Harish-Chandra Images in Representation Theory

• Harish-Chandra images are the canonical images of central or invariant elements that transition noncommutative structures to commutative geometric models.
• They encapsulate key techniques such as Fourier transforms, D-module formulations, and graded isomorphisms, linking analytic, algebraic, and geometric approaches.
• Their applications span classifying admissible modules, understanding quantum groups, and probing harmonic analysis on reductive groups through explicit invariants.

Harish-Chandra images constitute a central unifying concept in representation theory, noncommutative geometry, and harmonic analysis. The term refers to several interconnected objects: the images of central or invariant elements under canonical homomorphisms (such as the Harish-Chandra isomorphism for universal enveloping algebras, quantum groups, or D-modules), the supports or associated graded sheaves of distinguished modules (e.g., the Harish-Chandra D-module), or the explicit Fourier-analytic images of function spaces and Schwartz algebras governed by the Harish-Chandra transform. Across these settings, the Harish-Chandra image encapsulates the passage from noncommutative or equivariant data to a commutative or geometric model, encoding deep structural invariants.

1. The Harish-Chandra D-module and the Isospectral Commuting Variety

The Harish-Chandra D-module $M$ on %%%%1%%%% (for $g$ a complex reductive Lie algebra and $t \subset g$ a Cartan subalgebra) is defined as a holonomic module over $\mathcal{D}(g\times t)$: $$M = \mathcal{D}(g\times t) \Big/\Big(\mathcal{D}(g\times t)\cdot (\operatorname{ad}_g\otimes 1) + \mathcal{D}(g\times t)\cdot\{u\otimes 1 - 1\otimes \operatorname{rad}(u) \mid u\in \mathcal{D}(g)\}\Big)$$ where $\operatorname{rad}$ is the radial parts homomorphism, which identifies $\mathcal{D}(g)^G$ with $\mathcal{D}(t)^W$ via the Chevalley isomorphism. The associated graded module with respect to its canonical (Goodman–Kazhdan–Hodge) filtration is shown to be

$$\operatorname{gr}(M) \cong \mathcal{O}_{\mathfrak{X}_{\mathrm{norm}}}$$

as sheaves on $g\times t$, where $\mathfrak{X}_{\mathrm{norm}}$ is the normalization of the isospectral commuting variety: $$\mathfrak{X} = \{(x, y, t)\in g\times g\times t\mid [x,y]=0,\, t \text{ corresponds to } (x,y) \text{ via restriction}\}.$$ This result yields that $\mathfrak{X}_{\mathrm{norm}}$ is Cohen-Macaulay and Gorenstein—the latter confirming a conjecture of Haiman—since $\operatorname{gr}(M)$ is a Cohen-Macaulay, self-dual (up to shift) sheaf by Saito's theory of polarized Hodge modules. For $g = \mathfrak{gl}_n$, $M$ realizes a geometric link to the Hilbert scheme $\mathrm{Hilb}^n(\mathbb{C}^2)$: the Procesi bundle (crucial in the proof of Macdonald positivity) arises as the vector bundle corresponding to $\operatorname{gr}(M)$ (Ginzburg, 2010).

2. Explicit and Graded Harish-Chandra Isomorphism for Symmetric Superpairs

For symmetric pairs of strongly reductive Lie superalgebras of even type, a graded Harish-Chandra homomorphism is constructed, mapping the center of the universal enveloping algebra to Weyl group invariants on a Cartan subspace. The PBW (Poincaré–Birkhoff–Witt) theorem yields a decomposition

$$U(\mathfrak{g}) \cong U(\mathfrak{a}) \otimes U(\mathfrak{p}_1^\lambda) \otimes U(\mathfrak{n}_\lambda^\perp) \otimes U(\mathfrak{k})$$

leading to a well-defined Harish-Chandra projection that detects $k_\lambda$-invariants and their images solely from the lead terms in the decomposition. This extends classical results of Harish-Chandra and V. Kac–M. Gorelik (Alldridge, 2010).

3. Structural Images in Admissible Module Classification

For complex semisimple Lie groups and algebras, the Harish-Chandra isomorphism

$\gamma: Z(U(\mathfrak{g})) \to U(\mathfrak{h})^W$

associates to each irreducible admissible Harish-Chandra module a central character parametrized by a Weyl group orbit. The classification of irreducible modules is closely controlled by minimal $K$-type ("PRV component") and central character. The PRV determinants constructed in this context track the multiplicity and occurrence of composition factors in tensor products and annihilators of Verma modules, reflecting the detailed structure of Harish-Chandra images (Khare, 2012).

4. Geometric and Fourier-Analytic Harish-Chandra Images

The Harish-Chandra transform induces explicit isomorphisms in the context of harmonic analysis on real reductive groups: $f\mapsto \mathcal{H}f$ From the spherical Schwartz algebra $\mathcal{C}^p(G//K)$ (bi-K-invariant functions), the transform is a topological algebra isomorphism onto the algebra $Z(\mathfrak{F}^\epsilon)$ of $\mathfrak{w}$-invariant holomorphic functions in a tube domain, as established in the Trombi–Varadarajan Theorem and extended by Eguchi and others to non-spherical settings (Oyadare, 2017, Oyadare, 2019). The full Schwartz algebra's image can be described as an infinite matrix Fréchet algebra, realized blockwise along finite sets of $K$-types (Oyadare, 30 Jul 2024). The operator-valued Fourier transform $$\mathfrak{F}:\mathcal{C}^{p}(G)\rightarrow \mathcal{C}^{p}(\hat{G})$$ decomposes Schwartz functions into (countable) block matrices indexed by $K$-types, and the isomorphism is used to prove strong structural results in invariant harmonic analysis for all real-rank groups, establishing a comprehensive bridge between noncommutative convolution and commutative spectral data.

5. Harish-Chandra Images in Quantum and Braided Settings

For multi-parameter quantum groups $U_{q,G}(\mathfrak{g})$ of Okado–Yamane type, the Harish-Chandra homomorphism

$$\xi = \gamma^{(-\rho)}\circ\pi|_{Z(U)}: Z(U) \rightarrow U^0$$

(with $\gamma^{(-\rho)}$ a shift by the half-sum of positive roots) is shown to be an algebra isomorphism onto the $W$-invariant subalgebra of the "flat" Cartan part

$$U^0_{\mathrm{flat}} = \bigoplus_{\eta\in Q} \mathbb{K} \cdot (\omega'_\eta \omega_{-\eta}).$$

Each central element is thus uniquely determined (modulo a twist) by its Cartan component, and the isomorphism holds universally across all types and parameter matrices, extending previous results constrained by type and parity (Chen et al., 24 May 2025).

6. Harish-Chandra Images via Bimodule and Category Theoretic Methods

In the abstract representation–theoretic context, Harish-Chandra bimodules (e.g., for universal enveloping algebras or rational Cherednik algebras) admit functorial "images" in categories of Soergel bimodules and their singular/graded analogues. The functor

$$\bullet_\dag : HC(U^{\chi, \chi'}) \longrightarrow (Z^{\wedge_\chi}, Z^{\wedge_{\chi'}})\text{-bimod}^{C(G)}$$

constructed via completions and restriction functors (notably Losev's restriction via finite $W$-algebras and Rees algebras) is exact, monoidal, and fully faithful on projective objects. This setup allows one to generalize results of Soergel and Stroppel from integral blocks to all blocks, and to mirror the tensor structure of Harish-Chandra bimodules in the world of completed center bimodules. The indecomposable projectives correspond to direct summands labeled by double cosets in the affine Weyl group, and the entire framework provides an alternative, algebraic structure for understanding Harish-Chandra images in both classical and modular representation theory (Vu, 22 Jul 2025).

7. Geometric Realizations and Holomorphic Discrete Series

In the geometric setting, the Harish-Chandra highest weight modules (built via induction from a parabolic $\mathfrak{q}$ and a finite-dimensional $K$-type) have their infinitesimal characters—Harish-Chandra images of the center—explicitly described as values of the canonical homomorphism on a Cartan subalgebra symmetric under the Weyl group. Their global realizations as spaces of holomorphic sections over flag manifolds or "big cells" in symmetric spaces (as in the Borel–Weil–Bott or Harish-Chandra cell construction) realize these algebraic images as spectral invariants of differentials on homogeneous spaces (Fioresi et al., 2022).

Concurrently, the Harish-Chandra condition in the context of holomorphic discrete series—expressed as the convergence of explicit $L^2$ integrals weighted by (highest) weights—is shown to coincide with a precise inequality on Harish-Chandra images, namely the requirement that the image of the highest weight under the Harish-Chandra morphism be sufficiently negative compared to the structural parameters of the domain. This integrability condition, established via analytic methods such as reproducing kernel Hilbert spaces and polar coordinate integrals, directly governs which holomorphic discrete series representations exist for a given highest weight (Korányi, 2020, Koranyi, 2023).

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In synthesis, Harish-Chandra images—ranging from explicit homomorphic images of central and invariant elements, through sheaf-theoretic and graded D-modules, to operator-theoretic Fourier images—encode the passage from noncommutative or equivariant data to tractable commutative invariants. This transition underlies the structure and classification of representations, the geometry of associated varieties, and the explicit analytic and categorical tools used throughout modern Lie theory, quantum groups, and harmonic analysis.