Endoscopic transfer - Knowlpedia

Let $\mathfrak e$ be an endoscopic datum for a reductive group $G$ , with endoscopic group $H$ . A test function $f^H\in C_c^\infty(H(F))$ is an endoscopic transfer of $f\in C_c^\infty(G(F))$ if, for every strongly $G$ -regular element $\gamma_H\in H(F)$ ,

SO_{\gamma_H}(f^H) = \sum_{\delta\leftrightarrow\gamma_H} \Delta(\gamma_H,\delta)\,O_\delta(f).

The right side uses the transfer factor and runs over rational conjugacy classes in $G(F)$ matching $\gamma_H$ .

Matching functions

The local transfer theorem asserts, in the established endoscopic settings, that every test function on $G(F)$ has a transfer on $H(F)$ . The transfer is not unique as a function; its stable orbital integrals are the prescribed data. Variants cover archimedean fields, nonarchimedean fields, Lie algebras, and twisted endoscopy.

The fundamental lemma is the especially important unramified assertion that the unit of the spherical Hecke algebra transfers to the corresponding unit, with normalized measures.

Spectral transfer

Duality of invariant distributions converts geometric transfer into character identities. Stable distributions built from characters on $H(F)$ transfer to linear combinations of characters in $L$ -packets or $A$ -packets on $G(F)$ . This is the mechanism by which endoscopy describes packet structure.

Global use

For factorizable functions, compatible local transfers define an adelic transfer. Comparing the trace formula for $G$ with stable trace formulas for its endoscopic groups yields endoscopic classification and instances of Langlands functoriality.

Scope

Endoscopic transfer realizes a structured class of $L$ -homomorphisms. It does not encompass every case of functoriality. Ordinary, twisted, and weighted transfer are distinct assertions.

References

Jean-Loup Waldspurger, “Endoscopie et changement de caractéristique,” Journal of the Institute of Mathematics of Jussieu 5 (2006), 423–525. DOI.
Ngô Bảo Châu, “Survey on the fundamental lemma,” §§2–3. PDF.