The Duistermaat-Heckman formula with application to circle actions and Poincaré q-polynomials in twisted equivariant K-theory

In this paper, we deduce the sketch of proof of the Duistermaat-Heckman formula and investigate how the known Duistermaat-Heckman result could be specialized to the symplectic structure on the orbit space. The theorems of localization in equivariant cohomology not only provide us with beautiful mathematical formulas and stimulate achievements in algorithmic computations but also promote progress in theoretical and mathematical physics. We present the elliptic genera and the characteristic q-series for the circle actions and twisted equivariant K-theory, with the case of the symmetric group of n symbols separately analyzed. We show that the Poincaré q-polynomials admit presentation in terms of the Patterson-Selberg (or the Ruelle-type) spectral functions.