arXiv subjects
Publications and source records attributed to Hiroshi Ishimoto.
We give a local newform theory for the metaplectic group of rank 1 over a non-archimedean local field of characteristic 0 with odd residual characteristic. It was suggested by some results of Kohnen and Ueda on the classical theory of modular forms of half-integral weight.
In an earlier paper of W. Casselman, the theory of local newforms and conductors was initiated. Later, Roberts and Schmidt studied local newforms for the metaplectic group of rank 1. In this paper we define and calculate conductors of irreducible genuine representations of the metaplectic group of rank 1 over non-archimedean local field of characteristic zero and of odd residual characteristic. Moreover, we shall give an explicit formulae for dimensions of spaces of local newforms, and show a compatibility with the local theta correspondence.
In an earlier book of Arthur, the endoscopic classification of representations of quasi-split orthogonal and symplectic groups was established. Later Mok gave that of quasi-split unitary groups. After that, Kaletha, Minguez, Shin, and White gave that of non-quasi-split unitary groups for generic parameters. In this paper we prove the endoscopic classification of representations of non-quasi-split odd special orthogonal groups for generic parameters, following Kaletha, Minguez, Shin, and White.
In our previous paper we gave proofs of Ibukiyama's correspondences on holomorphic Siegel modular forms of degree 2 of half-integral weight and integral weight. In this paper, we formulate and prove similar correspondences on automorphic forms on $\operatorname{Mp}_4(\mathbb{A}_\mathbb{Q})$ or $\operatorname{SO}_5(\mathbb{A}_\mathbb{Q})$ generating large discrete series representations at the real components. In addition, we show that the correspondences can be described in terms of local theta correspondences.
We prove Ibukiyama's conjectures on Siegel modular forms of half-integral weight and of degree 2 by using Arthur's multiplicity formula on the split odd special orthogonal group $\SO_5$ and Gan-Ichino's multiplicity formula on the metaplectic group $\Mp_4$. In the proof, the representation theory of the Jacobi groups also plays an important role.
In this paper, we formulate and prove a local intertwining relation for metaplectic groups assuming the local intertwining relation for non-quasi-split odd special orthogonal groups.