hope to study ${\cal{D}}_{P(z)}(V)$ in this aspect in a future publication. During this research we have noticed that regular representations for affine Lie algebras have been studied in \[FP\] and that certain results of Peter-Weyl type on Kac-Moody groups have been obtained in \[KP\]. Presumably, \[FP\], \[KP\] (in the affine case) and the present paper are closely related. We hope to study the connections in a future publication.

In the next paper \[Li6\] we shall define and study ${\cal{D}}_{Q(z)}(W)$ for a given $V$-module $W$ and we shall study the connection between the regular representation ${\cal{D}}_{Q(z)}(V)$ and the modular invariance property of trace functions and correlation functions (\[Z\], \[DLM3\]).

This paper is organized as follows: In Section 2 we review the notion of the contragredient module and discuss certain related issues. In Section 3, we present $V\otimes V$-module ${\cal{D}}_{P(z)}(W)$. In Section 4, we present the analogue of the Peter-Weyl theorem.

[**Acknowledgments.**]{} We thank James Lepowsky for many interesting discussions on certain subtle issues in formal calculus. In this research we greatly benefit from studying Huang and Lepowsky’s papers \[H3\], \[HL0-3\] and we adopt many of their important viewpoints into this paper. We would like also to thank IAS for its financial 