1.1. A counting problem
Let n ≥ 1 be an integer. Consider the cuspidal automorphic representations π
of GLn over Q (see [GGPS66, Ch. 3],[BJ79, §4],[Cog04]) such that:
π ⊗ | ·
for some w ∈ Z,
(b) (conductor 1) πp is unramified for each prime p,
(c) (algebraicity) π∞ is algebraic and regular.
Our main aim in this paper is to give for small values of n, namely for n ≤ 8,
the number of such representations as a function of π∞. Recall that by the Harish-
Chandra isomorphism, the infinitesimal character of π∞ may be viewed following
Langlands as a semisimple conjugacy class in Mn(C) (see §3.4, §3.6). Condition
that the eigenvalues of this conjugacy class are distinct integers. The
opposite of these integers will be called the weights of π and we shall denote them
by k1 k2 · · · kn. When n ≡ 0 mod 4, we will eventually allow that kn/2 =
kn/2+1 but to simplify we omit this case in the discussion for the moment. If π
satisfies (a), the necessarily unique integer w ∈ Z such that
π ⊗ | ·
called the motivic weight of π, and denoted w(π).
Problem 1.1. For any n ≥ 1, determine the number N(k1,k2, · · · , kn) of cus-
pidal automorphic representations π of GLn satisfying (a), (b) and (c) above, and
of weights k1 k2 · · · kn.
An important finiteness result of Harish-Chandra ([HC68, Thm. 1.1]) asserts
that this number is indeed finite, even if we omit assumption (a). As far as we
know, those numbers have been previously computed only for n ≤ 2. For n = 1,
the structure of the id` eles of Q shows that if π satisfies (a), (b) and (c) then
w(π) = 2 k1 is even and π = | ·
. By considering the central character of π,
this also shows the relation n w(π) = 2
ki for general n. More interestingly,
classical arguments show that N(k − 1, 0) coincides with the dimension of the space
of cuspidal modular forms of weight k for SL2(Z), whose dimension is well-known
(see e.g. [Ser70]) and is
[k/12]. Observe that up to twisting π by
is no loss of generality in assuming that kn = 0 in the above problem. Moreover,
condition (a) implies for i = 1, · · · , n the relation ki + kn+1−i = w(π).
term algebraic here is in the sense of Borel [Bor77, §18.2], and is reminiscent to Weil’s
notion of Hecke characters of type A0: see §3.6. Langlands also uses the term of type Hodge, e.g.
in [Lan96, §5]. See also [BG], who would employ here the term L-algebraic, for a discussion of
other notions of algebraicity, as the one used by Clozel in [Clo90].
denote by [x] the floor of the real number x.