Fact-checked by Grok 2 weeks ago

Induced representation

In representation theory, an induced representation is a fundamental construction that extends a linear representation of a subgroup H of a finite or compact group G to a representation of the entire group G, typically by tensoring the group algebra of G over that of H with the original representation space.^[1] Specifically, if \phi: H \to \mathrm{GL}(W) is a representation on a vector space W, the induced representation \mathrm{Ind}_H^G(\phi) acts on the space V = \mathbb{C}[G] \otimes_{\mathbb{C}[H]} W, where the action of g \in G is given by g \cdot ( \sum a_i \otimes w_i ) = \sum g a_i \otimes w_i.^[2] This process can also be viewed categorically as the left adjoint functor to the restriction functor from G-representations to H-representations, with the right adjoint known as the co-induced representation, which coincides with the induced one when [G:H] is finite.^[3] Induced representations play a central role in decomposing representations of larger groups from smaller ones, enabling the study of irreducible representations through induction from subgroups.^[1] A key property is Frobenius reciprocity, which establishes a bijection between G-equivariant homomorphisms from the induced representation \mathrm{Ind}_H^G(W) to another G-representation U and H-equivariant homomorphisms from W to the restriction of U to H, formalized as \mathrm{Hom}_G(\mathrm{Ind}_H^G W, U) \cong \mathrm{Hom}_H(W, \mathrm{Res}_H^G U).^[3] This reciprocity extends to characters, where the inner product of characters satisfies \langle \mathrm{Ind}_H^G \chi, \psi \rangle_G = \langle \chi, \mathrm{Res}_H^G \psi \rangle_H for characters \chi of H and \psi of G.^[2] The character of an induced representation admits an explicit formula: for g \in G, \chi_{\mathrm{Ind}_H^G \phi}(g) = \sum_{\{r \in R \mid r^{-1} g r \in H\}} \chi_\phi(r^{-1} g r), where R is a set of coset representatives for G/H and \chi_\phi is the character of \phi, counting the number of fixed points in the induced permutation action adjusted by the subgroup representation.^[2] Notable theorems highlight their generative power; for instance, Artin's induction theorem states that the virtual characters of G form a lattice generated by inductions from cyclic subgroups, while Brauer's theorem shows they are spanned by inductions from p-elementary abelian subgroups for each prime p.^[1] In practice, inducing the trivial representation of H yields the permutation representation on the cosets G/H, which decomposes into irreducibles reflecting the group's action on the coset space.^[3] These tools underpin applications in character theory, modular representations, and broader areas like Lie groups and quantum field theory.^[4]

Introduction

Definition

In the context of representation theory over a field k, an induced representation arises from a subgroup H of a group G and a representation \sigma: H \to \mathrm{GL}(V) of H on a finite-dimensional vector space V over k. The induced representation \mathrm{Ind}_H^G(\sigma) is defined on the vector space W consisting of all functions f: G \to V that satisfy the transformation property f(gh) = \sigma(h^{-1}) f(g) for all g \in G and h \in H; this space W is equipped with the G-action given by (\mathrm{Ind}_H^G(\sigma)(k) f)(g) = f(g k^{-1}) for all k, g \in G.^[5]^[6] Equivalently, in the category of representations (or modules over the group algebra), W \cong k[G] \otimes_{k[H]} V as k[G]-modules, where the tensor product identifies the H-action on V with the left regular action on k[G]. If the index [G : H] is finite, the dimension satisfies \dim_k W = [G : H] \cdot \dim_k V.^[5]^[6] This construction motivates induced representations as a mechanism to extend a representation of H to one of G by effectively setting it to "zero" outside the left cosets of H in G, while ensuring compatibility with the group action across cosets.^[5]

Historical Development

The concept of induced representations traces its origins to the work of Ferdinand Georg Frobenius in the late 19th century, amid efforts to understand the structure of finite group representations, particularly for symmetric groups. In his 1896 paper "Über Gruppencharaktere," Frobenius introduced the notion of characters for finite groups, laying the groundwork for induction by considering how characters behave under group actions. He further developed this in 1897 with "Über die Darstellung der endlichen Gruppen durch lineare Substitutionen," where he explicitly formulated the induction of characters from a subgroup to the full group, applying it to decompose representations of symmetric groups and resolve key problems in their classification.^[7] Building on Frobenius's foundations, Issai Schur advanced the theory in the early 1900s by shifting focus from characters to complete representations. In his 1901 doctoral thesis under Frobenius and subsequent publications, including the two-part "Untersuchungen über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen" (1904 and 1907), Schur extended induction to full linear representations of finite groups over the complex numbers. He proved fundamental decomposition theorems, showing how induced representations decompose into irreducibles, and established orthogonality relations that underpin much of modern character theory.^[8]^[9] The mid-20th century saw a significant generalization to infinite-dimensional settings with George W. Mackey's contributions in 1951–1952. In "Induced Representations of Groups" (1951), Mackey began adapting induction to arbitrary groups, followed by his seminal "Induced Representations of Locally Compact Groups" (1952), which defined unitary induced representations for locally compact groups using measure-theoretic constructions. This work introduced the Mackey–Bargmann category, a framework categorizing induced representations and their equivalences, enabling systematic study of unitary representations beyond finite cases.^[10]^[11] In the 1950s and 1960s, Harish-Chandra extended these ideas to continuous groups, particularly semisimple Lie groups, through parabolic induction. In a series of papers starting with "Representations of Semisimple Lie Groups" (1951) and culminating in works on discrete and principal series representations (1950s–1960s), Harish-Chandra constructed irreducible unitary representations by inducing from parabolic subgroups, linking this process to the geometry of flag varieties and the analytic continuation of characters. This approach proved instrumental in developing the theory of automorphic forms, bridging representation theory with number theory.^[12]^[13] Key milestones include Frobenius's 1896 and 1897 papers in the Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften, which initiated induction for finite groups, and Mackey's 1952 paper in the Annals of Mathematics, which broadened the scope to locally compact groups.^[14]^[15]

Algebraic Constructions

General Construction

In the algebraic setting for finite groups, the induced representation \operatorname{Ind}_H^G(\sigma) of a representation \sigma: H \to \mathrm{GL}_k(V) of a subgroup H of a finite group G over a field k is constructed module-theoretically as follows. Let T be a set of representatives for the left cosets G/H. The underlying vector space is the direct sum

W = \bigoplus_{t \in T} V_t,

where each V_t is a copy of V. The group G acts on W by permuting the summands according to left multiplication on the cosets, with a twist by \sigma to ensure compatibility with the H-action on the fixed coset H. Specifically, for g \in G and v \in V_t, the coset tH maps to g t H = s H for unique s \in T and h \in H (so g t = s h), and

g \cdot v = \sigma(h^{-1}) v \in V_s.

This defines a kG-module structure on W, and \dim_k W = [G:H] \cdot \dim_k V.^[16] To describe the action explicitly on a basis, let \{v_1, \dots, v_d\} be a basis for V. A basis for W consists of the elements \{v_j^t \mid t \in T, \, 1 \leq j \leq d\}, where v_j^t denotes v_j placed in the summand V_t. For g \in G, if g t = s h with s \in T and h \in H, then

\pi(g) (v_j^t) = \sum_{i=1}^d a_{ij} v_i^s,

where the coefficients a_{ij} are determined by the matrix entries of \sigma(h^{-1}), i.e., \sigma(h^{-1}) v_j = \sum_i a_{ij} v_i. This permutation-twist action ensures the representation is well-defined independent of the choice of coset representatives.^[16] A key special case arises when \sigma is the trivial representation $1_H of H on V = k. Here, \operatorname{Ind}_H^G(1_H) is the permutation representation of G on the coset space G/H, with basis \{e_t \mid t \in T\} and action \pi(g) e_t = e_s, where s H = g t H. This representation has dimension [G:H] and corresponds to the action of G by left multiplication on the set of cosets.^[17] For finite groups, the induced representation coincides with the coinduced representation \operatorname{Coind}_H^G(\sigma), as the two constructions yield isomorphic modules. The functors \operatorname{Ind}_H^G and \operatorname{Res}_G^H form an adjoint pair, with the natural isomorphism

\operatorname{Hom}_G(\operatorname{Ind}_H^G(\sigma), \tau) \cong \operatorname{Hom}_H(\sigma, \operatorname{Res}_G^H(\tau))

for any G-representation \tau; this adjunction underpins Frobenius reciprocity without requiring a proof here. The explicit coset construction facilitates computations in character theory, tracing back to Frobenius' original development for induced characters.^[18]

Properties

Induced representations possess several key intrinsic properties in the algebraic setting for finite groups. One fundamental property is the transitivity of induction: for subgroups K \leq H \leq G, the induced representation satisfies \operatorname{Ind}_H^G(\operatorname{Ind}_K^H(\tau)) \cong \operatorname{Ind}_K^G(\tau), where \tau is a representation of K.^[19] This allows for iterative construction of representations across subgroup chains without altering the overall structure.^[19] The dimension of an induced representation \operatorname{Ind}_H^G(\sigma) from a representation \sigma of subgroup H in finite group G is given by \dim(\operatorname{Ind}_H^G(\sigma)) = [G:H] \cdot \dim(\sigma), reflecting the scaling by the index of the subgroup.^[19] Regarding decomposition, the multiplicity of an irreducible representation \tau in \operatorname{Ind}_H^G(\sigma) equals the multiplicity of \sigma in the restriction \operatorname{Res}_H^G(\tau), a property that follows from Frobenius reciprocity.^[19] Representations induced from one-dimensional representations are monomial, meaning they admit a basis permuted by the group action up to scalar multiples.^[20] Conversely, every monomial representation of a finite group decomposes as a direct sum of such induced one-dimensional representations.^[20] Brauer's induction theorem further characterizes the character ring: every irreducible character of G is an integer linear combination of characters induced from linear characters of the elementary subgroups of G, where an elementary subgroup is p-elementary for some prime p (a direct product of a p-group and a cyclic p'-group).^[21] Irreducibility of induced representations is governed by Mackey's criterion: for an irreducible representation \sigma of H, \operatorname{Ind}_H^G(\sigma) is irreducible if and only if, for every g \in G \setminus H, the inner product \langle \sigma, {}^g\sigma \rangle_{H \cap {}^g H} = 0, where {}^g\sigma(h) = \sigma(g^{-1} h g) restricted appropriately.^[20] This condition ensures no nontrivial intertwiners arise from conjugate actions on intersections. Specific examples illustrate these properties. In the symmetric group S_{n+1}, inducing the trivial representation from the subgroup S_n yields the permutation representation on cosets, which decomposes as the direct sum of the trivial representation and the irreducible standard representation of dimension n.^[20] For dihedral groups D_{2n}, inducing a one-dimensional representation from the cyclic rotation subgroup \langle r \rangle produces two-dimensional irreducible representations, demonstrating monomial structure and irreducibility under Mackey's criterion when the character values satisfy the necessary orthogonality.^[20]

Other Constructions

Analytic

In the analytic setting, the induced representation construction extends the algebraic notion to unitary representations of locally compact groups on Hilbert spaces, emphasizing measurable functions and integration with respect to quasi-invariant measures. For a locally compact group G with closed subgroup H, and a unitary representation \pi of H on a Hilbert space V, Mackey's analytic induction defines the induced representation \operatorname{Ind}_H^G(\pi) when G and H are unimodular, ensuring the existence of a G-invariant measure on G/H. The underlying Hilbert space consists of measurable functions \phi: G \to V satisfying the covariance condition \phi(gh) = \pi(h^{-1}) \phi(g) for all g \in G and h \in H, with the L^2-norm \|\phi\|^2 = \int_{G/H} \|\phi(g)\|_V^2 \, d\mu(gH) < \infty, where \mu is the invariant measure on G/H.^[22]^[23] The group G acts on this space by right translation, adjusted for unitarity: (\operatorname{Ind}_H^G(\pi)(k) \phi)(g) = \phi(k^{-1} g) for k \in G, which preserves the covariance and the inner product due to the invariance of \mu. For non-unimodular cases, the construction incorporates the modular function \Delta_G of G, modifying the action to (\operatorname{Ind}_H^G(\pi)(k) \phi)(g) = \Delta_G(k)^{1/2} \phi(k^{-1} g) to ensure unitarity, while the covariance twist remains \phi(gh) = \pi(h^{-1}) \Delta_H(h)^{-1/2} \phi(g) to account for the relative modular homomorphism \Delta_G|_H / \Delta_H. If \pi is unitary, then \operatorname{Ind}_H^G(\pi), realized as the completion of smooth compactly supported sections satisfying the covariance, is also unitary.^[22] In the special case of compact groups, Bargmann's version realizes the induced representation through a direct integral decomposition over the dual space, leveraging the Peter-Weyl theorem to express it as an integral of irreducible components weighted by multiplicities determined by the inducing representation.^[23]^[22]

Geometric

The induced representation \operatorname{Ind}_H^G(\sigma) of a representation \sigma: H \to \mathrm{GL}(V) of a subgroup H of a group G admits a geometric realization as the space of sections of the associated vector bundle E = G \times_H V over the homogeneous space G/H.^[24] This bundle is constructed by taking the quotient of G \times V under the equivalence relation (g, v) \sim (gh, \sigma(h^{-1})v) for h \in H, with the projection \pi: E \to G/H given by (g, v) \mapsto gH.^[4] The space of sections \Gamma(E) consists of maps s: G/H \to E such that \pi \circ s = \mathrm{id}_{G/H}, which can be identified with H-equivariant functions f: G \to V satisfying f(gh) = \sigma(h^{-1}) f(g) for all g \in G, h \in H.^[24] The group G acts on E by left translation, defined by g_0 \cdot (g, v) = (g_0 g, v), which descends to an action on G/H and induces a representation on \Gamma(E) via (g_0 \cdot f)(x) = f(g_0^{-1} x) for x \in G/H.^[25] This action makes \Gamma(E) the representation space for \operatorname{Ind}_H^G(\sigma), providing a transitive G-action on the base space that reflects the homogeneous structure.^[24] In the context of Lie groups, the bundle E is a homogeneous vector bundle over the manifold G/H, where smoothness ensures that sections are smooth maps compatible with the bundle structure.^[25] A representative example arises with line bundles induced from characters: for a semisimple Lie group G with Borel subgroup B containing a maximal torus T, and a one-dimensional representation of B given by a character \chi of T, the induced bundle G \times_B \mathbb{C}_\chi over the flag variety G/B has global sections realizing \operatorname{Ind}_B^G(\mathbb{C}_\chi), which by the Borel–Weil theorem is the irreducible representation of highest weight \chi if \chi is dominant integral (and zero otherwise).^[26] Induction can also be viewed geometrically as the pushforward \pi_*(\sigma) of the sheaf associated to \sigma under the projection \pi: G \to G/H, where for smooth cases this involves differential forms or connections on the bundle to ensure compatibility with the geometry.^[24] In sheaf cohomology settings, induced representations appear as cohomology groups of such pushforward sheaves on homogeneous spaces, linking algebraic induction to geometric invariants.^[27]

Key Theorems

Frobenius Reciprocity

Frobenius reciprocity is a fundamental theorem in representation theory that establishes an adjunction between the induction and restriction functors for representations of finite groups. For a finite group G and a subgroup H \leq G, let \sigma be a representation of H and \tau a representation of G. The theorem states that there is a natural isomorphism

\Hom_G(\Ind_H^G(\sigma), \tau) \cong \Hom_H(\sigma, \Res_H^G(\tau)).

This isomorphism can be constructed explicitly: given a G-equivariant map \phi: \Ind_H^G(\sigma) \to \tau, the corresponding H-equivariant map \sigma \to \Res_H^G(\tau) is the restriction of \phi to the copy of \sigma at the identity coset, which is automatically H-equivariant due to the G-equivariance of \phi; conversely, given an H-equivariant map \psi: \sigma \to \Res_H^G(\tau), it extends to a G-equivariant map \Ind_H^G(\sigma) \to \tau using the universal property of the induced representation.^[28] A proof outline proceeds via the categorical adjunction: the induced representation is given by \Ind_H^G(\sigma) = \mathbb{C}[G] \otimes_{\mathbb{C}[H]} \sigma, and the Hom-Tensor adjunction yields the isomorphism directly from the universal property of tensor products over rings. Alternatively, in the character-theoretic formulation for complex representations, the theorem equates the inner products of characters:

\langle \Ind_H^G(\chi_\sigma), \chi_\tau \rangle_G = \langle \chi_\sigma, \Res_H^G(\chi_\tau) \rangle_H,

where \langle \cdot, \cdot \rangle_K denotes the standard inner product on class functions for group K. This version follows from the orthogonality of irreducible characters and the formula for the induced character, providing a multiplicity interpretation: the multiplicity of an irreducible \tau in \Ind_H^G(\sigma) equals the multiplicity of \sigma in \Res_H^G(\tau).^[29] The theorem extends to compact groups, where representations are direct sums of finite-dimensional irreducibles by the Peter-Weyl theorem, allowing a similar decomposition and reciprocity for unitary representations. For a compact group G and closed subgroup H, the induced representation \Ind_H^G(\sigma) decomposes discretely, and the Hom isomorphism holds via integration over cosets in place of summation, leveraging the Peter-Weyl completeness for matrix coefficients.^[30] Frobenius reciprocity has significant implications, such as determining branching rules for representations of symmetric groups S_n restricted to S_{n-1}, where the multiplicity of an irreducible in the restriction equals its multiplicity in the induction of the dual branching. It also facilitates the decomposition of tensor products of representations by relating them to inductions from wreath product subgroups. The theorem originated in the work of Georg Frobenius on character theory of finite groups in 1896.^[31]^[29]

Clifford Theory

Clifford's theorem, established in 1937, describes the behavior of irreducible representations of a finite group G when restricted to a normal subgroup N \triangleleft G. Specifically, if \rho is an irreducible representation of G, then the restriction \operatorname{Res}_N^G \rho decomposes as a direct sum \bigoplus_{i=1}^r e \cdot \sigma_i, where each \sigma_i is a distinct irreducible representation of N, all \sigma_i have the same dimension, e is a positive integer denoting the common multiplicity, and r is the number of distinct conjugates of \sigma_1 under the action of G by conjugation. The group G acts transitively by conjugation on the set \{\sigma_1, \dots, \sigma_r\}, permuting the corresponding isotypic components e \cdot \sigma_i of the decomposition.^[32]^[33] In the context of induced representations, the theorem implies a precise block structure for \operatorname{Ind}_N^G \sigma when \sigma is an irreducible representation of N. Let I = I_G(\sigma) = \{ g \in G \mid {}^g \sigma \cong \sigma \} be the inertia subgroup of \sigma, which contains N and has index equal to the size of the G-orbit of \sigma. Then \operatorname{Ind}_N^G \sigma decomposes as a direct sum \bigoplus_{\psi} m_{\psi} \cdot \operatorname{Ind}_I^G \psi, where the sum runs over a set of irreducible representations \psi of I such that \operatorname{Res}_N^I \psi = m_{\psi} \cdot \sigma (isotypic of type \sigma), and each m_{\psi} is the multiplicity. The irreducible constituents of \operatorname{Ind}_N^G \sigma are precisely the \operatorname{Ind}_I^G \psi for such \psi, each appearing with multiplicity m_{\psi}. Moreover, the restriction \operatorname{Res}_N^G (\operatorname{Ind}_I^G \psi) is isotypic of type \sigma with multiplicity m_{\psi} \cdot [G : I].^[34]^[33] A key corollary establishes a Galois correspondence for the blocks of representations over normal subgroups. There is a bijection between the irreducible representations of G that contain \sigma in their restriction to N and the irreducible representations of the inertia subgroup I that are isotypic of type \sigma upon restriction to N, given by induction from I to G. This correspondence preserves the block structure and extends to chains of normal subgroups, mirroring the Galois theory of field extensions by relating orbits of characters under conjugation to subfields. Another corollary states that if \sigma is G-invariant (i.e., I = G), then \operatorname{Ind}_N^G \sigma contains irreducible constituents that extend \sigma, and the entire induced representation restricts isotypically to multiples of \sigma.^[33]^[34] The proof of Clifford's theorem relies on iterative application of Frobenius reciprocity, which equates the inner product \langle \operatorname{Res}_N^G \rho, \tau \rangle_N = \langle \rho, \operatorname{Ind}_N^G \tau \rangle_G for representations \rho of G and \tau of N, to analyze the multiplicities and conjugacy. Schur's lemma is then applied to the endomorphism algebra of the isotypic components, showing that the action of G on these components is equivalent to the regular representation of the quotient G / I, ensuring the transitive permutation and equal dimensions. This framework also justifies the block decomposition of the induced representation by identifying the minimal subrepresentations stabilized by the inertia group.^[32]^[33] Inertial subgroups play a central role in constructing the full set of irreducible representations of G from those of N. Starting from an irreducible \sigma of N, the inertia group I captures the stabilizer of \sigma under G-conjugation, and the irreducible representations of G "lying over" \sigma are obtained by inducing irreducible extensions (or projective extensions in modular cases) from I that restrict isotypically to \sigma. This process allows recursive construction of all irreducibles via chains of normal subgroups, reducing the representation theory of G to that of smaller inertia groups and quotients. For instance, if \sigma extends to an irreducible \psi of I, then \operatorname{Ind}_I^G \psi is irreducible and lies over \sigma.^[34]^[33]

Applications

Systems of Imprimitivity

A system of imprimitivity for a unitary representation U of a separable locally compact group G on a Hilbert space \mathcal{H} consists of a separable locally compact space M (the base space) and a projection-valued measure P that maps Borel subsets E \subseteq M to projections P_E in \mathcal{H}, satisfying the covariance relation U_g P_E U_g^{-1} = P_{g \cdot E} for all g \in G and Borel sets E, where P_M = I and the projections take values other than 0 or I except trivially.^[22] The system is transitive if the action of G on M is transitive, meaning G acts transitively via g \cdot E = \{ g \cdot m \mid m \in E \} for points m \in M. In Mackey's framework, such systems generalize the notion of imprimitive actions from finite group representations to the continuous setting, capturing decompositions of the representation space into G-equivalent "blocks" stabilized by a subgroup.^[35] Mackey's imprimitivity theorem establishes a fundamental equivalence: a transitive unitary representation \pi of G admits a system of imprimitivity with base space G/H (where H is a closed subgroup of G stabilizing a point in G/H) if and only if \pi is unitarily equivalent to the induced representation \operatorname{Ind}_H^G(\sigma) for some unitary representation \sigma of H.^[22] Specifically, if (U, P) is such a transitive system with stabilizer H, then (U, P) is unitarily equivalent to the pair generated by \sigma on the Hilbert space of \sigma and a quasi-invariant measure on G/H, with equivalence classes determined by unitary equivalence of the \sigma's.^[36] This theorem, originally formulated for locally compact groups, provides a bijection between transitive systems of imprimitivity (up to equivalence) and unitary representations of the stabilizer subgroup H.^[35] The construction linking systems of imprimitivity to induced representations proceeds via double cosets and intertwining operators. For \operatorname{Ind}_H^G(\sigma), the underlying Hilbert space consists of measurable sections over G/H transforming under \sigma along cosets, and the imprimitivity projections P_{gH} correspond to the characteristic functions of the cosets gH, which are G-equivalent under left translation.^[22] Intertwining operators between two such induced representations exist if and only if the original \sigma's are equivalent, often constructed by integrating over double cosets H \backslash G / H with a suitable kernel derived from the subgroup representations. This equivalence ensures that any transitive system arises canonically from an induced representation, with the blocks of imprimitivity precisely the orbits (cosets) under the transitive action.^[36] A canonical example is the quasi-regular representation on L^2(G/H), where G acts by left translation on the cosets G/H, inducing the regular representation of H (the trivial representation \sigma = 1). Here, the imprimitivity blocks are the individual cosets gH, each stabilized by the right action of H, and the system is transitive since G acts transitively on G/H. This construction appears in applications like the representation theory of semi-direct products, such as the Poincaré group, where orbits in the dual space yield imprimitivity systems over homogeneous spaces.^[22]^[35]

Representations of Lie Groups

In the representation theory of semisimple Lie groups, parabolic induction provides a fundamental method for constructing irreducible unitary representations from those of parabolic subgroups. For a semisimple Lie group G with a parabolic subgroup P = MAN in its Langlands decomposition—where M is the Levi factor, A is the vector part of the abelian factor, and N is the unipotent radical—the induced representation \operatorname{Ind}_P^G(\delta) is formed by tensoring a representation \delta = \sigma \otimes e^{\nu} \otimes 1 of M, A, and N respectively, where \sigma is a finite-dimensional representation of M, \nu is a character of A, and the trivial representation on N. This construction yields the principal series representations when P is a minimal parabolic subgroup, and more generally, it parametrizes tempered representations central to the Plancherel decomposition of L^2(G).^[37] Harish-Chandra realized these induced representations analytically as the space of smooth vectors in L^2(G), where the action of G is by right translation, ensuring compatibility with the Lie algebra \mathfrak{g} and a maximal compact subgroup K. The underlying Harish-Chandra module consists of K-finite vectors that are smooth under the \mathfrak{g}-action, and the center of the universal enveloping algebra \mathcal{Z}(\mathfrak{g}) acts via the Harish-Chandra isomorphism to characters determined by infinitesimal parameters. This framework allows the Bernstein center—arising from the action of the Hecke algebra on smooth representations—to distinguish blocks in the category of representations, facilitating the decomposition into supercuspidal and parabolically induced components.^[38] The Langlands classification theorem asserts that every irreducible unitary representation of a real reductive Lie group G arises as a Langlands quotient of a parabolically induced representation from a discrete series of the Levi subgroup M of a parabolic P = MAN, uniquely determined up to intertwining. Specifically, for each such representation \pi, there exists a standard module I(P, \sigma, \nu) that is the unique irreducible quotient of \operatorname{Ind}_P^G(\sigma \otimes e^{\nu} \otimes 1), with \sigma unitary and discrete series modulo center on M, and \nu in a certain positive chamber. This parametrization covers all tempered representations (when \operatorname{Re} \nu = 0) and extends to non-tempered ones, providing a complete classification without cohomological induction for real groups.^[39] A concrete example occurs for G = \mathrm{SL}(2, \mathbb{R}), where the principal series representations are obtained via Borel induction from the minimal parabolic subgroup B = AN (with M trivial), inducing characters e^{\nu} on A to yield irreducible unitaries on L^2(\mathbb{R}^\times) for \operatorname{Re} \nu = 0 and \nu \neq 0. In contrast, the discrete series representations arise from compact induction from the maximal compact subgroup K = \mathrm{SO}(2), inducing genuine characters of K to holomorphic or anti-holomorphic sections over the unit disk, which embed discretely into L^2(G) and parametrize the bottom layer of the unitary dual.^[40] To ensure unitarity of these induced representations, intertwining operators are normalized by factors involving the root data of G. For principal series, the standard intertwining operator T: \operatorname{Ind}_P^G(\delta) \to \operatorname{Ind}_{\bar{P}}^G(\bar{\delta}) (where \bar{P} is the opposite parabolic) is made unitary by multiplying by the absolute value of the Langlands-Sahlgren functional, which computes the constant term along N and equals a product over positive roots \prod_{\alpha > 0} \frac{1 - e^{-\langle \nu, \alpha^\vee \rangle}}{1 - e^{\langle \nu, \alpha^\vee \rangle}} at \operatorname{Re} \nu = 0, ensuring the operator is an isometry on K-finite vectors. This normalization is crucial for the analytic continuation and unitarity in the full unitary dual.^[41]

References

[1]
Induced Representation -- from Wolfram MathWorld
If a subgroup H of G has a group representation phi:H×W->W, then there is a unique induced representation of G on a vector space V.
[2]
[PDF] Induced Representations of Finite Groups - MIT Mathematics
Induction of Representations. References. Character Theory. Definition (Character of a representation). The character of a linear representation ρ : G → GL(V) ...
[3]
induced representation in nLab
May 28, 2025 · Over the complex numbers, the virtual representations of a finite group are all virtual combinations of induced representations of 1-dimensional ...Idea · Definition · Properties · Examples and Applications
[4]
[PDF] The Theory of Induced Representations in Field Theory
The theory of induced representations is a method of obtaining representations of a topological group starting from a representation of a subgroup. The classic ...
[5]
Representation Theory: A First Course | SpringerLink
Free delivery 14-day returnsThe primary goal of these lectures is to introduce a beginner to the finite dimensional representations of Lie groups and Lie algebras.
[6]
Linear Representations of Finite Groups - SpringerLink
Book Title: Linear Representations of Finite Groups · Authors: Jean-Pierre Serre · Series Title: Graduate Texts in Mathematics · Publisher: Springer New York, NY.
[7]
The origins of the theory of group characters | Archive for History of ...
Frobenius, G., 1897. Über die Darstellung der endlichen Gruppen durch lineare Substitutionen. S'ber. Akad. d. Wiss. Berlin, 994–1015. Google Scholar.Missing: Primfaktoren | Show results with:Primfaktoren
[8]
Frobenius, Schur, and the Berlin Algebraic Tradition - SpringerLink
Schur, I., Über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen, J. Reine Angew. Math. 127 (1904), 20–50. MATH Google Scholar.
[9]
https://zbmath.org/authors/?s=0&q=Schur%2C+Issai
[10]
A Generalization of Feit's Theorem on JSTOR
(1905), 77-91. 11. , Untersuchungen uber die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen, J. Reine Angew. Math.132 (1907), 85 ...
[11]
Induced Representations of Locally Compact Groups I - jstor
MACKEY, G. W., Induced representations of groups, Amer. J. Math. vol. 73 (1951) pp. 576-592. 21. MAUTNER, F. I., The completeness of the irreducible unitary ...
[12]
Representations of Semisimple Lie Groups: V - PMC
Harish-Chandra Representations of Semisimple Lie Groups on a Banach Space. Proc Natl Acad Sci U S A. 1951 Mar;37(3):170–173. doi: 10.1073/pnas.37.3.170 ...Missing: parabolic induction 1950s 1960s
[13]
HARISH-CHANDRA 11 OCTOBER 1923–16 OCTOBER 1983 ...
Harish-Chandra was one of the outstanding mathematicians of his generation, an alge- braist and analyst, and one of those responsible for transforming ...
[14]
[PDF] The origin of representation theory - UConn Math Department
[20] Frobenius, F. G. Über vertauschbare Matrizen. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin (1896) 601-614; also ...
[15]
Induced Representations of Locally Compact Groups II. The ... - jstor
MACKEY, G. W., Induced representations of locally compact groups I., Ann. of Math., vol. 55 (1952), pp. 101-139. 11. MAUTNER, F. I., Unitary representations ...
[16]
[PDF] m4/5p12 group representation theory mastery material: induced ...
See Chapter 9 of Serre 'Linear representations of finite groups' for the proof of this. This tells us that using induction (and allowing linear combinations of ...
[17]
[PDF] Math 210B. Induced representations and A5 character table
... representation V 0, k[G] ⊗k[G0] V 0 is described concretely as a direct sum of copies of V 0 indexed by coset representatives for G/G0. In this way, one ...
[18]
[PDF] Representations of finite groups - UBC Math
Nov 10, 2016 · An alternative definition of the induced representation ... There is an English translation, titled Linear representations of finite groups.
[19]
[PDF] Induced representations - Columbia Math Department
Induced representations construct representations of a group G from a subgroup H and a representation of H, using functions from G to W.
[20]
[PDF] Induced Representations - Alexander Rhys Duncan
Feb 27, 2023 · If V is a monomial representation of a finite group G, then. V is a direct sum of representations induced from one-dimensional represen- tations ...
[21]
[PDF] Refining Brauer's Induction Theorem
Aug 1, 2017 · A subgroup of G is called p-elementary if it is the direct product of a p-group and a cyclic p0-group. A subgroup of G is elementary if it ...
[22]
Imprimitivity for Representations of Locally Compact Groups I - PNAS
the parabolic elements of PSL2(K) do not form a single conjugate set in PSL2(K). IMPRIMITIVITY FOR REPRESENTATIONS OF LOCALLY. COMPACT GROUPS I. BY GEORGE W.
[23]
[PDF] Induced representations of locally compact groups
The purpose of this section is to make precise the relationship between our induced representation E r and the unitary induced representations of Mackey.
[24]
[PDF] Homogeneous Vector Bundles and Induced Representations 1
Spaces like this are called “homogeneous spaces”. At each point in G/H there is a subgroup of G such that left multiplication by elements in it leaves the point ...Missing: realization | Show results with:realization
[25]
[PDF] Induced representations and homogeneous vector bundles.
Let H be a subgroup of the group G, so that G acts transitively on. M:=G/H. We identified the space J(M) as the subspace of J(G).Missing: realization | Show results with:realization
[26]
[PDF] line bundles over flag varieties - UChicago Math
Aug 31, 2015 · The purpose of this paper is to state and prove the Borel–Weil Theorem. Doing so requires us to know about algebraic groups and some basic ...
[27]
[1207.6315] Localization of cohomological induction - arXiv
Jul 26, 2012 · We consider the tensor product of a D-module and a certain module associated with V, and prove that its sheaf cohomology groups are isomorphic ...
[28]
[PDF] Induced Representations and Frobenius Reciprocity
Theorem 1 (Frobenius reciprocity for finite groups). IndG. H is a left-adjoint functor to the restriction functor, coIndG. H is a right-adjoint. It turns out ...
[29]
[PDF] Introduction to representation theory by Pavel Etingof, Oleg Golberg ...
Historical interlude: Georg Frobenius's “Principle of. Horse Trade”. 79 ... So by Frobenius reciprocity, (χU|H ,χV ) = 0. This means that χU vanishes on ...<|control11|><|separator|>
[30]
A Generalization of the Frobenius Reciprocity Theorem - PNAS
consequence of the famousPeter-Weyl theory2 of compact groups. The classical Frobenius reciprocity theorem asserts in this case multiplicity of u in Mj(K) ...
[31]
[PDF] Representation Theory of Symmetric Groups - Lecture Notes
3.3.2 Branching Rule . ... This follows from Theorem 3.22 and Frobenius reciprocity noting that β ∈ α+ iff α ∈ β−.
[32]
Representations Induced in an Invariant Subgroup - jstor
REPRESENTATIONS INDUCED IN AN INVARIANT SUBGROUP. BY A. H. CLIFFORD. (Received January 19, 1937; revised April 30, 1937). The object of this paper is to ...
[33]
[PDF] Isaacs_Character_theory.pdf
' Preface. Character theory provides a powerful tool for proving theorems about finite groups. 1n fact.. there are some important results, such as Frobenius ...
[34]
(PDF) Clifford theory and applications - ResearchGate
Aug 6, 2025 · PDF | This is an introduction to Clifford theory of induced representations from normal subgroups of finite groups.Missing: Alfred | Show results with:Alfred
[35]
Systems of Imprimitivity | SpringerLink
This chapter discusses the notion of a system of imprimitivity for a group and, in some cases, describes all such systems to within unitary equivalence.
[36]
[PDF] C*-algebras and Mackey's theory of group representations
And Mackey's theory of induced representations and of representations of group extensions was fully developed. On the C*-algebra side, it was known that any ...
[37]
[PDF] Representations of Semisimple Lie Groups
Let us mention that unnormalized induction has a geometric interpretation in terms of vector bundles. Historically, induced representations of Lie groups.
[38]
[PDF] Unitary representations of real reductive groups - MIT Mathematics
The Langlands classification Theorem 6.1 realizes each irreducible representa- tion π1 of G as the unique irreducible quotient of an induced representation π1.
[39]
[PDF] Status of Classification of Irreducible Unitary Representations
J(P,T,V) is the unique irreducible. The Langlands representation J(P,T,v) quotient of the induced representation. = MAN. Ⓡ. U(P,T,V) ind("e 1) and is given as ...
[40]
[PDF] Representations of SL2(R) - UBC Math
Nov 1, 2020 · Harish-Chandra modules, are representations simultaneously of g and ... They occur discretely in the representation of G on L2(G), and they ...
[41]
[PDF] INTERTWINING OPERATORS FOR SL(n, R) - AW Knapp and EM ...
2Discrete series now refers to irreducible representations whose matrix coeffi- cients are square integrable. Bargmann allowed as well two other representations.