Derived Parabolic Induction

• Derived parabolic induction is a generalization of the classical parabolic induction functor, extending it to derived, operator-algebraic, and geometric frameworks for advanced representation theory.
• It employs structures such as Hilbert C*-modules, perverse sheaves, and dg categories to realize induction as bimodules and correspondences, facilitating precise adjunctions and dualities.
• This approach underpins modern advances in the Langlands program and geometric representation theory by enabling refined spectral sequence computations and module decomposition.

Derived parabolic induction refers to the family of cohomological and categorical constructions which extend the classical parabolic induction functor to derived, operator-algebraic, geometric, and categorical frameworks. Originating in the representation theory of reductive groups and the theory of operator algebras, derived parabolic induction realizes induction functors as bimodules or correspondences in various categories—Hilbert C*-modules, perverse sheaves, D-modules, triangulated or dg categories, and homotopy-theoretic categories. This generalization is not only crucial for the paper of higher extension groups, spectral sequences, and duality statements, but also underpins modern approaches to the Langlands program, geometric representation theory, and categorical structures of automorphic and admissible representations.

1. Operator-Algebraic Model: Hilbert C*-Modules and Module Decomposition

The reduced group C*-algebra $C_r^*(G)$ of a real reductive group admits a canonical decomposition as a finite direct sum over equivalence classes $[P,\sigma]$ of parabolic subgroups $P=LN$ and square-integrable (mod center) representations $\sigma$ of the compactly generated part $M$ of $L$: $$C_r^*(G) \cong \bigoplus_{[P,\sigma]} K\big(\operatorname{Ind}_P^G\,\sigma\big)^{W_\sigma},$$ where $K(-)$ denotes compact operators and $W_\sigma$ is a finite Weyl-type group determined by intertwining isomorphisms. Each summand is determined via the Plancherel formula and tempered representation theory of Harish-Chandra and Langlands (Clare et al., 2014).

Parabolic induction is realized by a Hilbert C*-bimodule $C^*_r(G/N)$, an explicit (left $C^*_r(G)$, right $C^*_r(L)$)-correspondence with the property: $$C^*_r(G/N) \otimes_{C^*_r(L)} \tau \cong \operatorname{Ind}_P^G \tau$$ for any tempered representation $\tau$ of $L$. This bimodule admits a direct sum decomposition pertinent to the combinatorics of induced representations. A secondary inner product is constructed on the adjoint bimodule $C^*_r(G/N)^*$; this secondary structure, defined via a wave packet integral involving the Plancherel measure, is essential for defining restriction, adjunctions, and the full t-structure of the derived category.

2. Adjunctions, Restriction Functors, and Dualities

The secondary inner product on $C^*_r(G/N)^*$ provides the structure necessary to define the functor of parabolic restriction, realized as

$$\operatorname{Res}^{G}_P H = C^*_r(G/N)^* \otimes_{C^*_r(G)} H,$$

for any Hilbert space $H$ carrying a tempered representation of $G$. This functor is, crucially, both left and right adjoint to parabolic induction, yielding natural isomorphisms: $$\operatorname{Hom}_G(\operatorname{Ind}_P^G \tau, \pi) \cong \operatorname{Hom}_L(\tau, \operatorname{Res}^G_P \pi),$$

$$\operatorname{Hom}_G(\pi, \operatorname{Ind}_P^G \tau) \cong \operatorname{Hom}_L(\operatorname{Res}^G_P \pi, \tau),$$

with all inner products and intertwining computed explicitly via the Plancherel formula and averaged over the appropriate Weyl group (Clare et al., 2014, Crisp et al., 2015).

In operator module categories, this adjunction generalizes to completely bounded maps with respect to Haagerup tensor products, and Frobenius reciprocity takes the form

$$\mathrm{CB}_{C^*(L)}\!\left(\operatorname{Res}_G X, Y\right) \cong \mathrm{CB}_{C^*(G)}\!\left(X, \operatorname{Ind}_G Y\right),$$

and an operator-algebraic formulation of Bernstein’s second adjoint theorem is given via explicit unit and counit maps (Crisp et al., 2015).

3. Geometric and Coherent Approaches: Perverse Sheaves and D-Modules

Derived parabolic induction admits geometric incarnations, particularly through equivalences relating graded and geometric categories:

• On the level of BGG category $\mathcal{O}$, parabolic induction lifts to the graded category $\mathcal{O}^{\mathrm{gr}}$, equivalent to the derived category of stratified mixed Tate motives on flag varieties, and is closely mirrored by the behavior of Soergel modules. Explicitly,

$$\operatorname{Ind}_P^G(M) \cong \bigoplus_{w \in W^P} B_w \otimes M,$$

with $B_w$ a Soergel bimodule (Eberhardt, 2016).

• Parabolic induction and restriction of equivariant D-modules on a reductive group $G$ correspond to explicit geometric functors between derived categories $\mathcal{D}(G)$ and $\mathcal{D}(L)$, constructed as pushforward-pullback along correspondences by the flag variety. When $L = T$ is a torus, the Harish-Chandra D-module $M = D(G \times T)/(\ldots)$ implements the induction kernel and is flat as a $D(T)_W$-module, ensuring exactness at the abelian and derived levels (Ginzburg, 2021).

4. Derived and DG-Category Frameworks

The derived category $D(G)$ of smooth representations (e.g., over mod $p$ or complex coefficients) is equivalent to the derived category of modules over an explicit dg Hecke algebra $H_G^\bullet$. Within this dg setting, the derived parabolic induction is realized via tensoring with an explicit $(H_G^\bullet, H_M^\bullet)$-bimodule (after resolving to the appropriate homotopy category), and admits both left and right adjoints: $$\operatorname{Ind}^{H_G}_{H_M}(Y) \cong I_{G,P} \otimes^{\mathbf{L}}_{H_M} Y, \quad \text{Right Adjoint:} \ Z \mapsto \mathbf{R}\operatorname{Hom}_{H_G}(I_{G,P}, Z).$$ Six-functor formalism in dg-categories ensures these functors obey the expected properties (derived tensor, derived Hom, compatibility with adjunctions) (Scherotzke et al., 2020).

In model category-theoretic approaches, such as with Gorenstein projective structures introduced by Hovey, one passes to a triangulated homotopy category $\mathrm{Ho}(\mathcal{H})$ of (pro-$p$ Iwahori) Hecke modules and demonstrates that parabolic induction (and its derived, or total derived, functors) facilitate a recollement that decomposes $\mathrm{Ho}(\mathcal{H})$ into subcategories capturing the images of parabolic induction from various Levi subgroups and their complements (Dupré, 2023).

5. Extensions, Deformations, and Spectral Sequences

The homological behavior of derived parabolic induction is revealed via calculations of extension groups and spectral sequences:

• Emerton’s $\delta$-functor computes the derived functors of ordinary parts (right adjoint of induction), yielding exact sequences

$$0 \to \mathrm{Ext}^1_L(\sigma, \operatorname{Ord}_P\pi) \to \mathrm{Ext}^1_G(\operatorname{Ind}_P^G \sigma, \pi) \to \mathrm{Hom}_L(\sigma, \mathrm{HOrd}^1_P \pi),$$

and dual exact sequences from derived Jacquet functors (Hauseux, 2016, Koziol, 2022).

• Under suitable genericity conditions, equivalence of deformation functors of smooth mod $p$ representations is proved: the universal deformation ring of a supersingular $\bar{\sigma}$ of $L$ is isomorphic to that of $\operatorname{Ind}_{P}^{G}\bar{\sigma}$, and any Banach lift factors uniquely through the induced module (Hauseux et al., 2016).
• In the derived framework, the left adjoint to parabolic induction, $L(U, -)$, is constructed (via Brown representability) and shown to preserve boundedness and global admissibility, with cohomology functors $H^i \circ L(U,-)$ vanishing in appropriate degrees and a spectral sequence

$$E_2^{i,j} = \mathrm{Ext}^i_{M}(L^{-j}(U, V), W) \Rightarrow \mathrm{Ext}^{i+j}_G(V, i_M^G W)$$

providing higher extension computations (Heyer, 2022).

6. Transitivity, Tensor Product Structures, and Globalization

At both the categorical and operator-algebraic levels, the transitivity of parabolic induction—i.e., induction in stages—is realized as associativity or compatibility of induced bimodules/constructions. In the setting of pro-$p$ Iwahori–Hecke algebras, the morphisms $\Theta_M^P$ and $\Xi_G^P$ (constructed via double coset decomposition) are shown to factor through a canonical subalgebra $\mathcal{H}_R(M,G)$, and the derived tensor product structure yields canonical isomorphisms for transitivity of induction and extension to derived categories (Heyer, 2020). This precise control enables new formulations of the Satake homomorphism and aids in understanding the structure of modular representations.

In the global (adelic) setting, the construction of restricted tensor products of Hilbert C*-modules produces global C*-correspondences from local data. When applied to local correspondences implementing parabolic induction, the global construction recovers adelic parabolic induction: for a restricted tensor product $\mathcal{E}_A = \otimes'_v \mathcal{E}(G_v/N_v)$, representations of Levi subgroups can be induced to representations of the global group via

$$\operatorname{Ind}_{P(\mathbb{A})}^{G(\mathbb{A})}(\pi) \cong \mathcal{E}_A \otimes_{C(L(\mathbb{A}))} H_\pi,$$

integrating parabolic induction in the analytic, operator-algebraic, and representation-theoretic senses (Goffeng et al., 3 Dec 2024).

7. Geometric, Cohomological, and Category-Theoretic Impact

Derived parabolic induction unifies discrete, geometric, and operator-algebraic methodologies:

• The construction of derived functors and adjoints (Jacquet modules, ordinary parts) elucidates the categorical behavior of representations and extensions.
• The precise control of module decompositions and inner products (including secondary structures) supports the analysis of adjunctions and dualities (derived second adjoint theorems, Frobenius reciprocity at the operator-algebraic level).
• Global-local compatibility is achieved through restricted tensor products, facilitating the passage from local representation theory to automorphic forms and globalization in arithmetic contexts.
• Derived frameworks allow for applications to mod $p$ representation theory, Satake isomorphism generalizations, and the modular Langlands program.

This cross-disciplinary synthesis continues to guide advances in representation theory, categorical harmonic analysis, and the analysis of automorphic spectra, with derived parabolic induction providing the foundational "lift" of induction functors to their most robust and geometric instantiations.