Tag - Representations of Lie algebras

Let Γ be a finite group acting on a simple Lie algebra 𝔤 and acting on an s-pointed projective curve (Σ, p = {p₁, . . . ,p_s}) faithfully (for s ≥ 1). Also, let an integrable highest weight module H_c(λ_i) of an appropriate twisted affine Lie algebra determined by the ramification at p_i with a fixed central charge c is attached to each p_i. We prove that the space of twisted conformal blocks attached to this data is isomorphic to the space associated to a quotient group of Γ acting on 𝔤 by diagram automorphisms and acting on a quotient of Σ. Under some mild conditions on ramification types, we prove that calculating the dimension of twisted conformal blocks can be reduced to the situation when Γ acts on 𝔤 by diagram automorphisms and covers of ℙ¹ with 3 marked points. Assuming a twisted analogue of Teleman's vanishing theorem of Lie algebra homology, we derive an analogue of the Kac-Walton formula and the Verlinde formula for general Γ-curves (with mild restrictions on ramification types). In particular, if the Lie algebra 𝔤 is not of type D₄, there are no restrictions on ramification types.

We will discuss the classification of simple modules with finite weight multiplicities over basic classical map superalgebras. Any such module is parabolically induced from a simple cuspidal bounded module over a cuspidal map subsuperalgebra. Moreover, any simple cuspidal bounded module is isomorphic to an evaluation module.

Simple Lie superalgebras over complex numbers are of two types: Classical type and Cartan type. In our earlier joint work with Christodoulopoulou and Wiesner, we have defined Whittaker modules for Lie superalgebras and proved some important properties of these modules. Recently I was able to extend some of these results to Cartan-type Lie superalgebras. In this talk, I will briefly summarize these results.