Abstract

We determine the number of level 1, polarized, algebraic regular, cuspidal

automorphic representations of GLn over Q of any given infinitesimal character,

for essentially all n ≤ 8. For this, we compute the dimensions of spaces of level 1

automorphic forms for certain semisimple Z-forms of the compact groups SO7, SO8,

SO9 (and G2) and determine Arthur’s endoscopic partition of these spaces in all

cases. We also give applications to the 121 even lattices of rank 25 and determinant

2 found by Borcherds, to level one self-dual automorphic representations of GLn

with trivial infinitesimal character, and to vector valued Siegel modular forms of

genus 3. A part of our results are conditional to certain expected results in the

theory of twisted endoscopy.

Received by the editor September 13, 2012 and, in revised form, July 18, 2013 and July 21,

2013.

Article electronically published on January 22, 2015.

DOI: http://dx.doi.org/10.1090/memo/1121

2010 Mathematics Subject Classification. Primary 11FXX; Secondary 11F46, 11F55, 11F70,

11F72, 11F80, 11G40, 11H06, 11R39, 11Y55, 22C05.

Key words and phrases. Automorphic representations, classical groups, compact groups, con-

ductor one, dimension formulas, endoscopy, invariants of finite groups, Langlands group of Z,

euclidean lattices, Sato-Tate groups, vector-valued Siegel modular forms.

The first author was supported by the C.N.R.S. and by the French ANR-10-BLAN 0114

project.

c

2015 American Mathematical Society

v