Criteria for the existence of cuspidal theta representations
Friedberg and Ginzburg Res. Number Theory
Criteria for the existence of cuspidal theta representations
Solomon Friedberg 0
David Ginzburg 1
0 Department of Mathematics, Boston College , Chestnut Hill, MA 02467-3806 , USA
1 School of Mathematical Sciences, Tel Aviv University , Ramat Aviv, 6997801 Tel Aviv , Israel
Theta representations appear globally as the residues of Eisenstein series on covers of groups; their unramified local constituents may be characterized as subquotients of certain principal series. A cuspidal theta representation is one which is equal to the local twisted theta representation at almost all places. Cuspidal theta representations are known to exist but only for covers of GLj, j ≤ 3. In this paper we establish necessary conditions for the existence of cuspidal theta representations on the r-fold metaplectic cover of the general linear group of arbitrary rank.
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canonical projection from Bn(r)(A) to Tn(r)(A), and further induce it to the group GL(nr)(A).
We abuse the notation slightly and write this induced representation IndBGn(Lr)(nr(A)(A))μs. It
follows from [12] that this construction is independent of the choice of A and of the extension
of ωs. Forming the Eisenstein series E(g, s) attached to this induced representation, it
folr+1
lows from [12] that when μs = δB2nr (with δBn the modular function of Bn), this Eisenstein
series has a nonzero residue representation. This is the representation (nr).
Let ν be a finite place for F such that |r|ν = 1. Defining similar groups over the local
field Fν , it follows from [12] that the local induced representation IndBGn(Lr)(nr(F)(νF)ν)δ r2+r1 has a
Bn
unique unramified subquotient which we again denote by (nr). This representation is
also the unique unramified subrepresentation of IndBGn(Lr)(nr(F)(νF)ν)δ r2−r1 . If χν denotes an
unramBn
1 r−1
ified character of Fν×, then the twisted induced representation IndBGn(Lr)(nr(F)(νF)ν)χνr δB2nr is also
reducible, and one can define the local twisted theta representation (nr,χ)ν as the unique
unramified subrepresentation. Here the twisting means that the induction is from a
genuine character of the group Z Tn(r)(Fν ) such that
The group {(tr , ζ ) | t ∈ Tn(Fν ), ζ ∈ μr } is in general a proper subgroup of Z Tn(r)(Fν ) ,
1
so this condition does not uniquely specify the twisting character χνr . Since the local
calculations below are independent of this choice, we do not indicate it in the notation.
Returning to the global case, we have
Definition 1 An automorphic representation π of GL(nr)(A) is called a theta
representation if for almost all places ν there are unramified characters χν such that the unramified
constituent of π is equal to (nr,χ)ν . If π is cuspidal, we say that π is a cuspidal theta
representation.
The interesting cases of such theta representations are when the local characters χν are
the unramified constituents of a global automorphic character χ . We shall write (nr,χ) for
such a representation.
Examples of such representations may be constructed as follows. Suppose that χ = χ r
1
for some global character χ1. Then one can construct theta representations (nr,χ) by means
of residues of Eisenstein series, by [12] (the case χ = 1 was described above). However,
these representations are never cuspidal.
In Flicker [6], a classification of all theta representations for the covering groups GL(2r)(A),
r ≥ 2, with c = 0 in the sense of [12] was given using the trace formula. The case n = r = 2
was also studied by Gelbart and Piatetski-Shapiro [7]. When n = r = 3, Patterson and
Piatetski-Shapiro [15] constructed a cuspidal theta representation (nr,χ) for any χ which is
not of the form χ13, again for the cover with c = 0. This construction applied the converse
theorem. This approach was used when n = r = 4 by Wang [18], and results were obtained
subject to certain technical hypotheses. No other examples of such representations are
known.
The basic problem is then to understand for what values of r and n, and for what
characters χ , there exists a cuspidal theta representation (nr,χ) . We shall give a necessary
condition for the existence of such a representation. However, we do not determine
whether or not these conditions are sufficient.
First, if r < n such cuspidal representations do not exist. This follows trivially since
every cuspidal automorphic representation of GL(nr)(A) must be generic, but the local
unramified representation (nr,χ)ν is not generic if r < n. Hence we may assume that r ≥ n.
Our main result is
Theorem 1 Fix a natural number r, and an automorphic character χ of GL1(A). Then
there is at most one natural number n such that there is a nonzero cuspidal theta
representation (nr,χ) . Moreover, if such n exists, then n divides r. If a cuspidal theta representation
(nr,χ) exists for some n which divides r, n ≥ 3, then χ = χ1r for any character χ1.
For n = 2 and twisting parameter c = 0, this result follows from [6]. This includes the
last assertion: if a cuspida (...truncated)