International Congress of Mathematicians

610 A. KlyachkoAgain our experience with the unitary triplet suggests that the exponential mapestablishes a Thompson's type correspondence between O'Shea-Sjamaar inequalitiesfor additive singular problem and that of for hyperbolic angles.5.3. P-adic spectral problemsThere is also a nonarchimedian counterpart of this theory, which deals withclassical Chevalley groups G p = SL(n,Q p ), SO(n,Q p ), or Sp(2n,Q p ) over p-adicfield Qp and their maximal compact subgroups Kp = SL(n,Z p ), SO(n,Z p ), orSp(2n, Z p ) respectively. Double coset Kp#Kp may be treated as a complete invariantof lattice L = gL 0 , L 0 = Z®" with respect to Kp. We call lattice L = gL 0unimodular, orthogonal or symplectic if respectively g £ SL(n,Q p ), g £ SO(n,Q p )or g £ Sp(2n,Q p ).It is commonly known that in the unimodular case there exists a basis e, ofL 0 such that ê, = p Qi e, form a basis of L for some a, £ Z. We define index (L : L 0 )by(L:L 0 ) = (p a \p a V..,p a »), ai>a 2 > •••> a„_3 > ... > az- n > Gi-n> an d a~i = —a«.Similarly, for symplectic lattice L we can choose symplectic basis e,, fj of L 0such that è, = p Qi e, and fj = p^aj fj form a basis of L. In this case we have(L:L 0 ) = (p a »,p a »- 1 ,...,p ai ,p- Ql ,...,p- a »- 1 ,p- a »), (5.3)with a n > a n -i >,... , > cti > 0.Notice that the spectra (5.1)-(5.3) have the same symmetry, as singular spectruma (A) of a matrix A e G in the corresponding classical complex group.Theorem 5.2. The following conditions are equivalent(1) There exists a sequence of (unimodular, orthogonal, symplectic) latticesLQ, LI, ... , Ljv-i, LM = LQof given indices ai = (L t : L,_i).(2) The indices ai satisfy the equivalent conditions of Theorem 5.1 for the correspondingcomplex group G.We'll give proof elsewhere. The theorem is known for the unimodular lattices,see [10].