A formula for the Betti numbers of compact locally symmetric Riemannian manifolds
Let \(G\) be a connected semisimple Lie group with finite center and \(K\) a maximal compact subgroup of \(G\). The orthogonal complement of the Lie algebra of \(K\) in the Lie algebra of \(G\) with respect to the Killing form of \(G\) is denoted by \(m\) while \(m^{\mathbb C}\) is the complexification of \(m\). The adjoint action of \(K\) in \(m\) induces a representation \(\text{ad}^p\) of \(K\) in the vector space \(\wedge^p(m^{\mathbb C})\), which can be written as a sum of irreducible representations \(\text{ad}^p = \tau_1^p + \dots + \tau_{s_p}^p\). Consider now a discrete subgroup \(\Gamma\) of \(G\) with compact quotient space \(\Gamma\backslash G\). Assume \(\Gamma\) acts freely on the symmetric space \(X = G/K\). The vector space of harmonic \(p\)-forms on \(X\) invariant by \(\Gamma\) is termed \(h^p(X,\Gamma)\). Let \(T\) be an irreducible unitary representation of \(G\), and \(T_K\) the restriction of \(T\) on \(K\). Finally \(N(T)\) denotes the multiplicity of \(T\) in the unitary representation \(U\) of \(G\) in the Hilbert space \(L^2(\Gamma\backslash G)\), and \(M(T_K; \tau_i^p)\) the multiplicity of the irreducible representation \(\tau_i^p\) of \(K\) in \(T\). Then the main result is the formula \[ \dim h^p(X,\Gamma) = \sum_{T\in D_0} N(T) \sum_{i=1}^{s_p} M(T_K;\tau_i^p)\] where \(D_0\) denotes the set of all irreducible unitary representations of \(G\) with vanishing Casimir operator. This is a formula for the \(p\)-th Betti number of the locally symmetric Riemannian space \(\Gamma\backslash G/K\). It is equivalent to the theorem of Cartan-Hodge when \(G\) is compact and \(\Gamma\) reduces to the identity. As an example, the author considers the 3-dimensional hyperbolic space \(X\) with the action of the group \(G= \mathrm{SL}(2,\mathbb C)\).
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