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Old forms on \(\text{GL}_n\). (English) Zbl 0758.11027

The author establishes results concerning vectors in a generic representation of \(\text{GL}_n\) over a local field, which may be used to obtain information about old and new automorphic forms.
Let \(\pi\) be an irreducible admissible generic representation of \(\text{GL}_n\) over a non-archimedean local field. In H. Jacquet, I. Piatetski-Shapiro and J. Shalika [Math. Ann. 256, 199–214 (1981; Zbl 0443.22013)] a decreasing family of compact open subgroups \(K(m)\), \(m\) a nonnegative integer, was defined with the following properties: for a certain integer \(c=c(\pi)\), the space of \(K(c)\)-invariant vectors of \(\pi\) is one dimensional, with a canonical generator (called a ‘new vector’), and there are no non-zero \(K(c-1)\)-fixed vectors. The author first studies the spaces of \(K(c+i)\)-fixed vectors of \(\pi\), \(i>0\). He gives a basis in terms of convolution with the \(\text{GL}_{n-1}\) Hecke algebra, which generalizes the Atkin-Lehner operators for \(\text{GL}_2\). In terms of automorphic forms, his result gives a basis for the space of old forms at a given level once the new forms of lower level are known. Next he gives more precise information about the new vector in the case of an unramified principal series, relating it to the \(L\) and \(\epsilon\) factors of the unique generic constituent of the representation. He concludes with some global applications. In particular, he gives lower bounds on the dimensions of the cuspidal cohomology associated with certain arithmetic subgroups of \(\text{SL}_3\) (viewed as an algebraic group over the rationals); these bounds are based on the computations of cuspidal cohomology of A. Ash, D. Grayson and P. Green [J. Number Theory 19, 412–436 (1984; Zbl 0552.10015)].

MSC:

11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E50 Representations of Lie and linear algebraic groups over local fields
11F75 Cohomology of arithmetic groups
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
11F60 Hecke-Petersson operators, differential operators (several variables)
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