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In this paper we deduce the sketch of proof of the Duistermaat-Heckmanformula and investigate how the known Duistermaat-Heckman result could bespecialized to the symplectic structure on the orbit space. The theorems oflocalization in equivariant cohomology not only provide us with beautifulmathematical formulas and stimulate achievements in algorithmic computations,but also promote progress in theoretical and mathematical physics. We presentthe elliptic genera and the characteristic $q$-series for the circle actionsand twisted equivariant K-theory, with the case of the symmetric group of $n$symbols separately analyzed. We show that the Poincar\'{e} $q$-polynomialsadmit presentation in terms of the Patterson-Selberg (or the Ruelle-type)spectral functions.

The Duistermaat-Heckman formula with application to circle actions and Poincar\'{e} $q$-polynomials in twisted equivariant K-theory