Finitely dimensional representations for Kac-Moody groups of type A₁⁽¹⁾
by
Yu Chen
Department of Mathematics, University of Torino, Via C.Alberto 10 10123 Torino, Italy

We describe here all finitely dimensional irreducible representations of Kac-Moody groups of type A₁⁽¹⁾.

Let H be a Kac-Moody group of tpye A₁⁽¹⁾ over a filed F of characteristic 0. Given a non trivial representation r: H ® GL₂(V), where V is a two dimensional vector space over a field K, for each positive integer n ³ 2 we construct a H-module of dimension n+1 as follows:

Let {u, v} be a base of V and K[u, v] the polynomials over u, v. The homogeneous polynomials of degree n form a subspace of K[u, v], which can be defined as a H-module by

h·(un-ivi) = (r(h)u)n-i (r(h)v)i,  "h Î H,  0 £ i £ n.

We denote this module by V_r (n).

We show that every finitely dimensional irreducible H-module is in fact isomorphic to a module of form Ä_i=1^r V_ri(n_i) for some positive integer r, where r_i is a non trivial two dimensional representation for H and n_i a positive integer greater than 2 for 1 £ i £ r.

To conclude the description, we determine at the end all two dimensional representations of H.

Date received: July 28, 2004

Copyright © 2004 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caoh-08.