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.

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. #### 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.

The infinitesimal character of the archimedean component of such a Π may be viewed as a semisimple conjugacy class in M_n(

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

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 :

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.

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.

(Application) When 2*O(w,v,u,0)+O^*(w,v,u) ≤ 1, the last table gives O^*(w,v,u).

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.

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. 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.

Denote by V(a,b,c) the irreducible representation of SO(7,**R**) with (standard) highest weight a ≥ b ≥ c ≥ 0.

The group G(**Z**) is the positive Weyl group of E_8. 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.

Denote by V(a,b,c,d) the irreducible representation of SO(8,**R**) with (standard) highest weight a ≥ b ≥ c ≥ d ≥ 0.

The group G(**Z**) is isomorphic to the Weyl group of E_8. 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.

Denote by V(a,b,c,d) the irreducible representation of SO(9,**R**) with (standard) highest weight a ≥ b ≥ c ≥ d ≥ 0.

The group G(**Z**) is a finite group of order 12096. 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.

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.