Ryshkov domains of reductive algebraic groups (Q461310)

In [Sov. Math., Dokl. 11, 1240--1244 (1970); translation from Dokl. Akad. Nauk SSSR 194, 514--517 (1970; Zbl 0229.10013)], \textit{S. S. Ryshkov} introduced a locally finite polyhedron in the cone of positive definite quadratic forms \(P_n\), given by the condition \(m(A) \geq 1\), where NEWLINE\[NEWLINEm(A) = \min_{0 \neq x \in \mathbb Z^n} x^t AxNEWLINE\]NEWLINE is the arithmetic minimum of \(A \in P_n\). Ryshkov's polyhedron is a reduction domain for \(P_n\), which is well matched with the Hermite invariant NEWLINE\[NEWLINEf : A \to m(A)/(\det A)^{1/n}NEWLINE\]NEWLINE on \(P_n\) in the sense that any local maximum of \(f\) is attained on its boundary.NEWLINENEWLINEIn the paper under review, the author introduces an analogue of Ryshkov's polyhedron in the scope of reduction theory of algebraic groups. Specifically, given a connected isotropic reductive algebraic group \(G\) over a number field \(k\) and a maximal \(k\)-parabolic subgroup \(Q\) of \(G\), he defines an open fundamental domain of the adele group \(G(\mathbb A) \) with respect to \(G(k)\), which is well matched (in an essentially analogous sense to Ryshkov's classical situation) with the arithmetic minimum function NEWLINE\[NEWLINEm_Q(g) = \min_{x \in Q(k) \backslash G(k)} H_Q(xg)NEWLINE\]NEWLINE on \(G(k) \backslash G(\mathbb A)\), where \(H_Q\) is a height function associated to \(Q\). This work is aimed at developing an analogue of classical reduction theory in this general setting and is related to the author's previous investigations of the generalized Hermite's constant [J. Lie Theory 10, No. 1, 33--52 (2000; Zbl 1029.11031); J. Math. Soc. Japan 55, No. 4, 1061--1080 (2003; Zbl 1103.11033)].