The $$\mathrm {PGSp}_{1,1}({\mathbb {A}})$$ PGSp 1 , 1 ( A ) -distinguished representations
- PDF / 415,882 Bytes
- 23 Pages / 439.37 x 666.142 pts Page_size
- 88 Downloads / 156 Views
Mathematische Zeitschrift
The PGSp1,1 (A)-distinguished representations Hengfei Lu1 Received: 8 September 2019 / Accepted: 4 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we use the regularized quaternionic Siegel–Weil formula to deal with the PGSp1,1 (A)-period problem for a cuspidal representation τ of PGSp4 (A E ), where E/F is a quadratic extension of number fields and E is contained in the 4-dimensional quaternion division algebra D of F with s D = 2. Keywords Siegel–Weil formula · Quaternionic Hermitian spaces · Period problems Mathematics Subject Classification 11F27 · 11F70 · 22E55
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 2 Global theta lifts for similitude groups . . . . . . . . . 2.1 Siegel–Weil formula . . . . . . . . . . . . . . . . . 3 Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . 4 Proof of Corollary 1.2 . . . . . . . . . . . . . . . . . . 5 Proof of Theorem 1.3 . . . . . . . . . . . . . . . . . . 6 Nongeneric tempered cuspidal representations of GSp4 6.1 Endoscopic case . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
1 Introduction Let F be a number field with an adele ring A. Assume that G is a reductive group defined over F. Let H be a closed subgroup of G. Given a cuspidal automorphic representation π of G(A), then π is said to be H (A)-distinguished or have a nonzero H (A)-period if the integral f (t)dt Z H (A)H (F)\H (A)
B 1
Hengfei Lu [email protected] Department of Mathematics, Weizmann Institute of Science, 234 Herzl St., P.O. Box 26, 76100 Rehovot, Israel
123
H. Lu
is nonzero for some vector f ∈ π, where Z H is the center of H . In many cases, it should be possible to characterize the H (A)-distinguished cuspidal representations as the images with respect to a functorial transfer to G from a third group G , which has been extensively studied by Sakellaridis and Venkatesh in [25] when G is split. This paper studies the pair (G, H ) = (Res E/F PGSp4 , PGSp4 ) and its inner forms, where E is a quadratic field extension of F and Res E/F denotes the Weil restriction of scalars. Fix a unitary character χ of F\A which in turn can be identified with a character on the center of GSp4 (A). Denote by L 2disc (GSp4 (F)\GSp4 (A), χ) the space of χ-equivariant, square integrable functions on GSp4 (F)\GSp4 (A). Let 2 (GSp4 (A), χ) be the set of equivalence classes of automorphic representations of GSp4 (A) that are constituents of L 2disc (GSp4 (F)\GSp4 (A), χ). According to t
Data Loading...
