1549380232-Automorphic_Forms_and_Applications__Sarnak_

!AS/Park City Mathematics Series Volume 12 , 2002 Automorphic Forms on Reductive Groups Armand Borel Introduction The goal of th ...

8 ARMAND BOREL, AUTOMORPHIC FORMS ON REDUCTIVE GROUPS The universal enveloping algebra U(g) is identified with the algebra of le ...

FIRST PROPERTIES OF AUTOMORPHIC FORMS 9 2.2. Remark. The notion of moderate growth (but not the exponent m) is independent of ...

10 ARMAND BOREL, AUTOMORPHIC FORMS ON REDUCTIVE GROUPS £ (resp. μ). Then a Casimir operator can be written Cg = -I: xr + I: yJ, ...

FIRST PROPERTIES OF AUTOMORPHIC FORMS 11 which proves (9). If C is a compact subset of G, then (10) The property ( nl) of JJ. ...

12 ARMAND BOREL, AUTOMORPHIC FORMS ON REDUCTIVE GROUPS Assuming (A3) expressed in this way, we let A(r, J, ~) be the space of au ...

REDUCTIVE GROUPS (REVIEW) 13 4.1.2. Parabolic subgroups. The general definition of parabolic subgroups will be recalled in ( 4 ...

14 ARMAND BOREL, AUTOMORPHIC FORMS ON REDUCTIVE GROUPS into F[G]. We let X(G) be the group of morphisms of G into px. If>. E ...

REDUCTIVE GROUPS (REVIEW) 15 4.3.1. Let G be a connected reductive F-group. Let S = Z(G)^0 be the identity component of its ce ...

16 ARMAND BOREL, AUTOMORPHIC FORMS ON REDUCTIVE GROUPS 4.4.3. The parabolic F-subgroups of G are, by definition, the groups of r ...

REDUCTIVE GROUPS (REVIEW) 17 4.4.6. Caution. The data constructed above depend on the choice of F. It would h ave b een more c ...

18 ARMAND BOREL, AUTOMORPHIC FORMS ON REDUCTIVE GROUPS representation C f-t a.c.a-^1. To determine them, it is convenient to wri ...

ARITHMETIC SUBGROUPS. REDUCTION THEORY 19 (where A+ is the positive Weyl chamber), also called a Cartan decomposition. Let p E ...

20 ARMAND BOREL, AUTOMORPHIC FORMS ON REDUCTIVE GROUPS consists of semisimple elements. The proofs of these statements reduces e ...

ARITHMETIC SUBGROUPS. REDUCTION THEORY 21 5.2.3. Lemma. Let I C J be subsets of .0. and 61 = 6 p 1 ,t,w a Siegel set with resp ...

22 ARMAND BOREL, AUTOMORPHIC FORMS ON REDUCTIVE GROUPS We can assume Ap to be in diagonal form. The entries are then weights of ...

ARITHMETIC SUBGROUPS. REDUCTION THEORY 23 5.6. Proposition. Let f be a smooth Z(g)-finite function on f\G. Then there exist ex ...

24 ARMAND BOREL, AUTOMORPHIC FORMS ON REDUCTIVE GROUPS for any .A E X(A). Write x = n.m.a.k as usual. Then a(x.c) = a(x).a(k.c). ...

CONSTANT TERMS. THE BASIC ESTIMATE 25 (b) We may take for H.S. norm on Lp the restriction of the H .S. norm on G for a given e ...

26 ARMAND BOREL, AUTOMORPHIC FORMS ON REDUCTIVE GROUPS the union of the translates etXj 6 for 0 ::::; t ::::; l. There exists a ...