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Indefinite orthogonal group

The indefinite orthogonal group, denoted O(p,q), is a Lie group consisting of all real n \times n matrices (with n = p + q) that preserve a non-degenerate quadratic form of signature (p,q) on \mathbb{R}^n, specifically those g satisfying g^T I_{p,q} g = I_{p,q}, where I_{p,q} is the diagonal matrix \operatorname{diag}(I_p, -I_q) with I_r the r \times r identity matrix.^[1] This group generalizes the classical orthogonal group O(n) by allowing indefinite metrics, making it non-compact for p, q > 0, and it acts as the isometry group for pseudo-Euclidean spaces of that signature.^[2] Unlike the compact orthogonal group O(n) = O(n,0), which preserves positive definite forms, O(p,q) has four connected components when p and q are both positive, determined by the signs of the determinant and the action on the forward light cone; its identity component is the special orthochronous group SO^+(p,q), often denoted SO(p,q) for brevity, which consists of transformations with determinant 1 preserving the time orientation.^[1] The Lie algebra \mathfrak{so}(p,q) comprises matrices X such that X^T I_{p,q} + I_{p,q} X = 0, forming a real form of the complex orthogonal Lie algebra \mathfrak{so}(n,\mathbb{C}), with dimension \binom{n}{2}.^[2] Notably, O(p,q) is isomorphic to O(q,p) via conjugation by a signature-flipping matrix, and low-dimensional cases exhibit close ties to other classical groups: for instance, O(2,1) is locally isomorphic to SL(2,\mathbb{R}), while SO(2,4) is the conformal group of (3+1)-dimensional Minkowski space.^[2]^[3] The indefinite orthogonal groups play a central role in the representation theory of reductive Lie groups, where their irreducible unitary representations, such as small unipotent ones, arise via theta correspondence with symplectic groups Sp(2n,\mathbb{R}) and are classified using derived functor modules with specific infinitesimal characters and lowest K-types.^[4] They also underpin geometric structures like hyperbolic spaces (via PO(1,n)) and symmetric spaces such as Grassmannians of isotropic subspaces, with applications in moduli problems, including Enriques surfaces, and in physics for Lorentzian geometries in special relativity.^[2]

Definition and Foundations

Quadratic Forms and Signatures

A quadratic form on a real vector space V of dimension n is a map Q: V \to \mathbb{R} given by Q(x) = x^T A x, where A is a real symmetric n \times n matrix and x \in V is identified with column vectors.^[5] For the indefinite orthogonal group, A is taken to be diagonal with p entries of +1 and q entries of -1, where p + q = n and both p \geq 1, q \geq 1; this yields the standard form Q(x) = \sum_{i=1}^p x_i^2 - \sum_{j=1}^q x_{p+j}^2.^[6] The pair (p, q) is called the signature of the quadratic form, which classifies it up to congruence over the reals.^[7] Associated to Q is the symmetric bilinear form B: V \times V \to \mathbb{R} defined by the polarization identity B(x, y) = \frac{Q(x + y) - Q(x) - Q(y)}{2}, which satisfies Q(x) = B(x, x) and captures the inner product structure preserved by the group.^[5] A quadratic form is indefinite if it takes both positive and negative values on nonzero vectors, corresponding to the presence of both positive and negative eigenvalues of A; this contrasts with positive-definite forms (signature (n, 0), all eigenvalues positive) and negative-definite forms (signature (0, n), all eigenvalues negative).^[7] Indefiniteness requires p, q \geq 1, ensuring the form is neither bounded above nor below.^[6] Sylvester's law of inertia states that for any real symmetric matrix A, there exists an invertible matrix P such that P^T A P is diagonal with exactly p entries of +1, q entries of -1, and the rest zeros, where p + q + r = n and r is the nullity; the triple (p, q, r) is invariant under congruence.^[5] For nondegenerate forms (where r = 0), the signature (p, q) uniquely determines the equivalence class of the quadratic form.^[7] The indefinite orthogonal group is denoted O(p, q) for the group preserving a quadratic form of signature (p, q), or alternatively O(n; \varepsilon) where \varepsilon = \operatorname{diag}(I_p, -I_q) is the signature matrix.^[6] The standard orthogonal group O(n) arises as the special case O(n, 0).^[5]

Group Axioms and Isometry Preservation

The indefinite orthogonal group O(p,q), with p + q = n, consists of all invertible n \times n real matrices g \in \mathrm{GL}(n, \mathbb{R}) satisfying g^T A g = A, where A = \diag(I_p, -I_q) is the diagonal matrix representing the indefinite quadratic form of signature (p,q). This defining relation ensures that elements of O(p,q) act as linear isometries on the pseudo-Euclidean space \mathbb{R}^{p,q}, preserving distances and angles as measured by the metric induced by A.^[1]^[8] The condition g^T A g = A is equivalent to the preservation of the associated nondegenerate symmetric bilinear form B(x,y) = x^T A y, meaning B(gx, gy) = B(x,y) for all x, y \in \mathbb{R}^n. Consequently, the quadratic form Q(x) = B(x,x) = x^T A x is also preserved under the action of g, i.e., Q(gx) = Q(x). This bilinear preservation characterizes the group axioms: closure under matrix multiplication and inversion follows from the fact that if g_1 and g_2 satisfy the relation, so does their product and inverse, with the identity matrix serving as the neutral element.^[8]^[1] Taking the determinant of the defining equation g^T A g = A implies \det(g)^2 \det(A) = \det(A), so \det(g)^2 = 1 and thus \det(g) = \pm 1. This determinant condition embeds O(p,q) as a subgroup of the orthogonal matrices in the indefinite metric, sharply distinguishing it from the full general linear group \mathrm{GL}(n, \mathbb{R}), whose elements have arbitrary nonzero determinants.^[8]^[1] The subgroup SO(p,q) = \{ g \in O(p,q) \mid \det(g) = 1 \}, known as the special indefinite orthogonal group, is an index-2 normal subgroup of O(p,q). Its connected component of the identity is SO^+(p,q), the proper orthochronous subgroup. In pseudo-Euclidean spaces, elements of SO(p,q) preserve the overall orientation of the space, as the positive determinant maintains the handedness of bases. When a distinguished time-like direction exists (e.g., in signatures like (1,n-1)), the orthochronous part SO^+(p,q) further preserves time-orientation by mapping future-directed time-like vectors to future-directed ones, which is crucial for physical applications such as causality in relativity. The full group O(p,q) comprises four connected components for p,q \geq 1, reflecting combinations of orientation and time-orientation reversals.^[8]

Algebraic Structure

Matrix Realizations

The indefinite orthogonal group O(p,q) is realized as the set of all (p+q) \times (p+q) real matrices g satisfying g^T \eta g = \eta, where \eta = \operatorname{diag}(I_p, -I_q) is the diagonal matrix with the p \times p identity matrix I_p in the top-left block and the negative q \times q identity matrix -I_q in the bottom-right block.^[9] This matrix equation defines the group elements as linear transformations that preserve the associated quadratic form x^T \eta x on \mathbb{R}^{p+q}.^[10] These matrices act on the pseudo-Euclidean space \mathbb{R}^{p,q}, equipped with the indefinite inner product \langle x, y \rangle = x^T \eta y, which has signature (p,q) consisting of p positive and q negative eigenvalues.^[11] The group O(p,q) thereby comprises the linear isometries of this space that fix the origin. To elucidate the structure, partition any g \in O(p,q) into blocks conforming to the signature: g = \begin{pmatrix} A & B \\ C & D \end{pmatrix}, where A \in \mathbb{R}^{p \times p}, B \in \mathbb{R}^{p \times q}, C \in \mathbb{R}^{q \times p}, and D \in \mathbb{R}^{q \times q}. Substituting into the defining relation g^T \eta g = \eta yields the system of equations

\begin{pmatrix} A^T A - C^T C & A^T B - C^T D \\ B^T A - D^T C & B^T B - D^T D \end{pmatrix} = \begin{pmatrix} I_p & 0 \\ 0 & -I_q \end{pmatrix}.

Thus, the blocks satisfy A^T A - C^T C = I_p, A^T B = C^T D, and B^T B - D^T D = -I_q.^[12] These relations highlight the non-compact nature of the group when both p > 0 and q > 0. Different matrix realizations of O(p,q) arise from changes of basis in \mathbb{R}^{p+q} that preserve the quadratic form, i.e., for any invertible P with P^T \eta P = \eta, the matrices transform as g' = P^{-1} g P, yielding an isomorphic group.^[10] Infinite-dimensional analogs exist, such as groups of operators on separable Hilbert spaces preserving indefinite inner products of signature (\mathfrak{p}, \mathfrak{q}) with infinite cardinals \mathfrak{p}, \mathfrak{q}, but their detailed study lies outside finite-dimensional Lie theory.^[13]

Lie Algebra

The Lie algebra of the indefinite orthogonal group O(p,q), denoted \mathfrak{so}(p,q), consists of all n \times n real matrices X with n = p + q satisfying X^T \eta + \eta X = 0, where \eta is the diagonal matrix with p entries of +1 and q entries of -1, preserving the indefinite quadratic form of signature (p,q).^[6] This condition implies that elements of \mathfrak{so}(p,q) are skew-symmetric with respect to the metric \eta. The dimension of \mathfrak{so}(p,q) is \frac{n(n-1)}{2}, identical to that of the compact orthogonal Lie algebra \mathfrak{so}(n).^[14] A basis for \mathfrak{so}(p,q) can be constructed from block-diagonal and off-diagonal components relative to the splitting of \mathbb{R}^n into positive and negative eigenspaces of the quadratic form. Specifically, it includes basis elements from \mathfrak{so}(p) and \mathfrak{so}(q) generating rotations within the positive and negative subspaces, respectively, each contributing dimensions \frac{p(p-1)}{2} and \frac{q(q-1)}{2}, along with boost generators forming a space isomorphic to \mathbb{R}^{p q} that mix the subspaces.^[15] The Lie bracket structure on this basis reflects the semisimple nature of \mathfrak{so}(p,q), with brackets between rotation generators yielding rotations, between boosts yielding rotations, and between rotations and boosts yielding boosts. The Killing form B(X,Y) = \operatorname{tr}(\operatorname{ad}_X \operatorname{ad}_Y) on \mathfrak{so}(p,q) is non-degenerate and indefinite, with signature determined by p and q, distinguishing it from the negative-definite form on compact Lie algebras. As a real form of a complex semisimple Lie algebra, \mathfrak{so}(p,q) admits a root system that is a non-compact real form of type D_r when n = 2r is even or B_r when n = 2r + 1 is odd, with roots corresponding to differences and sums of orthonormal basis vectors adjusted for the signature.^[6] The Cartan decomposition is \mathfrak{so}(p,q) = \mathfrak{k} \oplus \mathfrak{p}, where \mathfrak{k} = \mathfrak{so}(p) \oplus \mathfrak{so}(q) is the maximal compact subalgebra and \mathfrak{p} \cong \mathbb{R}^{p q} is the complementary space of symmetric matrices with respect to \eta in the off-diagonal blocks.^[15] The exponential map \exp: \mathfrak{so}(p,q) \to O(p,q) is defined by the power series \exp(X) = \sum_{k=0}^\infty \frac{X^k}{k!} and is surjective onto the connected component of the identity in O(p,q), though not necessarily a diffeomorphism globally due to the non-compactness.^[6] This surjectivity ensures that every element in the identity component arises as the flow of a one-parameter subgroup generated by an element of the Lie algebra.

Examples and Low-Dimensional Cases

O(1,1) and O(2,1)

The indefinite orthogonal group O(1,1) consists of $2 \times 2 real matrices g satisfying g^T \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} g = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, preserving the quadratic form of signature (1,1).^[1] These matrices have determinant \pm 1 and form a group of dimension 1, with four connected components corresponding to the choices of sign for the determinant and the preservation of the orientation of the two light cones.^[2] The connected component containing the identity, denoted SO^+(1,1), is generated by hyperbolic boosts and is isomorphic to the additive group \mathbb{R}.^[1] Elements of SO^+(1,1) admit an explicit parametrization in terms of hyperbolic functions: for a real parameter \theta, the boost matrix is

\begin{pmatrix} \cosh \theta & \sinh \theta \\ \sinh \theta & \cosh \theta \end{pmatrix},

which satisfies the group relation via the addition formulas for hyperbolic functions and preserves the Minkowski inner product.^[16] Reflections, such as \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} or \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}, generate the other components when combined with boosts, yielding an overall structure isomorphic to \mathbb{R} \rtimes \{\pm 1\} for the orthochronous subgroup including spatial reflections.^[17] The group O(2,1) comprises $3 \times 3 real matrices preserving the quadratic form of signature (2,1), given by the diagonal matrix \operatorname{diag}(1,1,-1), and has dimension 3 as a Lie group.^[1] Its connected component SO^+(2,1) consists of proper orthochronous transformations and is isomorphic to the projective special linear group \operatorname{PSL}(2,\mathbb{R}), via the adjoint representation of \operatorname{SL}(2,\mathbb{R}) or explicit matrix embeddings that map Möbius transformations to Lorentz boosts and rotations in the hyperbolic setting. Geometrically, SO^+(2,1) acts as the group of orientation-preserving isometries of the hyperbolic plane \mathbb{H}^2, where elements correspond to Möbius transformations z \mapsto \frac{az + b}{cz + d} with ad - bc = 1, preserving the hyperbolic metric in the upper half-plane model.^[18] The fundamental representation of O(2,1) is its defining action on the vector space \mathbb{R}^{2,1}, where group elements act linearly to preserve the indefinite inner product \langle x, y \rangle = x_1 y_1 + x_2 y_2 - x_3 y_3, enabling the realization of hyperbolic rotations and boosts that fix the light cone structure.^[1] This representation underpins the identification with isometries of \mathbb{H}^2, as the hyperboloid model embeds \mathbb{H}^2 in the null sheet of \mathbb{R}^{2,1}.^[19]

Lorentz Group O(1,3)

The Lorentz group O(1,3) is the group of all $4 \times 4 real matrices \Lambda that preserve the Minkowski metric \eta = \operatorname{diag}(1, -1, -1, -1), satisfying \Lambda^T \eta \Lambda = \eta, which corresponds to the quadratic form of signature (1,3). It is isomorphic to O(3,1), which preserves the metric \operatorname{diag}(-1, 1, 1, 1) of signature (3,1), via overall sign flip in the metric. This group acts linearly on four-dimensional Minkowski spacetime \mathbb{R}^{1,3}, transforming coordinates while maintaining the invariant spacetime interval ds^2 = dt^2 - dx^2 - dy^2 - dz^2.^[20]^[21] The group O(1,3) has four connected components, distinguished by the signs of the determinant and the effect on the time coordinate. The proper orthochronous component SO^+(1,3) consists of transformations with \det \Lambda = +1 that preserve the orientation of time, forming the connected component containing the identity. The full group includes additional components generated by parity inversion (spatial reflection, \det \Lambda = -1, orthochronous) and time reversal (reversing the sign of the time coordinate, \det \Lambda = +1, non-orthochronous), with the fourth component combining both. In special relativity, physical transformations are restricted to SO^+(1,3), ensuring preservation of causality and the direction of time.^[22]^[20] Elements of SO^+(1,3) comprise spatial rotations from SO(3), which act on the spatial coordinates while leaving time unchanged, and Lorentz boosts, which mix time and space coordinates via hyperbolic rotations. For instance, a boost along the x-axis corresponds to a hyperbolic rotation in the t-x plane, parameterized by rapidity \phi such that the transformation scales time and x by \cosh \phi and \sinh \phi, respectively, preserving the Minkowski norm. Parity and time-reversal extend the group but are not continuously connected to the identity. The Lie algebra \mathfrak{so}(1,3) is isomorphic to \mathfrak{sl}(2,\mathbb{C}) as complex Lie algebras.^[23]^[24] The group SO^+(1,3) is isomorphic to the projective special linear group \mathrm{PSL}(2,\mathbb{C}) = \mathrm{SL}(2,\mathbb{C})/\{\pm I\}, providing a double cover via the spin representation on two-component spinors, which maps to vector representations of spacetime transformations. This isomorphism highlights the group's structure as a double cover of the rotation group in three dimensions, with \mathrm{SL}(2,\mathbb{C}) acting on Hermitian matrices to induce Lorentz transformations. In the context of special relativity, elements of O(1,3) preserve the light cone structure—the set of null vectors with ds^2 = 0—thereby maintaining the causal structure of spacetime, where timelike intervals define possible influences between events and spacelike intervals forbid them.^[25]^[26]^[27]

Topological and Analytic Properties

Connected Components and Covering Groups

The indefinite orthogonal group O(p,q) for p,q \geq 1 has four connected components, distinguished by the sign of the determinant and the preservation or reversal of the time orientation (orthochronous or anti-orthochronous transformations).^[28] These components arise from the action on the maximal positive- and negative-definite subspaces of the underlying quadratic form, with the group of path components given by \pi_0(O(p,q)) \cong \mathbb{Z}_2 \times \mathbb{Z}_2.^[28] The connected component of the identity, denoted O^+(p,q), consists of the orthochronous transformations with positive determinant and is the unique component containing the identity matrix.^[28] The special indefinite orthogonal group SO(p,q) of determinant-1 elements has two connected components, with O^+(p,q) forming a normal subgroup of index 2 in SO(p,q) provided p,q \geq 2.^[28] As a Lie group, O(p,q) is non-compact, reflecting the indefinite signature of the quadratic form, and possesses the maximal compact subgroup K = O(p) \times O(q).^[29] The exponential map from the Lie algebra \mathfrak{o}(p,q) covers paths within the identity component, but the full group's topology requires accounting for the discrete \pi_0. The universal covering group of the identity component O^+(p,q) (equivalently SO^+(p,q)) is the spin group \mathrm{Spin}(p,q), which provides a double cover via the canonical projection.^[30] For instance, in the low-dimensional Lorentzian case, \mathrm{Spin}(3,1) \cong \mathrm{SL}(2,\mathbb{C}) double covers SO^+(3,1).^[30]

Fundamental Group and Homotopy

The identity component of the indefinite orthogonal group, denoted SO⁺(p,q) or O⁺(p,q), has fundamental group isomorphic to the product π₁(SO(p)) × π₁(SO(q)). This structure arises from the homotopy equivalence of SO⁺(p,q) to its maximal compact subgroup SO(p) × SO(q).^[1] For p, q ≥ 3, this yields π₁(SO⁺(p,q)) ≅ ℤ₂ × ℤ₂, while in lower dimensions such as p = q = 2, it is ℤ × ℤ. The nontrivial fundamental group relates to spin structures on manifolds equipped with indefinite metrics, where loops in SO⁺(p,q) correspond to obstructions for lifting to the spin bundle, often resolved via the double cover Spin(p,q). Higher homotopy groups of SO⁺(p,q) are given by the product π_k(SO(p)) × π_k(SO(q)) for all k ≥ 1, reflecting the deformation retract onto the maximal compact subgroup. In the stable range, specifically for k ≥ max(p,q), these are the product of the stable homotopy groups of SO(p) and SO(q).^[1] The stable homotopy groups of the orthogonal groups obey Bott periodicity with period 8: π_{k+8}(O) ≅ π_k(O) in the stable regime, and this periodicity holds for the stable homotopy groups of the compact factors SO(p) and SO(q). The double cover Spin(p,q) → SO⁺(p,q) provides the universal cover in cases where π₁(SO⁺(p,q)) ≅ ℤ₂, rendering Spin(p,q) simply connected for p + q ≥ 3 except specific low-dimensional exceptions like (2,2). For p, q > 2, Spin(p,q) has π₁ ≅ ℤ₂, corresponding to the remaining generator after covering one ℤ₂ factor. Compared to the definite orthogonal group O(n), the indefinite O(p,q) differs in its low-dimensional homotopy due to the noncompact topology, which deformation retracts to the compact SO(p) × SO(q) rather than SO(n).

Special Cases and Variants

Split Orthogonal Groups

The split orthogonal group \mathrm{SO}(n,n) is defined as the connected component of the identity in the indefinite orthogonal group \mathrm{O}(n,n), consisting of $2n \times 2n real matrices preserving the quadratic form of signature (n,n), typically represented by the diagonal matrix \operatorname{diag}(I_n, -I_n). Equivalently, it preserves the split quadratic form \sum_{i=1}^n x_i y_i on \mathbb{R}^{2n}, and is isomorphic to the group of matrices A \in \mathrm{GL}(2n, \mathbb{R}) satisfying A \begin{pmatrix} 0 & I_n \\ I_n & 0 \end{pmatrix} A^T = \begin{pmatrix} 0 & I_n \\ I_n & 0 \end{pmatrix}. This group realizes the split real form of the complex special orthogonal group \mathrm{SO}(2n, \mathbb{C}), characterized by maximal non-compactness among real forms of type D_n, with its maximal compact subgroup K = \mathrm{SO}(n) \times \mathrm{SO}(n) having dimension n(n-1) while the total dimension is n(2n-1).^[31]^[32]^[33] The Lie algebra \mathfrak{so}(n,n) admits a Cartan decomposition \mathfrak{so}(n,n) = \mathfrak{k} \oplus \mathfrak{p}, where \mathfrak{k} = \mathfrak{so}(n) \oplus \mathfrak{so}(n) is the Lie algebra of the maximal compact subgroup, and \mathfrak{p} consists of block-off-diagonal matrices of the form \begin{pmatrix} 0 & S \\ S^T & 0 \end{pmatrix} with S an arbitrary n \times n real matrix, of dimension n^2. This decomposition arises from the Cartan involution \theta(X) = -J X^T J, where J = \operatorname{diag}(I_n, -I_n), making the Killing form negative definite on \mathfrak{k} and positive definite on \mathfrak{p}.^[34]^[35]^[36] An Iwasawa decomposition of \mathrm{SO}(n,n) expresses every element uniquely as a product g = k a n with k \in K = \mathrm{SO}(n) \times \mathrm{SO}(n), a \in A a maximal split torus consisting of diagonal matrices with positive entries along the hyperbolic directions (isomorphic to (\mathbb{R}_+)^n), and n \in N an element of the unipotent radical, corresponding to strictly upper triangular matrices in a Chevalley basis adapted to the root system. This decomposition generalizes the Gram-Schmidt orthogonalization process and facilitates harmonic analysis on the group, with A abelian and exponentially embedded in \mathfrak{p}, and N nilpotent with Lie algebra generated by positive root spaces. The decomposition is unique and provides a diffeomorphism N \times A \times K \to \mathrm{SO}(n,n), useful for parametrizing representations and computing volumes of fundamental domains.^[35]^[37]^[38] Arithmetic subgroups of \mathrm{SO}(n,n), such as \mathrm{SO}(n,n; \mathbb{Z}) defined over the integer lattice preserving the form, are discrete subgroups commensurable with the integer points and act properly discontinuously on the symmetric space \mathrm{SL}(n, \mathbb{R})/\mathrm{SO}(n), yielding finite-volume locally symmetric spaces as quotients. These subgroups are analogous to \mathrm{SL}(n, \mathbb{Z}) in their action on the space of lattices up to rotation, and play a role in arithmetic geometry, particularly in the construction of Shimura varieties for orthogonal groups where additional Hodge structures are imposed, such as in the classification of polarized abelian varieties or K3 surfaces via period domains. Congruence subgroups, containing principal level-N kernels, ensure the quotients have good reduction properties modulo primes.^[39]^[40]^[41] In relation to other split classical groups, \mathrm{SO}(n,n) shares structural similarities with \mathrm{Sp}(2n, \mathbb{R}), both being split real forms of semisimple Lie groups of rank n, and in certain low-dimensional or projective settings, \mathrm{SO}(n,n) is isogenic to the projective symplectic group \mathrm{[PSp](/page/PSP)}(2n, \mathbb{R}), leading to analogous spaces of maximal representations and local systems. This connection highlights their common role in the study of Higgs bundles and character varieties. More generally, split orthogonal groups arise as \mathrm{O}(p,q) with p \approx q, balancing positive and negative eigenvalues to achieve the split torus.^[42]^[43]

Indefinite Unitary Groups

The indefinite unitary groups provide the complex counterparts to the indefinite orthogonal groups, extending the preservation of quadratic forms to sesquilinear Hermitian forms of indefinite signature over the complex numbers. These groups arise naturally in the study of complex symmetric spaces and holomorphic realizations of non-compact Lie groups.^[1] The indefinite unitary group U(p,q), where p + q = n, consists of all invertible complex n \times n matrices g \in \mathrm{GL}(n, \mathbb{C}) satisfying g^* \eta g = \eta, with \eta = \mathrm{diag}(I_p, -I_q) the diagonal matrix defining the Hermitian form of signature (p,q), and g^* denoting the conjugate transpose (adjoint with respect to complex conjugation).^[1] This condition ensures that U(p,q) preserves the indefinite sesquilinear form associated with \eta. The corresponding Lie algebra \mathfrak{u}(p,q) comprises the complex matrices X such that X^* \eta + \eta X = 0, capturing the infinitesimal generators of the group action.^[44] Unlike the definite case U(n,0) = U(n), which is compact, U(p,q) with p,q \geq 1 is non-compact. Regarding connected components, U(p,q) is connected, analogous to the connected component \mathrm{SO}^+(p,q) in the real orthogonal setting but without the disconnection arising from sign changes in the real case.^[44] A prominent low-dimensional example is U(1,1), whose special subgroup \mathrm{SU}(1,1) = \{ g \in U(1,1) \mid \det g = 1 \} is isomorphic to \mathrm{SL}(2,\mathbb{R}).^[45] This isomorphism highlights deep connections between complex and real Lie groups, and \mathrm{SU}(1,1) is particularly relevant in quantum mechanics, where it generates transformations for squeezed coherent states in quantum optics and describes dynamics in systems like the quantum Rabi model.^[46] Holomorphic realizations of U(p,q) involve its transitive action on bounded symmetric domains, such as the Siegel domain of type I consisting of p \times q complex matrices Z satisfying I_q - Z^* Z \succ 0 (positive definite), or more generally on tube domains in the context of Hermitian symmetric spaces.^[47] These domains provide models for the symmetric space U(p,q) / U(p) \times U(q), with the maximal compact subgroup U(p) \times U(q) acting as the isotropy group at the origin.^[44]

Subgroups and Representations

Maximal Compact Subgroups

The maximal compact subgroup K of the indefinite orthogonal group O(p,q) is isomorphic to O(p) \times O(q), realized via the block-diagonal embedding with respect to the orthogonal decomposition \mathbb{R}^{p+q} = V^+ \oplus V^-, where V^+ and V^- are the positive-definite and negative-definite eigenspaces of the quadratic form, respectively.^[48] This embedding preserves the quadratic form and ensures compactness, as each factor acts on its definite subspace.^[48] Any compact subgroup of O(p,q) is conjugate to a subgroup of this K, establishing its maximality.^[48] The Lie algebra \mathfrak{k} of K is the direct sum of the Lie algebras of O(p) and O(q), with dimension

\dim \mathfrak{k} = \frac{p(p-1)}{2} + \frac{q(q-1)}{2}.

^[48] The Lie algebra \mathfrak{g} = \mathfrak{so}(p,q) of O(p,q) decomposes via the Cartan decomposition \mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}, where \mathfrak{p} is the orthogonal complement to \mathfrak{k} with respect to the Killing form.^[10] This decomposition arises from the Cartan involution \sigma: X \mapsto -X^T on \mathfrak{g}, with \mathfrak{k} as the +1-eigenspace and \mathfrak{p} as the -1-eigenspace; it satisfies the relations

[\mathfrak{k}, \mathfrak{k}] \subset \mathfrak{k}, \quad [\mathfrak{k}, \mathfrak{p}] \subset \mathfrak{p}, \quad [\mathfrak{p}, \mathfrak{p}] \subset \mathfrak{k}.

^[10] The homogeneous space O(p,q)/K is a non-compact Riemannian symmetric space equipped with an invariant metric induced from the Killing form, possessing non-positive sectional curvature.^[49] This structure reflects the semisimple nature of the group and underpins applications in geometry and representation theory.^[49] Furthermore, K stabilizes the light cone \{ x \in \mathbb{R}^{p+q} \mid \langle x, x \rangle = 0 \}, the set of null directions preserved by the full group action.^[50]

Irreducible Representations

The irreducible representations of the indefinite orthogonal group O(p,q) encompass both finite-dimensional and infinite-dimensional cases, with the latter being unitary due to the non-compact nature of the group. The finite-dimensional irreducible representations coincide with those of the complex orthogonal group O(n,\mathbb{C}) where n = p + q, and are classified by dominant weights \lambda = (\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_m \geq 0) in the lattice of integral weights, subject to the condition that the representation is orthogonal (i.e., the highest weight satisfies \langle \lambda, \alpha^\vee \rangle \in \mathbb{Z} for simple coroots \alpha^\vee, and no half-integer spins unless including the spin group). These representations are non-unitary and can be realized as tensor powers of the standard representation with symmetrizers and antisymmetrizers, excluding those with invariant odd-degree factors under the center.^[51] The infinite-dimensional irreducible unitary representations form the unitary dual of O(p,q), classified within the Langlands framework for real reductive groups via parameters consisting of a tempered representation of the Langlands dual and a discrete choice for the cohomology degree. Seminal contributions by Vogan provide the algebraic structure through Harish-Chandra modules and associated varieties, determining unitarizability by conditions on the infinitesimal character and (\mathfrak{g},K)-cohomology, where K \cong O(p) \times O(q) is the maximal compact subgroup. The classification proceeds by inducing from parabolic subgroups, with explicit descriptions available for low-rank cases like O(2,1) \cong \mathrm{PSL}(2,\mathbb{R}) and O(3,1) (the Lorentz group), where representations are labeled by principal quantum numbers and helicities. For general p,q, the unitary dual remains partially explicit, relying on computational tools for higher ranks.^[51]^[52] Key classes of unitary irreducible representations include principal series, induced from characters of the minimal parabolic subgroup (Borel subgroup), which are unitary and irreducible for generic parameters; complementary series, filling gaps between principal series via analytic continuation for non-tempered parameters; and discrete series, existing when O(p,q) admits a compact Cartan subgroup (i.e., if at least one of p or q is even), realized on Hilbert spaces of square-integrable sections over the flag variety. No discrete series exist when both p and q are odd (e.g., O(2k+1,1) for k \geq 1), where representations are limits of discrete series or principal series.^[52] A distinguished example is the minimal unitary representation, unique up to isomorphism for p+q even and \geq 6 (with p,q \geq 2), realized on L^2 of a light cone in \mathbb{R}^{p+q-2} via a Schrödinger-like model, with Gelfand-Kirillov dimension p+q-3; it lacks a minimal K-type and arises in theta correspondences and conformal geometry, but does not exist when p+q is odd.^[53] Branching laws and restrictions to subgroups, such as from O(p,q) to O(p-1,q), further elucidate the structure, often yielding multiplicity-free decompositions for minimal representations into sums of irreducibles with specified highest weights. These representations underpin applications in quantum field theory, particularly for the Lorentz group O(3,1), where physical particles correspond to induced representations with Casimir eigenvalues matching masses and spins.^[54]

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