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,
Article electronically published on January 22, 2015.
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
2015 American Mathematical Society