×

zbMATH — the first resource for mathematics

On the Howe duality conjecture. (English) Zbl 0624.22011
The author shows that Howe’s global duality conjecture would follow from the local duality conjecture at all places, and he proves the latter in a number of new cases. Moreover, he conjectures a refinement in the global conjecture, concerning multiplicities.
Let K be a number field with adèle ring \({\mathbb{A}}\), Q a nondegenerate quadratic form in m variables defined over K. Let \(G=O(Q)\), and let \((\pi)\subset L^ 2_{cusp}(G(K)\setminus G({\mathbb{A}}))\) be the isotypic subspace for an irreducible cuspidal representation \(\pi\) of G(\({\mathbb{A}})\). Now \(G\times Sp_ r\subset Sp_{mr}\) for any positive integer r. By integrating (over G(K)\(\setminus G({\mathbb{A}}))\) every \(f\in (\pi)\) against every theta-kernel for the oscillator representation of \(Sp_{mr}({\mathbb{A}})\) one generates a complex subspace \({\mathcal A}_ r(\pi)\) of functions on \(Sp_ r(K)\setminus Sp_ r({\mathbb{A}}).\)
Let \(L_ r=L_ r(Q)\) be the subspace of \(L^ 2_{cusp}(G(K)\setminus G({\mathbb{A}}))\) orthogonal to all theta-kernels for \(Sp_{mr}\), and let \(R_ i=R_ i(Q)\) be the orthogonal complement of \(L_ 1\cap \cdot \cdot \cdot \cap L_ i\) in \(L_ 1\cap \cdot \cdot \cdot \cap L_{i- 1}.\)
Theorem: (1) \(L^ 2_{cusp}(G(K)\setminus G({\mathbb{A}}))=R_ 1\oplus \cdot \cdot \cdot \oplus R_ m\), orthogonal direct sum.
(2) If \(\pi \in R_ i\), then \({\mathcal A}_ i(\pi)\cap L^ 2_{cusp}(Sp_ i(K)\setminus Sp_ i({\mathbb{A}}))\neq (0).\)
Let \({\mathcal A}^ 0(\pi)\) denote the intersection in (2). Then the author states Howe’s global duality conjecture in the following form: Conjecture: Given \(\pi\) occuring in \(R_ i\), there is an irreducible representation \({\mathcal B}(\pi)\) of \(Sp_ i({\mathbb{A}})\) occuring in \(L^ 2_{cusp}(Sp_ i(K)\setminus Sp_ i({\mathbb{A}}))\) such that \({\mathcal A}^ 0(\pi)\) is a multiple of \({\mathcal B}(\pi)\). Moreover, the map \(\pi\mapsto {\mathcal B}(\pi)\) is injective. To this he adds a refinement: Conjecture: The multiplicity of \(\pi\) in \(R_ i\) is equal to that of \({\mathcal B}(\pi)\) in \({\mathcal A}^ 0(\pi).\)
The refinement would follow from a local multiplicity one supplement to Howe’s local duality conjecture at all places. The author verifies the local multiplicity one conjecture in certain cases. Besides the orthogonal decomposition of \(L^ 2_{cusp}\) of O(Q) there is an analogous one for \(L^ 2_{cusp}\) of \(Sp_ n\). For \(n=2\), the Saito- Kurakawa subspace defined by Piatetski-Shapiro is properly contained in one piece of the orthogonal decomposition.

MSC:
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
22E50 Representations of Lie and linear algebraic groups over local fields
PDF BibTeX Cite
Full Text: Numdam EuDML
References:
[1] Arthur, J. : Eisenstein series and the trace formula . Proc. of Symposium in Pure Math. 33 (1979) 253-274. · Zbl 0431.22016
[2] Asmuth, C. : Weil representations of symplectic p-adic groups . Amer. Jour. of Math. 101(4) (1979) 885-909. · Zbl 0421.22011
[3] Bernshtein, I.N. and Zelevinski, A.V. : Representations of the group GL2(F) where F is a non-Archimedean Local field . Russian Math. Surveys 31(3) (May-June, 1976) 1-68. · Zbl 0348.43007
[4] Borel, A. and Jacquet, H. : Automorphic forms and automorphic representations . Proc. of Symposium in Pure Math. 33 (1979) 185-202. · Zbl 0414.22020
[5] Cartier, P. : Representations of p-adic groups: a survey . Proc. of Symposium in Pure Math. 33 (1979) 111-157. · Zbl 0421.22010
[6] Flath, D. : Decomposition of representations into tensor products . Proc. of Symposium in Pure Math. 33 (1979) 179-184. · Zbl 0414.22019
[7] Godement, R. and Jacquet, H. : Zeta functions of simple algebras . Springer Lecture Notes 260 (1972). · Zbl 0244.12011
[8] Gustafson, R. : The Degenerate Principal Series for Sp(2n) . Ph.D. Thesis, Yale (May, 1979). · Zbl 0482.22013
[9] Harish Chandra : Harmonic analysis on reductive p-adic groups (notes by G. van Dijk) . Springer Lecture Notes 162 (1970). · Zbl 0202.41101
[10] Howe, R. : \theta -series and invariant theory . Proc. of Symposium in Pure Math. 33 (1979) 275-287. · Zbl 0423.22016
[11] Howe, R. and Piatetski-Shapiro, I.I. : A counterexample to the ”generalized Ramanujan conjecture” for quasi-split groups . Proc. of Symposium in Pure Math. 33 (1979) 315-322. · Zbl 0423.22018
[12] Howe, R. and Piatetski-Shapiro, I.I. : Some examples of automorphic forms on Sp4 , Duke Math. Jour. 50(1) (1983) 55-106. · Zbl 0529.22012
[13] Jacquet, H. , Piatetski-Shapiro, I.I. and Shalika, J.A. : Automorphic forms on Gl(3) . Annals of Math. 109 (1979) 213-258. · Zbl 0401.10037
[14] Kashiwara, M. and Vergne, M. : On the Segal-Shale-Weil representation and harmonic polynomials . Inventiones Math. 44 (1978) 1-47. · Zbl 0375.22009
[15] Novodvorsky, M. : New unique models of representations of unitary groups . Comp. Math. 33 (1976) 289-295. · Zbl 0337.22018
[16] O’Meara, T. : Introduction to quadratic forms . Die Grundlehren der Math. Wiss., bd 117, Springer Verlag (1963). · Zbl 0107.03301
[17] Piatetski-Shapiro, I.I. : L-functions for GSp2 , preprint. · Zbl 1001.11020
[18] Piatetski-Shapiro, I.I. : On the Saito Kurokawa Lifting , preprint. · Zbl 0515.10024
[19] Rallis, S. : Langlands Functoriality and the Weil Representation . Amer. Jour. of Math. 104(3) (1982) 469-515. · Zbl 0532.22016
[20] Rallis, S. : On a relation between SL2 cusp forms and automorphic forms on orthogonal groups . Proc. of Symposium in Pure Math. 33 (1979) 297-314. · Zbl 0425.10034
[21] Rallis, S. and Schiffmann, G. : Distributions invariantes par le group orthogonal . Springer Lecture Notes 497 (1975) 494-642. · Zbl 0329.10016
[22] Rallis, S. and Schiffmann, G. : Représentations supercuspidales der groupe métaplectique . Kyoto Math. Journal 17 (1977) 567-603. · Zbl 0398.22023
[23] Shalika, J. : The Multiplicity One Theorem for GLn . Annals of Math. 100(1) (1974) 171-193. · Zbl 0316.12010
[24] Warner, G. : Harmonic analysis on semisimple Lie groups I . Grund. Math. Wiss. 188, Springer Verlag (1972). · Zbl 0265.22020
[25] Weil, A. : Sur certaines groupes d’operateurs unitares . Acta Math. 111 (1964) 143-211. · Zbl 0203.03305
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.