Descent for transfer factors.

*(English)*Zbl 0743.22009
The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. II, Prog. Math. 87, 485-563 (1990).

[For the entire collection see Zbl 0717.00009.]

Transfer from a reductive group \(G\) over a local field to an endoscopic group \(H\) for \(G\) relates orbital integrals for \(G\) to stable orbital integrals for \(H\). The coefficients in these relations are called transfer factors. They were determined by the authors, which means that, if transfer is possible, the transfer factors must be as they define them. The existence of transfer however has not yet been proved in general. In the present paper the proof of existence of transfer is reduced to that of local transfer at the identity, namely transfer will exist for \(G\) if it exists locally at 1 for all connected centralizers of semisimple elements of \(G\). This result is deduced from the compatibility of the transfer factors for \(G\) and those for the centralizers. The proof of this compatibility is very complicated and occupies the greater part of the paper.

Transfer from a reductive group \(G\) over a local field to an endoscopic group \(H\) for \(G\) relates orbital integrals for \(G\) to stable orbital integrals for \(H\). The coefficients in these relations are called transfer factors. They were determined by the authors, which means that, if transfer is possible, the transfer factors must be as they define them. The existence of transfer however has not yet been proved in general. In the present paper the proof of existence of transfer is reduced to that of local transfer at the identity, namely transfer will exist for \(G\) if it exists locally at 1 for all connected centralizers of semisimple elements of \(G\). This result is deduced from the compatibility of the transfer factors for \(G\) and those for the centralizers. The proof of this compatibility is very complicated and occupies the greater part of the paper.

Reviewer: J.G.M.Mars (Utrecht)

##### MSC:

22E35 | Analysis on \(p\)-adic Lie groups |

22E50 | Representations of Lie and linear algebraic groups over local fields |