論文

Howe and Tan (Bull Am Math Soc 28:1–74, 1993) investigated a degenerate principal series representation of indefinite orthogonal groups O(b+,b-) and explicitly described its composition series. In particular it contains a unique unitarizable irreducible submodule Π, which is isomorphic to a cohomological representation. In this paper we construct orthogonal automorphic forms locally corresponding to Π as theta liftings of holomorphic Mp2(R) cusp forms by using the Borcherds’ method (Invent Math 132:491–562, 1998). We propose a special choice of Schwartz functions to define the liftings, which yields precise descriptions of their Fourier expansions.

© 2018 World Scientific Publishing Co. Pte Ltd. All rights reserved. Let H G/K be the symmetric tube domain associated with the Jordan algebra Hermr(F), F = R, C, or H, and X = G/P be its Shilov boundary. Also, let IGP (χ) be a degenerate principal series representation of G. Then we investigate the Bessel integrals assigned to functions in general K-types of IG P (χ). We give individual upper bounds of their supports, when IGP(χ) is reducible. We also use the upper bounds to give a partition for the set of all K-types in IGP (χ), that turns out to explain the G-module structure of IGP (χ). Thus, our results concretely realize a relationship observed by Kashiwara and Vergne [K-types and singular spectrum, in Noncommutative Harmonic analysis, Lecture Notes in Mathematics, Vol. 728 (Springer, 1979), pp. 177-200] between the Fourier supports and the asymptotic K-supports assigned to G-submodules in IGP (χ).

© 2020 World Scientific Publishing Company. By extending arguments by Gritsenko, we construct a lifting of skew-holomorphic Jacobi cusp forms of odd weight k + 1 and index N to Siegel paramodular forms of degree 2 with L2-integrability.