1549380232-Automorphic_Forms_and_Applications__Sarnak_

82 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMS

is a congruence subgroup and assume an E A(Q) and

deg(an ) = deg(TaJ --* oo

where Tan =Kan Kc G(A1 ), and K is associated tor (see§ 3.3). For x E f\G(JR)

we can consider Ta,. x = ~ <\nx where (§ 3.3)

r an r = Il r oi,n.

Theorem 4.3. - If deg(an) _, oo, the sets Tanx become equidistributed on f\G

for the invariant measure, i.e., for f E Cc(f\G),

d ~ ) L f(oi,nX) _, r f(g)dμ.

eg an i Jr\G

See [17, Thm. 1.6]. In fact the rate of convergence can be estimated for suitably

differentiable functions f: see [19a, cor. 8.3] and [17, Thm. 1.7].

We note that lower bounds for the eigenvalues of the Laplacian would also

follow from this method, because Oh's results about restriction are also proved for

real groups. For instance, for Sp(g), an explicit, strictly positive and optimal lower

bound for the Laplacian on f\G/K= for r a congruence subgroup will follow from

[17, 3.3 Theorem].

Finally, we note that results such as Theorem 4.2 (or rather its analogue for

GL( n), see [17]) yield the best possible estimates for certain classical equidistribu-

tion problems, as noticed earlier by Sarnak [50] - see also Chiu [14]. For example,

one obtains the equidistribution, in the space SL( n, Z) \SL( n , JR) of lattices modulo

similitude, of the full Hecke transforms T(m)Ao (Ao a fixed lattice) for m _, oo;

moreover, since we obtain the best rate of convergence in the quantitative version

of Thm. 4.3, we also get the best possible estimate for the remainder term. See

[17, end of Sec. 1] as well as [19a, 19b].^2

4.2 The tau conjecture ...

In [17] results similar to Theorems 4.1 and 4.2 are obtained for a large variety

of groups G/Q. They fail, however, when G is anisotropic over Q, p (or v if we

consider a number field instead of Q) is a prime where G has rank 1, and the Hecke

operator is a Hecke operator "at p" -i.e., essentially belongs to the Hecke algebra

of G(Qp), cf. § 3.3. This was precisely the case studied by Burger and Sarnak for

a real prime -where eigenvalues of Hecke operators are replaced by eigenvalues of

the Laplacian, see § 4.1.

The excluded cases were studied in [15c]. The final result is the following.

Assume G is a semi-simple, simply connected, "absolutely quasi-simple" group over

Q (i.e., G does not factor as a group over Q)^3. Assume v (= p or oo) is a prime.

We can consider the set G\,aut of "automorphic" representations of Gv = G(Qv)

(§ 3.3). This is a subset of Gv which obviously contains the trivial representation.

Theorem 4.4 ([15c]). - The trivial representation is isolated in Gv,aut.

(^2) For a group associated to self duality such as Sp(g), the estimates for standard Hecke operators
are better tha n for GL(n) (compare Examples 3.1and3.2 in [17]). Is there an abstract reason for
this?
(^3) In fact [15c] deals with the general case where ((Ji is replaced by a number field.