Endoscopic Transfer on Real Lie Algebras

• Endoscopic transfer on real Lie algebras is the process by which invariant distributions, orbital integrals, and representation data are systematically matched between a real reductive Lie algebra and its endoscopic group.
• Transfer factors, defined via Galois cohomology and combinatorial formulas, enable precise matching by balancing orbital integrals and ensuring compatibility with local analytic invariants.
• The method commutes with the Fourier transform, underpinning its role in harmonic analysis and the Langlands program while facilitating applications in representation theory and spectral transfer.

Endoscopic transfer on real Lie algebras is the process by which invariant distributions, orbital integrals, and representation-theoretic data are matched or transferred between a real reductive Lie algebra and the Lie algebra of its endoscopic group. This mechanism, foundational to harmonic analysis and the Langlands program, requires precise definitions of transfer factors, compatibility with Fourier transforms, and intricate classification of orbits and representations. The theory extends techniques from automorphic forms over adele groups to the archimedean, Lie algebraic context, incorporating local methods, analytic invariants, and explicit combinatorial formulas.

1. Foundational Principles of Endoscopic Transfer

Endoscopic transfer arises from the need to relate invariants (such as orbital integrals and stable characters) on a real reductive group $G$ with those on its endoscopic group $H$. The construction begins by selecting an endoscopic datum $(H, s, \xi)$ compatible with $G$, together with a quasi-split inner form $G^*$. For real Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$, one defines matching functions via transfer factors $\Delta(X_H, X_G)$ assigned to pairs $(X_H, X_G)$ of regular semisimple elements.

The transfer process involves balancing orbital integrals through these transfer factors: $J_{G,H}(X_H, f_G) = J_H^{st}(X_H, f_H),$ where $f_G \in \mathcal{S}(\mathfrak{g})$ and $f_H \in \mathcal{S}(\mathfrak{h})$ are Schwartz functions, and $J_{G,H}$ combines stable conjugacy and transfer factors. The Langlands functoriality principle underpins the correspondence of these parameters, embedding the transfer mechanism into a broader framework.

2. Transfer Factors and Their Structural Role

Transfer factors $\Delta(\cdot,\cdot)$ are complex-valued functions used to mediate between orbital integrals or stable characters on $\mathfrak{g}$ and $\mathfrak{h}$. Their precise structure incorporates Galois cohomology, Tate–Nakayama pairings, and products of terms $A_I$, $A_{II}$, and $A_{III}$: $$\Delta' = (A_I A_{III})^{-1} A_{II}, \quad \Delta_D = A_I A_{III} A_{II}^{-1}, \quad \Delta_D = (\Delta')^{-1},$$ as detailed in (Shelstad, 2014). Depending on normalization—classical or renormalized—these factors may differ by explicit constants (e.g., determinants evaluated at $-1$ in Whittaker normalization).

This formalism ensures that matching of stable orbital integrals: $$\Delta'(\delta_1, \delta) \cdot O_\delta(f) = SO_{\delta_1}(f_1),$$ is preserved across variants of the local Langlands correspondence, and analogous formulas exist for matching stable traces on representations.

3. Fourier Transform Compatibility

A pivotal insight confirmed in (Chen et al., 6 Aug 2025) is that endoscopic transfer for real Lie algebras commutes with the Fourier transform. For a nondegenerate bilinear form $\langle \cdot, \cdot \rangle$ and Schwartz function $f$,

$$\mathcal{F}_\psi(f)(X) = \int_{\mathfrak{g}} f(Y) \psi(\langle X, Y \rangle) dY,$$

with $\psi(x) = \exp(2\pi i x)$, the Fourier transform intertwines orbital integrals and distributions $j^G$, $i^G$ through

$$J_G(X, \mathcal{F}_\psi(f)) = \int_{\mathfrak{g}} f(Y) j^G(X, Y) dY, \quad i^G(X, Y) = \Delta_G(Y)^{1/2} j^G(X, Y).$$

The transfer distributions

$${}_G,_H(X_H, X_G) = \gamma_\psi^{\mathfrak{g}} \sum_{X'_G} \Delta(X_H, X'_G) i^G(X'_G, X_G),$$

and

$$\tilde{{}_G,_H}(X_H, X_G) = \gamma_\psi^{\mathfrak{h}} \sum_{X'_H} w(X'_H)^{-1} \Delta(X'_H, X_G) i^H(X'_H, X_H),$$

coincide for all regular semisimple elements. This yields that matching functions have matching Fourier transforms, a fact established locally over $\mathbb{R}$ through reduction to elliptic elements, application of Rossmann’s formula, and Harish-Chandra’s uniqueness theorem.

4. Nilpotent Orbits, Subalgebra Classification, and Carrier Algebras

Understanding the fine structure of nilpotent orbits and regular subalgebras is crucial for the parameterization of invariant distributions subject to endoscopic transfer. Algorithms in (Dietrich et al., 2014) generate "carrier algebras" via Cartan- and grading-based procedures, organizing nilpotent orbits into conjugacy classes using SL$_2$-triples and combinatorial invariants. These carrier algebras provide a stratification of orbital integrals, facilitating their comparison and transfer. In type $A_\ell$, inner ideals take the form

$$B_{ \{ s,t \} }, \quad \dim B_{ \{ s,t \} } = s(\ell+1-t),$$

and are classified via adapted subsets $I$ in the Satake diagram for the real Lie algebra (Draper et al., 2022).

5. Spectral Transfer and Representation Theory

On the spectral side, endoscopic transfer matches stable traces and packet structures for representations, particularly limits of discrete series and nontempered packets. Spectral transfer factors, often normalized via Whittaker data and c-Levi group splittings, ensure: $$\mathrm{St} \text{–} \mathrm{Trace}_{\pi_1}(f_1) = \sum A(T_1, T) \cdot \mathrm{Trace}[\pi(f) \cdot r(\theta, \omega)],$$ where $A(T_1, T)$ is a spectral transfer factor and $r(\theta, \omega)$ a twisted intertwining operator (Shelstad, 2014). These dual spectral formulas are compatible with the geometric side, up to explicit constants.

Dirac cohomology provides a mechanism for character lifting via the Dirac index

$I(X) = (X \otimes S^+) - (X \otimes S^-),$

and the transfer factor can be expressed as

$$\Delta(\gamma) = \mathrm{ch}\, S^+(\gamma) - \mathrm{ch}\, S^-(\gamma),$$

where $S^\pm$ are spin module components (Huang, 2020). This structure enables explicit lifting of tempered characters in the real Lie algebra context.

6. Analytic Frameworks: Integration, Germs, and Local Homogeneity

Integration of representations and endoscopic transfer in the Lie algebra setting relies on geometric and analytic tools. Integrable representations are realized as vector fields on homogeneous spaces; non-integrable ones are embedded into commutative coordinate algebras, often expressible as power series via the coproduct structure of the universal enveloping algebra (Le-Bert, 2012). Shalika germs, originally defined for asymptotic expansions of orbital integrals, have motivic analogs and admit transfer principles exploitable for field-independent identities (Gordon et al., 2015).

Radiality properties for smooth functions in multidimensional restricted root spaces, proven via orthogonal decompositions

$$\mathfrak{g}_\lambda = \mathbb{R} X \oplus [\mathfrak{m}, X],$$

demonstrate how symmetry and decomposition in real Lie algebra settings can inform analysis and facilitate transfer calculations (Korvers, 2018).

7. Computational and Structural Aspects

Effective computation of endoscopic transfer in real Lie algebras draws on tools such as the CoReLG package (GAP implementation) for the automation of carrier algebra listings, nilpotent orbit representatives, and invariant calculations (Dietrich et al., 2014). Vessel theory for non-selfadjoint representations establishes joint characteristic functions serving as unitary invariants, enabling analytic transfer between different contexts (Shamovich et al., 2012). The explicit transfer of invariant differential operators and analysis of kernel structures for twisted Dirac operators serve as representation-theoretic bridges between larger groups and their endoscopic subgroups (Mehdi et al., 2021).

Conclusion

Endoscopic transfer on real Lie algebras integrates local analytic techniques, deep combinatorial structures, and transfer factor normalizations to precisely match invariant distributions, representations, and orbital integrals across related groups. The commutation of the endoscopic transfer with the Fourier transform—established using purely local methods—ensures robustness in harmonic analysis and supports applications to the stabilization of the trace formula and advanced comparison theorems in the Langlands program. Ongoing research continues to extend these compatibilities to less regular elements, broader classes of groups, and finer spectral decompositions, driving deeper understanding in both representation theory and arithmetic geometry.