Unitary structure in representations of infinite-dimensional groups and a convexity theorem

In this paper, we show that a K a c M o o d y algebra fl(A) associated to a symmetr izable generalized Car tan matr ix A carries a cont ravar ian t Hermi t i an form which is positive-definite on all root spaces. We deduce that every integrable highest weight g(A)-module L(A) carries a cont ravar ian t positivedefinite Hermi t i an form. This allows us to define the m o m e n t m a p and prove a general izat ion of the Schu r -Horn -Kos t an t -Heckman-At iyah -P re s s l ey convexity theorem. The proofs are based on an identity which also gives est imates for the act ion of 9(A) on fl(A) and L(A). We hope that the main idea behind the paper is apparent : it is to use the interplay between the coadjoint and the highest weight representat ions. We are grateful to V. Gui l lemin for an in t roduct ion to the m o m e n t map.