1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 3. KNOWN BOUNDS FOR THE CUSPIDAL SPECTRUM 67 Of course now this is weaker than the results of Kim and Shahidi, or Kim-Sa ...

68 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMS 18.1, 18 .6]. Thus it has a natural Borel structure; we will consider measure ...

LECTURE 3. KNOWN BOUNDS FOR THE CUSPIDAL SPECTRUM 69 This follows from theorems of Harish-Chandra (in the real case) and J. Bern ...

70 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMS We also note an obvious corollary of Theorem 3.4. Lemma 3.6. -Assume H C G is ...

LECTURE 3. KNOWN BOUNDS FOR THE CUSPIDAL SPECTRUM 71 First of all, an easy argument [19a,§ 5.1] shows that we may increase Sin o ...

72 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMS an adelic Hecke operator on the left-hand side. This is simply the action, b ...

LECTURE 3. KNOWN BOUNDS FOR THE CUSPIDAL SPECTRUM 73 The rest of the proof proceeds as in Burger-Sarnak [11]. Write r,r L i=l, . ...

74 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMS in the representation theory of local semi-simple groups over real or p-adic ...

LECTURE 3. KNOWN BOUNDS FOR THE CUSPIDAL SPECTRUM 75 We let the reader formulate an analogue of Conjecture 7. We now have the fo ...

76 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMS Since the eigenvalues ti of ta,p verify p-l/2+c(a)::::; ltil::::; pl/2+c(a) ( ...

LECTURE 3. KNOWN BOUNDS FOR THE CUSPIDAL SPECTRUM 77 3.6. A soft^6 proof of the Gelbart-Jacquet estimates. To illustrate the str ...

78 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMS Now return to the restriction (3.15). It follows from base change that 7rE,oo ...

Lecture 4. Applications: control of the spectrum In this lecture we will try to describe some results (beyond those of§ 3.1) whi ...

80 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMS class, the class of regular unipotent elements. This corresponds to the trivi ...

LECTURE 4. APPLICATIONS: CONTROL OF THE SPECTRUM 81 For Sp(g) this approach yields the following result. We work in fact with th ...

82 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMS is a congruence subgroup and assume an E A(Q) and deg(an ) = deg(TaJ --* oo w ...

LECTURE 4. APPLICATIONS: CONTROL OF THE SPECTRUM 83 A few remarks are in order. Recall t hat K azhda n 's property T for Gv is t ...

84 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMS consider the action of we on L~(r\G(JE.)/ K 00 ). Then the spectrum ofwc is b ...

LECTURE 4. APPLICATIONS: CONTROL OF THE SPECTRUM 85 Now assume 7r E G occurs in r\G, and 7r verifies (i)- (iii). Assume moreover ...

86 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMS cycles tend to be homologically non-trivial. For some examples see [5] and § ...