2 9 M ay 2 00 2 REPRESENTATIONS OF DOUBLE AFFINE LIE ALGEBRAS

The representation theory of Kac–Moody algebras, and in particular that of affine Lie algebras, has been extensively studied over the past twenty years. The representations that have had the most applications are the integrable ones, so called because they lift to the corresponding group. The affine Lie algebra associated to a finite-dimensional complex simple Lie algebra g is the universal (one–dimensional) central extension of the Lie algebra of polynomial maps from C * → g. There are essentially two kinds of interesting integrable representations of this algebra: one is where the where the center acts as a positive integer, or positive energy representations; and the other is where the center acts trivially, or the level zero. Both kinds of representations have interesting applications: representations of the first kind have connections with number theory through the Rogers–Ramanujam identities, the theory of vertex algebras and conformal field theory; representations of the second kind are connected with the six–vertex and XXZ–model [11] and the references therein, the Kostka polynomials and the fermionic formulas of Kirillov and Reshetikhin, [18]. The study of such level zero representations was begun in [4] and continued in [5],[6]. More recently a geometric approach to such representations was developed in [21] for the corresponding quantum algebras. The affine Lie algebras admit an obvious generalization. Namely, we can consider central extensions of the polynomial maps (C *) ℓ → g. Not surprisingly, these algebras are a lot more complicated, for instance the central extension is now infinite—dimensional. A systematic study of such algebras can be found in [2] and the representation theory has been studied in [3], [9], [10], [20]. In general, interesting theories have been found for the quotients of this algebra by a central ideal of finite–codimension. One such quotient is the double affine algebra, this algebra is obtained from the affine algebra in the same way that the affine algebra is obtained from the finite–dimensional algebra. One can also define a corresponding quantum object, and representations of these have been studied in [21], [22], [23]. However, relatively little is still known about the integrable representations of the double affine algebras. In this paper we study representations of the double affine algebra g tor when one of the centers acts trivially; this is also the situation studied in the quantum case mentioned above. The category of such representations is not semisimple and our interest is in indecomposable integrable …