General SectionGamma factors and converse theorems for classical groups over finite fields☆
Introduction
Local gamma factors play essential roles in the theories of automorphic forms and representations of p-adic groups, especially in the local Langlands correspondence conjecture, Langlands functoriality conjecture, and local converse theorems. Local gamma factors can usually be defined using Langlands-Shahidi method or Rankin-Selberg method, at least for generic representations. In this paper, we prove multiplicity one theorems of certain Fourier-Jacobi models (analogs for Bessel models were proved in [GGP12b]) over finite fields and define the finite fields analogue of local gamma factors for irreducible generic cuspidal representations of quasi-split classical groups , using the Rankin-Selberg method. We also obtain explicit formulas for these gamma factors in terms of corresponding Bessel functions. These gamma factors provide important invariants for generic cuspidal representations and are expected to play important roles in the representation theory of these groups over finite fields. There are interesting questions that how these invariants are related to the Deligne-Lusztig theory on virtual characters ([DL76], [L84]), and to the finite fields analogue of the Gan-Gross-Prasad conjecture ([GGP12b]).
Over p-adic fields, the uniqueness of Bessel and Fourier-Jacobi models for classical groups were proved in [AGRS10] and [Su12] respectively. But over finite fields, the general statement of uniqueness of Bessel and Fourier-Jacobi models is not true for all irreducible representations of , see [GGP12b, §4,5] for example. This suggests that we cannot have a uniform proof using distribution theory as in the p-adic fields case. To conquer these difficulties, we link the finite fields case with the p-adic fields case. To do so, we need to restrict our representations of to irreducible cuspidal representations and use the theory of depth zero representations of over p-adic fields. The Bessel model case has been carried out in [GGP12b]. The main difficulty in the Fourier-Jacobi model case is to connect the Weil representations over finite fields and p-adic fields. To make this connection, we use the generalized lattice models for Weil representations.
Over finite fields, the Bessel and Fourier-Jacobi models are special cases of Generalized Gelfand-Graev models considered by Kawanaka, see [K85], [K86]. There are many results on the computation of such multiplicities in more general settings, for example see [L92], [Gec99], [GeH08]. See also [LZ19], [LZ21] for certain uniqueness results of Fourier-Jacobi models for , , and the split exceptional group of type over finite fields.
As applications of the gamma factors defined above, we prove the converse theorems for these groups, namely, -twisted gamma factors, , will uniquely determine irreducible generic cuspidal representations of . Therefore, these -twisted gamma factors form complete sets of invariants for irreducible generic cuspidal representations of .
Over finite fields, gamma factors and converse theorems have been defined and considered for general linear groups and the split exceptional group of type . Roditty [Ro10] defined gamma factors for cuspidal representations of over finite fields. Nien in [Ni14] proved the converse theorem for cuspidal representations of , using special properties of Bessel functions and the twisted gamma factors defined by Roditty. The authors defined the -twisted gamma factors for and proved the converse theorem for generic cuspidal representations of the split exceptional group in [LZ21]. Gamma factors over finite fields were defined in a more general context in [BK00]. In [NZ21], Nien and Zhang verified a converse theorem for Gauss sum of characters of finite fields and showed that such a character is determined by Gauss sum twisted by characters of , for , or for in the appendix by Zhiwei Yun.
Similar to local fields cases, it is expected that -twisted gamma factors for irreducible generic cuspidal representations of can also be defined using Langlands-Shahidi method. In future work, the authors plan to define -twisted gamma factors using Langlands-Shahidi method and verify the consistency with those defined in this paper using Rankin-Selberg method.
This paper does not include the case of , which is a work in progress of a student of the first named author. The converse theorem for is expected to be more subtle.
Following is the structure of this paper. For , we prove a multiplicity one theorem in Section 2, define the GL-twisted gamma factors in Section 3, and prove the converse theorem in Section 4. Cases of will be considered in Sections 5, 6, 7, respectively.
The authors would like to thank James Cogdell, Clifton Cunningham, Dihua Jiang and Freydoon Shahidi for their interest, constant support and encouragement. The authors also would like to thank the anonymous referee for careful reading and many useful suggestions.
Section snippets
A multiplicity one theorem for
Let F be a p-adic field with odd residue characteristic. Let be the ring of integers of F, be the maximal ideal of , and be a fixed generator. Let be the residue field. Let be the natural projection. Let ψ be a fixed unramified additive character of F, and let be the character of k defined by
Generic representations and Bessel functions
In this subsection, we introduce the notion of Bessel functions for generic representations of . In [PS83], Bessel functions was used to study representations of over a finite field k as an analogy of representations of over p-adic fields. See [Co14] for a nice survey. Many constructions were extended to in [Ro10].
Let be the upper triangular unipotent subgroup of . Let be the generic character of U defined by Let π be an
A converse theorem for
In this section, we still let k be a finite field with odd characteristic. The purpose of this section is to prove the following Theorem 4.1 Let π and be two irreducible -generic cuspidal representations of with the same central character. If for all irreducible generic representations τ of and for all n with , then .
Notice that in the above theorem π and are assumed to be generic with respect to the same generic character. A p-adic version of Theorem 4.1
Gamma factors and a converse theorem for
The technique used in the previous sections can also be used to define gamma factors for for and then give a proof of the local converse theorem for , where k is a finite field of odd characteristic, is the quadratic extension of k, and is the quasi-split unitary group of size 2r associated with the extension . Since the proof is quite similar, we just give a sketch in this section and highlight the differences in the proof.
Let F be a p-adic field with
Gamma factors and a converse theorem for
Let k be a finite field of odd characteristic and be the quadratic extension of k. Let be the nontrivial Galois element in . Let . As in previous sections, for a positive integer m, set Let Note that, if is even, then the definition of is a little bit different from that defined in §5. In fact, the unitary group in §5 was defined by a skew-Hermitian form and the group considered in this
A converse theorem for
In this section, let k be a finite field without any restriction on the characteristic.
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The first-named author is partially supported by NSF Grants DMS-1702218, DMS-1848058, and start-up funds from the Department of Mathematics at Purdue University. The second-named author is partially supported by NSFC grant 11801577.