}.

We define the k-rational characters α_{Q} ∈ X_{k}^{∗}(Z_{Q}) and α̂_{Q} ∈ X_{k}^{∗}(M_{Q}) as follows:

α_{Q}(diag(λI_{m}, μI_{n−m})) = λμ^{−1}

for diag(λI_{m}, μI_{n−m}) ∈ Z_{Q}(k) and

α̂_{Q}(diag(a, b)) = Nr_{M_{m}(D)/k}(a)^{(n−m)/gcd(m,n−m)}Nr_{M_{n−m}(D)/k}(b)^{−m/gcd(m,n−m)}

for diag(a, b) ∈ M_{Q}(k). Then α_{Q} (resp. α̂_{Q}) is trivial on Z_{G} and forms a Z-basis of the module X_{k}^{∗}(Z_{G}\Z_{Q}) (resp. X_{k}^{∗}(Z_{G}\M_{Q})).

Define the unimodular subgroups G(A)^{1}, M_{Q}(A)^{1} and Q(A)^{1} as follows:

G(A)^{1} = {g ∈ G(A) : |Nr_{M_{n}(D)/k}(g)|_{A} = 1},
M_{Q}(A)^{1} = {diag(a, b) ∈ M_{Q}(A) : |Nr_{M_{m}(D)/k}(a)|_{A} = |Nr_{M_{n−m}(D)/k}(b)|_{A} = 1},
Q(A)^{1} = U_{Q}(A)M_{Q}(A)^{1}.

The height function H_{Q} : G(A) ⟶ R_{+} is well defined by

H_{Q}(u · diag(a, b) · h) = |α̂_{Q}(diag(a, b))|_{A}^{−1}

for u ∈ U_{Q}(A), diag(a, b) ∈ M_{Q}(A) and h ∈ K, and this is left Z_{G}(A)Q(A)^{1} and right K invariant. We set X_{Q} = Q(k)\G(k) and Y_{Q} = Q(A)^{1}\G(A)^{1}. Then X_{Q} is a subset of Y_{Q} and the natural map Y_{Q} ⟶ (Z_{G}(A)Q(A)^{1})\G(A) is injective. Thus H_{Q} is restricted to Y_{Q}. Then the Hermite constants γ(G, Q, k) and γ̃(G, Q, k) are defined to be

γ(G, Q, k) = max_{g∈G(A)^{1}} min_{x∈X_{Q}} H_{Q}(xg).

We write γ_{n,m}(D) for γ(G, Q_{m}, k), and especially γ_{n}(D) for γ(G, Q_{1}, k) since it is an analogue of Hermite–Rankin’s constant.

4.2 An explicit lower bound of γ_{n,m}(D)

Since Q = Q_{m} is maximal, there is a positive constant ê_{Q} such that δ_{Q}(g) = |α̂_{Q}(g)|ê_{Q} ; A holds for all g ∈ M_{Q}(A). It was proved in [Wa] that

(\frac{D_{G,Q} · E_{Q}}{C_{G,Q}} · \frac{τ(G)}{τ(Q)})^{1/ê_{Q}} ≤ γ(G, Q, k) ,  (10)

where D_{G,Q} and E_{Q} are given as follows with the notations in §1.1:

D_{G,Q} = \{ \begin{matrix}[X_{k}^{∗}(Z_{G}) : X_{k}^{∗}(G)]/[X_{k}^{∗}(Z_{Q}) : X_{k}^{∗}(M_{Q})] & (ch(k) = 0), \\ d_{G}^{∗}/d_{M_{Q}}^{∗} & (ch(k) > 0), \end{matrix}

E_{Q} = \{ \begin{matrix}ê_{Q}[X_{k}^{∗}(Z_{Q}/Z_{G}) : X_{k}^{∗}(M_{Q}/Z_{G})] & (ch(k) = 0). \\ (1 − q_{0}^{−ê_{Q}}) & (ch(k) > 0). \end{matrix}

Here, q_{0} > 1 stands for the generator of the subgroup |α̂_{Q}(M_{Q}(A)∩G(A)^{1})|_{A} of the cyclic group q^{Z}. The inequality (10) is strict if ch(k) > 0. It is easy to see

[X_{k}^{∗}(Z_{G}) : X_{k}^{∗}(G)] = dn, [X_{k}^{∗}(Z_{Q}) : X_{k}^{∗}(M_{Q})] = d^{2}m(n − m),
[X_{k}^{∗}(Z_{Q}/Z_{G}) : X_{k}^{∗}(M_{Q}/Z_{G})] = dm(n − m)/gcd(m, n − m),  ê_{Q} = d · gcd(m, n − m)
d_{G}^{∗} = log q,  d_{M_{Q}}^{∗} = (log q)^{2},  q_{0} = q^{n/gcd(m,n−m)}.

Therefore,

D_{G,Q} · E_{Q} = \{ \begin{matrix}dn & (ch(k) = 0). \\ (1 − q^{−dn})/(log q) & (ch(k) > 0). \end{matrix}

Since τ(G) = τ(Q) = 1 is known, Cororally 1 gives the following.

Theorem 3 If ch(k) = 0, then

\{dn · μ_{D_{A}}(D_{A}/D)^{m(n−m)} · ρ_{D}^{−1} · \frac{∏_{j=n−m+1}^{n} Z_{D}(jd)}{∏_{j=2}^{m} Z_{D}(jd)}\}^{\frac{1}{d·gcd(m,n−m)}} ≤ γ_{n,m}(D) .

If ch(k) > 0, then

\{\frac{1 − q^{−dn}}{log q} · μ_{D_{A}}(D_{A}/D)^{m(n−m)} · ρ_{D}^{−1} · \frac{∏_{j=n−m+1}^{n} Z_{D}(jd)}{∏_{j=2}^{m} Z_{D}(jd)}\}^{\frac{1}{d·gcd(m,n−m)}} < γ_{n,m}(D) .

For example, if D is a quaternion division algebra over Q and m = 1, then one has ρ_{Q} = 1, μ_{D_{A}}(D_{A}/D) = N𝔡_{D/Q}^{1/2} = ∏_{p∈𝔙_{f,2}} p and hence

\{\frac{12n(2n − 1)^{r_{3}}}{π^{2n+1/2}} ζ_{Q}(2n)ζ_{Q}(2n − 1)Γ(n)Γ(n − \frac{1}{2}) ∏_{p∈𝔙_{f,2}} p^{n−1} (\frac{1 − p^{−(2n−1)}}{1 − p^{−1}} )\}^{1/2} ≤ γ_{n}(D) ,

where r_{3} = 1 or 0 according as D is definite or indefinite. We denote the value of the left-hand side by [n, D]. For a square-free integer N > 1, let D_{N} be a quaternion algebra over Q such that N𝔡_{D_{N} /Q}^{1/2} = N, e.g., D_{2} = (−1, −1), D_{3} = (−1, −3), D_{5} = (−2, −5), D_{6} = (−1, 3), D_{7} = (−1, −7) and D_{10} = (−2, 5), where (a, b) stands for the quaternion algebra generated by i and j with i^{2} = a, j^{2} = b and ij = −ji. The following tables give numerical examples of [n, D_{N}]:

\begin{matrix}n & [n, D_{2}] & [n, D_{3}] & [n, D_{5}] & [n, D_{7}] \\ 2 & 1.297258519 & 1.443456027 & 1.726586552 & 1.978704389 \\ 3 & 1.515273677 & 1.995775367 & 3.042255888 & 4.115273864 \\ 4 & 2.530418525 & 4.040765897 & 7.938578156 & 12.70444456 \\ 5 & 5.393737367 & 10.52001705 & 26.67683122 & 50.51365650 \\ 6 & 13.94246428 & 33.28151972 & 108.9521040 & 244.1035544 \\ 7 & 42.33203429 & 123.7370964 & 522.9445997 & 1386.303048 \\ 8 & 147.6045644 & 528.3922475 & 2882.945637 & 9042.800847 \\ 9 & 581.1565361 & 2547.947350 & 17947.12248 & 66607.84112 \\ 10 & 2549.878172 & 13691.81879 & 124505.8889 & 546744.5241 \end{matrix}

By [C-W], it is known γ_{2}(D_{2}) = 2, γ_{2}(D_{3}) = 3 and γ_{2}(D_{5}) = 5.

\begin{matrix}n & [n, D_{6}] & [n, D_{10}] & [n, D_{14}] & [n, D_{15}] \\ 2 & 1.559110703 & 1.864926623 & 2.137245010 & 2.075098781 \\ 3 & 2.484720294 & 3.787578034 & 5.123474644 & 4.988640043 \\ 4 & 6.085153489 & 11.95502729 & 19.13213909 & 19.09070223 \\ 5 & 19.81735311 & 50.25316799 & 95.15640162 & 98.01444678 \\ 6 & 80.25844451 & 262.7381944 & 588.6561594 & 627.1722287 \\ 7 & 388.2457592 & 1640.825823 & 4349.756821 & 4796.155594 \\ 8 & 2182.851359 & 11909.79207 & 37356.88820 & 42634.46615 \\ 9 & 13982.96635 & 98492.61985 & 365539.4219 & 431818.2696 \\ 10 & 100515.7012 & 914034.6441 & 4013813.651 & 4907997.900 \end{matrix}

There is no example of the exact value of γ_{n}(D) for indefinite quaternion algebras.

4.3 The asymptotic distribution of rational points on Y_{Q}

Let Q = Q_{m}, X_{Q} = Q(k)\G(k) and Y_{Q} = Q(A)^{1}\G(A)^{1} be the same as in §4.1. The projective variety Q\G is a k-form of Grassmannian and is called the Brauer–Severi variety. The set X_{Q} is considered as the set of k-rational points of Q\G. For a positive real number T, let us define the subset B_{T} of Y_{Q} by

B_{T} = {y ∈ Y_{Q} : H_{Q}(y) ≤ T } .

For g ∈ G(A)^{1}, the subset B_{T} g is the translation of B_{T} by g. The constant γ_{n,m}(D) measures the existence of rational points in B_{T} g, i.e., we have B_{T} g ∩ X_{Q} ≠ ∅ for every g ∈ G(A)^{1} if γ_{n,m}(D) ≤ T. In the case that k is an algebraic number field, the cardinality of B_{T} g ∩ X_{Q} is increasing to proportion to the volume of B_{T} as T → ∞. More precisely, it was proved in [Wa2] that

lim_{T →∞} ♯(B_{T} g ∩ X_{Q}) · \frac{D_{G,Q} · E_{Q}}{C_{G,Q}} T^{−ê_{Q}} = \frac{τ(Q)}{τ(G)} .

Therefore, we obtain the following.

Theorem 4 We assume k is an algebraic number field. Then the asymptotic behavior

♯(B_{T} g ∩ X_{Q}) ∼ \frac{T^{d·gcd(m,n−m)}}{dn|D_{k}|^{d^{2}(m(n−m)+1)/2}N𝔡_{D/k}^{(m(n−m)+1)/2}} \frac{∏_{j=2}^{m} Z_{D}(jd)}{∏_{j=n−m+1}^{n} Z_{D}(jd)} as T → ∞

holds for all g ∈ G(A)^{1}.

For example, if k = Q, m = 1 and D = D_{N} as defined above, then we have

♯(B_{T} g ∩ X_{Q}) ∼ \frac{T^{2}}{[n, D_{N}]^{2}} as T → ∞.

References

[B-T] Borel, A and Tits, J., Groupes réductifs, Publ. Math. I.H.E.S. 27 (1965) 55 - 150. [C-W] Coulangeon, R. and Watanabe, T., Hermite constant and Voronoï theory over a

quaternion skew field, preprint, 2003.

[Ic] Ichino, A., A regularized Siegel–Weil formula for unitary groups, Math. Z. to appear.

[Ik] Ikeda, T., On the residue of the Eisenstein series and the Siegel–Weil formula, Comp. Math. 103 (1996) 183 - 218.

[K] Knapp, A., Representation Theory of Semisimple Groups, Princeton Univ. Press, Princeton, 1986.

[L] Lai, K. F., Tamagawa numbers of reductive algebraic groups, Comp. Math. 41 (1980) 153 -188.

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

[R] Reiner, I., Maximal Orders, Academic Press, London etal., 1975.

[T] Turner, S., Zeta-functions of central simple algebras over global fields, An. Acad. Brasil Ciênc. 48 (1976) 171 - 186.

[V] Vignéras, M.-F., Arithmétique des algébres de quaternions, Springer-Verlag, Lecture Note in Mathematics 800, Heidelberg, 1980.

[Wa] Watanabe, T., Fundamental Hermite constants of linear algebraic groups, J. Math. Soc. Japan 55 (2003) 1061 - 1080.

[Wa2] Watanabe, T,. The Hardy–Littlewood property of flag varieties, Nagoya Math. J. 170 (2003) 185 - 211.

[We] Weil, A., Basic Number Theory, Springer-Verlag, Berlin etal., 1974.

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

nakamura@gaia.math.wani.osaka-u.ac.jp ; watanabe@math.wani.osaka-u.ac.jp