Harish-Chandra integrals as nilpotent integrals

Ha rish-Chandr a integrals as nilp otent integrals M. Bertola ‡ ,♯ 1 2 , A. Prats F errer ♯ 3 . ‡ Dep artment of Mathematics and Statistics, Conc or dia University 1455 de Maisonneuve W., Montr´ eal, Qu´ eb e c, Canada H3G 1M8 ♯ Centr e de r e cher ches math ´ ematiques, Universit´ e de Montr ´ eal 2920 Chemin de la tour, Montr ´ eal, Qu´ eb e c, Canada H3T 1J4. Abstract Recently the co r relation functions of the so–c a lled Itzyks on-Zub er/Harish-Chandr a integrals w er e co mputed (by one of the authors a nd collab orators ) for all classical gro ups using a n in tegra tio n form ula that r elates in tegrals over compact groups with respect to the Haar measur e and Gaussian integrals ov er a maximal nilpotent Lie subalgebra of their complexifica tion. Since the integration formula a p osteriori had the same for m for the class ical series , a conjecture w as formulated that such a formula should hold for ar bitrary semisimple Lie gro ups. W e prov e this conjecture using an a bs tract Lie–theoretic approach. 1 In tro duction and setting In ra ndom matr ix theory [ 1 , 2 ] a particula rly impor tan t r ole is play ed b y the so– called Itzykson–Zub er/Har ish- Chandra measure. Its integral is [ 3 , 4 ] Z U ( N ) d U e tr ( X U Y U † ) = C N det(e x i y j ) i,j ∆( X )∆( Y ) (1.1) where d U is the Haa r mea sure on U ( N ), X , Y are diago nal matrices and ∆( X ) = Q i 0 α ( H ) α ( J ) Z n + d N F ( H + N , J w + N † )e − h N ,N † i , (1.2) wher e J w stands for the action of the Weyl gr oup on h , ǫ w is the usual sign homomorphism and C is a su itable c onstant dep ending only on the Lie algebr a un der c onsider ation. (Al l t he symb ols wil l b e define d mor e in detail later) 1 W ork supp orted in part b y th e Natural Sciences and Engineering Researc h Council of Canada (NSERC). 2 bertola@crm. umon treal.ca 3 pratsferrer@crm.umontreal.ca 1 Such co njecture w as verified a p osterior i fo r all the classical series but the authors of [ 5 ] failed to provide a general pr oo f that w o uld apply also to exceptiona l Lie alg ebras. This ident ity was the main initial s tep tow a r ds an effective computation of a ll correla tio n functions for spherical integrals ov er the compa ct forms of the classical groups, S U ( N ) , S O (2 n , R ) , S O (2 n + 1 , R ) , S p ( n, R ). In this short note w e provide a Lie–alg ebro–theoretica l pro of of Conjecture 1 (Thm. 5.1 ) that do es not rely on any specific ity of the Lie algebra as long as it is semisimple and provides a pr ecise v a lue for the prop ortiona lit y constant C . W e will need to prove a slight generaliza tion of W eyl–integration for m ula (whic h may well b e known in the literature but w e could not find in any of the standard refer ences). The pro of of Co njecture 1 is con tained in Thm. 5.1 . W e will liber ally use k nown facts about (semi)-simple Lie a lg ebras, all of which can be found in standa rd reference bo oks like [ 7 ]. Let g b e a complex semisimple Lie alg ebra o ver C , G the corr e sponding simply connected group, G = exp( g ). Let h = h C ⊂ g be a Cartan subalgebra (o ver C unless otherwise specified) and R ⊂ h ∨ C be the set o f ro ots. Let an ordering of the ro ots b e chosen: it fixes the set of po sitiv e roo ts R + . The set of s imple pos itiv e ro ots with respect to this ordering will be denoted by Φ. W e will use the Chev alley ba sis { E α , H α } α ∈ R (called ro ot ve ctors and c or o ots resp ectiv e ly ) of g where E α spans g α and the set H α for α ∈ Φ spans h . Suc h a basis has the prop erties [ E α , E − α ] = H α , [ H α , E ± α ] = ± 2 E ± α , α ∈ R . Her e and in the following w e will use the notatio ns h R := X α ∈ Φ R { H α } , h := h C := X α ∈ Φ C { H α } , n + = X α> 0 C { E α } , b + = h C + n + . (1.3) The compact form k ⊂ g will b e chosen as the span of k = i h R + X α> 0 R { X α , Y α } , X α := ( E α − E − α ) , Y α := i ( E α + E − α ) (1.4) On each o f the ab ov e Lie alg ebras we will use the Leb esgue meas ure such that the unit cub e in the co or dina tes given b y the sp ecified basis has unit volume. W e finally re c all that any s emisimple Lie algebra admits a decomp osition g = k + a + n + where k is a compa ct Lie algebra, a ⊂ h R is Abelian. This decompo sition subtends the Iwasawa de c omp osition of the Lie gro up G = K A N . 2 Sc h ur decomp ositions The fir st fact we need is a generalizatio n of the Sch ur decompos ition to arbitrary Lie a lg ebras: Sch ur deco mposition is a widely k nown decomp osition of matr ices and states that any co mplex square matrix M can b e wr itten in the form U T U † with U ∈ U ( N ) and T a n upp er semi- tr iangular matrix. Theorem 2. 1 Any ad-regular element M ∈ g is K –c onjugate to an element in b + = h C + n + . Pro of. It is known [ 7 ] that an y ad-regula r element in g is conjugate to an elemen t in h , namely M = Ad g H ∈ h C . There are (generica lly) | W | s uch wa ys of representing M . Using the Iwasa wa decomp osition g = k an we hav e immediately M = Ad k ( H + ( Ad an H − H )) = Ad k B (2.1) with B = Ad an H ∈ b + and Ad an H − H ∈ n + . Q .E.D. 2 Since we will b e concer ned with int egration formulæ, the a bove theorem suffices s ince the set of ad-regula r elements is a Zar iski op en se t, dense in g : in particular nonr egular elements are a set of Leb esgue measur e zero. How ever, the fo llo wing more genera l theorem ca n a lso be prov e d (but we will not pr o ve it here in the interest o f conciseness and also b ecause it is co mpletely irrelev ant to our ma in purp ose). Theorem 2. 2 Any element M ∈ g is c onjugate d by an element of the maximal c omp act sub gr oup K t o an element H + N with H ∈ h , N ∈ n + 3 Complex W eyl i nte gration form ula The goa l of this section is to write an integral form ula for functions on g in terms of integrals on b and K . Define M := ( K × b ) /T (3.1) where the action o f the Car tan torus is t · ( k , V ) := ( kt − 1 , tV t − 1 ), V ∈ b k ∈ K . W e will write V = H + N w ith iH ∈ h C and N ∈ n + . The a ction of T is then t · ( k , H , N ) = ( k t − 1 , H , tN t − 1 ) so that we ca n also think o f M as M = h × ( K × n + ) /T . The tangent space to M at [( k , V )] is identified with k /i h R + b by [( k s , V s )] := [( k e sX , V + sW )]. Consider the map π : M − → g [ k , V ] 7→ k V k − 1 (3.2) The top ological degree of π is the ca rdinality of the W eyl gro up W : to see this it is s ufficien t to note that Ad K h C ⊂ g has the advoca ted deg ree and then use a contin uity argument . The differential of the map π : M → g at a p oint [( k, V )] is then computed as d d s Ad k e sX ( V + sW )   s =0 = k ([ X , V ] + W ) k − 1 , X ∈ k /i h R , W ∈ b . (3.3) In order to write a matr ix representation o f the a bov e map we write it in the natur al basis of T M g = n − + b + dπ : T [ k,V ] M ∼ k / i h R ⊕ b + − → n − + b + ( X, W ) 7→ d π ( X, W ) = Ad k ([ X, V ] + W ) (3.4) W e compute the determina nt of the a bove map witho ut the Ad k term (which do es not change its v alue) and we think of n − as a vector space o ver R with a real basis provided b y n − := P α< 0 R { E α } + i P α< 0 R { E α } ad H + N ( X α ) = ℜ ( α ( H )) E − α + i ℑ ( α ( H )) E − α + X − β > − α C { E − β } mo d b + (3.5) ad H + N ( Y α ) = −ℑ ( α ( H )) E − α + i ℜ ( α ( H )) E − α + X − β > − α C { E − β } mo d b + (3.6) V := H + N It app ears that the matrix has a blo ck-upper triangular s hape and these “upp er triang ular” parts do not contribute to the determinant. The latter b ecomes then the pro duct of the determinant s o f the above 2 × 2 blo cks whic h ar e simply | α ( H ) | 2 . 3 The Jaco bian o f d π at the p oint [ k, V ] ∈ M ( V = H + N ) is thus J ( k, V ) = Y α ∈ R + | α ( H ) | 2 =: | ∆( H ) | 2 . (3.7) The no tation ∆( H ) := Q α ∈ R + α ( H ) is used in analo gy with the case of g = sl ( n, C ) where it reduces to the V andermonde determinant. A well kno wn pro perty is that ∆( H w ) = ( − ) w ∆( H ) (3.8) where w ∈ W and H w stands for the action of the W eyl gr oup on h C and the notation ( − ) w means the parity of the W eyl-transforma tion (i.e. the parity of the n umber of elementary reflec tio ns along walls of W eyl chambers in which w can be decomp o sed). Co lle c ting these pieces of information we have pro ved the follo wing Theorem 3. 1 (Comple x-W eyl i n tegration form ul a) L et F : g → C b e a smo oth inte gr able function invariant under t he adjoint action of K . Then Z g d M F ( M ) = c k Z h C × n + d H d N F ( H + N ) | ∆( H ) | 2 , c k := µ ( K ) /µ ( T ) | W | wher e d M , d H , d N ar e the L eb esgue me asu re s on g , h C , n + r esp e ctively define d ab ove, µ ( K ) and µ ( T ) ar e t he induc e d me asur es o n the c omp act gr ou p K and the maximal t orus T , W is the Weyl gr oup and | W | is its c ar dinality. There is one mor e piece of information that we can extract from the ab o ve and is contained in the follo wing Corollary 3.1 F or any Ad K –invariant smo oth inte gr able fun ction F : g → C t he function b F ( H ) := Z n + d N F ( H + N ) : h C → C (3.9) is Weyl–invari ant . Pro of. By the gene r alized Sch ur dec ompos itio n (Thm. 2.1 ) a regula r elemen t M ca n b e represented modulo the Ad K action as H + N ∈ h + n + or H w + e N where H w is in the same W – orbit through H and e N ∈ n + is so me other element in the same nilpotent subalgebra n + . In gener al the dep endence of e N on N , H is a complicated expressio n. Consider a small ball M ∈ U ⊂ g co nsisting of regular e le men ts. This ball can b e mapp ed diffe omorphic al ly to some neighborho o d H × L in h C × ( K × n + ) /T with H lying in a s uitable W eyl chamber and c o n taining H . Cho o sing another W eyl cham b er H w we hav e a distinct diffeomo rphism b et ween U and H w × e L . Since the Jaco bian computed ab o ve is alw ays | ∆( H ) | 2 = | ∆( H w ) 2 | and indep endent of N , w e conclude that the transformation N 7→ e N preser v es the Leb esgue measure of n + , namely d e N / d N = 1. Since regular elements ar e o pen and dense (and the complement has zero measure) we can then write b F ( H w ) = Z n + d e N F ( H w + e N ) = Z n + d e N F ( Ad k ( H w + e N )) = Z n + F ( H + N ) d N = b F ( H ) . (3.10) Q.E.D. The integration for m ula of Thm. 3.1 should b e considered a mild generaliza tion of the standard W eyl integration formula whic h we s tate here for functions on the Lie algebra of a compact Lie gr oup K . 4 Theorem 3. 2 (W eyl in tegration form ul a) L et F : k → C b e a smo oth function inte gr able with r esp e ct to t he L eb esgue me asur e. Then Z k d X F ( X ) = c k Z i h R d H | ∆( H ) | 2 Z K d k F ( Ad k ( H )) (3.11) In c ase of an Ad K –invariant function the ab ove r e duc es t o Z k d X F ( X ) = c k Z i h R d H | ∆( H ) | 2 F ( H ) (3.12) wher e we ne e d t o put the absolute–value sign b e c ause ∆( H ) 2 may b e ne gative on i h R if dim n + is o dd ( c k has the same me aning and value as in Thm. 3.1 ). Remark 3.1 The explicit value of µ ( K ) /µ ( T ) was c ompute d by Mac donald [ 8 ] for a slightly differ ent choic e of normalization for the L eb esgue me asur e. The actual value of this c onst ant is irr elevant for our purp ose and do es not dir e ctly enter the c omputation of the pr op ortionality c onstant in the c onje ctur e. 4 Gaussian in tegrals Let V R be a real vector space a nd h , i : V R × V R → R a p ositive definite bilinear pairing (a n inner pro duct). Let V C := V R ⊗ C b e its complexification. Denote b y d x the Leb esgue mea sure on V R and d z the Leb esgue measure on V C (the normaliz a tions of which are irre lev ant at this po in t). The inner pro duct h , i extends to a n inner -pro duct (linearly ov er C ) on V C . Mo reov er the real form V R ⊂ V C defines also a natural co njuga tion z → z which fixes V R . Lemma 4. 1 With t he notations and definitions ab ove, for any p olynomial function F on V C × V C define < F > R := 1 Z R Z V R Z V R d x d y e − a h x,x i− c h y ,y i− 2 b h x,y i F ( x, y ) < F > C := 1 Z C Z V C d z e − a h z,z i− c h z ,z i− 2 b h z,z i F ( z , z ) (4.1) wher e Z R and Z C ar e determine d 4 by the r e quir ement that < 1 > = 1 . W hile the c onver genc e of t he inte gr als in the two c ases imp oses differ ent c onditions on the numb ers a, b, c , nevertheless (i) b oth < F > R , C ar e p olynomials in a/δ, b/ δ, c/δ , δ := ac − b 2 and (ii) as p olynomials they c oincide. Pro of . The key is in showing that the genera ting functions for the moments of the tw o in teg rals in either cases are iden tica l, namely that for A, B ∈ V C G R ( A, B ) := D e h x,A i + h y,B i E R = exp  1 4  c δ h A, A i + a δ h B , B i − 2 b δ h A, B i  = G C ( A, B ) := D e h z ,A i + h z ,B i E C (4.2) for then < F > R = F ( ∂ A , ∂ B ) G R ( A, B )   A =0= B = < F > C , which proves b oth p oints of the lemma at the sa me time. In order to show ( 4.2 ) w e use an orthonormal coordina te basis for h , i so that – writing A = ( α 1 , . . . , α n ) and B = ( β 1 , . . . , β n ) in this bas is – the in tegra l G R factorizes as G R = Q G 1 ( α j , β j ) with G 1 ( α, β ) := 1 Z 1 , R Z R d x Z R d y exp  − ( x, y ) M  x y  + xα + y β  , M :=  a b b c  (4.3) 4 It is an easy exercise that we lea ve to the interested reader to v eri fy that Z R = (2 π ) n δ n 2 , Z C = (2 π ) n ( − δ ) n 2 , n = dim R V R = dim C V C , δ = ac − b 2 . These precise expressions are nev ertheless irrelev ant f or our purposes. 5 Define ( x ′ , y ′ ) = ( x, y ) − 1 2 ( α, β ) M − 1 , where no w the con tour s of integration ma y be some lines pa r allel to the real axis in the complex x ′ and y ′ planes. Ho wev er the ensuing integrals c a n b e defo rmed (by Cauch y theorem) back to the real axis and the integral yields G 1 ( α, β ) = g ( α, β ) := exp 1 4  c δ α 2 + a δ β 2 − 2 b δ αβ  . F or the second case w e hav e G C = Q e G 1 ( α j , β j ) with e G 1 ( α, β ) given b elow: we need to express the integration in the real/ imaginary pa rt of z = x + iy e G 1 ( α, β ) := 1 Z 1 , C Z C d 2 z exp  − ( z , z )  a b b c   z z  + z α + z β  = (4.4) = Z R Z R d x d y exp  − ( x, y )  a + c + 2 b i ( a − c ) i ( a − c ) 2 b − a − c   x y  + x ( α + β ) + iy ( α − β )  (4.5) In this ca se w e p erfor m the shift ( x ′ , y ′ ) = ( x, y ) − 1 2 ( α + β , i ( α − β ))  a + c + 2 b i ( a − c ) i ( a − c ) 2 b − a − c  − 1 , followed by deforming back the integration contours on the rea l x ′ and y ′ axes. Str a igh tforward linear algebra g iv es the s ame result g ( α, β ) as a bove. Q.E. D. 5 Pro of of Conjecture 1 Let ϑ : g → g b e the Cartan inv ol ution (antilinear) ϑ ( cE α ) = − cE − α , ϑ ( cH α ) = − cH α . (5.1) Definition 5.1 F or a (semi)simple Lie algebr a g over C , given t he Cartan involution ϑ define d ab ove, we wil l denote by M † = − ϑ ( M ) , and by M ϑ = ϑ ( M ) . Remark 5.1 The notation M † has b e en define d t o c oincide with t he u sual hermitian c onjugate in the st andar d fundamental r epr esentation of sl ( n, C ) . The Carta n inv olution ϑ (or † ) defines a real for m of g whic h is precisely k , the c omp act r e al form a s the 1–eigenspa ce of ϑ . Two prop erties are immediate ( h , i is the Killing form): •  M , M ϑ  ≤ 0 is a negativ e definite sesquili near quadr atic form for M ∈ g ; •  X , X †  = −  X , X ϑ  = − h X , X i ≥ 0 is a p ositive definite quadra tic for m for X ∈ k (as a real v ecto r s pace). Consider the following quadratic form on g × g Q A ( X, Y ) := a h X , X i + 2 b h X, Y i + c h Y , Y i , A :=  a b b c  (5.2) W e leav e to the re a der to v erify the following easy Lemma 5. 1 Ther e exist t wo op en domains D k and D ı g for t he p ar ameters a, b, c su ch that • if ( a, b, c ) ∈ D k then ℜ Q A ( X, Y ) is p ositive definite on k × k ; • if ( a, b, c ) ∈ D ı g then ℜ Q A ( M , M ϑ ) is p ositive definite on g . 6 The spec ific form o f these domains is lar gely ir relev ant for our considerations and in the interest of conciseness we will not sp ecify them further. Definition 5.2 Define Z k := Z k Z k dX d Y e − Q A ( X,Y ) , ( a, b, c ) ∈ D k , Z g := Z g dM e − Q A ( M ,M ϑ ) , ( a, b, c ) ∈ D ı g . (5.3) Then, for any p olynomial function F on g × g we define < F > k := 1 Z k Z k Z k dX d Y F ( X, Y )e − Q A ( X,Y ) , < F > g := 1 Z g Z g dM F ( M , M ϑ )e − Q A ( M ,M ϑ ) (5.4) As an applica tion of Lemma 4.1 with V R = k , V C = k ⊗ C = g and ϑ as the inv olution leaving k inv ar ian t, we hav e Prop osition 5. 1 F or any p olynomial function F on g × g b oth < F > k and < F > g ar e p ol ynomials in a/δ, b/ δ, c/δ with δ = ac − b 2 . As p olynomials they coincid e The next theor em con tains the pr oo f of Conjecture 1 with prec is e v alues of the pr opor tionality constants. Theorem 5. 1 L et F b e an Ad K invariant (p olynomial) funct ion on g × g , wher e the action of Ad K is the diagonal action F ( X, Y ) = F ( Ad k ( X ) , Ad k ( Y )) , ∀ X, Y ∈ g (5.5) Then, for any H , J ∈ h C Z K d k F ( H , Ad k ( J ))e γ h H, Ad k ( J ) i = C g | W | X w ∈ W e γ h H,J w i ∆( H )∆( J w ) R n + d N F ( H + N , J w + N ϑ )e γ h N ,N ϑ i R n + d N e γ h N ,N ϑ i (5.6) The normalization c onstant C g is given by C g = | W | dim h Y j =1 m j ! Y α> 0 h α, α i 2 γ . (5.7) wher e m j ar e the exp onents of the Weyl gr oup. Mor e over we have Z n + d N e γ h N ,N ϑ i = Y α> 0 π h α, α i 2 γ (5.8) Remark 5.2 By simplifying the values of t he c onst ants (note that dim n + is the numb er of p ositive r o ots) we obtain Z K d k F ( H , Ad k ( J ))e γ h H, Ad k ( J ) i = Q dim h j =1 m j ! π dim n + X w ∈ W e γ h H,J w i ∆( H )∆( J w ) Z n + d N F ( H + N , J w + N ϑ )e γ h N ,N ϑ i (5.9) Remark 5.3 Note that the c onver genc e of the Gaussian inte gr al in the formula demands ℜ ( γ ) > 0 , but the identity is one b etwe en analytic fun ctions of γ . 7 Pro of . W e star t fro m the pro of of ( 5.8 ): since h E α , E β i = 2 h α,α i δ α, − β 5 , w r iting N = P α> 0 n α E α the integral is recast in to the form Z n + d N e γ h N ,N ϑ i = Y α> 0 Z C d 2 n α e − 2 γ h α,α i | n α | 2 = Y α> 0 π h α, α i 2 γ ( ℜ ( γ ) > 0) . (5.10) The v alue of C g is co mputed by ev alua ting explicitly the integrals on b oth sides for F ≡ 1, which reduces the formula to the famous Harish–Chandra expression Z K d k e γ h H,Ad k ( J ) i = C g | W | X w ∈ W e γ h H, J w i ∆( H )∆( J w ) (5.11) In this case the eq ua lit y w as established in (Thm. 2, pag 104 [ 3 ]) 6 where the v alue of the constant C g was given by C g = ( γ ) − dim n +  ∆ , ∆  . The brack et  p ( H ) , q ( H )  was defined ibidem for any po lynomials p , q ov er h b y writing them in a orthonormal basis h ω j , ω k i = δ j k 7 p ( H ) := X ~ n a ~ n dim( h ) Y ℓ =1 ω n ℓ ℓ ( H ) , q ( H ) := X ~ n b ~ n dim( h ) Y ℓ =1 ω n ℓ ℓ ( H ) (5.12)  p, q  := X ~ n a ~ n b ~ n dim( h ) Y ℓ =1 m ℓ ! (5.13) The num b er  ∆ , ∆  (∆ = Q α> 0 α ) has b een computed in [ 8 ] 8 and is given b y  ∆ , ∆  = 2 dim h − dim g 2 | W | n Y j =1 m i ! Y α> 0 h α, α i (5.14) which proves the e x pression for the constants noticing that dim g − d im h 2 = dim n + . W e now turn to the pro of of the equality: consider first the integral ov er k × k 1 Z k Z k Z k d X d Y F ( X , Y )e − a h X,X i− c h Y , Y i− 2 b h X,Y i = = c 2 k Z k Z i h R Z i h R d H d J ∆( H ) 2 ∆( J ) 2 e − a h H,H i− c h J,J i Z K d k F ( H , k J k − 1 )e − 2 b h H,kJ k − 1 i | {z } =: I ( H ,J ) (5.15) where we hav e used W eyl in tegr ation formula (Thm. 3.2 ) twice. On the other hand for the integral ov er g , using the complex W eyl integration formula (Thm. 3.1 ) we ha ve T ( a, b, c ) := 1 Z g Z g d M F ( M , M ϑ )e − a h M ,M i− c h M ϑ ,M ϑ i − 2 b h M ,M ϑ i = = c k Z g Z h C d Z | ∆( Z ) | 2 e − a h Z,Z i− c h Z ϑ ,Z ϑ i Z n + d N F ( Z + N , Z ϑ + N ϑ )e − 2 b h Z + N ,Z ϑ + N ϑ i = 5 Indeed, h [ E α , E − α ] , H i = h E α , [ E − α , H ] i = α ( H ) h E α , E − α i . On the other hand [ E α , E − α ] = H α and th us α ( H ) h E α , E − α i = h H α , H i . Ev al uating on H = H α ( α ( H α ) = 2, h H α , H α i = 4 h α,α i ) we get the assertion. 6 In lo c. cit. the exp onen t has a plus sign and no constan t γ , whic h means that we ha ve to map H 7→ γ H in Harish–Chandra’s formula, thus yielding the factor ( γ ) − d im n + due to the homogeneit y of ∆. 7 W e denote by the s ame sym b ol h , i the induced inner product on h ∨ . 8 The formula is quoted as rep orted in an app endix of a paper of Harder cited ibidem, due to a pri v ate comm unication of Stein b erg. 8 = c k Z g Z h C d Z | ∆( Z ) | 2 e − a h Z,Z i− c h Z ϑ ,Z ϑ i − 2 b h Z,Z ϑ i Z n + d N F ( Z + N , Z ϑ + N ϑ )e − 2 b h N ,N ϑ i (5.16) Here we ha ve used that f ( M ) := F ( M , M ϑ )e − Q A ( M ,M ϑ ) is Ad K –inv aria nt (but not Ad G –inv aria nt!) and then the simple fact that  Z + N , Z ϑ + N ϑ  =  Z, Z ϑ  +  N , N ϑ  . W e now p oint out that T ( a, b, c ) is a polyno mial in a/δ, b/δ, c/ δ by Lemma 4.1 . Since | ∆( Z ) | 2 = ( − ) n + ∆( Z )∆( Z ϑ ) (where Z ϑ = − Z † is again the natural conjugation w.r.t. i h R ), applying once more Lemma 4.1 with V R = i h R and V C = h C we o bta in T ( a, b, c ) = ( − ) n + c k Z g Z i h R d H Z i h R d J ∆( H )∆( J )e − a h H, H i− c h J,J i− 2 b h H ,J i Z n + d N F ( H + N , J + N ϑ )e − 2 b h N ,N ϑ i = = ( − ) n + c k Z n + Z g Z i h R d H Z i h R d J ∆( H ) 2 ∆( J ) 2 e − a h H,H i− c h J ,J i e − 2 b h H,J i ∆( H )∆( J ) 1 Z n + Z n + d N F ( H + N , J + N ϑ )e − 2 b h N ,N ϑ i | {z } =: G ( H,J ) (5.17) where the last line is just a different w ay of rewriting the previous line with Z n + = R n + exp − 2 b  N , N ϑ  d N . Compariso n of formulæ ( 5.15 ) and ( 5.17 ) suggests the na ¨ ıve observ atio n that I ( H , J ) = ( − ) n + c k Z k Z n + c 2 k Z g G ( H, J ) but this cannot p ossibly b e the ca se since I ( H , J ) is W eyl–inv a riant in b oth v ariables while in general G ( H, J ) is not. What will b e shown instead is that the symmetrization of G ( H , J ) is pr opor tional to I ( H, J ), namely I ( H , J ) = ( − ) n + c k Z k Z n + c 2 k Z g 1 | W | X w ∈ W G ( H, J w ) (5.18) which is precisely the ass ertion of our theorem. Note that G ( H w , J w ): indeed from Corollar y 3.1 the in teg r al 9 f ( Z, Z ϑ ) := Z n + d N F ( Z + N , Z ϑ + N ϑ )e − 2 b h N ,N ϑ i (5.19) is a polynomia l (since F is a poly no mial in both v ariable s ) with inv aria nce f ( Z w , Z ϑ w ) = f ( Z, Z ϑ ). The polynomial f ( H, J ) can b e written as f ( H, J ) = e H ∂ Z e J ∂ Z ϑ f ( Z, Z ϑ )   Z =0= Z ϑ (5.20) where H ∂ Z , J ∂ Z ϑ stand for the vector–fields H ∂ Z Z = H , H ∂ Z Z ϑ = 0 and s imilarly J ∂ Z ϑ Z = 0 , J ∂ Z ϑ Z ϑ = J . Therefore it is sufficient to symmetrize G ( H, J ) with resp ect to –say– J in o rder to obta in a co mpletely W eyl– inv ariant function. The symmetrization can b e ca rried under the in tegral sign without c hang ing its v alue since the measures d H , d J and the exponential fa ctors that prece de G in ( 5.17 ) are all W –in v aria nt. W e ha ve th us o bta ined c k Z n + ( − ) n + Z g Z i h R × i h R d H d J ∆( H ) 2 ∆( J ) 2 e − a h H,H i − c h J,J i X w ∈ W G ( H, J w ) | W | = c 2 k Z k Z i h R × i h R d H d J ∆( H ) 2 ∆( J ) 2 e − a h H,H i− c h J ,J i I ( H , J ) Of co urse this e q ualit y p er se do es not imply eq . 5.18 . How ever we can use the following arg umen t. W e replace the inv ariant polynomial F ( X , Y ) b y h ( X ) g ( Y ) F ( X , Y ) with h, g a rbitrary Ad K inv ariant p olynomia ls over k 9 The prefactor of whic h already has the ad vocated in v ariance. 9 Note that g ( H + N ) = g ( H ) (and so for h ): indeed any Ad K –inv aria nt p olynomial on k is automatically Ad G – inv ariant (on k ⊗ C = g ) and for a generic H , H + N is Ad G –conjugate to H itself since a d H + N is semis imple (in the a djoin t repres en tatio n). Thus, in eq. 5.17 , the tw o extra factors h, g will b e indepe nden t of N and th us factorizable outside of the integral ov er the nilpo ten t a lgebra n + . On the other hand, in eq. 5.15 they cle a rly a nd immediately factor out of the K -integral th us y ielding the ident ity Z i h R d H Z i h R d J ∆( H ) 2 ∆( J ) 2 e − a h H,H i− c h J ,J i h ( H ) g ( J ) c k ( − ) n + Z n + Z g 1 | W | X w ∈ W G ( H, J w ) − c 2 k Z k I ( H , J ) ! | {z } =: R ( J,H ) = 0 (5.21) v alid for arbi t r ary W eyl–inv a riants p olynomia ls h, g o n h R . Note that in H := L 2 ( i h R × i h R , d H d J e − a h H,H i− c h J ,J i ) the set of all polyno mia ls is de ns e and that the br ack et e x pression above b elongs to this space (we can ta ke a, c ∈ R − for this computation). The pro jector onto the subspace of W –inv ariant functions H W := { f ( H, J ) ∈ H : f ( H w , J w ′ ) = f ( H, J ) , ∀ w , w ′ ∈ W } (5.22 ) is self-adjoint and hence the range is a closed subspa ce, to whic h R ( J, H ) b elongs. The space of W eyl in v ar ian t po lynomials form a basis in this space a nd in par ticular are dense. Thus the v anishing of eq. 5.21 says that R ( J, H ) is orthog onal to such a dense set, thus is identically v anishing. The last detail is that the identit y so far has be en prov ed only for H , J ∈ i h R ; how ever, being an identit y b etw een p o lynomials, it must hold for its complexification as well, na mely on the whole h C . The theorem is proved and so is Conjecture 1 , with γ = − 2 b . Q.E.D. References [1] P . Di F rancesco , P . Ginsparg , and J. Z inn- Justin. 2D gravity and r a ndom matrices. Phys. R ep. , 2 5 4(1-2):133 , 1995. [2] J.-M. Daul, V. A. Kazakov, and I. K. Kostov. Ra tio nal theo ries o f 2d gravity from the tw o- ma trix mo del. Nucle ar Phys. B , 409(2):3 1 1–338, 19 93. [3] Harish-Cha ndra. Differential o pera tors on a semisimple Lie algebr a. Amer. J. Math. , 79:87– 120, 195 7. [4] C. Itzykso n and J. B. Zuber . The planar a pproximation. I I. J. Math. Phys. , 21(3):41 1–421, 19 80. [5] A. Pra ts F err e r , B. E ynard, P . Di F rancesco, and J.-B. 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