User:William Sarem/sandbox

Definition
A function f defined on the domain D and with values in $$\mathbb{C}$$ is said to be holomorphic at a point $$z\in D$$ if it is complex-differentiable at this point, in the sense that there exists a complex linear map $$L:\mathbb{C}^n \to \mathbb{C}$$ such that

$$ f(z+h) = f(z) + L(h) + o(\lVert h\rVert) $$

The function f is said to be holomorphic if it is holomorphic at all points of its domain of definition D.

If f is holomorphic, then all the partial maps :

$$z \mapsto f(z_1,\dots,z_{i-1},z,z_{i+1},\dots,z_n) $$

are holomorphic as functions of one complex variable : we say that f is holomorphic in each variable separately. Conversely, if f is holomorphic in each variable separately, then f is in fact holomorphic : this is known as Hartog's theorem, or as Osgood's lemma under the additional hypothesis that f is continuous.

In mathematics, Matsushima's formula, introduced by, is a formula for the Betti numbers of a quotient of a symmetric space G/H by a discrete group, in terms of unitary representations of the group G. The Matsushima–Murakami formula is a generalization giving dimensions of spaces of automorphic forms, introduced by.

Statement of the formula in the case of a compact quotient
Let G be a Lie group, and K a connex compact subgroup of G. Denote $$\mathcal{R}_G$$ the set of all isomorphism classes of irreducible unitary representations $$\pi : G \to U(H_\pi)$$, where $$H_\pi$$ is a complex separable Hilbert space and $$\pi$$ a group morphism. For every cocompact lattice $$\Gamma$$ of G, denote $$L^2(\Gamma\backslash G)$$ the Hilbert space of square-integrable complex-valued functions on $$\Gamma\backslash G$$, endowed with the hermitian product associated to the Haar measure on $$\Gamma\backslash G$$. Note that the existence of a cocompact lattice forces G to be unimodular. Then, there is a unitary representation $$\pi : G \to U(L^2(\Gamma\backslash G))$$, defined by the following formula:

$$\forall g\in G, \forall f \in L^2(\Gamma\backslash G), \pi(g)(f) = f(\cdot g).$$

By ... theorem, the representation $$\pi$$ can be decomposed into irreducible unitary representations, each one appearing at most a finite number of times:

$$L^2(\Gamma\backslash G) = \widetilde{\bigoplus_{\pi \in R_G}}m_\pi(\Gamma)H_\pi$$

In this decomposition, the sum is a Hilbertian direct sum, and at most a countable number of terms are nonzero, because $$L^2(\Gamma\backslash G)$$ is separable. It is a part of the theorem that all multiplicities $$m_\pi(\Gamma)$$ are finite.

In this context, the Matsushima formula is the following decomposition of the complex De Rham cohomology of the manifold $$\Gamma\backslash G/K$$:

$$H^i_{dR}(\Gamma\backslash G/K,\mathbb{C}) \simeq \bigoplus_{\pi \in \mathcal{R}_G}m_\pi(\Gamma)H^i(\mathfrak{g},\mathfrak{k},H_\pi^{\infty})$$

In this formula the multiplicities $$m_\pi(\Gamma)$$ are exactly the one appearing in the previous decomposition of $$L^2(\Gamma\backslash G)$$ into irreducibles representations. The symbols $$\mathfrak{g},\mathfrak{k}$$ respectively denotes the Lie algebras associated with G and K, and $$H_\pi^{\infty}\subset H_\pi$$ is the subspace of $$C^\infty$$-vectors for the representation $$\pi$$, which is also a $$\mathfrak{g}$$-module. Finally, $$H^*(\mathfrak{g},\mathfrak{k},H_\pi^{\infty})$$ is the cohomology of the Chevalley–Eilenberg complex associated to $$\mathfrak{g},\mathfrak{k},H_\pi^{\infty}$$. Remark that, since the compact manifold $$\Gamma\backslash G/K$$ has finite cohomology spaces, the sum on the right must have finitely many non-zero terms, and thus only a finite number of irreducible sub-representations of $$\pi : G \to U(L^2(\Gamma\backslash G))$$ have non-vanishing cohomology.