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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. ^[1] The Matsushima–Murakami formula is a generalization giving dimensions of spaces of automorphic forms, introduced by Matsushima & Murakami (1968).^[2]

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.

References

^ Matsushima, Yozô (1967), "A formula for the Betti numbers of compact locally symmetric Riemannian manifolds", Journal of Differential Geometry, 1 (1–2): 99–109, doi:10.4310/jdg/1214427883, ISSN 0022-040X, MR 0222908, S2CID 117292003
^ Matsushima, Yozô; Murakami, Shingo (1968), "On certain cohomology groups attached to Hermitian symmetric spaces. II", Osaka Journal of Mathematics, 5: 223–241, ISSN 0030-6126, MR 0266238

[1] Matsushima, Yozô (1967), "A formula for the Betti numbers of compact locally symmetric Riemannian manifolds", Journal of Differential Geometry, 1 (1–2): 99–109, doi:10.4310/jdg/1214427883, ISSN 0022-040X, MR 0222908, S2CID 117292003

[2] Matsushima, Yozô; Murakami, Shingo (1968), "On certain cohomology groups attached to Hermitian symmetric spaces. II", Osaka Journal of Mathematics, 5: 223–241, ISSN 0030-6126, MR 0266238

[1]

[2]

@@ Line 2: / Line 2: @@
 <!-- EDIT BELOW THIS LINE -->
+=== Definition ===
-In mathematics, hyperbolic complex space is a Hermitian manifold which is the equivalent of the real hyperbolic space in the context of complex manifolds. It is characterised by being the only [[simply connected]] [[Hermitian manifold]] whose [[holomorphic sectional curvature]] is constant equal to -1. Its underlying Riemannian manifold has non-constant negative curvature, pinched between -4 and -1 (or -1 and -1/4, according to the choice of a normalization of the metric): in particular, it is a [[CAT(k) space|CAT(-1) space]]. Complex hyperbolic spaces are [[Kähler manifold]]s.
+A function ''f'' defined on the domain ''D'' and with values in <math>\mathbb{C}</math> is said to be holomorphic at a point <math>z\in D</math> if it is complex-differentiable at this point, in the sense that there exists a complex linear map <math>L:\mathbb{C}^n \to \mathbb{C}</math> such that
-Complex hyperbolic space are also the [[symmetric space]]s associated with the [[Lie group|Lie group]]s <math>PU(n,1)</math>. They constitute one of the three families of rank one symmetric spaces of noncompact type, together with real and quaternionic hyperbolic spaces, classification to which must be added one exceptionnal space, the Cayley plane.
+<math>
-== Construction of the complex hyperbolic space ==
+f(z+h) = f(z) + L(h) + o(\lVert h\rVert)
+</math>
+The function ''f'' is said to be holomorphic if it is holomorphic at all points of its domain of definition ''D''.
-=== Projective model ===
-Let <math>\langle u,v\rangle := -u_1\overline{v_1} + u_2\overline{v_2} + \dots + u_{n+1}\overline{v_{n+1}}</math> be a [[Sesquilinear form|pseudo-Hermitian form]] of signature <math>(n,1)</math> in the complex vector space <math>\mathbb{C}^{n+1}</math>. The projective model of the complex hyperbolic space is the [[Complex projective space|projectivized space]] of all negative vectors for this form: <math>\mathbb{H}^n_\mathbb{C} =  \{[\xi] \in \mathbb{CP}^n | \langle \xi,\xi\rangle <0\}.</math>
+If ''f'' is holomorphic, then all the partial maps :
-As an opet set of the complex projective space, this space is endowed with the structure of a [[Complex manifold|complex manifold]]. It is [[Biholomorphism|biholomorphic]] to the unit ball of <math>\mathbb{C}^n</math>, as one can see by noting that a negative vector must have non zero first coordinate,  and has a unique representant with first coordinate equal to 1 in the [[Complex projective space|projective space]]. The condition <math>\langle \xi,\xi\rangle<0 </math> when <math>\xi=(1,x_1,\dots,x_{n+1})</math> is equivalent to <math>\sum_{i=1}^{n} |x_i|^2 < 1</math>. The map sending the point <math>(x_1,\dots,x_n)</math> of the unit ball of <math>\mathbb{C}^n</math> to the point <math>[1:x_1:\dots:x_n]</math> of the projective space thus defines the required biholomorphism.
+<math>z \mapsto f(z_1,\dots,z_{i-1},z,z_{i+1},\dots,z_n)
-This model is the equivalent of the Poincaré disk model. Contrary to the real hyperbolic space, the complex projective space cannot be defined as a sheet of the hyperboloid <math>\langle x,x\rangle = -1</math>, because the projection of this hyperboloid on the projective model has connected fiber <math>\mathbb{S}^1</math> (the fiber being <math>\mathbb{Z}/2\mathbb{Z}</math> in the real case).
+</math>
+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 [[Hartogs's theorem on separate holomorphicity|Hartog's theorem]], or as [[Osgood's lemma]] under the additional hypothesis that ''f'' is [[continuous function|continuous]].
-A [[Hermitian metric]] is defined on <math>\mathbb{H}^n_\mathbb{C}</math> in the following way: if <math>p\in \C^{n+1}</math> belongs to the cone <math>\langle p,p\rangle=-1</math>, then the restriction of <math>\langle\cdot,\cdot\rangle </math> to the orthogonal space <math>(\C p)^{\perp} \subset \C^{n+1}</math> defines a definite positive hermitian product on this space, and because the tangent space of <math>\mathbb{H}^n_\mathbb{C}</math> at the point <math>[p]</math> can be naturally identified with <math>(\C p)^{\perp}</math>, this defines a hermitian inner product on <math>T_{[p]}\mathbb{H}^n_\mathbb{C}</math>. As can be seen by computation, this inner product does not depend on the choice of the representant <math>p</math>. This metric is a [[Kähler metric]].
-===Siegel model===
+In [[mathematics]], '''Matsushima's formula''', introduced by {{harvs|txt|last=Matsushima|year=1967|authorlink= Yozô Matsushima}}, is a formula for the [[Betti number]]s of a quotient of a [[symmetric space]] ''G''/''H'' by a [[discrete group]], in terms of [[unitary representation|unitary]] [[group representation|representations]] of the [[group (mathematics)|group]] ''G''.
-=Group of holomorphic isometries and symmetric space=
+<ref>{{Citation | last1=Matsushima | first1=Yozô | title=A formula for the Betti numbers of compact locally symmetric Riemannian manifolds |mr=0222908 | year=1967 | journal=Journal of Differential Geometry | issn=0022-040X | volume=1 | issue=1–2 | pages=99–109| doi=10.4310/jdg/1214427883 | s2cid=117292003 | doi-access=free }}</ref> The '''Matsushima–Murakami formula''' is a generalization giving dimensions of spaces of [[automorphic form]]s, introduced by {{harvtxt|Matsushima|Murakami|1968}}.<ref>{{Citation | last1=Matsushima | first1=Yozô | last2=Murakami | first2=Shingo | title=On certain cohomology groups attached to Hermitian symmetric spaces. II |mr=0266238 | year=1968 | journal=Osaka Journal of Mathematics | issn=0030-6126 | volume=5 | pages=223–241}}</ref>
+== Statement of the formula in the case of a compact quotient==
-The group of holomorphic isometries of the complex hyperbolic space is the [[Lie group]] <math>PU(n,1)</math>. This group acts transitively on the complex hyperbolic space, and the stabilizer of a point is isomorphic to the unitary group <math>U(n)</math>. The complex hyperbolic space is thus homeomorphic to the [[Homogeneous space|homogeneous space]] <math>PU(n,1)/U(n)</math>. The stabilizer <math>U(n)</math> is the [[Maximal compact subgroup|maximal compact subgroup]] of <math>PU(n,1)</math>.
+Let ''G'' be a [[Lie group]], and K a connex [[Compact group|compact subgroup]] of G. Denote <math>\mathcal{R}_G</math> the set of all [[Representation_theory#Equivariant_maps_and_isomorphisms |isomorphism]] classes of [[Irreducible representation|irreducible]] [[Unitary representation|unitary representations]] <math>\pi : G \to U(H_\pi)</math>, where <math>H_\pi</math> is a complex separable [[Hilbert space]] and <math>\pi</math> a group morphism.
-=Curvature=
+For every cocompact [[Lattice (discrete subgroup)|lattice]] <math>\Gamma</math> of G, denote <math>L^2(\Gamma\backslash G)</math> the Hilbert space of square-integrable complex-valued functions on <math>\Gamma\backslash G</math>, endowed with the hermitian product associated to the [[Haar measure]] on <math>\Gamma\backslash G</math>. Note that the existence of a cocompact lattice forces G to be [[Unimodular group|unimodular]]. Then, there is a unitary representation <math>\pi : G \to U(L^2(\Gamma\backslash G))</math>, defined by the following formula:
+<math>\forall g\in G, \forall f \in L^2(\Gamma\backslash G), \pi(g)(f) = f(\cdot g).</math>
-The group of holomorphic isometries <math>PU(n,1)</math> acts [[Group action|transitively]] on the tangent complex lines of the hyperbolic complex space. This is why this space has constant [[holomorphic sectional curvature]], that can be computed to be equal to -4 (with the above normalization of the metric). This property characterizes the hyperbolic complex space : up to isometric biholomorphism, there is only one [[Simply connected space|simply connected]] complete [[Kähler manifold]] of given constant [[holomorphic sectional curvature]]<ref name=":0">{{Cite book |last=Kobayashi |first=Shōshichi |url=https://www.worldcat.org/oclc/34259751 |title=Foundations of differential geometry, vol. 2 |last2=Nomizu |first2=Katsumi |date=1996 |publisher=Wiley |others= |isbn=0-471-15733-3 |edition= |location=New York |oclc=34259751}}</ref>.
+By ... theorem, the representation <math>\pi</math> can be decomposed into irreducible unitary representations, each one appearing at most a finite number of times:
-Furthermore, when a Hermitian manifold has constant holomorphic sectional curvature equal to <math>k</math>, the sectional curvature of every real tangent plane <math>\Pi</math> is completely determined by the formula :
-<math>K(\Pi) = \frac{k}{4}\left(1+3\cos^2(\alpha(\Pi)\right)</math>
+<math>L^2(\Gamma\backslash G) = \widetilde{\bigoplus_{\pi \in R_G}}m_\pi(\Gamma)H_\pi</math>
+In this decomposition, the sum is a [[hilbert space#Direct_sums|Hilbertian direct sum]], and at most a countable number of terms are nonzero, because <math>L^2(\Gamma\backslash G)</math> is [[Separable space|separable]]. It is a part of the theorem that all multiplicities <math>m_\pi(\Gamma)</math> are finite.
-where <math>\alpha(\Pi)</math> is the angle between <math>\Pi</math> and <math>J\Pi</math>, ie the infimum of the angles between a vector in <math>\Pi</math> and a vector in <math>J\Pi</math><ref name=":0" />. This angle equals 0 if and only if <math>\Pi</math> is a complex line, and equals <math>\pi/2</math> if and only if <math>\Pi</math> is totally real. Thus the sectional curvature of the complex hyperbolic space varies from -4 (for complex lines) to -1 (for totally real planes).
+In this context, the Matsushima formula is the following decomposition of the complex [[De Rham cohomology]] of the manifold <math>\Gamma\backslash G/K</math>:
-In complex dimension 1, every real plane in the tangent space is a complex line: thus the hyperbolic complex space of dimension 1 has constant curvature equal to -1, and by the [[Uniformization theorem|uniformization theorem]], it is isometric to the real hyperbolic plane. Hyperbolic complex spaces can thus be seen as another high-dimensional generalization of the hyperbolic plane, less standard than the real hyperbolic spaces. A third possible generalization is the [[Homogeneous space|homogeneous space]] <math>SL_n(\mathbb{R})/SO_n(\mathbb{\R})</math>, which for <math>n=2</math> again coincides with the hyperbolic plane, but becomes a symmetric space of rank greater than 1 when <math>n\ge 3</math>.
+<math>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})</math>
-=Totally geodesic subspaces=
+In this formula the multiplicities <math>m_\pi(\Gamma)</math> are exactly the one appearing in the previous decomposition of <math>L^2(\Gamma\backslash G)</math> into irreducibles representations. The symbols <math>\mathfrak{g},\mathfrak{k}</math> respectively denotes the [[Lie algebra|Lie algebras]] associated with ''G'' and ''K'', and <math>H_\pi^{\infty}\subset H_\pi</math> is the subspace of <math>C^\infty</math>-vectors for the representation <math>\pi</math>, which is also a <math>\mathfrak{g}</math>-module. Finally, <math>H^*(\mathfrak{g},\mathfrak{k},H_\pi^{\infty})</math> is the cohomology of the [[Chevalley–Eilenberg complex]] associated to <math>\mathfrak{g},\mathfrak{k},H_\pi^{\infty}</math>. Remark that, since the compact manifold <math>\Gamma\backslash G/K</math> 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 <math>\pi : G \to U(L^2(\Gamma\backslash G))</math> have non-vanishing cohomology.
-Every totally geodesic submanifold of the complex hyperbolic space of dimension n is one of the following :
-* a copy of a complex hyperbolic space of smaller dimension
-* a copy of a real hyperbolic space of real dimension smaller than <math>n</math>
-In particular, there is no codimension 1 totally geodesic subspace of the complex hyperbolic space.
-== References ==
+==References==
+<references/>