Fundamental Hermite Constants of Linear Algebraic Groups

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(GLn, 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 attached to an absolutely irreducible strongly k-rational representation π : G −→ GL(Vπ) of G. More precisely, if G(A) denotes the adele group of G and G(A) the unimodular part of G(A), it is defined by γ π = max g∈G(A)1 min γ∈G(k) ||π(gγ)xπ|| , where xπ is a non-zero k-rational point of the highest weight line in the representation space Vπ and || · || is a height function on the space GL(Vπ(A))Vπ(k). This constant is called a generalized Hermite constant by the reason that, in the case when k = Q, G = GLn and π = πd is the d-th exterior representation of GLn, γ GLn πd is none other than the Hermite Rankin constant ([R]): γn,d = max g∈GLn(R) min x1,··· ,xd∈Z n x1∧···∧xd 6=0 det(xi ggxj)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