Level one algebraic cusp forms of classical groups
By Gaetan Chenevier and David Renard
This site is a companion to the paper Level one algebraic cusp forms of classical groups of small rank by the
authors (to appear in Memoirs of the AMS).
You will find various data and tables relative to this paper, we hope to add new data soon.
We consider cuspidal automorphic representations Π of GL_n over the field of rational numbers Q such that :
Π^* = Π ⊗ |·|^w for some integer w,
Π is unramified at each finite place,
Π is algebraic regular at the infinite place (see loc. cit.).
The infinitesimal character of the archimedean component of such a Π may
be viewed as a semisimple conjugacy class in M_n(C). The opposite of
its eigenvalues are by definition
distinct integers k_1 > k_2 > … > k_n called the weights of Π.
When
n = 0 mod 4 we actually allow the possibility k_{n/2+1}=k_{n/2}.
Moreover, up to twist by |·|^{k_n}, we may assume that k_n=0, so that
k_1=k_i+k_{n+i-1}=w for each i.
We define the Hodge weights of Π as the integers w_1 > w_2 > … w_r
≥ 0, where r=[n/2] and w_i=2k_i-w. The eigenvalues of the infinitesimal character of
Π⊗|·|^{w/2} are thus the ± w_i/2, plus 0 if n is
odd.
The selfdual representation Π⊗|·|^{w/2} is either symplectic
(case w_i odd) or orthogonal (case w_i even) in the sense of Arthur. We also say that Π is symplectic or orthogonal accordingly.
If it is symplectic then n is even. If it is orthogonal and n is even then n ≡ 0 mod 4.
We denote by :
S(w_1,…,w_r) the number
of symplectic Π as above for GL(2r) with odd Hodge weights w_1>…>w_r.
O(w_1,…w_r) the number of orthogonal Π as above for GL(2r) with even Hodge
weights w_1>…>w_r.
O^*(w_1,…,w_r) the number of (necessarily orthogonal) Π as above for GL(2r+1) with even Hodge
weights w_1>…>w_r>0.
For instance, it is well-known that S(w) is the dimension of the space of cusp forms of weight w+1 for SL(2,Z), namely S(w)=[(w+1)/12], unless w >1 and w ≡ 1 mod 12 in which case S(w)=[(w+1)/12]-1.
Problem : Compute those numbers.
Our main result is a solution of this problem whenever n ≤ 8 (and all Hodge
weights).
WARNING : Part of these results is still conditional, see the paper.
As explained in the paper, orthogonal Π satisfying the assumptions above
do not exist in rank n ≡ 2 mod 4 or if the half sum of the w_i is not congruent to [(r+1)/2] mod 2. We shall thus always assume this extra congruence in the formulas below.
O^*(w)=S(w/2) (symmetric square lift).
The table of the [w,v,S(w,v)] for w≤ 99 (following Arthur and
Tsushima's formula).
O(w,v)=S((w+v)/2)S((w-v)/2) if v>0, and O(w,0)=S(w/2)(S(w/2)-1)/2 (tensor product
lift).
O^*(w,v)=S((w+v)/2,(w-v)/2) (reduced lambda square lift).
The table of the [w,v,u,S(w,v,u)] for w ≤ 149.
The table of the [w,v,u,S(w,v,u)] such that S(w,v,u) is nonzero, for w ≤ 149.
The table of the [w,v,u] such that S(w,v,u)=1 for w ≤ 149 .
The table of the
[w,v,u,2*O(w,v,u,0)+O^*(w,v,u)] for w ≤ 40.
The table of the
[w,v,u,2*O(w,v,u,0)+O^*(w,v,u)] with 2*O(w,v,u,0)+O^*(w,v,u) nonzero for w
≤ 40.
(Application) When 2*O(w,v,u,0)+O^*(w,v,u) ≤ 1, the last table
gives O^*(w,v,u).
Commented table of [w,v,u,m(w,v,u)], where m(w,v,u)
is the dimension the space of vector valued Siegel modular forms of genus 3 and
infinitesimal character ± w/2, ± v/2, ± u/2, deduced from this list when w ≤ 30
(including the endoscopic classification of all the eigenforms).
Bergstrom, Faber and Van der Geer's conjectural table of the [w,v,u,m(w,v,u)] with m(w,v,u) nonzero for
w<48 (see this paper, we are grateful to
these authors for sharing this list).
The table of the [w,v,u,t,O(w,v,u,t)] for 0 < t < w ≤ 40.
The table of the [w,v,u,t,O(w,v,u,t)] such that O(w,v,u,t) is nonzero for 0 < t < w ≤ 40.
The table of the [w,v,u,t,S(w,v,u,t)] for w ≤ 53 .
The table of the [w,v,u,t,S(w,v,u,t)] such that S(w,v,u,t) is nonzero, for w ≤ 53.
The table of the [w,v,u,t] such that S(w,v,u,t)=1 for w ≤ 53.
Let G_2(w,v) be the conjectural number of orthogonal Π of GL(7) with Hodge weights
w+v>w>v which are lifts of G_2.
The table of the [w,v,G_2(w,v)] for w+v ≤ 100.
Our results above depend on the (non conjectural) computation of the dimension of the spaces of automorphic forms of level 1 for certain
semisimple groups G over the integers whose real points are compact, namely a certain form of SO(7), SO(8), SO(9) and G_2. We give below
tables for those dimensions.
The group G(
Z) is the positive Weyl group of E_7.
Denote by V(a,b,c) the irreducible representation of SO(7,R) with (standard) highest weight a ≥ b ≥ c ≥ 0.
This table contains the list of [a,b,c, m(a,b,c)] where
a ≥ b ≥ c is a dominant weight and m(a,b,c) is the dimension of the invariants of G(Z) in the space
V(a,b,c), for a ≤ 72.
The same table but in the infinitesimal character parameterization : the list of [2a+5,2b+3,2c+1,m(a,b,c)].
The source of the computer program, for PARI/GP 2.5.0.
The group G(
Z) is the positive Weyl group of E_8.
Denote by V(a,b,c,d) the irreducible representation of SO(8,R) with (standard) highest weight a ≥ b ≥ c ≥ d ≥ 0.
This table contains the list of [a,b,c,d, m(a,b,c,d)] where
a ≥ b ≥ c ≥ d is a dominant weight and m(a,b,c,d) is the dimension of the invariants of G(Z) in the space
V(a,b,c,d), for a ≤ 17.
The same table but in the infinitesimal character parameterization : the list of
[2a+6,2b+4,2c+2,2d,m(a,b,c,d)].
The source of the computer program, for PARI/GP 2.5.0.
The group G(
Z) is isomorphic to the Weyl group of E_8.
Denote by V(a,b,c,d) the irreducible representation of SO(9,R) with (standard) highest weight a ≥ b ≥ c ≥ d ≥ 0.
This table contains the list of [a,b,c,d, m(a,b,c,d)] where
a ≥ b ≥ c ≥ d is a dominant weight and m(a,b,c,d) is the dimension of the invariants of G(Z) in the space
V(a,b,c,d), for a ≤ 23.
The same table but in the infinitesimal character parameterization : the list of
[2a+7,2b+5,2c+3,2d+1,m(a,b,c,d)].
The source of the computer program, for PARI/GP 2.5.0.
The group G(
Z) is a finite group of order 12096.
Here are its elements as a subgroup of GL(7,Z[1/2]), following Cohen, Nebe, Plesken.
Let ω_1=2α+β and ω_2=3α+2β denote the standard fundamental weights of G(R).
Denote by V(a,b) the irreducible representation of G(R) with highest weight a ω_1 + b ω_2.
This table contains the list of [a,b,m(a,b)] where
a, b ≥ 0 and m(a,b) is the dimension of the invariants of G(Z) in the space
V(a,b), for a+b ≤ 50.
The same table but in the infinitesimal character
parameterization, viewed as a conjugacy class in so(7) : the list of
[2a+4b+6, 2a+2b+4,2b+2,m(a,b)].
The source of the computer program, for PARI/GP 2.5.0.