Transfer of orbital integrals and division algebras. (English) Zbl 0722.11028
A gigantic leap forward, marking the transition from the study – by means of a trace formula – of automorphic forms on \(\mathrm{GL}(2)\) [H. Jacquet and R. P. Langlands, Automorphic forms on \(\mathrm{GL}(2)\), Lect. Notes Math. 114 (1970; Zbl 0236.12010)] to those on \(\mathrm{GL}(n)\), was made in the celebrated – but unpublished – work of Deligne-Kazhdan, of the late 1970’s. Their local lifting result is as follows.
Let \(F_ u\) be a local non-archimedean field, \(G_ u\) the multiplicative group of a division algebra \(D_ u\) central of rank \(n\) over \(F_ u\), and \(G'_ u=\mathrm{GL}(n,F_ u)\). There is an embedding of the set of conjugacy classes \(\gamma\) in \(G_ u\) as the set of elliptic conjugacy classes \(\gamma '\) in \(G'_ u\), defined by \(p_{\gamma}=p_{\gamma '}\); here \(p_{\gamma}\) is the characteristic polynomial of \(\gamma\) ; \(p_{\gamma '}\) is that of \(\gamma '\).
Theorem. There is a bijection from the set of equivalence classes of irreducible \(G_ u\)-modules \(\pi_ u\) to the set of equivalence classes of irreducible square-integrable \(G'_ u\)-modules \(\pi '_ u\), defined by the character relation \(\chi_{\pi '_ u}(\gamma ')=(-1)^{n- 1}\chi_{\pi_ u}(\gamma)\) for every regular \(\gamma\) in \(G_ u\) with image \(\gamma '\) in \(G'_ u\). Here \(\chi_{\pi '_ u}\) denotes the character of \(\pi '_ u\), and \(\chi_{\pi_ u}\) that of \(\pi_ u.\)
The new ideas introduced there include a simple form of the trace formula (a stronger version later appeared in [the author and D. Kazhdan, J. Anal. Math. 50, 189–200 (1988; Zbl 0666.10018)]) and applications of the Bernstein center (later developed in [D. Kazhdan, J. Anal. Math. 47, 1–36 (1986; Zbl 0634.22009)]).
It was mistakenly assumed by later writers not sufficiently familiar with the original work that transfer of orbital integrals between \(G_ u\) and \(G'_ u\) was necessary for the proof. The purpose of the present note is to prove the Theorem (making use of Hecke theory) and show that transfer of orbital integrals is a consequence of the Theorem, not a prerequisite, thus dissipating some misconceptions concerning the difficulty of this case.
The Theorem is used as the first step in the inductive proof of an extension to the general case where \(D_ u\) is any simple – not necessarily division – algebra; see [J.-N. Bernstein, P. Deligne, D. Kazhdan, M.-F. Vigneras (editors), Représentations des groupes réductifs sur un corps local (1984; Zbl 0544.00007)] as completed in [the author, J. Anal. Math. 49, 135–202 (1987; Zbl 0656.10024)].
Let \(F_ u\) be a local non-archimedean field, \(G_ u\) the multiplicative group of a division algebra \(D_ u\) central of rank \(n\) over \(F_ u\), and \(G'_ u=\mathrm{GL}(n,F_ u)\). There is an embedding of the set of conjugacy classes \(\gamma\) in \(G_ u\) as the set of elliptic conjugacy classes \(\gamma '\) in \(G'_ u\), defined by \(p_{\gamma}=p_{\gamma '}\); here \(p_{\gamma}\) is the characteristic polynomial of \(\gamma\) ; \(p_{\gamma '}\) is that of \(\gamma '\).
Theorem. There is a bijection from the set of equivalence classes of irreducible \(G_ u\)-modules \(\pi_ u\) to the set of equivalence classes of irreducible square-integrable \(G'_ u\)-modules \(\pi '_ u\), defined by the character relation \(\chi_{\pi '_ u}(\gamma ')=(-1)^{n- 1}\chi_{\pi_ u}(\gamma)\) for every regular \(\gamma\) in \(G_ u\) with image \(\gamma '\) in \(G'_ u\). Here \(\chi_{\pi '_ u}\) denotes the character of \(\pi '_ u\), and \(\chi_{\pi_ u}\) that of \(\pi_ u.\)
The new ideas introduced there include a simple form of the trace formula (a stronger version later appeared in [the author and D. Kazhdan, J. Anal. Math. 50, 189–200 (1988; Zbl 0666.10018)]) and applications of the Bernstein center (later developed in [D. Kazhdan, J. Anal. Math. 47, 1–36 (1986; Zbl 0634.22009)]).
It was mistakenly assumed by later writers not sufficiently familiar with the original work that transfer of orbital integrals between \(G_ u\) and \(G'_ u\) was necessary for the proof. The purpose of the present note is to prove the Theorem (making use of Hecke theory) and show that transfer of orbital integrals is a consequence of the Theorem, not a prerequisite, thus dissipating some misconceptions concerning the difficulty of this case.
The Theorem is used as the first step in the inductive proof of an extension to the general case where \(D_ u\) is any simple – not necessarily division – algebra; see [J.-N. Bernstein, P. Deligne, D. Kazhdan, M.-F. Vigneras (editors), Représentations des groupes réductifs sur un corps local (1984; Zbl 0544.00007)] as completed in [the author, J. Anal. Math. 49, 135–202 (1987; Zbl 0656.10024)].
Reviewer: Yuval Z. Flicker
MSC:
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 11F72 | Spectral theory; trace formulas (e.g., that of Selberg) |
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
| 11S45 | Algebras and orders, and their zeta functions |
| 11S80 | Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.) |
