Biorthogonal Systems of Theta Functions and Macdonald Denominators

Abstract

The fact $$\lim _{p \rightarrow 0} \theta (\zeta ; p)=1-\zeta $$ , $$\zeta \in \mathbb {C}^{\times }$$ suggests that the theta function $$\theta (\zeta ; p)$$ is an elliptic analogue of a linear function of $$\zeta $$ . What is the elliptic analogue of a polynomial of $$\zeta $$ ? Rosengren and Schlosser gave seven kinds of answers to this fundamental question by introducing seven infinite series of spaces of theta functions associated with the irreducible reduced affine root systems, $$R_n=A_{n-1}$$ , $$B_n$$ , $$B^{\vee }_n$$ , $$C_n$$ , $$C^{\vee }_n$$ , $$BC_n$$ , $$n \in \mathbb {N}$$ , and $$D_n$$ , $$n \in \{2, 3, \dots \}$$ . Here n indicates the degree of the elliptic analogues of polynomials. The basis functions for the function spaces are called the $$R_n$$ theta functions and are denoted by $$\{\psi ^{R_n}_j\}_{j=1}^n$$ . It was proved that the determinants consisting of $$\{\psi ^{R_n}_j\}_{j=1}^n$$ provide the Macdonald denominator formulas, which are the elliptic extensions of the Weyl denominator formulas. In this chapter, first we give a brief review of the theory of Rosengren and Schlosser. Then we introduce appropriate inner products and prove the biorthogonality relations for the $$R_n$$ theta functions of Rosengren and Schlosser.