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Jones %Copyright: Copyright (c) 1997, 2009 American Mathematical Society %Copyright: (), with Reserved Font Name CMR5. % This Font Software is licensed under the SIL Open Font License, Version 1.1. % This license is in the accompanying file OFL.txt, and is also % available with a FAQ at: http://scripts.sil.org/OFL. %%EndComments FontDirectory/CMR5 known{/CMR5 findfont dup/UniqueID known{dup /UniqueID get 5000788 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /CMR5 def /FontBBox {-10 -250 1304 750 }readonly def /PaintType 0 def /FontInfo 9 dict dup begin /version (003.002) readonly def /Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050\051, with Reserved Font Name CMR5.) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def 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Jones %Copyright: Copyright (c) 1997, 2009 American Mathematical Society %Copyright: (), with Reserved Font Name CMTI8. % This Font Software is licensed under the SIL Open Font License, Version 1.1. % This license is in the accompanying file OFL.txt, and is also % available with a FAQ at: http://scripts.sil.org/OFL. %%EndComments FontDirectory/CMTI8 known{/CMTI8 findfont dup/UniqueID known{dup /UniqueID get 5000826 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /CMTI8 def /FontBBox {-35 -250 1190 750 }readonly def /PaintType 0 def /FontInfo 9 dict dup begin /version (003.002) readonly def /Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050\051, with Reserved Font Name CMTI8.) readonly def /FullName (CMTI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def 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Jones %Copyright: Copyright (c) 1997, 2009 American Mathematical Society %Copyright: (), with Reserved Font Name CMTI10. % This Font Software is licensed under the SIL Open Font License, Version 1.1. % This license is in the accompanying file OFL.txt, and is also % available with a FAQ at: http://scripts.sil.org/OFL. %%EndComments FontDirectory/CMTI10 known{/CMTI10 findfont dup/UniqueID known{dup /UniqueID get 5000828 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /CMTI10 def /FontBBox {-35 -250 1124 750 }readonly def /PaintType 0 def /FontInfo 9 dict dup begin /version (003.002) readonly def /Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050\051, with Reserved Font Name CMTI10.) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def /UnderlinePosition -100 def /UnderlineThickness 50 def end readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 52 /four put dup 58 /colon put dup 59 /semicolon put dup 65 /A put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 76 /L put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 127 /dieresis put readonly def currentdict end currentfile eexec 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Jones %Copyright: Copyright (c) 1997, 2009 American Mathematical Society %Copyright: (), with Reserved Font Name CMR7. % This Font Software is licensed under the SIL Open Font License, Version 1.1. % This license is in the accompanying file OFL.txt, and is also % available with a FAQ at: http://scripts.sil.org/OFL. %%EndComments FontDirectory/CMR7 known{/CMR7 findfont dup/UniqueID known{dup /UniqueID get 5000790 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /CMR7 def /FontBBox {-27 -250 1122 750 }readonly def /PaintType 0 def /FontInfo 9 dict dup begin /version (003.002) readonly def /Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050\051, with Reserved Font Name CMR7.) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def 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7B1E0E20406023E7E7CE8619F12E7D0F5560 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark {restore}if %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.0: CMR10 003.002 %%Title: CMR10 %Version: 003.002 %%CreationDate: Mon Jul 13 16:17:00 2009 %%Creator: David M. Jones %Copyright: Copyright (c) 1997, 2009 American Mathematical Society %Copyright: (), with Reserved Font Name CMR10. % This Font Software is licensed under the SIL Open Font License, Version 1.1. % This license is in the accompanying file OFL.txt, and is also % available with a FAQ at: http://scripts.sil.org/OFL. %%EndComments FontDirectory/CMR10 known{/CMR10 findfont dup/UniqueID known{dup /UniqueID get 5000793 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /CMR10 def /FontBBox {-40 -250 1009 750 }readonly def /PaintType 0 def /FontInfo 9 dict dup begin /version (003.002) readonly def /Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050\051, with Reserved Font Name CMR10.) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark {restore}if %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.0: CMR8 003.002 %%Title: CMR8 %Version: 003.002 %%CreationDate: Mon Jul 13 16:17:00 2009 %%Creator: David M. Jones %Copyright: Copyright (c) 1997, 2009 American Mathematical Society %Copyright: (), with Reserved Font Name CMR8. % This Font Software is licensed under the SIL Open Font License, Version 1.1. % This license is in the accompanying file OFL.txt, and is also % available with a FAQ at: http://scripts.sil.org/OFL. %%EndComments FontDirectory/CMR8 known{/CMR8 findfont dup/UniqueID known{dup /UniqueID get 5000791 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /CMR8 def /FontBBox {-36 -250 1070 750 }readonly def /PaintType 0 def /FontInfo 9 dict dup begin /version (003.002) readonly def /Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050\051, with Reserved Font Name CMR8.) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def 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668 y Ft(:)e Fr(S)g Fq(!)23 b(B)s Ft(\()p Fq(H)q Ft(\))29 b Fl(is)h(a)g(unitary)g(r)l(epr)l (esentation.)494 767 y(\(ii\))42 b(If)30 b(the)g(r)l(epr)l(esentation)g Fr(\031)1467 779 y Fp(a)1537 767 y Fl(is)h(irr)l(e)l(ducible,)g(then)f (the)g(action)g(of)h Fr(S)j Fl(on)c Fr(X)36 b Fl(is)30 b(er)l(go)l(dic.)468 867 y(\(iii\))43 b(If)30 b Fr(S)35 b Fl(is)30 b(a)g(top)l(olo)l(gic)l(al)i(gr)l(oup)e(and)g(one)g(has)549 1045 y Ft(lim)540 1097 y Fp(s)p Fo(!)p Fg(1)689 1045 y Fr(\026)p Ft(\()p Fr(E)19 b Fq(4)14 b Ft(\()p Fr(E)5 b(:s)p Ft(\)\))23 b(=)33 b(lim)1274 1097 y Fp(s)p Fo(!)p Fg(1)1526 932 y Fi(Z)1404 1174 y Fp(E)15 b Fo(4)c Fm(\()p Fp(E)s(:s)p Fm(\))1725 1045 y Fq(j)p Fr(j)5 b Ft(\()p Fq(\001)p Fr(;)14 b(s)p Ft(\))1950 1011 y Fm(1)p Fp(=)p Fm(2)2074 1045 y Fq(\000)k Ft(1)p Fq(j)2222 1011 y Fm(2)2258 1045 y Ft(d)p Fr(\026)24 b Ft(=)32 b(lim)2465 1097 y Fp(s)p Fo(!)p Fg(1)2614 932 y Fi(Z)2611 1170 y Fp(E)2697 1045 y Ft(\()p Fr(a)p Ft(\()p Fr(s;)14 b Fq(\001)p Ft(\))19 b Fq(\000)f 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Fq(1)p Ft(,)28 b(and)f(this)h(completes)f(the)h(pro)r(of.)1437 b Fj(\003)555 2559 y Ft(F)-7 b(or)33 b(the)h(follo)n(wing)f(theorem)g (w)n(e)g(recall)f(that)i(a)f Fl(multiplicity-fr)l(e)l(e)k(r)l(epr)l (esentation)j Ft(is)34 b(a)456 2659 y(unitary)27 b(represen)n(tation)f (whose)h(comm)n(utan)n(t)g(is)g(comm)n(utativ)n(e.)456 2809 y Fw(Theorem)33 b(2.4.)41 b Fl(Assume)31 b(the)g(setting)g(of)i (De\014nition)e(2.1,)i(wher)l(e)f Ft(\()p Fr(X)r(;)14 b(\026)p Ft(\))32 b Fl(satis\014es)g(either)456 2909 y(of)e(the)g(c)l(onditions)h(in)f(L)l(emma)g(1.1,)h(and)f(let)1241 3045 y Fr(S)1292 3057 y Fm(0)1353 3045 y Ft(:=)22 b Fq(f)p Fr(s)h Fq(2)g Fr(S)28 b Fq(j)23 b Fr(x:s)h Ft(=)e Fr(x)30 b Fl(for)h(a.e.)40 b Fr(x)23 b Fq(2)h Fr(X)7 b Fq(g)p Fr(:)456 3181 y Fl(If)37 b(the)g(set)g Fq(f)p Fr(a)p Ft(\()p Fq(\001)p Fr(;)14 b(s)p Ft(\))37 b Fq(j)f Fr(s)g Fq(2)h Fr(S)1395 3193 y Fm(0)1432 3181 y Fq(g)g Fl(sp)l(ans)g(a)h Fr(w)1884 3151 y Fo(\003)1923 3181 y Fl(-dense)f(line)l(ar)h(subsp)l (ac)l(e)f(of)h Fr(L)2933 3151 y Fo(1)3003 3181 y Ft(\()p Fr(X)r(;)14 b(\026)p Ft(\))p Fl(,)40 b(then)456 3281 y Fr(\031)503 3293 y Fp(a)552 3281 y Ft(:)29 b Fr(S)h Fq(!)c(B)s Ft(\()p Fq(H)q Ft(\))31 b Fl(is)h(a)g(multiplicity-fr)l(e)l (e)g(r)l(epr)l(esentation)g(and)g(mor)l(e)l(over)g(the)g(fol)t(lowing)h (as-)456 3380 y(sertions)d(ar)l(e)g(e)l(quivalent:)519 3498 y(\(i\))42 b(The)31 b(action)f(of)h Fr(S)j Fl(on)c Ft(\()p Fr(X)r(;)14 b(\026)p Ft(\))30 b Fl(is)g(er)l(go)l(dic.)494 3598 y(\(ii\))42 b(The)31 b(r)l(epr)l(esentation)f Fr(\031)1412 3610 y Fp(a)1482 3598 y Fl(is)g(irr)l(e)l(ducible.)456 3748 y(Pr)l(o)l(of.)43 b Ft(Recall)31 b(that)h Fq(H)g Ft(=)d Fr(L)1407 3718 y Fm(2)1444 3748 y Ft(\()p Fr(X)r(;)14 b(\026)p Ft(\))32 b(and)g(for)f(an)n(y)g Fr( )j Fq(2)c Fr(L)2386 3718 y Fo(1)2456 3748 y Ft(\()p Fr(X)r(;)14 b(\026)p Ft(\))32 b(w)n(e)g(denote)g(b)n(y)f Fr(M)3309 3760 y Fp( )3389 3748 y Fq(2)456 3848 y(B)s Ft(\()p Fq(H)q Ft(\))26 b(b)r(e)i(the)f(op)r(erator)f(of)h(m)n(ultiplication)g(b)n(y)g Fr( )s Ft(.)36 b(By)27 b(Lemma)g(1.1,)f(the)i(op)r(erator)d(algebra) 1303 3984 y Fq(A)e Ft(:=)g Fq(f)p Fr(M)1626 3996 y Fp( )1698 3984 y Fq(j)h Fr( )i Fq(2)d Fr(L)1960 3949 y Fo(1)2030 3984 y Ft(\()p Fr(X)r(;)14 b(\026)p Ft(\))p Fq(g)23 b(\022)f(B)s Ft(\()p Fq(H)q Ft(\))456 4120 y(is)32 b(a)f(maximal)h(self-adjoin)n(t)g (subalgebra)e(of)i Fq(B)s Ft(\()p Fq(H)q Ft(\).)50 b(T)-7 b(o)32 b(pro)n(v)n(e)e(that)j Fr(\031)2765 4132 y Fp(a)2838 4120 y Ft(is)f(a)f(m)n(ultiplicit)n(y-)456 4219 y(free)i(represen)n (tation,)g(w)n(e)g(will)h(sho)n(w)f(that)g Fr(\031)1929 4231 y Fp(a)1970 4219 y Ft(\()p Fr(S)5 b Ft(\))2090 4189 y Fo(0)2146 4219 y Fq(\022)33 b(A)p Ft(.)55 b(Hence)34 b(w)n(e)f(m)n(ust)g(pro)n(v)n(e)f(that)i(if)456 4319 y Fr(T)40 b Fq(2)29 b(B)s Ft(\()p Fq(H)q Ft(\))i(and)g Fr(T)12 b(\031)s Ft(\()p Fr(s)p Ft(\))29 b(=)f Fr(\031)s Ft(\()p Fr(s)p Ft(\))p Fr(T)43 b Ft(for)31 b(all)g Fr(s)e Fq(2)g Fr(S)5 b Ft(,)32 b(then)f Fr(T)40 b Fq(2)30 b(A)p Ft(.)48 b(In)31 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b(sp)l(ac)l(e)i(of)f(r)l(e)l(al)g(me)l(asur)l(able)g(functions)f(on)h Fr(X)7 b Fl(,)29 b(with)h(the)g(c)l(orr)l(esp)l(onding)h(r)l(epr)l (esenta-)456 1855 y(tion)e Fr(\025)9 b Ft(:)29 b Fr(G)23 b Fq(!)g Ft(End)14 b(\()p Fq(F)8 b Ft(\))p Fl(,)31 b Fr(\025)1329 1867 y Fp(g)1368 1855 y Ft(\()p Fr(f)9 b Ft(\))23 b(:=)f Fr(f)28 b Fq(\016)17 b Fr(\013)1796 1871 y Fp(g)1830 1854 y Fe(\000)p Fh(1)1913 1855 y Fl(,)30 b(and)g(de\014ne)g(the)g(unitary)f(r)l(epr)l(esentation)815 2012 y Fr(\031)12 b Ft(:)29 b Fq(F)d Fn(o)1077 2024 y Fp(\025)1139 2012 y Fr(G)d Fq(!)g(B)s Ft(\()p Fr(L)1480 1977 y Fm(2)1516 2012 y Ft(\()p Fr(X)r(;)14 b(\026)p Ft(\)\))p Fr(;)99 b Ft(\()p Fr(\031)s Ft(\()p Fr(f)t(;)14 b(g)s Ft(\))p Fr(')p Ft(\)\()p Fr(x)p Ft(\))26 b(=)c(e)2510 1977 y Fm(i)p Fp(f)7 b Fm(\()p Fp(x)p Fm(\))2662 2012 y Fr(')p Ft(\()p Fr(\013)2801 2027 y Fp(g)2835 2011 y Fe(\000)p Fh(1)2917 2012 y Ft(\()p Fr(x)p Ft(\)\))p Fr(:)456 2155 y Fl(If)35 b(the)g(line)l(ar)h(dual)f(sp)l(ac)l(e)h(of)g Fr(X)41 b 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b(BEL)-5 b(TIT)-38 b(\030)1800 236 y(\025)1792 251 y(A)22 b(AND)g(D.)h(BEL)-5 b(TIT)-38 b(\030)2378 236 y(\025)2370 251 y(A)555 450 y Ft(Hence)34 b Fr(\031)k Ft(is)c(as)f(in)h(Prop)r(osition)e(3.1.)55 b(In)35 b(order)d(to)i(apply)g(that)g(prop)r(osition)f(w)n(e)g(m)n(ust) 456 550 y(c)n(hec)n(k)22 b(that)h(the)h(linear)e(span)h(of)g(the)g(set) h Fq(f)p Ft(exp)o(\(i)p Fr(\036)p Ft(\))g Fq(j)f Fr(\036)h Fq(2)f(F)8 b(g)23 b Ft(is)g Fr(w)2557 520 y Fo(\003)2596 550 y Ft(-dense)f(in)i Fr(L)2995 520 y Fo(1)3064 550 y Ft(\()p Fr(G)p Ft(\),)h(hence)456 649 y(that)31 b(if)h Fr( )g Fq(2)e Fr(L)947 619 y Fm(1)984 649 y Ft(\()p Fr(G)p Ft(\))i(and)1310 582 y Fi(R)1304 728 y Fp(G)1379 649 y Fr( )17 b Ft(exp\(i)p Fr(\036)p Ft(\))30 b(=)f(0)i(for)g(all)g Fr(\036)f Fq(2)f(F)8 b Ft(,)33 b(then)e(necessarily)f Fr( )i Ft(=)d(0.)48 b(T)-7 b(o)456 807 y(this)33 b(end,)i(using)e(the)h (ab)r(o)n(v)n(e)e(condition)h(for)g Fr(\036)g Ft(=)f Fr(\030)26 b Fq(\016)c Ft(log)2338 827 y Fp(G)2427 807 y Ft(with)34 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Fr(\030)t Ft(\))p Fr(:)456 2193 y Ft(Then)27 b(the)h(follo)n(wing)f(assertions)f(hold:) 522 2316 y(\(i\))43 b(The)27 b(group)1074 2295 y Fi(e)1058 2316 y Fr(G)c Ft(:=)g Fr(G)c Fn(n)1406 2328 y Fm(Ad)1513 2316 y Fd(g)28 b Ft(is)f(nilp)r(oten)n(t)h(and)g(its)g(cen)n(ter)f(is)g Fr(Z)d Fq(\002)18 b Fd(z)p Ft(.)499 2426 y(\(ii\))45 b Fi(e)-48 b Fr(\031)31 b Ft(is)c(a)g(unitary)g(irreducible)g(represen) n(tation)f(of)2240 2405 y Fi(e)2224 2426 y Fr(G)p Ft(.)555 2553 y(W)-7 b(e)26 b(recall)f(that)h(the)g(m)n(ultiplication)g(in)g (the)g(semi-direct)g(pro)r(duct)f(group)2986 2532 y Fi(e)2969 2553 y Fr(G)h Ft(is)g(giv)n(en)f(b)n(y)1184 2688 y(\()p Fr(g)1256 2700 y Fm(1)1293 2688 y Fr(;)14 b(Y)1378 2700 y Fm(1)1416 2688 y Ft(\))19 b Fq(\001)f Ft(\()p Fr(g)1580 2700 y Fm(2)1617 2688 y Fr(;)c(Y)1702 2700 y Fm(2)1740 2688 y Ft(\))23 b(=)g(\()p Fr(g)1955 2700 y Fm(1)1992 2688 y Fr(g)2032 2700 y Fm(2)2068 2688 y Fr(;)14 b(Y)2153 2700 y Fm(1)2209 2688 y Ft(+)k(Ad)2401 2700 y Fp(G)2457 2688 y Ft(\()p 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Fd(z)18 b Fq(\002)g Fd(z)p Ft(.)555 3290 y(T)-7 b(o)27 b(see)h(that)i Fi(e)-48 b Fr(\031)31 b Ft(is)c(a)g(represen)n(tation)f (w)n(e)h(need)h(to)f(c)n(hec)n(k)g(that)h(the)g(function)1254 3431 y Fr(a)9 b Ft(:)28 b Fq(O)21 b(\002)1544 3410 y Fi(e)1528 3431 y Fr(G)i Fq(!)g Fn(T)p Fr(;)97 b(a)p Ft(\()p Fr(\030)t(;)14 b Ft(\()p Fr(g)s(;)g(Y)19 b Ft(\)\))24 b(:=)e(e)2464 3396 y Fm(i)p Fo(h)p Fp(\030)r(;Y)14 b Fo(i)456 3575 y Ft(is)26 b(a)g(co)r(cycle)f(in)i(the)g(sense)f(of)g (De\014nition)h(2.1.)36 b(In)26 b(fact,)h(using)f(the)h(righ)n(t)e (action)h(of)3187 3554 y Fi(e)3170 3575 y Fr(G)h Ft(on)f Fq(O)456 3675 y Ft(giv)n(en)g(b)n(y)971 3780 y Fq(O)21 b(\002)1157 3759 y Fi(e)1141 3780 y Fr(G)i Fq(!)g(O)r Fr(;)97 b Ft(\()p Fr(\030)t(;)14 b Ft(\()p Fr(g)s(;)g(Y)19 b Ft(\)\))24 b Fq(7!)f Fr(\030)f Fq(\016)c Ft(Ad)2232 3792 y Fp(G)2288 3780 y Ft(\()p Fr(g)s Ft(\))23 b(=)f(Ad)2614 3743 y Fo(\003)2614 3801 y Fp(G)2670 3780 y Ft(\()p Fr(g)2745 3746 y Fo(\000)p Fm(1)2834 3780 y Ft(\))p Fr(\030)t(;)345 b Ft(\(3.2\))456 3898 y(it)28 b(follo)n(ws)e(b)n(y)j(\(3.1\))e(and)g (the)h(ab)r(o)n(v)n(e)f(de\014nition)h(of)f Fr(a)h Ft(that)941 4024 y Fr(a)p Ft(\()p Fr(\030)t(;)14 b Ft(\()p Fr(g)1166 4036 y Fm(1)1203 4024 y Fr(;)g(Y)1288 4036 y Fm(1)1326 4024 y Ft(\)\()p Fr(g)1430 4036 y Fm(2)1467 4024 y Fr(;)g(Y)1552 4036 y Fm(2)1589 4024 y Ft(\)\))24 b(=)f Fr(a)p Ft(\()p Fr(\030)t(;)14 b Ft(\()p Fr(g)1990 4036 y Fm(1)2027 4024 y Fr(g)2067 4036 y Fm(2)2104 4024 y Fr(;)g(Y)2189 4036 y Fm(1)2245 4024 y Ft(+)k(Ad)2436 4036 y Fp(G)2492 4024 y Ft(\()p Fr(g)2564 4036 y Fm(1)2601 4024 y Ft(\))p Fr(Y)2681 4036 y Fm(2)2719 4024 y Ft(\)\))1677 4169 y(=)23 b(e)1802 4135 y Fm(i)p Fo(h)p Fp(\030)r(;Y)1939 4143 y Fh(1)1970 4135 y Fm(+Ad)2107 4143 y Ff(G)2155 4135 y Fm(\()p Fp(g)2213 4143 y Fh(1)2246 4135 y Fm(\))p Fp(Y)2311 4143 y Fh(2)2344 4135 y Fo(i)1677 4314 y Ft(=)g(e)1802 4280 y Fm(i)p Fo(h)p Fp(\030)r(;Y)1939 4288 y Fh(1)1970 4280 y Fo(i)2001 4314 y Ft(e)2038 4280 y Fm(i)p Fo(h)p Fp(\030)r Fo(\016)p Fm(Ad)2236 4288 y Ff(G)2284 4280 y Fm(\()p Fp(g)2342 4288 y Fh(1)2375 4280 y Fm(\))p Fp(;Y)2460 4288 y Fh(2)2492 4280 y Fo(i)1677 4443 y Ft(=)g Fr(a)p Ft(\()p Fr(\030)t(;)14 b Ft(\()p Fr(g)1990 4455 y Fm(1)2027 4443 y Fr(;)g(Y)2112 4455 y Fm(1)2149 4443 y Ft(\)\))p Fr(a)p Ft(\()p Fr(\030)t(:)p Ft(\()p Fr(g)2424 4455 y Fm(1)2462 4443 y Fr(;)g(Y)2547 4455 y Fm(1)2585 4443 y Ft(\))p Fr(;)g Ft(\()p Fr(g)2726 4455 y Fm(2)2763 4443 y Fr(;)g(Y)2848 4455 y Fm(2)2885 4443 y Ft(\)\))p Fr(:)456 4583 y Ft(The)31 b(prop)r(ert)n(y)f Fr(a)p Ft(\()p Fr(\030)t(;)14 b Fw(1)p Ft(\))29 b(=)f Fr(\030)35 b Ft(for)c(al)f Fr(\030)j Fq(2)c(O)34 b Ft(is)c(clear)g (from)h(the)h(de\014nition)f(of)g Fr(a)p Ft(.)47 b(Also)31 b(note)456 4683 y(that)d(the)h(Liouville)f(measure)g(on)g Fq(O)j Ft(is)d(in)n(v)-5 b(arian)n(t)27 b(under)i(the)g(group)e(action) h(\(3.2\).)39 b(It)29 b(then)456 4782 y(follo)n(ws)d(b)n(y)i(Prop)r (osition)d(2.3)i(that)j Fi(e)-48 b Fr(\031)31 b Ft(is)d(a)f(con)n(tin)n (uous)f(unitary)h(represen)n(tation.)555 4882 y(Moreo)n(v)n(er,)34 b(to)h(see)g(that)j Fi(e)-49 b Fr(\031)39 b Ft(is)c(irreducible)f(w)n (e)h(will)g(use)g(Corollary)e(3.3.)58 b(T)-7 b(o)35 b(this)g(end,)456 4981 y(\014rst)26 b(note)h(that)g(the)h(group)d(action)i(\(3.2\))f(is)h (transitiv)n(e,)f(hence)h(ergo)r(dic)f(\(see)h(Remark)f(1.3\).)456 5081 y(F)-7 b(urthermore,)26 b(recall)h(that)h(the)g(mapping)1599 5216 y Fq(O)e(!)d Fd(g)1839 5181 y Fo(\003)1839 5236 y Fp(e)1877 5216 y Fr(;)97 b(\030)27 b Fq(!)c Fr(\030)t Fq(j)2229 5228 y Fc(g)2264 5236 y Ff(e)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 1124 251 a Fm(REPRESENT)-5 b(A)g(TIONS)28 b(OF)h(NILPOTENT)g(LIE)g(GR)n(OUPS)635 b(7)456 450 y Ft(is)35 b(a)f(global)g(c)n(hart)g(whic)n(h)h(tak)n(es)f(the)i (Liouville)e(measure)g(of)h Fq(O)j Ft(to)c(a)h(Leb)r(esgue)g(measure) 456 550 y(on)i Fd(g)623 520 y Fo(\003)623 570 y Fp(e)661 550 y Ft(,)i(where)e Fr(e)g Ft(is)g(the)h(jump)g(index)g(set)f(of)g Fq(O)j Ft(with)e(resp)r(ect)f(to)g(some)g(Jordan-H\177)-42 b(older)456 649 y(basis)24 b(in)i Fd(g)f Ft(\(see)g(for)g(instance)g ([BB10a)n(]\).)36 b(Then)26 b(w)n(e)f(can)g(use)g(the)g(F)-7 b(ourier)25 b(transform)f(to)h(see)456 752 y(that)33 b(the)h(linear)f(subspace)f(generated)h(b)n(y)g Fq(f)p Ft(e)1964 722 y Fm(i)p Fo(h)p Fp(Y)t(;)p Fo(\001i)2155 752 y Fq(j)f Fr(Y)52 b Fq(2)33 b Fd(g)p Fq(g)g Ft(is)g(w)n(eak)2785 722 y Fo(\003)2822 752 y Ft(-dense)g(in)h Fr(L)3242 722 y Fo(1)3311 752 y Ft(\()p Fq(O)r Ft(\))456 852 y(\()p Fq(')23 b Fr(L)633 822 y Fm(1)669 852 y Ft(\()p Fq(O)r Ft(\))801 822 y Fo(\003)840 852 y Ft(\).)38 b(Therefore)26 b(w)n(e)h(can)g(use)h(Corollary)d(3.3)i(to)g(obtain)h(that)i Fi(e)-49 b Fr(\031)31 b Ft(is)d(irreducible.)555 951 y(In)19 b(addition)f(to)h(the)g(ab)r(o)n(v)n(e)e(prop)r(erties)h(of)i Fi(e)-48 b Fr(\031)22 b Ft(w)n(e)c(also)g(recall)f(some)h(additional)g (information)456 1051 y(on)35 b(the)i(irreducible)e(represen)n(tation)h Fi(e)-48 b Fr(\031)39 b Ft(that)d(w)n(as)f(obtained)h(in)g([BB10a)n(,)j (Prop.)60 b(5.1\(2\)].)456 1151 y(Firstly)-7 b(,)35 b(the)e(space)g(of) h(smo)r(oth)f(v)n(ectors)f(for)h(the)h(represen)n(tation)g Fi(e)-48 b Fr(\031)37 b Ft(is)c Fq(S)6 b Ft(\()p Fq(O)r Ft(\).)56 b(Moreo)n(v)n(er,)456 1250 y(select)27 b(an)n(y)g (Jordan-H\177)-42 b(older)25 b(basis)i Fr(X)1660 1262 y Fm(1)1697 1250 y Fr(;)14 b(:)g(:)g(:)g(;)g(X)1951 1262 y Fp(n)2023 1250 y Ft(in)28 b Fd(g)f Ft(and)h(de\014ne)1216 1453 y Fi(e)1195 1474 y Fr(X)1264 1486 y Fp(j)1322 1474 y Ft(=)1409 1332 y Fi(\()1569 1418 y Ft(\(0)p Fr(;)14 b(X)1749 1430 y Fp(j)1784 1418 y Ft(\))111 b(for)27 b Fr(j)h Ft(=)22 b(1)p Fr(;)14 b(:)g(:)g(:)f(;)h(n;)1476 1537 y Ft(\()p Fr(X)1577 1549 y Fp(j)s Fo(\000)p Fp(n)1705 1537 y Fr(;)g Ft(0\))111 b(for)27 b Fr(j)h Ft(=)22 b Fr(n)d Ft(+)f(1)p Fr(;)c(:)g(:)g(:)f(;)h Ft(2)p Fr(n:)456 1721 y Ft(Then)694 1700 y Fi(e)672 1721 y Fr(X)741 1733 y Fm(1)778 1721 y Fr(;)g(:)g(:)g(:)g(;)984 1700 y Fi(e)963 1721 y Fr(X)1032 1733 y Fm(2)p Fp(n)1137 1721 y Ft(is)28 b(a)f(Jordan-H\177)-42 b(older)25 b(basis)h(in)2136 1717 y Fi(e)2138 1721 y Fd(g)i Ft(and)f(the)h(corresp)r(onding)d(predual)i (for)456 1827 y(the)h(coadjoin)n(t)e(orbit)1182 1806 y Fi(e)1162 1827 y Fq(O)f(\022)1338 1823 y Fi(e)1341 1827 y Fd(g)1383 1797 y Fo(\003)1448 1827 y Ft(asso)r(ciated)i(with)h (the)g(represen)n(tation)g Fi(e)-48 b Fr(\031)30 b Ft(is)1638 1977 y Fi(e)1640 1981 y Fd(g)1683 1994 y Fb(e)-39 b Fp(e)1740 1981 y Ft(=)23 b Fd(g)1870 1993 y Fp(e)1924 1981 y Fq(\002)18 b Fd(g)2049 1993 y Fp(e)2108 1981 y Fq(\022)2193 1977 y Fi(e)2195 1981 y Fd(g)p Fr(;)456 2140 y Ft(where)28 b Fi(e)-47 b Fr(e)27 b Ft(is)g(the)h(set)g(of)g(jump)g(indices)g(for) 1847 2119 y Fi(e)1827 2140 y Fq(O)r Ft(.)780 2365 y(4.)41 b Fs(Applica)-6 b(tion)31 b(to)h(Ga)n(ussian)f(measures)g(on)g(Hilber) -6 b(t)32 b(sp)-6 b(a)n(ces)555 2514 y Ft(W)f(e)28 b(\014rst)g(recall)e (here)h(a)g(few)h(facts)g(from)f([BB10b)o(])h(and)f([BBM15)o(].)456 2675 y Fw(De\014nition)35 b(4.1.)42 b Ft(If)32 b Fq(V)38 b Ft(is)32 b(a)f(real)f(Hilb)r(ert)i(space,)g Fr(A)e Fq(2)g(B)s Ft(\()p Fq(V)7 b Ft(\))31 b(with)h(\()p Fr(Ax)e Fq(j)g Fr(y)s Ft(\))f(=)g(\()p Fr(x)i Fq(j)e Fr(Ay)s Ft(\))456 2774 y(for)d(all)h Fr(x;)14 b(y)26 b Fq(2)d(V)7 b Ft(,)27 b(and)g(moreo)n(v)n(er)d(Ker)13 b Fr(A)23 b Ft(=)g Fq(f)p Ft(0)p Fq(g)p Ft(,)i(then)j(the)f Fl(Heisenb)l(er)l(g)i (algebr)l(a)35 b Ft(asso)r(ciated)456 2874 y(with)c(the)h(pair)e(\()p Fq(V)7 b Fr(;)14 b(A)p Ft(\))32 b(is)f(the)h(real)e(Hilb)r(ert)i(space) e Fd(h)p Ft(\()p Fq(V)7 b Fr(;)14 b(A)p Ft(\))30 b(=)f Fq(V)e Fj(u)20 b Fq(V)28 b Fj(u)20 b Fn(R)32 b Ft(endo)n(w)n(ed)e(with) 456 2973 y(the)f(Lie)f(brac)n(k)n(et)g(de\014ned)h(b)n(y)f([\()p Fr(x)1540 2985 y Fm(1)1578 2973 y Fr(;)14 b(y)1656 2985 y Fm(1)1693 2973 y Fr(;)g(t)1760 2985 y Fm(1)1797 2973 y Ft(\))p Fr(;)g Ft(\()p Fr(x)1945 2985 y Fm(2)1983 2973 y Fr(;)g(y)2061 2985 y Fm(2)2098 2973 y Fr(;)g(t)2165 2985 y Fm(2)2202 2973 y Ft(\)])26 b(=)e(\(0)p Fr(;)14 b Ft(0)p Fr(;)g Ft(\()p Fr(Ax)2703 2985 y Fm(1)2765 2973 y Fq(j)25 b Fr(y)2854 2985 y Fm(2)2891 2973 y Ft(\))20 b Fq(\000)f Ft(\()p Fr(Ax)3168 2985 y Fm(2)3231 2973 y Fq(j)25 b Fr(y)3320 2985 y Fm(1)3357 2973 y Ft(\)\).)456 3073 y(The)h(corresp)r(onding)e Fl(Heisenb)l(er)l(g)29 b(gr)l(oup)j Fn(H)p Ft(\()p Fq(V)7 b Fr(;)14 b(A)p Ft(\))23 b(=)f(\()p Fd(h)p Ft(\()p Fq(V)7 b Fr(;)14 b(A)p Ft(\))p Fr(;)g Fq(\003)p Ft(\))27 b(is)f(the)h(Lie)f(group)f(whose)456 3173 y(underlying)i(manifold)g(is)h Fd(h)p Ft(\()p Fq(V)7 b Fr(;)14 b(A)p Ft(\))28 b(and)g(whose)e(m)n(ultiplication)i(is)g (de\014ned)g(b)n(y)502 3326 y(\()p Fr(x)581 3338 y Fm(1)619 3326 y Fr(;)14 b(y)697 3338 y Fm(1)734 3326 y Fr(;)g(t)801 3338 y Fm(1)838 3326 y Ft(\))19 b Fq(\003)f Ft(\()p Fr(x)1028 3338 y Fm(2)1066 3326 y Fr(;)c(y)1144 3338 y Fm(2)1180 3326 y Fr(;)g(t)1247 3338 y Fm(2)1285 3326 y Ft(\))23 b(=)g(\()p Fr(x)1507 3338 y Fm(1)1563 3326 y Ft(+)18 b Fr(x)1693 3338 y Fm(2)1731 3326 y Fr(;)c(y)1809 3338 y Fm(1)1864 3326 y Ft(+)k Fr(y)1988 3338 y Fm(2)2025 3326 y Fr(;)c(t)2092 3338 y Fm(1)2148 3326 y Ft(+)k Fr(t)2261 3338 y Fm(2)2316 3326 y Ft(+)g(\(\()p Fr(Ax)2572 3338 y Fm(1)2634 3326 y Fq(j)23 b Fr(y)2721 3338 y Fm(2)2758 3326 y Ft(\))c Fq(\000)f Ft(\()p Fr(Ax)3033 3338 y Fm(2)3094 3326 y Fq(j)23 b Fr(y)3181 3338 y Fm(1)3218 3326 y Ft(\)\))p Fr(=)p Ft(2\))456 3480 y(for)k(\()p Fr(x)662 3492 y Fm(1)700 3480 y Fr(;)14 b(y)778 3492 y Fm(1)815 3480 y Fr(;)g(t)882 3492 y Fm(1)919 3480 y Ft(\))p Fr(;)g Ft(\()p Fr(x)1067 3492 y Fm(2)1105 3480 y Fr(;)g(y)1183 3492 y Fm(2)1220 3480 y Fr(;)g(t)1287 3492 y Fm(2)1324 3480 y Ft(\))23 b Fq(2)h Fn(H)p Ft(\()p Fq(V)7 b Fr(;)14 b(A)p Ft(\).)555 3641 y(Let)37 b Fq(V)764 3653 y Fo(\000)857 3641 y Ft(b)r(e)g(a)g(real) f(Hilb)r(ert)h(space)g(and)f(\()p Fq(\001)j(j)g(\001)p Ft(\))2137 3653 y Fo(\000)2230 3641 y Ft(b)r(e)e(its)h(scalar)d(pro)r (duct.)64 b(F)-7 b(or)37 b(ev)n(ery)456 3740 y Fr(a)23 b Fq(2)g(V)652 3752 y Fo(\000)735 3740 y Ft(and)k(ev)n(ery)f (symmetric,)h(nonnegativ)n(e,)f(injectiv)n(e,)h Fl(tr)l(ac)l(e-class)35 b Ft(op)r(erator)25 b Fr(K)33 b Ft(on)26 b Fq(V)3388 3752 y Fo(\000)456 3840 y Ft(there)h(is)g(a)h(unique)f(probabilit)n(y)g (Borel)g(measure)f Fr(\015)32 b Ft(on)c Fq(V)2301 3852 y Fo(\000)2384 3840 y Ft(with)1077 4035 y(\()p Fq(8)p Fr(x)c Fq(2)g(V)1356 4047 y Fo(\000)1412 4035 y Ft(\))1545 3922 y Fi(Z)1522 4160 y Fo(V)1564 4168 y Fe(\000)1646 4035 y Ft(e)1683 4001 y Fm(i\()p Fp(x)p Fo(j)p Fp(y)r Fm(\))1848 4009 y Fe(\000)1900 4035 y Ft(d)p Fr(\015)5 b Ft(\()p Fr(y)s Ft(\))23 b(=)f(e)2249 4001 y Fm(i\()p Fp(a)p Fo(j)p Fp(x)p Fm(\))2414 4009 y Fe(\000)2462 4001 y Fo(\000)2524 3978 y Fh(1)p 2524 3987 29 3 v 2524 4021 a(2)2562 4001 y Fm(\()p Fp(K)t(x)p Fo(j)p Fp(x)p Fm(\))2770 4009 y Fe(\000)456 4304 y Ft(and)27 b Fr(\015)32 b Ft(is)c(called)f (the)h Fl(Gaussian)i(me)l(asur)l(e)g(with)g(the)g(me)l(an)g Fr(a)f Fl(and)i(the)e(varianc)l(e)i Fr(K)6 b Ft(.)555 4403 y(No)n(w)30 b(assume)f(that)h Fr(a)d Ft(=)f(0)j(and)h(let)g Fq(V)1788 4415 y Fm(+)1870 4403 y Ft(:=)d(Ran)13 b Fr(K)36 b Ft(and)29 b Fq(V)2468 4415 y Fm(0)2532 4403 y Ft(:=)e(Ran)13 b Fr(K)2886 4373 y Fm(1)p Fp(=)p Fm(2)3020 4403 y Ft(b)r(e)30 b(endo)n(w)n(ed)456 4505 y(with)24 b(the)h(scalar)e(pro)r(ducts)h(giv)n (en)f(b)n(y)h(\()p Fr(K)6 b(x)23 b Fq(j)g Fr(K)6 b(y)s Ft(\))2058 4517 y Fm(+)2136 4505 y Ft(:=)23 b(\()p Fr(x)g Fq(j)h Fr(y)s Ft(\))2472 4517 y Fo(\000)2552 4505 y Ft(and)g(\()p Fr(K)2819 4474 y Fm(1)p Fp(=)p Fm(2)2923 4505 y Fr(x)g Fq(j)f Fr(K)3117 4474 y Fm(1)p Fp(=)p Fm(2)3220 4505 y Fr(y)s Ft(\))3296 4517 y Fm(0)3357 4505 y Ft(:=)456 4604 y(\()p Fr(x)k Fq(j)g Fr(y)s Ft(\))688 4616 y Fo(\000)744 4604 y Ft(,)j(resp)r(ectiv)n(ely)-7 b(,)30 b(for)f(all)g Fr(x;)14 b(y)30 b Fq(2)d(V)1803 4616 y Fo(\000)1859 4604 y Ft(,)j(whic)n(h)g(turn)g(the)g(linear)f(bijections)h Fr(K)14 b Ft(:)29 b Fq(V)3279 4616 y Fo(\000)3361 4604 y Fq(!)456 4707 y(V)507 4719 y Fm(+)589 4707 y Ft(and)f Fr(K)828 4677 y Fm(1)p Fp(=)p Fm(2)941 4707 y Ft(:)f Fq(V)1042 4719 y Fo(\000)1121 4707 y Fq(!)d(V)1279 4719 y Fm(0)1343 4707 y Ft(in)n(to)k(isometries.)36 b(W)-7 b(e)28 b(th)n(us)f(obtain)g(the)h(real)f(Hilb)r(ert)h(spaces)1661 4861 y Fq(V)1712 4873 y Fm(+)1790 4861 y Fr(,)-14 b Fq(!)23 b(V)1956 4873 y Fm(0)2017 4861 y Fr(,)-14 b Fq(!)23 b(V)2183 4873 y Fo(\000)456 5016 y Ft(where)35 b(the)h(inclusion)g(maps)f(are)g (Hilb)r(ert-Sc)n(hmidt)i(op)r(erators,)f(since)f Fr(K)2891 4986 y Fm(1)p Fp(=)p Fm(2)3032 5016 y Fq(2)i(B)s Ft(\()p Fq(V)3265 5028 y Fo(\000)3320 5016 y Ft(\))g(is)456 5116 y(a)d(Hilb)r(ert-Sc)n(hmidt)i(op)r(erator.)56 b(Also,)37 b(the)e(scalar)e(pro)r(duct)i(of)g Fq(V)2624 5128 y Fm(0)2696 5116 y Ft(extends)g(to)f(a)h(dualit)n(y)456 5216 y(pairing)26 b(\()p Fq(\001)e(j)f(\001)p Ft(\))917 5228 y Fm(0)964 5216 y Ft(:)k Fq(V)1065 5228 y Fo(\000)1140 5216 y Fq(\002)18 b(V)1274 5228 y Fm(+)1352 5216 y Fq(!)23 b Fn(R)p Ft(.)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 456 251 a Fm(8)992 b(I.)22 b(BEL)-5 b(TIT)-38 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Fw(De\014nition)35 b(4.2.)43 b Ft(Let)32 b Fq(V)1303 873 y Fm(+)1389 861 y Ft(b)r(e)h(a)e(real)g(Hilb)r(ert)h(space)f(with)h(the)h(scalar)d(pro) r(duct)h(denoted)456 961 y(b)n(y)26 b(\()p Fr(x;)14 b(y)s Ft(\))24 b Fq(7!)f Ft(\()p Fr(x)h Fq(j)f Fr(y)s Ft(\))1117 973 y Fm(+)1172 961 y Ft(.)37 b(Also)26 b(let)i Fr(A)9 b Ft(:)28 b Fq(V)1711 973 y Fm(+)1789 961 y Fq(!)23 b(V)1946 973 y Fm(+)2028 961 y Ft(b)r(e)k(a)g(nonnegativ)n(e,)e(symmetric,)i (injectiv)n(e,)456 1060 y(trace-class)g(op)r(erator.)40 b(Let)29 b Fq(V)1433 1072 y Fm(0)1500 1060 y Ft(and)g Fq(V)1714 1072 y Fo(\000)1799 1060 y Ft(b)r(e)h(the)g(completions)e(of) i Fq(V)2665 1072 y Fm(+)2749 1060 y Ft(with)g(resp)r(ect)f(to)g(the)456 1160 y(scalar)d(pro)r(ducts)1285 1271 y(\()p Fr(x;)14 b(y)s Ft(\))23 b Fq(7!)g Ft(\()p Fr(x)h Fq(j)f Fr(y)s Ft(\))1831 1283 y Fm(0)1891 1271 y Ft(:=)g(\()p Fr(A)2096 1237 y Fm(1)p Fp(=)p Fm(2)2201 1271 y Fr(x)h Fq(j)f Fr(A)2380 1237 y Fm(1)p Fp(=)p Fm(2)2484 1271 y Fr(y)s Ft(\))2560 1283 y Fm(+)456 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Fq(V)1079 2044 y Fm(+)1134 2032 y Fr(;)14 b(A)p Ft(\))23 b Fq(!)g(B)s Ft(\()p Fr(L)1541 2001 y Fm(2)1577 2032 y Ft(\()p Fq(V)1660 2044 y Fo(\000)1717 2032 y Fr(;)14 b(\015)5 b Ft(\)\))27 b(is)h(de\014ned)g(b)n(y)1089 2188 y Fr(\031)s Ft(\()p Fr(x;)14 b(y)s(;)g(t)p Ft(\))p Fr(\036)25 b Ft(=)d Fr(\032)1602 2200 y Fp(x)1644 2188 y Ft(\()p Fq(\001)p Ft(\))1731 2154 y Fm(1)p Fp(=)p Fm(2)1836 2188 y Ft(e)1873 2154 y Fm(i\()p Fp(t)p Fm(+\()p Fo(\001j)p Fp(y)r Fm(\))2122 2162 y Fh(0)2153 2154 y Fm(+)2214 2131 y Fh(1)p 2214 2140 V 2214 2174 a(2)2252 2154 y Fm(\()p Fp(x)p Fo(j)p Fp(y)r Fm(\))2398 2162 y Fh(0)2430 2154 y Fm(\))2460 2188 y Fr(\036)p Ft(\()p Fq(\000)p Fr(x)d Ft(+)f Fq(\001)p Ft(\))456 2332 y(for)27 b(\()p Fr(x;)14 b(y)s(;)g(t)p Ft(\))23 b Fq(2)h Fn(H)p Ft(\()p Fq(V)1092 2344 y Fm(+)1146 2332 y Fr(;)14 b(A)p Ft(\))29 b(and)e Fr(\036)d Fq(2)f Fr(L)1675 2302 y Fm(2)1712 2332 y Ft(\()p Fq(V)1795 2344 y Fo(\000)1851 2332 y Fr(;)14 b(\015)5 b Ft(\).)456 2487 y Fw(Prop)s(osition)36 b(4.3.)44 b Fl(The)36 b(r)l(epr)l(esentation)f Fr(\031)12 b Ft(:)30 b Fn(H)p Ft(\()p Fq(V)2153 2499 y Fm(+)2208 2487 y Fr(;)14 b(A)p Ft(\))33 b Fq(!)f(B)s Ft(\()p Fr(L)2634 2456 y Fm(2)2670 2487 y Ft(\()p Fq(V)2753 2499 y Fo(\000)2810 2487 y Fr(;)14 b(\015)5 b Ft(\)\))35 b Fl(fr)l(om)g(De\014ni-)456 2586 y(tion)29 b(4.2)i(is)g(irr)l(e)l(ducible.)456 2741 y(Pr)l(o)l(of.)43 b Ft(See)27 b(for)g(instance)h(from)f([BB10b)o(,)h (Rem.)37 b(3.6])27 b(or)f([BBM15)o(].)718 b Fj(\003)456 2904 y Fw(Corollary)29 b(4.4.)38 b Fl(In)26 b(the)i(ab)l(ove)g (setting,)f(the)h(action)f(by)h(tr)l(anslations)f(of)h Fq(V)2904 2916 y Fm(+)2986 2904 y Fl(on)f Ft(\()p Fq(V)3185 2916 y Fo(\000)3241 2904 y Fr(;)14 b(\015)5 b Ft(\))27 b Fl(is)456 3003 y(er)l(go)l(dic.)456 3158 y(Pr)l(o)l(of.)43 b Ft(In)24 b(the)g(presen)n(t)f(framew)n(ork,)g(the)h(represen)n (tation)e Fr(\031)27 b Ft(is)d(the)g(unitary)f(represen)n(tation)456 3258 y(asso)r(ciated)i(to)h(the)h(measure)e(space)h(\()p Fq(V)1718 3270 y Fo(\000)1774 3258 y Fr(;)14 b(\015)5 b Ft(\))26 b(acted)g(on)g(b)n(y)g(the)h(additiv)n(e)f(group)f(\()p Fq(V)3141 3270 y Fm(+)3197 3258 y Fr(;)14 b Ft(+\))26 b(b)n(y)456 3357 y(translations.)35 b(The)27 b(co)r(cycle)g(of)g(that)g (measurable)f(dynamical)h(system)g(whic)n(h)g(giv)n(es)f(rise)h(to)456 3457 y(the)h(represen)n(tation)d Fr(\031)31 b Ft(is)d(giv)n(en)f(b)n(y) 1369 3609 y Fr(a)p Ft(\()p Fq(\001)p Fr(;)14 b Ft(\()p Fr(x;)g(y)s(;)g(t)p Ft(\)\))24 b(=)e(e)1944 3574 y Fm(i\()p Fp(t)p Fm(+\()p Fo(\001j)p Fp(y)r Fm(\))2193 3582 y Fh(0)2225 3574 y Fm(+)2285 3552 y Fh(1)p 2285 3561 V 2285 3594 a(2)2324 3574 y Fm(\()p Fp(x)p Fo(j)p Fp(y)r Fm(\))2470 3582 y Fh(0)2501 3574 y Fm(\))456 3753 y Ft(for)29 b(all)h(\()p Fr(x;)14 b(y)s(;)g(t)p Ft(\))28 b Fq(2)g Fn(H)p Ft(\()p Fq(V)1221 3765 y Fm(+)1275 3753 y Fr(;)14 b(A)p Ft(\).)46 b(The)30 b(conclusion)f(follo)n(ws)g(b)n(y)h(Prop)r(ositions)e(4.3)i (and)g(2.3\(ii\))456 3852 y(for)d(the)h(righ)n(t)f(group)f(action)1106 3996 y Fq(V)1157 4008 y Fo(\000)1231 3996 y Fq(\002)18 b Fn(H)p Ft(\()p Fq(V)1462 4008 y Fm(+)1517 3996 y Fr(;)c(A)p Ft(\))23 b Fq(!)g(V)1828 4008 y Fo(\000)1884 3996 y Fr(;)97 b Ft(\()p Fr(v)s(;)14 b Ft(\()p Fr(x;)g(y)s(;)g(t)p Ft(\)\))25 b Fq(7!)e(\000)p Fr(x)18 b Ft(+)g Fr(v)456 4140 y Ft(and)27 b(w)n(e)g(are)g(done.)2308 b Fj(\003)1708 4336 y Fs(References)456 4468 y Fv([BB09])111 b(I.)38 b(Beltit)-29 b(\030)n(\025)-35 b(a,)42 b(D.)37 b(Beltit)-29 b(\030)o(\025)-35 b(a,)41 b Fk(Magnetic)d(pseudo-di\013er)l(ential)i(Weyl)f(c)l(alculus)h(on)g (nilp)l(otent)f(Lie)777 4551 y(gr)l(oups)p Fv(.)25 b(Ann.)e(Global)h (Anal.)f(Geom.)h(36)g(\(2009\),)h(no.)f(3,)f(293{322.)456 4634 y([BB10a])76 b(I.)26 b(Beltit)-29 b(\030)o(\025)-35 b(a,)27 b(D.)f(Beltit)-29 b(\030)n(\025)-35 b(a,)27 b Fk(Smo)l(oth)j(ve)l(ctors)e(and)h(Weyl-Pe)l(dersen)f(c)l(alculus)i(for) e(r)l(epr)l(esentations)777 4717 y(of)e(nilp)l(otent)g(Lie)f(gr)l(oups) p Fv(.)f(Ann.)g(Univ.)f(Buc)n(har.)h(Math.)f(Ser.)g(1\(LIX\))j (\(2010\),)f(no.)e(1,)h(17{46.)456 4800 y([BB10b])72 b(I.)19 b(Beltit)-29 b(\030)n(\025)-35 b(a,)19 b(D.)f(Beltit)-29 b(\030)o(\025)-35 b(a,)19 b Fk(On)i(Weyl)f(c)l(alculus)j(in)e (in\014nitely)f(many)h(variables)p Fv(.)e(In:)29 b(P)-6 b(.)18 b(Kielano)n(wski,)777 4883 y(V.)g(Buc)n(hstab)r(er,)i(A.)e (Odzijewicz,)h(M.)e(Sc)n(hlic)n(henmaier,)j(Th.)e(V)-6 b(orono)n(v)19 b(\(eds.\),)h Fk(XXIX)h(Workshop)777 4967 y(on)k(Ge)l(ometric)h(Metho)l(ds)g(in)e(Physics)p Fv(,)f(AIP)g(Conf.)g (Pro)r(c.,)f(1307,)i(Amer.)e(Inst.)h(Ph)n(ys.,)g(Melville,)777 5050 y(NY,)g(2010,)h(pp.)g(19{26.)456 5133 y([BBM15])46 b(I.)37 b(Beltit)-29 b(\030)o(\025)-35 b(a,)40 b(D.)c(Beltit)-29 b(\030)o(\025)-35 b(a,)40 b(M.)c(M\025)-35 b(an)n(toiu,)41 b Fk(On)d(Wigner)g(tr)l(ansforms)h(in)f(in\014nite)g(dimensions)p Fv(.)777 5216 y(Preprin)n(t)23 b(arXiv:1501.05404)i([math.R)-6 b(T].)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 1124 251 a Fm(REPRESENT)-5 b(A)g(TIONS)28 b(OF)h(NILPOTENT)g(LIE)g(GR)n(OUPS)635 b(9)456 450 y Fv([Di69])137 b(Dixmier,)29 b(J.,)h Fk(L)l(es)h(algebr)l(es)g(d'op)n (\023)-35 b(er)l(ateurs)34 b(dans)e(l'esp)l(ac)l(e)g(hilb)l(ertien)e (\(alg)n(\022)-35 b(ebr)l(es)33 b(de)e(von)g(Neu-)777 533 y(mann\))p Fv(.)41 b(Deuxi)n(\022)-33 b(eme)40 b(\023)-33 b(edition,)44 b(revue)e(et)f(augmen)n(t)n(\023)-33 b(ee.)42 b(Cahiers)f(Scien)n(ti\014ques,)46 b(F)-6 b(asc.)40 b(XXV.)777 616 y(Gauthier-Villars)22 b(Editeur,)i(P)n(aris,)e(1969.)456 699 y([Is96])158 b(R.S.)21 b(Ismagilo)n(v,)i Fk(R)l(epr)l(esentations)i (of)f(in\014nite-dimensional)h(gr)l(oups)p Fv(.)e(T)-6 b(ranslations)22 b(of)g(Mathe-)777 782 y(matical)i(Monographs,)g(152.)g (American)f(Mathematical)i(So)r(ciet)n(y)-6 b(,)25 b(Pro)n(vidence,)f (RI,)g(1996.)456 865 y([SS08])133 b(A.M.)28 b(Sinclair,)h(R.R.)f (Smith,)i Fk(Finite)g(von)h(Neumann)g(algebr)l(as)h(and)f(masas)p Fv(.)f(London)g(Math-)777 948 y(ematical)j(So)r(ciet)n(y)h(Lecture)g (Note)g(Series,)g(351.)f(Cam)n(bridge)g(Univ)n(ersit)n(y)g(Press,)h (Cam)n(bridge,)777 1031 y(2008.)456 1114 y([T)-6 b(a03])131 b(T)-6 b(ak)n(esaki,)19 b(M.,)e Fk(The)l(ory)j(of)g(op)l(er)l(ator)h (algebr)l(as.)g(III)p Fv(.)c(Encyclopaedia)i(of)e(Mathematical)h (Sciences,)777 1197 y(127.)32 b(Op)r(erator)f(Algebras)g(and)h (Non-comm)n(utativ)n(e)h(Geometry)-6 b(,)34 b(8.)d(Springer-V)-6 b(erlag,)32 b(Berlin,)777 1280 y(2003.)555 1436 y Fu(Institute)i(of)f (Ma)l(thema)l(tics,)i(\\Simion)d(Stoilo)o(w",)k(of)d(the)g(R)n(omanian) f(A)n(cademy,)k(P.O.)e(Bo)o(x)456 1519 y(1-764,)24 b(Bucharest,)i(R)n (omania)555 1602 y Fk(E-mail)g(addr)l(ess)5 b Fv(:)33 b Fa(ingrid.beltita@gmail.c)q(om)555 1743 y Fu(Institute)h(of)f(Ma)l (thema)l(tics,)i(\\Simion)d(Stoilo)o(w",)k(of)d(the)g(R)n(omanian)f(A)n (cademy,)k(P.O.)e(Bo)o(x)456 1826 y(1-764,)24 b(Bucharest,)i(R)n (omania)555 1909 y Fk(E-mail)g(addr)l(ess)5 b Fv(:)33 b Fa(beltita@gmail.com)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------1512070534203--