Complexity of group actions and stability of root numbers. With an appendix by Wentang Kuo. (English) Zbl 1183.22010
Ginzburg, David (ed.) et al., Automorphic forms and \(L\)-functions II. Local aspects. A workshop in honor of Steve Gelbart on the occasion of his sixtieth birthday, Rehovot and Tel Aviv, Israel, May 15–19, 2006. Providence, RI: American Mathematical Society (AMS); Ramat Gan: Bar-Ilan University. (ISBN 978-0-8218-4708-4/pbk). Contemporary Mathematics 489. Israel Mathematical Conference Proceedings, 259-283 (2009).
We begin the review with a brief historical account. A key tool in establishing Langlands functoriality has been the stability of root numbers under ramified twists: if \(G=\text{GL}_n\) and \(\pi_1\), \(\pi_2\) are irreducible, admissible representations (with the same central character) of \(G(k)\), where \(k\) is a \(p\)-adic local field with additive character \(\psi\), then it was shown by H. Jacquet and J. Shalika [Math. Ann. 271, 319–332 (1985; Zbl 0541.12010)] that \(\gamma(s,\pi_1\times\chi,\psi)=\gamma(s,\pi_2\times\chi,\psi)\) for all characters \(\chi\) of \(\text{GL}_1(k)\) which are sufficiently highly ramified. This result was extended to \(G=\text{SO}_{2n+1}\) by J. W. Cogdell and I. I. Piatetski-Shapiro [Manuscr. Math. 95, No.4, 437–461 (1998; Zbl 0959.22011)] by representing each gamma factor \(\gamma(s,\pi\times\chi,\psi)\) as a Mellin transform of a certain incomplete (partial) Bessel function associated to the representation.
Now let \(G\) be a quasi-split, connected, reductive algebraic group over \(k\), and \(M\) a Levi subgroup of a self-associate parabolic \(P=MN\). F. Shahidi [Int. Math. Res. Not. 2002, No. 39, 2075–2119 (2002; Zbl 1025.22014)] constructed, using a densely defined map \(\phi:M\to N\), Bessel functions whose Mellin transforms represent the local coefficients \(C_\psi(s,\pi)\) associated to a representation of \(M(k)\), thereby generalising Cogdell and Piatetski-Shapiro’s approach. Stability of root numbers for all groups \(G\) of interest in functoriality was then established by a number of authors by building gamma factors out of local coefficients and analysing the asymptotics of the relevant Bessel functions (e.g. J. W. Cogdell, H. H. Kim, I. I. Piatetski-Shapiro and F. Shahidi [Publ. Math., Inst. Hautes Étud. Sci. 99, 163–233 (2004; Zbl 1090.22010)] treat stability for \(G=\text{SO}_{2n+1}\), \(\text{SO}_{2n}\), and \(\text{Sp}_{2n}\)).
Now we turn to the paper at hand. Quoting from the introduction, “This paper is an attempt to put the problem of general stability of root numbers [...] in the context of appropriate group actions and their complexity as well as the underlying algebraic geometry. Our aim is modest and we claim no “theorems”, rather we establish some basic facts about the ingredients which go into our expressions for local coefficients which must be stabilized.”
The paper works in the generality of a quasi-split, connected, reductive algebraic group \(G\) and Levi subgroup \(M\) as above. In Proposition 2.4.7 it is shown that \(M_\phi\), (the closure of) the image of \(\phi\), is an irreducible closed subvariety of \(M\) and that it is stable under conjugation by an appropriate subgroup \(H\leq M\). Following D. Luna and T. Vust [Comment. Math. Helv. 58, 186–245 (1983; Zbl 0545.14010)], Shahidi recalls the definition of the complexity of an action of an affine algebraic group \(H\) on an irreducible variety \(X\): \(c(X):=\dim X-\max \{\dim Bx : x\in X\}\), where \(B\) is a Borel subgroup of \(H\). In the case at hand, \(H\) acts by conjugation and we therefore have complexities \(c(M_\phi)\), \(c(N)\); in Proposition 3.11 it is shown that, in most cases, \(c(N)-c(M_\phi)=\deg\phi\).
Combining this proposition with a deep result of R. Sundaravaradhan [Int. Math. Res. Not. 2008, No. 2, article ID rnm141, 1–22 (2008; Zbl 1232.22009)], the complexity \(c(N)\) can be calculated with relative ease. Importantly, this complexity is exactly the dimension of the quotient space \(H\backslash N\), over whose \(k\)-points one must integrate to form the Mellin transforms of the Bessel functions associated to \(G\). When \(G\) is a classical group, this complexity is zero; conversely, for the pair \((G,M)=(\text{SO}_{2n}, \text{GL}_n)\), which gives the exterior square \(L\)-function of a representation of \(\text{GL}_n(k)\), the complexity \(c(N)\) is non-zero and the problem of stability is open. Quoting the final remark, “It therefore appears that so far we have been only able to prove the stability when \(c(N)=0\).”
The appendix, by W. Kuo, contains calculations concerning dimension/rank inequalities for certain orbit spaces associated to \((G,M)\). These inequalities are related to the complexity ideas in the main article and they show that the partial Bessel functions associated to representations of \(M(k)\) are rarely traditional Bessel functions. In particular, these calculations are used to show that \(c(N)>0\) in the case of the exterior square representation of the previous paragraph.
For the entire collection see [Zbl 1167.11002].
Now let \(G\) be a quasi-split, connected, reductive algebraic group over \(k\), and \(M\) a Levi subgroup of a self-associate parabolic \(P=MN\). F. Shahidi [Int. Math. Res. Not. 2002, No. 39, 2075–2119 (2002; Zbl 1025.22014)] constructed, using a densely defined map \(\phi:M\to N\), Bessel functions whose Mellin transforms represent the local coefficients \(C_\psi(s,\pi)\) associated to a representation of \(M(k)\), thereby generalising Cogdell and Piatetski-Shapiro’s approach. Stability of root numbers for all groups \(G\) of interest in functoriality was then established by a number of authors by building gamma factors out of local coefficients and analysing the asymptotics of the relevant Bessel functions (e.g. J. W. Cogdell, H. H. Kim, I. I. Piatetski-Shapiro and F. Shahidi [Publ. Math., Inst. Hautes Étud. Sci. 99, 163–233 (2004; Zbl 1090.22010)] treat stability for \(G=\text{SO}_{2n+1}\), \(\text{SO}_{2n}\), and \(\text{Sp}_{2n}\)).
Now we turn to the paper at hand. Quoting from the introduction, “This paper is an attempt to put the problem of general stability of root numbers [...] in the context of appropriate group actions and their complexity as well as the underlying algebraic geometry. Our aim is modest and we claim no “theorems”, rather we establish some basic facts about the ingredients which go into our expressions for local coefficients which must be stabilized.”
The paper works in the generality of a quasi-split, connected, reductive algebraic group \(G\) and Levi subgroup \(M\) as above. In Proposition 2.4.7 it is shown that \(M_\phi\), (the closure of) the image of \(\phi\), is an irreducible closed subvariety of \(M\) and that it is stable under conjugation by an appropriate subgroup \(H\leq M\). Following D. Luna and T. Vust [Comment. Math. Helv. 58, 186–245 (1983; Zbl 0545.14010)], Shahidi recalls the definition of the complexity of an action of an affine algebraic group \(H\) on an irreducible variety \(X\): \(c(X):=\dim X-\max \{\dim Bx : x\in X\}\), where \(B\) is a Borel subgroup of \(H\). In the case at hand, \(H\) acts by conjugation and we therefore have complexities \(c(M_\phi)\), \(c(N)\); in Proposition 3.11 it is shown that, in most cases, \(c(N)-c(M_\phi)=\deg\phi\).
Combining this proposition with a deep result of R. Sundaravaradhan [Int. Math. Res. Not. 2008, No. 2, article ID rnm141, 1–22 (2008; Zbl 1232.22009)], the complexity \(c(N)\) can be calculated with relative ease. Importantly, this complexity is exactly the dimension of the quotient space \(H\backslash N\), over whose \(k\)-points one must integrate to form the Mellin transforms of the Bessel functions associated to \(G\). When \(G\) is a classical group, this complexity is zero; conversely, for the pair \((G,M)=(\text{SO}_{2n}, \text{GL}_n)\), which gives the exterior square \(L\)-function of a representation of \(\text{GL}_n(k)\), the complexity \(c(N)\) is non-zero and the problem of stability is open. Quoting the final remark, “It therefore appears that so far we have been only able to prove the stability when \(c(N)=0\).”
The appendix, by W. Kuo, contains calculations concerning dimension/rank inequalities for certain orbit spaces associated to \((G,M)\). These inequalities are related to the complexity ideas in the main article and they show that the partial Bessel functions associated to representations of \(M(k)\) are rarely traditional Bessel functions. In particular, these calculations are used to show that \(c(N)>0\) in the case of the exterior square representation of the previous paragraph.
For the entire collection see [Zbl 1167.11002].
Reviewer: Matthew Morrow (Nottingham)
MSC:
| 22E35 | Analysis on \(p\)-adic Lie groups |
| 14L30 | Group actions on varieties or schemes (quotients) |
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
