Jacobi-Hecke algebras and a rationality theorem for a formal Hecke series
In the paper under review the author develops a Hecke theory for the Jacobi group of degree \((1,m)\) over a local field of characteristic \(\neq 2\). In this case the symplectic group is always \(SL(2)\). Under a certain maximality condition the Hecke algebra is shown to decompose into a direct product of two algebras. The irregular component is finite-dimensional and semisimple. The Satake isomorphism maps the regular component onto the algebra of Laurent polynomials in one variable invariant under the Weyl group of \(SL(2,F)\). Finally a rationality theorem is derived for the Hecke series along the lines of \textit{S. Böcherer}'s proof [Abh. Math. Sem. Univ. Hamb. 56, 35-47 (1986; Zbl 0613.10026)].
scientific article; zbMATH DE number 1334555