On the de Rham Cohomology Group of a Compact Kähler Symplectic Manifold

Abstract

Representations of U(n) and Sp(n) (cf.[29; VII]): The set of equivalence classes of irreducible complex representations of the unitary group U(n) (resp.the unitary symplectic group Sp(n» is in natural bijective correspondence with the set of non-increasing sequences of integers (resp.nonnegative integers) f".If PL is an irreducible repre-e, e C, leil=I}.Let Xl be the character of PI: Then X'[ID(n) is a Laurent polynomial of e u .. " en, i.e., a finite linear combination of the "Laurent monomials" eft .. •e~", ia e Z. On the set of such Laurent monomials one puts the lexicographical order with respect to their indices I=(il' .. " in).Then the highest nonzero term which appear in the polynomial expression for XI is just e{t .. •e~n.This fact indeed characterizes PI' -Representations of Sp(l) and H X (cf.[5; II.5]): -Irreducible complex representations of U(1) (resp.Sp(l» are up to equivalences indexed by integers (resp.nonnegative integers), say r.In the case of U(1)= {e e C; lel=I} these are I-dimensional and given by Pr(e)=e r .In the case of Sp(1) they are described as follows.Let VI be the underlying complex vector space of the right H-module H with respect to the inclusion C={a+bi}~H.VI is then naturally a complex Sp(l)-module, and the induced Sp(l)-modules are all irreducible, where sr denote the symmetric products.We let Vo=C be the trivial Sp(1)-module.Then these Vr give the desired repre-Compact Kiihler Symplectic Manifold 109 sentations pr' They are all self-conjugate in the sense that pr is equivalent to its complex conjugate (ir (cf.(1.7) (1.8)).Consider C* and HX naturally as real algebraic groups Then any rational complex representation of C* = U(I) X z2 Rx (resp.H X =Sp(l) X z2RX) is equivalent to exactly one of the following representations Pk,,, r, k E Z, r >0, k=r (2);(1.5) Pk,r(g) = tkpr(e), g=(e, t) E U(1)X z2 RX (resp.Sp(l)Xz2RX).We shall denote by Vk,r the corresponding HX-module.Real representations ofSp(l) and HX: i) When r =2s-l, s > 1, is