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 Matsushima (1967), 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 Matsushima & Murakami (1968).

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 $\Gamma \backslash G/K$:

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

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.