We find a counterpart of the classical fact that the regular representation R(G) of a simple complex group G is spanned by the matrix elements of all irreducible representations of G. Namely, the algebra of functions on the big cell G 0 ⊂ G of the Bruhat decomposition is spanned by matrix elements of big projective modules from the category O of representations of the Lie algebra g of G, and has the structure of a g ⊕ g-module.We extend both regular representations to the affine groupĜ, and show that the loop form of the Bruhat decomposition ofĜ yields modified versions of R(Ĝ). They involve pairings of positive and negative level modules, with the total value of the central charge required for the existence of non-trivial semi-infinite cohomology. In this paper we consider in detail the case G = SL(2, C), the corresponding finite-dimensional and affine Lie algebras, and the closely related to them Virasoro algebra.Using the Fock space realization, we show that both types of modified regular representations for the affine and Virasoro algebras become vertex operator algebras, whereas the ordinary regular representations have instead the structure of conformal field theories. We identify the inherited algebra structure on the semi-infinite cohomology when the central charge is generic. We conjecture that for the integral values of the central charge the semiinfinite cohomology coincides with the Verlinde algebra and its counterpart associated with the big projective modules.
The structure of tensor representations of the classical finite-dimensional Lie algebras was described by H. Weyl. In this paper we extend Weyl's results to the classical infinite-dimensional locally finite Lie algebras gl ∞ , sl ∞ , sp ∞ and so ∞ , and study important new features specific to the infinite-dimensional setting.Let g be one of the above locally finite Lie algebras and let V be the natural representation of g. The tensor representations of g have the form V ⊗d for the cases g = sp ∞ , so ∞ , and the form V ⊗p ⊗ V ⊗q * for the cases g = gl ∞ , sl ∞ , where V * is the restricted dual of V. In contrast with the finite-dimensional case, these tensor representations are not semisimple. We explicitly describe their Jordan-Hölder constituents, socle filtrations, and indecomposable direct summands.
Let g be a simple Lie algebra, and let M λ be the Verma module over g with highest weight λ. For a finite-dimensional g-module U we introduce a notion of a regularizing operator, acting in U , which makes the meromorphic family of intertwining operators Φ : M λ+µ → M λ ⊗ U holomorphic, and conjugates the dynamical Weyl group operators A w (λ) ∈ End(U ) to constant operators. We establish fundamental properties of regularizing operators, including uniqueness, and prove the existence of a regularizing operator in the case g = sl 3 .
Abstract. In this paper we construct examples of commutative rings of difference operators with matrix coefficients from representation theory of quantum groups, generalizing the results of our previous paper [ES] to the q-deformed case. A generalized Baker-Akhiezer function is realized as a matrix character of a Verma module and is a common eigenfunction for a commutative ring of difference operators.In particular, we obtain the following result in
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