Quantum-mechanical calculations in the algebraic group theory

Quantum oscillators on simple Lie algebras satisfying the special symmetry conditions are considered. Statsums, the Witten index and some simple correlators are calculated. The relations between these expressions and orders of algebraic groups over finite fields and degrees of some their representations are established under the condition that the temperatureT of systems is equal toT=/lnq. We consider the conformal limit of the theories where ranks of groups go to infinity. Also we discuss the relation between the adelic limit of the theories and the Tamagawa numbers.